COMPARATIVE RESULTS ON SOME POLYLOCAL LIMIT PROBLEMS FOR LINEAR DIFFERENTIAL EQUATIONS

by
OLEG ARAMĂ
Paper presented at the Colloquium on the Theory of Partial Differential Equations,
organized by the RPR Academy
and the RPR Mathematical and Physical Sciences Society
between 21-26 Sept. 1959, Bucharest

Given a linear and homogeneous differential equation

y^{(n)}+a_{1}(x)y^{(n-1)}+\ldots+a_{n-1}(x)y^{\prime}+a_{n}(x)y=0

(1)

In his memoir [24], Ch. J. de la Vallée Poussin established the following theorem:

Assuming that the functions $a_{i}(x),(i=1,2,\ldots,n)$ are continuous in an interval $[a,b]$ , either $L_{i}=\max_{x\in[a,b]}\left|a_{i}(x)\right|$ and either $h_{0}$ the positive root of the equation

L_{n}\frac{h^{n}}{n!}+L_{n-1}\frac{h^{n-1}}{(n-1)!}+\ldots+L_{1}\frac{h}{1}-1=0

Then whatever choice is made $n$ puncture $M_{i}\left(x_{i},y_{i}\right),(i=1,2,\ldots,n)$ from the xOy plane, so that $a\leqq x_{1}<x_{2}<\ldots<x_{n}\leqq b,x_{n}-x_{1}\leqq h_{0}$ , for the chosen path, there is one and only one integral curve of equation (1), which passes through the points $M_{i}\left(x_{i},y_{i}\right).^{1)}$

As specified in the cited memorandum, this theorem also extends to the case when some of the nodes $x_{1},x_{2},\ldots,x_{n}$ are confused into groups, as follows:

Given a system of $m$ NUMBERS $p_{1},p_{2},\ldots,p_{m}$ , satisfying the condition $p_{1}+p_{2}+\cdots+p_{m}=n$ : If the coefficients of the differential equation (1) are

⁰⁰footnotetext:¹⁾ An alternative to this theorem was established by S. Zaidman in [29].

continuous functions in the interval $[a,b]$ , then m nodes would be chosen anyway $x_{1}<x_{2}<\ldots<x_{m}$ from the range $[a,b]$ , so that $x_{m}-x_{1}\leqq h_{0}$ and no matter which real number system is chosen $\left\{y_{1}^{(0)},\ldots,y_{1}^{\left(p_{1}-1\right)}\right\},\left\{y_{2}^{(0)},\ldots,y_{2}^{\lef t(p_{2}-1\right)}\right\},\ldots\ldots,\left\{y_{m}^{(0)},\ldots,y_{m}^{\left(p_{m}-1\right)}\right\}$ , for such a choice, the differential equation (1) admits one and only one integral $y(x)$ , satisfying the conditions:

y\left(x_{k}\right)=y_{k},\quad y^{\prime}\left(x_{k}\right)=y_{k}^{(1)},\ldots,y^{\left(p_{k}-1\right)}\left(x_{k}\right)=y_{k}^{\left(p_{k}-1\right)},\quad(k=1,2,\ldots,m).

(2)

In the following, we will note with $H_{p_{1},p_{2},\ldots,p_{m}}$ upper limit of positive numbers $h$ , satisfying the inequality $h\leqq ba$ and which have the property that no matter what is chosen $m$ knots $x_{1}<x_{2}<\ldots<x_{m}$ from the range $[a,a+h]$ and no matter which real number system is chosen $\left\{y_{k}^{(0)},\ldots,y_{k}^{\left(p_{k}-1\right.}\right\}$ , $(k=1,2,\ldots,m)$ , for such a choice, there is one and only one integral of the differential equation (1) that satisfies the conditions (2). The previously stated theorem shows that the set of these numbers $h$ is not empty. It is easily seen that the family of integrals of the differential equation (1) possesses the interpolation property (2) in the semiclosed interval $\left[a,a+H_{p_{1}},p_{2},\ldots,p_{m}\right)$ , and also that this interval has a maximal character in $[a,b]$ .

Next, let us consider all possible natural number systems. $p_{1},p_{2},\ldots,p_{m}$ satisfying the condition $p_{1}+p_{2}+\ldots+p_{m}=n$ Each such system will correspond to a number for the same differential equation. $H_{p_{1},p_{2},\ldots,p_{m}}$ During a working meeting of Section Ia of the Computing Institute in Cluj, Prof. T. Popoviciu raised the issue of developing a comparative study of numbers $H_{p_{1}},p_{2},\ldots,p_{m}$ for the same differential equation. This problem was posed in order to obtain necessary and sufficient conditions regarding the coefficients of the differential equation - conditions that ensure the existence and uniqueness of the solution of the polylocal boundary value problem with simple nodes, in a given interval.

The present research is situated within this problem. Before proceeding to its exposition, we would like to recall that the existence and uniqueness theorems of solutions to polylocal boundary value problems for linear differential equations have formed the subject of many works, of which we mention in the bibliography at the end only those that have a closer connection with the present research.

We will first assume that the given differential equation (1) has the coefficients $a_{i}(x),(i=1,2,\ldots,n)$ continue in an open interval $(a,b)$ We will denote by $Y_{n}$ the set of integrals of this equation in the interval ( $a,b$ ). We begin by giving a few definitions, which will be relevant in the following exposition.

Definition 1. It is said that the family $Y_{n}$ own the property $I_{n}(a,b)$ (i.e. it is an interpolator of the order $n$ on simple nodes in the interval ( $a,b$ )), if any of them are n distinct nodes $x_{1},x_{2},\ldots,x_{n}$ , located in the interval ( $a,b$ ), and whatever the real values are $y_{1},y_{2},\ldots,y_{n}$ , there is one and only one integral $y(x)\in Y_{n}$ , which satisfies the conditions $y\left(x_{i}\right)=y_{i},(i=1,2,\ldots,n)$ .

Definition 2. Given a system of $m$ natural numbers $p_{1},p_{2},\ldots,p_{m}$ , satisfying the condition $p_{1}+p_{2}+\ldots+p_{m}=n$ We say that the family $Y_{n}$ own the property $I_{p_{1}},p_{2},\ldots,p_{m}(a,b)$ , if any m distinct nodes $x_{1},x_{2},\ldots,x_{m}$ from the range $(a,b)$ and whatever $be$ real number systems

\left\{y_{1}^{(0)},y_{2}^{(1)},\ldots,y_{1}^{\left(p_{1}-1\right)}\right\},\left\{y_{2}^{(0)},y_{2}^{(1)},\ldots,y_{2} ^{\left(p_{2}-1\right)}\right\},\ldots\ldots,\left\{y_{m}^{(0)},y_{m}^{(1)},\ldots,y_{m}^{\left(p_{m}-1\right)}\right\}

there is one and only one integral $y(x)\in Y_{n}$ , which satisfies the conditions

y\left(x_{k}\right)=y_{k}^{(0)},y^{\prime}\left(x_{k}\right)=y_{k}^{(1)},\ldots,y^{\left(p_{k}+1\right)}\left(x_{k}\right)=y_{k}^{\left(p_{k}-2\right)}\quad(k=1,2,\ldots,m)

Definition 3. We say that the family $Y_{n}$ own the property $I_{n}^{*}(a,b)$ , if that family owns the properties $I_{p_{1},p_{2},\ldots,p_{m}}(a,b)$ , whatever the natural number system $p_{1},p_{2},\ldots,p_{m}$ , satisfying the condition $p_{1}+p_{2}+\ldots++p_{m}=n$ .

Note. In the adopted notations, the properties $I_{n}(a,b)$ and $I_{\underbrace{1,1,\ldots,1}_{n\text{ times }}}^{I_{\text{cid. }}}(a,b)$ , coincide.

We will establish in the following the following theorem:
theorem 1. If the family $Y_{n}$ has the property $I_{n}(a,b)$ , then it also has the property $I_{n}^{*}(a,b)$ .

To make it easier to demonstrate the proof of this theorem, we will first state a few lemmas.
$\mathrm{L}_{1}\mathrm{e}\mathrm{m}\mathrm{a}$ 1. Given m natural numbers, $p_{1},p_{2},\ldots,p_{m}$ satisfying the equality $p_{1}+p_{2}+\ldots+p_{m}=n$ The necessary and sufficient condition that the family $Y_{n}$ to have the property $I_{p_{1},p_{2},\ldots,p_{m}}(a,b)$ , is that the differential equation (1) does not admit any non-identically zero integral, which has in the interval ( $a,b$ ), m distinct roots $x_{1},x_{2},\ldots,x_{m}$ , having the orders of multiplicity ² ) greater or at least equal to the numbers respectively $p_{1},p_{2},\ldots,p_{m}$ .

The proof of this lemma is immediate. From this lemma, as particular cases, the 1st, 2nd and 3rd lemmas stated below follow:

Le ma 2. The necessary and sufficient condition that the family $Y_{n}$ to have ownership $I_{n}(a,b)$ , is that the differential equation (1) does not admit any non-identically zero integral, which vanishes for n distinct values in the interval $(a,b)$ .

Le ma 3. The necessary and sufficient condition that the family $Y_{n}$ to have the property $I_{n}^{*}(a,b)$ , is that the differential equation (1) does not admit any non-zero integral that has n roots in the interval ( $a,b$ ), each root being counted as many times as its order of multiplicity.

⁰⁰footnotetext:² ) By order of multiplicity of a root

x_{0}

of a function

y(x)

we understand the strict order of multiplicity; thus,

x_{0}

is a multiple root of order

k

for the function

y(x)

, if relationships take place

y\left(x_{0}\right)=y^{\prime}\left(x_{0}\right)=\ldots=y^{(k-1)}\left(x_{0}\right)=0,y^{(k)}\left(x_{0}\right)\neq 0

They are 4. If the family $Y_{n}$ has the property $I_{n}(a,b)$ , then any non-identical integral is zero $y(x)\in Y_{n}$ , which is cancelled for $n-1$ distinct values in the range ( $a,b$ ), has all simple roots (i.e. of order 1) in this interval.

Proof. The property formulated in this lemma is obvious for $n=2$ We will therefore consider $n\geqq 3$ We assume that $Y_{n}$ has the property $I_{n}(a,b)$ . Either $y_{0}(x)$ a non-zero non-identical integral of equation (1), which has $n-1$ distinct roots $x_{1},x_{2},\ldots,x_{n-1}$ in the interval ( $a,b$ ). We note from the beginning that the integral $y_{0}(x)$ cannot have other distinct roots in the interval ( $a,b$ ), since otherwise the property would be contradicted $I_{n}(a,b)$ of the family $Y_{n}$ . We will first show that none of these roots can have an even order of multiplicity. Indeed, let us suppose by

absurd that among those $n-1$ roots of the integral $y_{0}(x)$ , there would be a root $x_{i}$ , having an even order of multiplicity, that is

y_{0}\left(x_{i}\right)=y_{0}^{\prime}\left(x_{i}\right)=\ldots=y_{0}^{(2k-1)}\left(x_{i}\right)=0;y^{(2k)}\left(x_{i}\right)\neq 0,\quad(k\geq 1).

Since the roots of any non-zero integral of equation (1) are isolated points, it follows that there will be a sufficiently small neighborhood $\left(x_{i}-\delta,x_{i}+\delta\right)$ of the point $x_{i}$ , in which the function $y_{0}(x)$ will keep a constant sign, except for the point $x=x_{i}$ , in which it cancels out. For the sake of clarity, let us assume that $y_{0}(x)$ is positive in the intervals $\left(x_{i}-\delta,x_{i}\right)$ and $\left(x_{i},x_{i}+\delta\right)$ (fig. 1). Let $\eta(x)$ an integral of equation (1), which cancels out for all roots of the integral $y_{0}(x)$ I except the point $x=x_{i}$ , in which it takes the value 1:

	$\displaystyle\eta\left(x_{1}\right)=\eta\left(x_{2}\right)=\ldots=\eta\left(x_{i-1}\right)=0$
	$\displaystyle\eta\left(x_{i}\right)=1$		(3)
	$\displaystyle\eta\left(x_{i+1}\right)=\eta\left(x_{i+2}\right)=\ldots=\eta\left(x_{n-1}\right)=0$

Such an integral $\eta(x)$ , which satisfies conditions (3), exists, since by hypothesis the family $Y_{n}$ has the property $I_{n}(a,b)$ . Then either $\xi_{1}$ and $\xi_{2}$ two arbitrary numbers, respectively satisfying the inequalities $x_{i}-\delta<\xi_{1}<x_{i}<<\xi_{2}<x_{i}+\delta$ Obviously, inequalities will occur. $y_{0}\left(\xi_{1}\right)>0$ and $y_{0}\left(\xi_{2}\right)>0$ We consider the function $\eta_{\lambda}(x)=\lambda\eta(x)$ , where $\lambda_{\text{ouste }}^{*}$ a positive factor, small enough for the inequalities to occur simultaneously

\eta_{\lambda}\left(\xi_{1}\right)<y_{0}\left(\xi_{1}\right);\quad\eta_{\lambda}\left(\xi_{2}\right)<y_{0}\left(\xi_{2}\right)

(4)

How $\eta\left(x_{i}^{n}\right)=1$ , it follows that $\eta_{\lambda}\left(x_{i}\right)=\lambda>0$ , and how $y_{0}\left(x_{i}\right)=0$ , the inequality results $\eta_{\lambda}\left(x_{i}\right)>y_{0}\left(x_{i}\right)$ From this inequality, as well as from (4), it follows that in the interval ( $x_{i}-\delta,x_{i}+\delta$ ) equation curves $y=y_{0}(x)\stackrel{{\scriptstyle s}}^{\mathrm{i}}$

We will now show further that all roots in the interval ( $a,b$ ) of such an integral $y_{0}(x)$ are simple (i.e. of order 1). Indeed, let us suppose by absurdity that a non-zero non-identical integral $y_{0}(x)\in Y_{n}$ , which has $n-1$ distinct roots in the interval ( $a,b$ ), would have among them at least one of order greater than or at least equal to 3 . Either $x_{i}$ such a root. So

y_{0}\left(x_{i}\right)=y_{0}^{\prime}\left(x_{i}\right)=y_{0}^{\prime\prime}\left(x_{i}\right)=0.

(5)

Be it also $\eta_{\varepsilon}(x)$ an integral of equation (1), which verifies at the point $x_{i}$ the following Cauchy conditions:

	$\displaystyle\eta_{\varepsilon}\left(x_{i}\right)=\eta_{\varepsilon}^{\prime}\left(x_{i}\right)=0$
	$\displaystyle\eta_{\varepsilon}^{\prime\prime}\left(x_{i}\right)=\varepsilon,(\varepsilon>0)$
	$\displaystyle\eta_{\varepsilon}^{\prime\prime\prime}\left(x_{i}\right)=y_{0}^{\prime\prime\prime}\left(x_{i}\right)$
	$\displaystyle\cdots\cdots\cdots\cdot\cdot\cdot\cdot\cdot$		(6)
	$\displaystyle\eta_{\varepsilon}^{(n-1)}\left(x_{i}\right)=y_{0}^{(n-1)}\left(x_{i}\right)$

this will correspond to a threshold $E(\delta)$ , so that for any $\varepsilon$ satisfying the inequality $0<\varepsilon<E(\delta)$ , for relationships to take place

y_{0}(x)-\delta\leqq\eta_{\mathrm{e}}(x)\leqq y_{0}(x)+\delta,

(7)

whatever $x\in\left[a_{1},b_{1}\right]$ Let the numbers be $M_{i}$ defined as follows:

	$\displaystyle M_{i}=\max_{x\in\left[x_{i},x_{i+1}\right]}\left\|y_{0}(x)\right\|,(i=1,2,\ldots,n-2);M_{0}=\max_{x\in\left[a_{1},x_{1}\right]}\left\|y_{0}(x)\right\|;$
	$\displaystyle M_{n-1}=\max_{x\in\left[x_{n-1},b\right]}\left\|y_{0}(x)\right\|.\text{ Considerăm în (7) numărul }\delta\text{ astfel încît }$
	$\displaystyle 0<\delta<\min_{i=0,1,\ldots,n-1}\left\{M_{i}\right\}=M.$		(8)

Taking now $\varepsilon$ so as to satisfy the inequality $0<\varepsilon<E(M)$ and taking into account inequalities (7), it can be seen in figure 2 that in the interval ( $a_{1},b_{1}$ ), the equation curve $y=\eta_{\varepsilon}(x)$ , corresponding to the number $\varepsilon$ chosen, will cross the axis $Ox$ , at least by $n-1$ times, and therefore the integral $\eta_{\mathrm{B}}(x)$ will be cancelled in the interval ( $a_{1},b_{1}$ ), for at least $n$ distinct values. But as can be seen from (6), whatever the value of the parameter $\varepsilon$ , integral $\eta_{\varepsilon}(x)$ has a double root, namely $x=x_{i}$ . Also from (6) it is seen that the integral $\eta_{\varepsilon}(x)$ cannot be identically null, since it was assumed that $\varepsilon>0$ These results, however, contradict a previously established fact, namely that in the hypothesis that $Y_{n}$ has the property $I_{n}(a,b)$ , any non-identically zero integral of equation (1), which vanishes in $n-1$ distinct points in ( $a,b$ ), has all roots in this odd interval. In conclusion, the integral $y_{0}(x)$ previously considered cannot have in the interval ( $a,b$ ) no root of order greater than or equal to three, and thus the lemma is proven.

I give you 5. If $Y_{n}$ has the property $I_{n}(a,b)$ , then $Y_{n}$ has the properties $I_{p_{1}},p_{2},\ldots,p_{m}(a,b)$ , where $p_{1},p_{2},\ldots,p_{m}$ are arbitrary natural numbers, satisfying the conditions $p_{1}+p_{2}+\ldots+p_{m}=n$ and $\max\left\{p_{1},p_{2},\ldots,p_{m}\right\}=2$ .

Demonstration. Suppose that $Y_{n}$ has the property $I_{n}(a,b)$ We note from the beginning that to prove this lemma, we can assume that
at least two of the numbers $p_{1},p_{2},\ldots,p_{m}$ are greater than the number 1. Indeed, from the hypothesis $\max\left\{p_{1},p_{2},\ldots,p_{m}\right\}=2$ , which occurs in the statement of lemma 5, it follows that at least one of the numbers $p_{i}$ is equal to 2. Then, if only one of the numbers $p_{i}$ would be greater than 1, we would have $m=n-1$ , and the corresponding property $I_{p_{1}},p_{2^{\prime}},\ldots,p_{m}(a,b)$ would immediately follow from the successive application of Lemmas 1 and 4. Indeed, assuming by absurdity that $Y_{n}$ would not have that property $I_{p_{1}},p_{2},\ldots,p_{m}(a,b)$ , it would follow according to Lemma 1 that the differential equation (1) would admit a non-identically zero integral, which would have in the interval ( $a,b$ ), $n-1$ distinct roots, at least one of these roots having a multiplicity order greater than or equal to 2. This ciscondition would, however, contradict the statement of lemma 4.

We will therefore assume for the proof of Lemma 5, that at least two of the numbers $p_{i}$ are equal to 2 , hence, taking into account the condition $p_{1}+p_{2}+\ldots+p_{m}=n$ , the inequality results $m<n-1$ .

So be it. $p_{1},p_{2},\ldots,p_{m}$ , some kind of system $m$ natural numbers satisfying the conditions:

	$\displaystyle m<n-1$
	$\displaystyle p_{1}+p_{2}+\ldots+p_{m}=n$		(9)
	$\displaystyle\max\left\{p_{1},p_{2},\ldots,p_{m}\right\}=2$

This number system is arbitrary, but once chosen, we assume it is fixed for what follows.

With these clarifications, let us assume contrary to the statement of Lemma 5, that $Y_{n}$ would not have the property $I_{p_{1}},p_{2},\ldots,p_{m}(a,b)$ , where $p_{1},p_{2},\ldots,p_{m}$ are natural numbers chosen in compliance with conditions (9). Then, according to Lemma 1, it follows that equation (1) will admit at least one non-zero non-identical integral $y_{0}(x)$ , which has in the interval ( $a,b$ ), $m$ distinct roots $x_{1},x_{2},\ldots,x_{m}$ , having respectively the multiplicity orders $\pi_{1},\pi_{2},\ldots,\pi_{m}$ , satisfying the inequalities

\pi_{1}\geqq p_{1},\quad\pi_{2}\geqq p_{2},\ldots,\quad\pi_{m}\geqq p_{m}.

(10)

Let's note with $i_{1},i_{2},\ldots,i_{\alpha}$ clues $i$ , for which $\pi_{i}$ represents an even number and with $j_{1},j_{2},\ldots,j_{\beta}$ clues $j$ , for which $\pi_{j}$ is an odd number. Without restricting the generality of the reasoning, we can assume that the roots $x_{1},x_{2},\ldots,x_{m}$ of the integral $y_{0}(x)$ are consecutive and satisfy the inequalities

x_{1}<x_{2}<\ldots<x_{m}

(11)

Let us consider among these roots, those that correspond to the indices $j_{1},j_{2},\ldots,j_{\beta}$ , that is, those that represent odd-order roots for the integral $y_{0}(x)$ These roots are in number $\beta$ and we will note them respectively $x_{j_{1}},x_{j_{2}},\ldots,x_{j_{\beta}}$ We will take in the intervalu1 ( $x_{m},b$ ) some distinct nodes $\xi_{1}<<\xi_{2}<\ldots<\xi_{n-\beta-1}$ in number of $n-\beta-1$ . It is found, taking into account the first relation in (9), that $n-\beta-1>0$ We choose these nodes in such a way that none of them coincides with any root of the function $y_{0}(x)$ , which would possibly be in the interval ( $x_{m},b$ ).

Be it now $\eta(x)$ a non-zero non-identical integral of equation (1), which verifies the conditions:

	$\displaystyle\eta\left(x_{j_{1}}\right)=\eta\left(x_{j_{2}}\right)=\ldots=\eta\left(x_{j_{\beta}}\right)=0$		(12)
	$\displaystyle\eta\left(\xi_{1}\right)=\eta\left(\xi_{2}\right)=\ldots=\eta\left(\xi_{n-\beta-1}\right)=0.$

The existence of such an integral $\eta(x)$ , non-identical null in the interval ( $a,b$ ), results from the hypothesis that the family $Y_{n}$ has the property $I_{n}(a,b)$ , taking into account the fact that the number of cancellation conditions in (12) is $n-1$ We will show in the following that under the adopted hypotheses, for sufficiently small positive values of the parameter $\varepsilon$ , at least one of the integrals $\varepsilon\eta(x)$ , or - $\varepsilon\eta(x)$ , will take at least $n$ distinct points in the interval ( $a,b$ ), common values with the integral $y_{0}(x)$ , without coinciding identically with $y_{0}(x)$ , which will bring - contradiction of property $I_{n}(a,b)$ of the family $Y_{n}$ .

Indeed, because the $n-1$ conditions in (12) refer to $n-1$ distinct nodes in the interval ( $a,b$ ) and because by integral assumption $\eta(x)$ is not identically null in ( $a,b$ ), results according to the property $I_{n}(a,b)$ of the family $Y_{n}$ , that the integral $\eta(x)$ cannot have in the range ( $a,b$ ) roots other than $x_{j_{1}},x_{j_{2}},\ldots,x_{j_{\beta}}$ and $\xi_{1},\xi_{2},\ldots,\xi_{n-\beta-1}$ Then it also follows according to Lemma 4 that all these roots in the interval $(a,b)$ of the integral $\eta(x)$ are simple (of order 1), and therefore, if the variable $x$ grows in $\bmod$ continuously from the value $a$ at value $b$ , then the integral $\eta(x)$ alternately change the sign next to each value in the string:

x_{j_{1}},x_{j_{2}},\ldots,x_{j_{\beta}};\xi_{1},\xi_{2},\ldots,\xi_{n-\beta-1}.

(13)

Taking into account the fact that in the interval $\left[x_{1},x_{m}\right]$ , all odd roots of the integral $y_{0}(x)$ there are odd roots for $\eta(x)$ - and conversely it follows that if the variable $x$ grows from $x_{1}$ TO $x_{m}$ , then for one of the integrals $\eta(x)$ or $-\eta(x)$ , the sense of change of its sign in front of each of these odd roots $x_{j_{1}},x_{j_{2}},\ldots,x_{j_{\beta}}$ , will coincide with the direction of change of the sign of the integral $y_{0}(x)$ Let us denote by $\bar{\eta}(x)$ that of the integrals $\eta(x)$ and $-\eta(x)$ , for which this desired result is achieved, that is, that integral, for which in sufficiently small neighborhoods of the numbers $x_{j_{1}},x_{j_{2}},\ldots,x_{j_{\beta}}$ , the following equalities occur:

\operatorname{sgn}\{\bar{\eta}(x)\}=\operatorname{sgn}\left\{y_{0}(x)\right\},\left\{\begin{array}[]{l}x\in\left[x_{j_{1}}-\varepsilon_{1},x_{j_{1}}+\varepsilon_{1}\right],\text{ sau }\\ \cdots\cdots\cdots\cdots\cdots\\ x\in\left[x_{j_{\beta}}-\varepsilon_{\beta},x_{j_{\beta}}+\varepsilon_{\beta}\right]\end{array}\right.

Here the intervals $\left[x_{j_{1}}-\varepsilon_{1},x_{j_{1}}+\varepsilon_{1}\right],\ldots,\left[x_{j_{\beta}}-\varepsilon_{\beta},x_{j_{\beta}}+\varepsilon_{\beta}\right]$ are chosen small enough so that they are contained in the interval ( $a,b$ ) and not contain other roots of the integral $y_{0}(x)$ , than respectively $x_{j_{1}},x_{j_{2}},\ldots,x_{j_{\beta}}$ For the integral $\bar{\eta}(x)$ , the relation (14) will also take place in sufficiently small neighborhoods of the numbers $x_{i_{1}},x_{i_{2}},\ldots,x_{i_{\alpha}}$ , except for the means of these neighborhoods, since at these points the integral $y_{0}(x)$ is canceled, while $\bar{\eta}(x)$ it cannot be canceled.

Let us denote by ( $\Gamma$ ) equation curve $y=y_{0}(x)$ and with ( $\Gamma_{\varepsilon}$ ) equation curve $y=\varepsilon\bar{\eta}(x)$ , where $\varepsilon$ is a positive parameter. We will now examine how the curves ( $\Gamma$ ) and ( $\Gamma_{\varepsilon}$ ) between them, when the parameter s is small. We first observe that these curves cannot coincide identically in the interval $(a,b)$ , since in the additional nodes $\xi_{1},\xi_{2},\ldots,\xi_{n-\beta-1}$ (whose number is greater than zero, as specified previously), the integral $\bar{\eta}(x)$ is canceled, while $y_{0}(x)$ is different from zero.

Let us consider again the set formed by the indices $j_{1},j_{2},\ldots,j_{\beta}$ , for which $\pi_{j}$ is an odd number. We will divide this set into two subsets as follows: we will denote by $j_{k_{1}},j_{k_{2}},\ldots,j_{k_{\gamma}}$ , those indices $j_{k}$ , for which $\pi_{j_{k}}=1$ , and with $j_{l_{1}},j_{l_{2}},\ldots,j_{l_{8}}$ , clues $j_{l}$ , for which $\pi_{f_{l}}\geqq 3$ .

Regarding points 1a $x_{j_{k_{1}}},x_{j_{k_{2}}},\ldots,x_{j_{k_{Y}}}$ we find that they are (by hypothesis) simple roots for the function $y_{0}(x)$ , that is

		$\displaystyle y_{0}\left(x_{j_{k_{1}}}\right)=y_{0}\left(x_{j_{k_{2}}}\right)=\ldots=y_{0}\left(x_{j_{k_{\gamma}}}\right)=0$
		$\displaystyle y_{0}^{\prime}\left(x_{j_{k_{1}}}\right)\neq 0,y_{0}^{\prime}\left(x_{j_{k_{3}}}\right)\neq 0,\ldots,y_{0}^{\prime}\left(x_{j_{k_{\gamma}}}\right)\neq 0.$

But as previously shown, the numbers $x_{j_{k_{1}}},x_{j_{k_{2}}},\ldots,x_{j_{k_{\gamma}}}$ represent simple roots and for the integral $\bar{\eta}(x)$ , and therefore also for $\varepsilon\bar{\eta}(x)$ , whatever the value of the parameter $\varepsilon>0$ ;

		$\displaystyle\bar{\eta}\left(x_{j_{k_{1}}}\right)=\bar{\eta}\left(x_{j_{k_{2}}}\right)=\ldots=\bar{\eta}\left(x_{j_{k_{Y}}}\right)=0$
		$\displaystyle\bar{\eta}^{\prime}\left(x_{j_{k_{1}}}\right)\neq 0,\bar{\eta}^{\prime}\left(x_{j_{k_{2}}}\right)\neq 0,\ldots,\bar{\eta}^{\prime}\left(x_{j_{k_{Y}}}\right)\neq 0.$

Taking into account the fact that the functions $y_{0}^{\prime}(x)$ and $\bar{\eta}^{\prime}(x)$ does not cancel for the values $x_{f_{k_{1}}},x_{j_{k_{2}}},\ldots,x_{f_{k_{\gamma}}}$ , it follows that if the parameter $\varepsilon$ takes positive values, below a certain threshold, then the relationships will take place

y_{0}^{\prime}\left(x_{j_{k_{1}}}\right)\neq\varepsilon\bar{\eta}^{\prime}\left(x_{j_{k_{1}}}\right),\ldots,y_{0}^{\prime}\left(x_{j_{k_{\gamma}}}\right)\neq\varepsilon\bar{\eta}^{\prime}\left(x_{j_{k_{\gamma}}}\right).

It follows from this that for such values of the parameter $\varepsilon$ , the curves $(\Gamma)$ and $\left(\Gamma_{\varepsilon}\right)$ intersect at the points $x_{j_{k_{1}}}x_{j_{k_{1}}},\ldots,x_{j_{k_{\lambda}}}$ , crossing each other at these points.

Then, regarding the points $x_{j_{1}},x_{j_{2}},\ldots,x_{j_{l_{\delta}}}$ , the following property occurs: If the parameter $\varepsilon$ takes positive values, lower than a certain threshold, then in each of the sufficiently small neighborhoods, given:

\left(x_{j_{l_{1}}}-\varepsilon_{l_{1}},x_{j_{l_{1}}}+\varepsilon_{l_{1}}\right),\quad\left(x_{j_{l_{2}}}-\varepsilon_{l_{2}},x_{j_{l_{9}}}+\varepsilon_{l_{2}}\right),\ldots,\left(x_{j_{l_{\delta}}}-\varepsilon_{l_{\delta}},x_{j_{l_{\delta}}}+\varepsilon_{l_{\delta}}\right),

cURVES $(\Gamma)$ and $\left(\Gamma_{\epsilon}\right)$ will intersect at least in three distinct points. Indeed, the numbers $x_{j_{l_{1}}},x_{j_{l_{2}}},\ldots,x_{j_{l_{\delta}}}$ represent simple roots for the integral $\bar{\eta}(x)$ (as shown previously), and multiple roots of order

odd greater than or at least equal to 3, for the integral $y_{0}(x)$ (this by hypothesis). Let $x_{j_{l_{1}}}$ one of these points. Let us assume for the sake of clarity that in a neighborhood of this point, the function $y_{0}(x)$ is increasing (fig. 3). Then let $\zeta_{1}$ and $\zeta_{2}$ two numbers, satisfying the inequalities:

x_{j_{l_{1}}}-\varepsilon_{l_{1}}<\zeta_{1}<x_{j_{l_{1}}}<\zeta_{2}<x_{j_{l_{1}}}+\varepsilon_{l_{1}}.

Since equalities (14) hold, it follows that there is a threshold $E_{1}$ , so that for $\varepsilon<E_{1}$ inequalities to occur:

	$\displaystyle y_{0}\left(\zeta_{1}\right)\leqq\varepsilon\bar{\eta}\left(\zeta_{1}\right)$
	$\displaystyle y_{0}\left(\zeta_{2}\right)\geqq\varepsilon\bar{\eta}\left(\zeta_{2}\right).$		(15)

On the other hand, taking into account the fact that $x_{j_{l_{1}}}$ is a common root of the functions $y_{0}(x)$ and $\bar{\eta}(x)$ , it follows that the curves $(\Gamma)$ and $(\Gamma_{\varepsilon})$ intersect at the point $x_{j_{l_{1}}}$ , regardless of the value of the parameter $\varepsilon$ .

Then, developing the functions $y_{0}(x)$ and $\varepsilon\bar{\eta}(x)$ in Taylor series at the point $x_{j_{l_{1}}}$ and taking into account the orders of multiplicity of the root $x_{j_{l_{1}}}$ in relation to the two functions, it is found that, for any $\varepsilon>0$ , there is a sufficiently small subneighborhood of the point $x_{j_{l_{1}}}$ so that the curve $(\Gamma)$ to be below the curve $(\Gamma_{\varepsilon})$ for the values $x>x_{j_{l_{1}}}$ from this sub-neighborhood and above the curve $(\Gamma_{\varepsilon})$ for $x<x_{j_{l_{1}}}$ (Fig. 3).

From this and from relation (15) it follows that, for any $\varepsilon$ satisfying inequalities $0<\varepsilon<E_{1}$ , the curves $(\Gamma)$ and $(\Gamma_{\varepsilon})$ intersects in the interval

\left(x_{j_{l_{1}}}-\varepsilon_{l_{1}},\,x_{j_{l_{1}}}+\varepsilon_{l_{1}}\right)

at least three distinct points, crossing each other at these points.

Analogous conclusions are formulated for each of the roots $x_{j_{1}}$ , $x_{j_{l3}},\ldots,x_{j_{l\delta}}$ .

Finally, it is also observed that in sufficiently small neighborhoods of the points $x_{i_{1}},x_{i_{2}},\ldots,x_{i_{\delta}}$ , for sufficiently small positive values of the parameter $\varepsilon$ , the curves $(\Gamma)$ and $\left(\Gamma_{\mathrm{g}}\right)$ they intersect in at least two points, crossing each other at these points (fig. 4).

We finally obtain the following result:

the intersection points of the curves ( $\Gamma$ ) and ( $\Gamma_{\varepsilon}$ ) will be greater than or at least equal to $N=2\alpha+\gamma+3\delta$ . Taking into account $2-\mathrm{a}$ and the 3rd relation in (9), as well as the inequalities (10), we deduce that

2\alpha+\gamma+3\delta\geqq p_{1}+p_{2}+\ldots+p_{m}=n.

It follows from this that the number $N$ of distinct intersection points of the curves ( $\Gamma$ ) and ( $\Gamma_{\varepsilon}$ ) from the interval ( $a,b$ ), satisfies the inequality $N\geqq n$ and hence that, for positive and sufficiently small values of the parameter $\varepsilon$ , integral $\tilde{y}_{\epsilon}(x)=y_{0}(x)-\varepsilon\bar{\eta}(x)$ of equation (1), it cancels out for $N\geqq n$ distinct values in the range ( $a,b$ ). But $\widetilde{y}_{\epsilon}(x)$ cannot be identically null in the interval ( $a,b$ ) since in the additional nodes $\xi_{1},\xi_{2},\ldots\ldots,\xi_{n-\beta-1}$ , whose number is greater than zero, as follows from the first relation (9), the integration $\bar{\eta}(x)$ is canceled, while $y_{0}(x)$ is different from zero.

In conclusion, the particular integral $\widetilde{y}_{\varepsilon}(x)$ of equation (1) is not identically zero and when $\varepsilon>0$ is small enough, it cancels out for $N\geqq n$ distinct values in the range ( $a,b$ ). According to Lemma 2, this result contradicts the property $I_{n}(a,b)$ of the family $Y_{n}$ and from this follows the assertion of lemma 5.

I have 6. If $y_{0}(x)$ is a non-zero integral of equation (1), which in $m$ distinct points $x_{1},x_{2},\ldots,x_{m}$ from ( $a,b$ ), satisfies the conditions

	$\displaystyle y\left(x_{1}\right)=y^{\prime}\left(x_{1}\right)=\ldots=y^{\left(p_{1}-1\right)}\left(x_{1}\right)=0$
	$\displaystyle y\left(x_{2}\right)=y^{\prime}\left(x_{2}\right)=\ldots=y^{\left(p_{2}-1\right)}\left(x_{2}\right)=0$		(16)
	$\displaystyle y\left(x_{m}\right)=y^{\prime}\left(x_{m}\right)=\ldots=y^{\left(p_{m}-1\right)}\left(x_{m}\right)=0,$

where $p_{1},p_{2},\ldots,p_{m}$ are natural numbers, satisfying the conditions:

	$\displaystyle p_{1}+p_{2}+\ldots+p_{m}=n-1$		(17)
	$\displaystyle\max\left\{p_{1},p_{2},\ldots,p_{m}\right\}=2,$

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then, assuming that the family $Y_{n}$ has the property $I_{n}(a,b)$ , the relations result:

y^{\left(p_{1}\right)}\left(x_{1}\right)\neq 0,\quad y^{\left(p_{2}\right)}\left(x_{2}\right)\neq 0,\quad\ldots,y^{\left(p_{m}\right)}\left(x_{m}\right)\neq 0.

(18)

Demonstration. Let $y_{0}(x)$ a non-zero non-identical integral of the differential equation (1), which at the points $x_{1},x_{2},\ldots,x_{m}$ satisfies conditions (16) and (17). Let us denote by $\pi_{1},\pi_{2},\ldots,\pi_{m}$ , the multiplicity orders of the roots $x_{1},x_{2},\ldots,x_{m}$ of the integral $y_{0}(x)$ We will prove that under the assumptions of Lemma 6, the following equalities hold: $\pi_{1}=p_{1},\pi_{2}=p_{2},\ldots,\pi_{m}=p_{m}$ To this end, we note from the beginning that the integral $y_{0}(x)$ considered, cannot have in the interval ( $a,b$ ) roots other than $x_{1},x_{2},\ldots,x_{m}$ , since otherwise it would contradict the statement of Lemma 5. With this clarification, we will first show that the numbers $\pi_{1},\pi_{2},\ldots,\pi_{m}$ and $p_{1},p_{2},\ldots,p_{m}$ are respectively of the same parity.

We consider the function $\tilde{y}_{\varepsilon}(x)=y_{0}(x)-\varepsilon\bar{\eta}(x)$ , where $y_{0}(x)$ is the integral that occurs in the statement of lemma 6, and $\bar{\eta}(x)$ is also an integral of the differential equation (1), constructed according to the procedure indicated in the proof of Lemma 5, relative to the roots $x_{1},x_{2},\ldots,x_{m}$ of the integral $y_{0}(x).^{3}$ Just as there, it is shown that if any root $x_{i}$ from the group $x_{1},x_{2},\ldots,x_{m}$ has the order of multiplicity with respect to $y_{0}(x)$ , par, then no matter how small a neighborhood of this root may be $x_{i}$ , for sufficiently small positive values of the parameter $\varepsilon$ , integral $\widetilde{y}_{\varepsilon}(x)$ will have in the considered neighborhood at least two distinct roots; then, if $x_{j}$ is an odd root with respect to $y_{0}(x)$ , then no matter how small a neighborhood of this root may be for positive and sufficiently small values of the parameter $\varepsilon$ , integral $\tilde{y}(x)$ will have in that vicinity one root or at least three distinct roots, as $\pi_{j}=1$ or $\pi_{j}>1$ . It follows that if any root $x_{j}$ from the interval ( $a,b$ ), of the integral $y_{0}(x)$ , has odd order of multiplicity, then this order is necessarily equal to 1. Otherwise, in the neighborhood of this root $x_{j}$ , integral $\widetilde{y}_{\varepsilon}(x)$ will have three distinct roots, and taking into account the relations (17), it will result that in the interval ( $a,b$ ) integral $\widetilde{y}_{\varepsilon}(x)$ has at least $(n-1)+2=n+1$ distinct roots (if the parameter $\varepsilon$ takes positive, sufficiently small values). This would contradict the property $I_{n}(a,b)$ of the family $Y_{n}$ , given that the integral $\widetilde{y}_{\varepsilon}(x)$ is not identically null in the interval $(a,b).{}^{4}$ ) The following property ultimately results:

\left.\begin{array}[]{l}\text{ Dacă în conditiile lemei }6,\pi_{j}\text{ este impar, atunci }\pi_{j}=1,\\ \text{ şi deci }\pi_{j}=p_{j}\text{. }\end{array}\right\}

⁰⁰footnotetext: ³ ) However, unlike those presented in the proof of Lemma 5, the inequality

n-\beta

-1>0

results in the present case as follows: It is first noted that in the hypotheses of lemma 6, the inequality cannot occur

m\geqq n-1

, since such an inequality together with relations (16) and (17) would contradict the statement of Lemma 4. So necessarily

m<n-1

. Hence, taking into account the obvious inequality

\beta\leqq m

, the inequality results

n-\beta-1>0

We recall that the number

n-\beta-1

represents the number of auxiliary nodes

\xi_{1},\xi_{2},\ldots,\xi_{n-\beta-1}

, which intervene through formulas (12) in the definition of the integral

\bar{\eta}(x)

.
⁴ ) Proof of the property that the integral

\tilde{y}_{\varepsilon}(x)

is not identically null in the interval (

a,b

), is done exactly as in the case of Lemma 5.

Either now $x_{i}$ a root of the integral $y_{0}(x)$ , from the interval ( $a,b$ ), having an even order of multiplicity, $\pi_{i}$ We will first prove that the following property holds:

\left.\begin{array}[]{l}\text{ Dacă în conditiile lemei }6,\pi_{i}\text{ este par, atunci și numărul }\\ p_{i}\text{ este de aseme nea par. }\end{array}\right\}

Indeed, assuming absurdly that the number $p_{i}$ is odd, then taking into account the second equality in (17), it would result that $p_{i}=1$ On the other hand, the root $x_{i}$ having an even order of multiplicity with respect to the integral $y_{0}(x)$ , it follows that this root will correspond to the integral $\widetilde{y}_{8}(x)$ two distinct roots, located in however small a neighborhood of the point $x_{i}$ , if of course the parameter takes sufficiently small positive values. This would give the result: For positive and sufficiently small values of the parameter $\varepsilon$ , integral $\tilde{y}_{\varepsilon}(x)$ has in the range $(a,b)$ at least $(n-1)+1=n$ distinct roots. This result would contradict the property $I_{n}(a,b)$ of the family $y_{n}$ , given that the integral $\widetilde{y}_{\epsilon}(x)$ is not identically null in the interval $\left.(a,b).{}^{4}\right)$ The property (20) finally results.

We will now show more, namely that under the assumptions of Lemma 6, whatever the even root is $x_{i}$ from the interval ( $a,b$ ), of the integral $y_{0}(x)$ , for this root the equality holds $\pi_{i}=p_{i}$ Indeed, the inequality is obvious. $\pi_{i}\geqq p_{i}$ Let us suppose by absurdity that there exists in the interval ( $a,b$ ), at least one pear root $x_{i_{0}}$ of the integral $y_{0}(x)$ , whose order $\pi_{i_{0}}$ satisfies the strict inequality $\pi_{i_{0}}>p_{i_{0}}$ . We will show that such an assumption leads to an absurdity. Indeed, we first observe that from the inequality $\pi_{i_{0}}>p_{i_{0}}$ , taking into account the fact that the numbers $\pi_{i_{0}}$ and $p_{i_{0}}$ have the same parity, the relationship results

\pi_{i_{0}}\geqq p_{i_{0}}+2=4

(21)

$O$ such a relationship for the case when $n\leqq 4$ , is absurd. Next we assume that $n\geqq 5$ .

We will divide the set of roots $x_{1},x_{2},\ldots,x_{m}$ of the integral $y_{0}(x)$ , from the interval ( $a,b$ ), in two subsets. In the first subset we will consider the even-order roots, which we will denote by $x_{i_{1}},x_{i_{2}},\ldots,x_{i_{\alpha}}$ , and in the second subset we will consider the roots of odd order, and we will denote them by $x_{j_{1}},x_{j_{2}},\ldots,x_{j_{\beta}}$ Obviously that $\alpha+\beta=m$ We will distinguish two cases:

Case 1: $\alpha=1,\quad\beta=m-1,(n\geqq 5)$ We
assume that the differential equation (1) has in the interval ( $a,b$ ) a single even root, $x_{i_{0}}$ , and that the order $\pi_{i_{0}}$ of this root satisfies inequality (21). All other roots in the interval ( $a,b$ ), of the integral $y_{0}(x)$ , being assumed odd, they will necessarily be simple, according to property (19), established previously. Taking into account property (19), as well as equalities (17), we deduce that in the considered case, the number of these roots is $\beta=n-3$ In each of them, the equation curve $y=y_{0}(x)$ will cross the axis $Ox$ .

Whether $\varepsilon$ a positive number and let $\mu_{\varepsilon}(x)$ the integral of the differential equation (1), which satisfies at the point $x_{i_{6}}$ the following Cauchy conditions:

	$\displaystyle\mu_{\varepsilon}\left(x_{i_{0}}\right)=y_{0}\left(x_{i_{0}}\right)=0,\quad\mu_{\varepsilon}^{\prime}\left(x_{i_{0}}\right)=y_{0}^{\prime}\left(x_{i_{0}}\right)=0,\quad\mu_{\varepsilon}^{\prime\prime}\left(x_{i_{0}}\right)=y_{0}^{\prime\prime}\left(x_{i_{0}}\right)=0$
	$\displaystyle\mu_{\varepsilon}^{\prime\prime\prime}\left(x_{i_{0}}\right)=\varepsilon,$		(22)
	$\displaystyle\mu_{\varepsilon}^{(\mathrm{IV})}\left(x_{i_{0}}\right)=y_{0}^{(\mathrm{IV})}\left(x_{i_{0}}\right),\ldots,\mu_{\varepsilon}^{(n-1)}\left(x_{i_{0}}\right)=y_{0}^{(n-1)}\left(x_{i_{0}}\right).$

From these formulas it is seen that the integral $\mu_{\mathrm{e}}(x)$ satisfy at point $x_{i_{0}}$ the same Cauchy conditions, as $y_{0}(x)$ , except for the 3rd order derivative, which at this point takes the value $\varepsilon$ .

If the parameter $\varepsilon$ takes sufficiently small positive values, then the Cauchy conditions, which the integral satisfies $\mu_{\varepsilon}(x)$ , are close to the Cauchy conditions that the integral satisfies $y_{0}(x)$ , and if $\varepsilon$ tends to zero, then the function $\mu_{\varepsilon}(x)$ will tend uniformly towards $y_{0}(x)$ , in any closed subinterval $\left[a_{1},b_{1}\right]$ contained in ( $a,b$ ).

Since the equation curve $y=y_{0}(x)$ crosses the axis $Ox$ in each of the points $x_{j_{1}},x_{j_{2}},\ldots,x_{j_{\beta}}$ , it follows that no matter how small the neighborhoods of these points are chosen, there exists a threshold for them $E$ , so that whatever the positive number is $\varepsilon<E$ , the corresponding integral curve $y=\mu_{\varepsilon}(x)$ to cross the axis $Ox$ in each of the chosen neighborhoods, at one point. We will note the abscissas of these crossing points respectively with $\bar{x}_{j_{1}},\bar{x}_{j_{2}},\ldots,\bar{x}_{j_{\beta}}$ . Apart from these roots, the integral $\mu_{\varepsilon}(x)$ still admits the root $x_{i_{0}}$ , with multiplicity order 3, which is evident from formulas (22). It ultimately follows that the integral $\mu_{\mathrm{e}}(x)$ has in the interval ( $a,b$ ), the triple root $x_{i_{0}}$ , and in addition other $\beta$ distinct roots $\bar{x}_{j_{1}},\bar{x}_{j_{2}},\ldots,\bar{x}_{j_{\beta}}$ , different from $x_{i_{0}}$ and having odd orders of multiplicity. For these $\beta+1$ roots of the integral $\mu_{\varepsilon}(x)$ , conditions (16) and (17) are satisfied if we choose $\bar{p}_{i_{0}}=2$ and $\bar{p}_{j_{1}}=\bar{p}_{j_{2}}==\ldots=\bar{p}_{j_{\beta}}=1$ , and if it is taken into account that in case 1 considered, $\beta=n-3$ It is also found that the integral $\mu_{\varepsilon}(x)$ cannot have in the interval ( $a,b$ ) other distinct roots than $\bar{x}_{j_{1}},\bar{x}_{j_{2}},\ldots,\bar{x}_{j_{\beta}}$ and $x_{i_{0}}$ Indeed, Otherwise, the total number of distinct roots it would have in the interval ( $a,b$ ) the non-identical integral is zero $\mu_{\varepsilon}(x)$ , would be greater than or at least equal to $n-1$ , and according to Lemma 4 it would follow that all its roots are simple. This would contradict the fact that the root $x_{i_{0}}$ of the integral $\mu_{\varepsilon}(x)$ , is triple. Under these conditions, it is valid for the integral $\mu_{\varepsilon}(x)$ property (19), which states that all odd-order roots of such a non-identically zero integral must be simple. This property, however, contradicts the existence for the integral $\mu_{\varepsilon}(x)$ of the triple root $x_{i_{0}}$ . It ultimately follows that the root $x_{i_{0}}$ , of the integral $y_{0}(x)$ , cannot have a multiplicity order greater than 2, and hence that $\pi_{i}=p_{i},q.e.d$ .

Case 2: $\alpha\geqq 2,(n\geqq 5)$ .

In this case, taking into account equalities (17), as well as properties (19) and (20), it follows that $\beta\leqq n-5$ . Now let the number $\nu=(n-1)-\beta-2$ . Taking into account that $\beta\leqq n-5$ , the inequality results $\nu\geqq 2$ We will assume that
the roots $x_{1},x_{2},\ldots,x_{m}$ are written in ascending order; $x_{1}<x_{2}<\ldots<x_{m}$ We choose in the interval ( $x_{m},b$ ), $\nu$ distinct nodes $\xi_{1},\xi_{2},\ldots,\xi_{\nu}$ . Since the integral $y_{0}(x)$ does not have in the interval ( $a,b$ ) roots other than $x_{1},x_{2},\ldots,x_{m}$ (previously established fact), it results that none of the chosen nodes $\xi_{1},\xi_{2},\ldots,\xi_{v}$ cannot represent any root of the integral $y_{0}(x)$ .

Whether $\mu(x)$ a non-zero non-identical integral of equation (1), which satisfies the conditions:

	$\displaystyle\mu\left(x_{i_{0}}\right)=\mu\left(x_{i_{0}}\right)=0$
	$\displaystyle\mu\left(x_{j_{1}}\right)=\mu\left(x_{j_{2}}\right)=\ldots=\mu\left(x_{j_{\beta}}\right)==0$		(23)
	$\displaystyle\mu\left(\xi_{1}\right)=\mu\left(\xi_{2}\right)=\ldots=\mu\left(\xi_{\nu}\right)=0.$

The number of these cancellation conditions being $2+\beta+\nu=n-1$ , it follows from Lemma 5 that there is such a non-identically zero integral, satisfying conditions (23). This integral cannot have in the interval ( $a,b$ ) other distinct roots than $x_{j_{1}},x_{j_{2}},\ldots,x_{j_{\beta}},\xi_{1},\xi_{2},\ldots,\xi_{v}$ , and $x_{i_{0}}$ Indeed, otherwise the total number of distinct roots that the non-identical integral would have in this interval would be zero. $\mu(x)$ , would be greater than or at least equal to $n-1$ , and according to Lemma 4 it would follow that all its roots are simple. This would contradict the first series of equalities in (23). On the other hand, we note that the conditions (23), which are satisfied by the non-identical zero integral $\mu(x)$ , have the form of conditions (16) and (17) which are satisfied by the integral $y_{0}(x)$ In these circumstances they are valid for the whole $\mu(x)$ properties (19) and (20), based on which all the roots of the integral $\mu(x)$ , except for the root $x_{i_{0}}$ , are simple, and the root $x_{i_{0}}$ has an even order of multiplicity. Thus we arrive at the conclusion that the integral $\mu(x)$ satisfies conditions analogous to the conditions that the integral satisfied $y_{0}(x)$ in case 1 previously treated. Based on the results obtained when treating case 1, it can be stated that the root seems $x_{i_{0}}$ , of the integral $\mu(x)$ , must necessarily have order 2:

\mu\left(x_{i_{0}}\right)=\mu^{\prime}\left(x_{i_{0}}\right)=0,\quad\mu^{\prime\prime}\left(x_{i_{0}}\right)\neq 0.

(24)

From the above it follows that the integral curve of equation $y=\mu(x)$ crosses the axis $Ox$ in each of the roots $x_{j_{1}},x_{j_{2}},\ldots,x_{j_{\beta}},\xi_{1},\xi_{2},\ldots,\xi_{v}$ , and are located on the same side of the axis $Ox$ in a sufficiently small neighborhood of the root $x_{i_{0}}$ .

On the other hand, referring to the integral $y_{0}(x)$ , also based on property (19) we can state that the odd roots $x_{j_{1}},x_{j_{2}},\ldots,x_{j_{\beta}}$ must be simple for $y_{0}(x)$ Celelaïte roots $x_{i_{1}},x_{i_{2}},\ldots,x_{i_{\alpha}}$ , have even orders of multiplicity and among these roots is the root $x_{i_{0}}$ , which has the order $\pi_{i_{0}}$ satisfying inequality (21). It follows from this that the integral curve of equation $y=y_{0}(x)$ , crosses the axis $Ox$ in each of the roots $x_{j_{1}},x_{j_{2}},\ldots,x_{j_{\beta}}$ and are located on the same side of the axis $Ox$ in the vicinity of each of the roots $x_{i_{1}},x_{i_{2}},\ldots,x_{i_{\alpha}}$ . T, Taking into account these properties established for the integrals $\mu(x)$ and $y_{0}(x)$ , as well as the fact that all the roots $\xi_{1},\xi_{2},\ldots,\xi_{v}$ of the integral $\mu(x)$ are located outside the range $\left[x_{1},x_{m}\right]$ , it follows that if the variable $x$ increases from the value $a$
at the value $b$ , then for at least one of the integrals $\mu(x)$ or - $\mu(x)$ , the sense of change of its sign, next to each of the roots $y_{j_{1}},x_{j_{2}},\ldots,x_{j_{\beta}}$ , will coincide with the direction of change of the sign of the integral $y_{0}(x)$ next to each of these roots. Let us note with $\bar{\mu}(x)$ that of the integrals $\mu(x)$ and $-\mu(x)$ for which this desideratum is achieved, that is, for which - in sufficiently small neighborhoods of the numbers $x_{j_{1}},x_{j_{2}},\ldots,x_{j_{\beta}}-$ the following equalities occur:

\operatorname{sgn}\{\bar{\mu}(x)\}=\operatorname{sgn}\left\{y_{0}(x)\right\},\quad\left\{\begin{array}[]{l}x\in\left[x_{j_{1}}-\varepsilon_{1},x_{j_{1}}+\varepsilon_{1}\right],\text{ sau }\\ \ldots\ldots\ldots\ldots\ldots\ldots..\\ x\in\left[x_{j_{\beta}}-\varepsilon_{\beta},x_{j_{\beta}}+\varepsilon_{\beta}\right]\end{array}\right.

Taking into account the previously established fact that the integral $\bar{\mu}(x)$ does not have in the interval ( $a,b$ ) other distinct roots than $x_{j_{1}},x_{j_{2}},\ldots,x_{j_{\beta}},\xi_{1},\xi_{2},\ldots,\xi_{v}$ (of order 1), as well as the pear root $x_{i_{0}}$ , it follows that equality (25) will also hold in sufficiently small neighborhoods of the apparent roots $x_{i_{1}},x_{i_{2}},\ldots,x_{i_{\alpha}}$ of the integral $y_{0}(x)$ , with the possible exception of the centers of these neighborhoods. Let us denote by ( $\Gamma$ ) equation curve $y=y_{0}(x)$ and with ( $\Gamma_{\varepsilon}$ ) equation curve $y=\varepsilon\bar{\mu}(x)$ , where $\varepsilon$ is a parameter taking positive values.

It is easy to see that there is a threshold $E_{1}$ , so that if $\varepsilon<E_{1}$ then at each of the points $x_{j_{1}},x_{j_{2}},\ldots,x_{j_{\beta}}$ , the curves $(\Gamma)$ and ( $\Gamma_{\varepsilon}$ ) to intersect, crossing each other. (This statement results from the fact that the numbers $x_{j_{1}},x_{j_{2}},\ldots,x_{j_{\beta}}$ are simple roots for each of the integrals $y_{0}(x)$ and $\bar{\mu}(x))$ .

Then, choosing sufficiently small neighborhoods of the roots, it seems $x_{i_{1}},x_{i_{2}},\ldots,x_{i_{\alpha}}$ , except for the root $x_{i_{u}}$ , there is a threshold for them $E_{2}$ so that whatever it is $\varepsilon$ satisfying inequalities $0<\varepsilon<E_{2}$ , the curves ( $\Gamma$ ) and ( $\Gamma_{i}$ ) intersect in each of the considered neighborhoods, in two distinct points, crossing each other at these points.

Regarding the point $x_{i_{0}}$ , taking into account relations (24) and (21), we find the following: The equation curve $y=\bar{\mu}(x)$ has in $x_{i_{0}}$ a first order contact with the axis $Ox$ ,

\bar{\mu}\left(x_{i_{0}}\right)=\mu^{\prime}\left(x_{i_{0}}\right)=0,\quad\bar{\mu}^{\prime\prime}\left(x_{i_{0}}\right)\neq 0,

(

\prime

)

while the equation curve $y=y_{0}(x)$ has in $x_{i_{0}}$ an odd-order contact $\geqq 3$ with the axis $Ox$ .

	$\displaystyle y_{0}\left(x_{i_{0}}\right)=y_{0}^{\prime}\left(x_{i_{0}}\right)=y_{0}^{\prime\prime}\left(x_{i_{0}}\right)=y_{0}^{\prime\prime\prime}\left(x_{i_{0}}\right)=\ldots=y_{0}^{\left(\pi i_{0}-1\right)}\left(x_{i_{0}}\right)=0$
	$\displaystyle y_{0}^{\left(\pi i_{i_{0}}\right)}\left(x_{i_{0}}\right)\neq 0$		(26)

Let be a closed interval $\left[x_{i_{0}}-\varepsilon_{i_{0}},x_{i_{0}}+\varepsilon_{i_{0}}\right]$ , chosen small enough to be contained in ( $a,b$ ) and not contain any other root of the integral $y_{0}(x)$ , or of the integral $\bar{\mu}(x)$ . By the way the integral was chosen $\bar{\mu}(x)$ between the integrals $\bar{\mu}(x)$ and - $\mu(x)$ , it follows that if the neighborhood $\left[x_{i_{0}}-\varepsilon_{i_{0}},x_{i_{0}}+\varepsilon_{i_{0}}\right]$ of the point $x_{i_{0}}$ is small enough, then the equations $y=y_{0}(x)$ and $y=\bar{\mu}(x)$ they will be located on the same side of the axis $Ox$ , that is, the equality will occur

\operatorname{sgn}\left\{y_{0}(x)\right\}=\operatorname{sgn}\{\bar{\mu}(x)\},\quad x\in\left[x_{i_{0}}-\varepsilon_{i_{0}},x_{i_{0}}+\varepsilon_{i_{0}}\right].

(

\prime

)

Let us assume, for the sake of clarity, that $\bar{\mu}^{\prime\prime}\left(x_{i_{0}}\right)>0$ . Hence, based on the relationships $\left(24^{\prime}\right)$ and $\left(25^{\prime}\right)$ it results that $\operatorname{sgn}\left\{y_{0}(x)\right\}=\operatorname{sgn}\{\bar{\mu}(x)\}=1$ , when $x\in\left[x_{i_{0}}-\varepsilon_{i_{0}},x_{i_{0}}\right)$ and when $x\in\left(x_{i_{0}},x_{i_{0}}+\varepsilon_{i_{0}}\right]$ (fig. 5). We will demonstrate in the following that there is a threshold $E_{3}$ , so that if $\varepsilon$ satisfies inequalities $0<\varepsilon<E_{3}$ , then in the vicinity of $\left[x_{i_{0}}-\varepsilon_{i_{0}},x_{i_{0}}+\varepsilon_{i_{0}}\right]$ equation curves $y==y_{0}(x)$ and $y=\varepsilon\bar{\mu}(x)$ , apart from the point $x_{i_{0}}$ in which they present a contact of order 1, they intersect in two more distinct points, crossing each other in the latter. Indeed, either $\bar{\xi}_{1}$ and $\bar{\xi}_{2}$ real numbers, satisfying the inequalities:

x_{i_{0}}-\varepsilon_{i_{0}}<\bar{\xi}_{1}<x_{i_{0}}<\bar{\xi}_{2}<x_{i_{0}}+\varepsilon_{i_{0}}.

Whether $\bar{\varepsilon}$ a sufficiently small positive number so that the inequalities are satisfied (fig. 6):

	$\displaystyle\bar{\varepsilon}\bar{\mu}\left(\overline{\xi_{1}}\right)<y_{0}\left(\bar{\xi}_{1}\right)$
	$\displaystyle\bar{\varepsilon}\bar{\mu}\left(\overline{\xi_{2}}\right)<y_{0}\left(\overline{\xi_{2}}\right)$		(27)

Developing functions $y_{0}(x)$ and $\varepsilon\bar{\mu}(x)$ according to Taylor's formula at the point $x_{i_{0}}$ and taking into account the relations (24'), (26), (25') as well as the fact that $\pi_{i_{0}}>2$ , it is found that whatever $\varepsilon$ positive, in a sufficiently small subneighborhood of the point $x_{i_{0}}$ , the curve $y=y_{0}(x)$ will be below the curve $y=\varepsilon\bar{\mu}(x)$ (fig. 6). T, Taking into account this observation, as well as the inequalities (27), it follows that for any $\varepsilon$ positive, satisfying the inequality $\varepsilon<\bar{\varepsilon}=E_{3}$ , the equation curves $y=y_{0}(x)$ and $y=\varepsilon\bar{\mu}(x)$ will intersect at two distinct points in $\left(\bar{\xi}_{1},\bar{\xi}_{2}\right)$ , crossing each other at these points and also presenting at the point $x_{i_{0}}$ a contact of order 1 (fig. 6). In conclusion, taking into account what was shown previously, it results that if $\varepsilon$ satisfies the inequality

0<\varepsilon<\min\left\{E_{1},E_{2},E_{3}\right\},

then the integral $\tilde{y}_{\varepsilon}(x)=y_{0}(x)-\varepsilon\bar{\mu}(x)$ will have in the interval ( $a,b$ ) roots, which come as follows:

To each root $x_{i}$ of even order, of the integral $y_{0}(x)$ , except for the root $x_{i_{0}}$ , $\hat{i}$ and will correspond to the integral $\widetilde{y}_{e}(x)$ two distinct roots $\bar{x}_{i}$ and $\bar{x}_{i}$ .

root $x_{i_{0}}$ , multiple of even order $\geqq 4$ of the integral $y_{0}(x)$ , will correspond to the integral $\tilde{y_{\mathrm{s}}}(x)$ a double root $x_{i_{0}}$ and two other simple roots $\bar{x}_{i_{0}},\bar{x}_{i_{0}}$ .

Finally, the roots $x_{j}$ odd (simple) order of the integral $y_{0}(x)$ are simple roots and for $\tilde{y}_{\varepsilon}(x)$ .

It ultimately follows that the integral $\widetilde{y_{\epsilon}}(x)$ will have in the interval $(a,b)$ a root of order 2 and others $2(\alpha-1)+2+\beta=2\alpha+\beta$ simple roots. But since $2\alpha+\beta=n-1$ , which is deduced from (16) and (17), taking into account properties (19) and (20) it results that $\tilde{y}_{6}(x)$ has in the interval ( $a,b$ ) at least $(n-1)+1=n$ distinct roots (one of which is double). Then, since by hypothesis the family $Y_{n}$ has the property $I_{n}(a,b)$ , it follows from Lemma 2 that $\tilde{y}_{\varepsilon}(x)\equiv 0$ in the interval $(a,b)$ This identity, however, contradicts the fact that $\tilde{y}_{\varepsilon}(x)=y_{0}(x)-\varepsilon\bar{\mu}(x)$ does not cancel in additional nodes $\xi_{1},\xi_{2},\ldots,\xi_{v}$ , whose number $\nu$ , as shown previously, satisfies the inequality $\nu\geqq 2$ .

It ultimately follows that inequality (21) cannot hold and according to property (20), the equality must hold $\pi_{i_{0}}=p_{i_{0}}$ , what

We will now move on to the actual proof of Theorem 1.

Proof of Theorem 1.

To simplify the exposition, we first give the following definitions:
Definition 4. We say that the family $Y_{n}$ has the property $I_{n}^{(k)}(a,b)$ (where $k$ is a natural number satisfying the inequality $k\leqq n$ ), if that family $Y_{n}$ has the properties $I_{p_{1}},p_{2},\ldots,p_{m}(a,b)$ relating to all natural number systems $m,p_{1},p_{2},\ldots,p_{m}$ , which satisfy the conditions:

	$\displaystyle p_{1}+p_{2}+\ldots+p_{m}=n$
	$\displaystyle\max\left(p_{1},p_{2},\ldots,p_{m}\right)=k.$		(28)

Definition 5. We say that the family $Y_{n}$ own the property $P_{n}^{(k)}(a,b)$ (where $k$ is a natural number satisfying the inequality $k\leqq n-1$ ), if any non-identical integral of the equation is zero, which in m distinct points $x_{1},x_{2},\ldots,x_{m}$ FROM $(a,b)$ satisfies the conditions:

	$\displaystyle y\left(x_{1}\right)=y^{\prime}\left(x_{1}\right)=\ldots=y^{\left(p_{1}-1\right)}\left(x_{1}\right)=0$
	$\displaystyle y\left(x_{2}\right)=y^{\prime}\left(x_{2}\right)=\ldots=y^{\left(p_{2}-1\right)}\left(x_{2}\right)=0$		(29)
	$\displaystyle y\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots$
	$\displaystyle y\left(x_{m}\right)=y^{\prime}\left(x_{m}\right)=\ldots=y^{\left(p_{m}-1\right)}\left(x_{m}\right)=0$

where $p_{1},p_{2},\ldots,p_{m}$ are arbitrary natural numbers, satisfying the conditions

	$\displaystyle p_{1}+p_{2}+\ldots+p_{m}=n-1$
	$\displaystyle\max\left\{p_{1},p_{2},\ldots,p_{m}\right\}=k$		(30)

then from the realization of conditions (29) and (30) it results

y^{\left(p_{1}\right)}\left(x_{1}\right)\neq 0,\quad y^{\left(r_{2}\right)}\left(x_{2}\right)\neq 0,\ldots,y^{\left(p_{m}\right)}\left(x_{m}\right)\neq 0.

(31)

To prove Theorem 1, we will show step by step that under the assumptions of this theorem, the family $Y_{n}$ has the properties

	$\displaystyle\left\{I_{n}^{(1)}(a,b),\quad P_{n}^{(1)}(a,b)\right\},\quad\left\{I_{n}^{(2)}(a,b),\quad P_{n}^{(2)}(a,b)\right\},\ldots,\left\{I_{n}^{(k)}(a,b),\quad P_{n}^{(k)}(a,b)\right\}$
	$\displaystyle\ldots\ldots,\quad\left\{I_{n}^{(n-1)}(a,b),\quad P_{n}^{(n-1)}(a,b)\right\},I_{n}^{(n)}(a,b).$		(32)

For this purpose, we use the principle of induction relative to the upper index $k$ First of all, we note that for the family $Y_{n}$ , properties $\left\{I_{n}^{(1)}(a,b)\right.$ , $\left.P_{n}^{(1)}(a,b)\right\},\left\{I_{n}^{(2)}(a,b),P_{n}^{(2)}(a,b)\right\}$ , which corresponds to the values $k=1$ and $k=2$ , are true based on the statements of Lemmas 4, 5 and 6. We now assume that for the family $Y_{n}$ the following properties occur:

\left\{I_{n}^{(1)}(a,b),P_{n}^{(1)}(a,b)\right\},\left\{I_{n}^{(2)}(a,b),P_{n}^{(2)}(a,b)\right\},\ldots,\left\{I_{n}^{(k-1)}(a,b),P_{n}^{(k-1)}(a,b)\right\}

(33)

which correspond respectively to the values $1,2,\ldots,k-1$ of the upper index. Let us prove that in this hypothesis, the family $Y_{n}$ also has the properties $I_{n}^{(k)}(a,b)$ and $P_{n}^{(k)}(a,b)$ We will assume, in what follows, $k>2$ We begin by establishing the following lemma:

Le ma 7. If the family $Y_{n}$ has all the properties indicated in (33), then that family also has the property $I_{n}^{(k)}(a,b)$ .

Demonstration. To establish the property $I_{n}^{(k)}(a,b)$ , we will use the proof procedure indicated when establishing Lemma 5.

Let us suppose by absurdity that in the hypotheses of lemma 7, the family $Y_{n}$ would not have the property $I_{n}^{(k)}(a,b)$ Then, according to Lemma 1, it would follow that the differential equation (1) admits at least one non-zero non-identical integral $y_{0}(x)$ , which has in the interval ( $a,b$ ), $m$ distinct roots $x_{1},x_{2},\ldots,x_{m}$ , having multiplicity orders $\pi_{1},\pi_{2},\ldots,\pi_{m}$ , so that

\pi_{1}\geqq p_{1},\quad\pi_{2}\geqq p_{2},\ldots,\pi_{m}\geqq p_{m},

(34)

where $p_{1},p_{2},\ldots,p_{m}$ represent natural numbers, satisfying conditions (28).

We note from the beginning that to prove the property $I_{n}^{(k)}(a,b)$ , we can assume that at least two of the numbers $p_{1},p_{2},\ldots,p_{m}$ are equal to the number $k$ Indeed, from the hypothesis $\max\left\{p_{1},p_{2},\ldots,p_{m}\right\}=k$ , which intervenes in the definition of the property $I_{n}^{(k)}(a,b)$ , it follows that at least one of the numbers $p_{1},p_{2},\ldots,p_{m}$ is equal to $k$ . Then if only one of the numbers $p_{1},p_{2},\ldots,p_{m}$ would be equal to $k$ , and all the others would be smaller than $k$ , then the existence of an integral $y_{0}(x)$ non-zero, satisfying conditions (29), is in contradiction with the property $P_{n}^{(k-1)}(a,b)$ , also assumed to be true by hypothesis. In conclusion, to prove the property $I_{n}^{(k)}(a,b)$ , we can therefore assume that at least two of the numbers $p_{1},p_{2},\ldots,p_{m}$ are equal to $k$ .

We will associate numbers $p_{1},p_{2},\ldots,p_{m}$ respectively the numbers $p_{1}^{*},p_{2}^{*},\ldots p_{m}^{*}$ as follows:

p_{i}^{*}=\begin{cases}p_{i},&\text{ dacă }\pi_{i}\text{ şi }p_{i}\text{ sînt de aceeaşi paritate }\\ p_{i}+1,&\text{ dacă }\pi_{i}\text{ şi }p_{i}\text{ nu sînt de aceeaşi paritate }\\ &(i=1,2,\ldots,m)\end{cases}

We further consider the sequence of numbers $p_{1}^{**},p_{2}^{**},\ldots,p_{m}^{**}$ , as follows:

p_{i}^{**}=\left\{\begin{array}[]{ll}p_{i}^{*}-2,&\text{ dacă }p_{i}^{*}>1\\ p_{i}^{*},&\text{ dacă }p_{i}^{*}=1\end{array}\quad(i=1,2,\ldots,m)\right.

Obviously, the number sequences $\left\{\pi_{1},\pi_{2},\ldots,\pi_{m}\right\},\quad\left\{p_{1}^{*},p_{2}^{*},\ldots,p_{m}^{*}\right\}$ , $\left\{p_{1}^{**},p_{2}^{**},\ldots,p_{m}^{**}\right\}$ are such that their terms of the same rank have the same parity, and the last row may also contain the number zero.

Since, by hypothesis, at least two of the numbers $p_{1},p_{2},\ldots,p_{m}$ are equal to $k$ , and since also by hypothesis $k>2$ , it follows, taking into account (35) and (36) that for at least two indices $i$ the strict inequality occurs $p_{i}^{**}<p_{i}$ . Hence, taking into account the inequalities $p_{i}^{**}\leqq p_{i}(i=1,2,\ldots,m)$ , which also results from (35) and (36), and then from the first relation in (28), the inequality results:

\sum_{i=1}^{m}p_{i}^{**}\leqq n-2

(37)

We will divide the set of numbers $p_{1}^{**},p_{2}^{**},\ldots,p_{m}^{**}$ into two subsets as follows: we will denote by $i_{1},i_{2},\ldots,i_{a}$ those indices $i$ for which $p_{i}^{**}$ takes the value zero, and with $j_{1},j_{2},\ldots,j_{\beta}$ , clues $j$ for which $p_{j}^{**}$ takes values greater than zero (if of course such indices exist).

Let's note with $\nu$ , the sum of these numbers $p_{j}^{**}$ Taking into account (37), the inequality obviously results

v=p_{j_{1}}^{**}+p_{j_{2}}^{**}+\ldots+p_{j_{\beta}}^{**}\leqq n-2

(

\prime

)

Without restricting the generality of the proof, we can assume that the roots $x_{1},x_{2},\ldots,x_{m}$ of the integral $y_{0}(x)$ , are consecutive and satisfy the inequalities $x_{1}<x_{2}<\ldots<x_{m}$ (this is because the natural number $m$ it can be any).

We will choose in the interval ( $x_{m},b$ ), $n-v-1$ distinct nodes $\xi_{1},\xi_{2},\ldots,\xi_{n-\nu-1}$ , so that none of these nodes coincides with any root of the integral $y_{0}(x)$ , which would possibly be in the interval ( $x_{m},b$ ). Taking into account ( $37^{\prime}$ ), it is found that the number $n-v-1$ which represents the number of additional nodes $\xi_{1},\xi_{2},\ldots,\xi_{n-\nu-1}$ , is greater than or at least equal to 1, so the set of these nodes is not empty.

Let us now consider the roots $x_{j_{1}},x_{j_{2}},\ldots,x_{j_{\beta}}$ , corresponding to the numbers $p_{j_{1}}^{**},p_{j_{2}}^{**},\ldots,p_{j_{\beta}}^{**}$ previously highlighted, and either $\eta(x)$ a non-zero non-identical integral of equation (1), which verifies the conditions:

	$\displaystyle\eta\left(x_{j_{1}}\right)=\eta^{\prime}\left(x_{j_{1}}\right)=\ldots\ldots=\eta^{\left(p_{j_{1}-1}^{**}\right)}\left(x_{j_{1}}\right)=0$		(38)
	$\displaystyle\eta\left(x_{j_{2}}\right)=\eta^{\prime}\left(x_{j_{2}}\right)=\ldots\ldots=\eta^{\left(p_{j_{2}-1}^{**}\right)}\left(x_{j_{2}}\right)=0$
	$\displaystyle\cdots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\left(p_{j_{\beta}^{}-1}^{*}\left(x_{j_{\beta}}\right)=0\right.$
	$\displaystyle\eta\left(x_{j_{\beta}}\right)=\eta^{\prime}\left(x_{j_{\beta}}\right)=\ldots\ldots\ldots\left(\xi_{n-\nu-1}\right)=0$
	$\displaystyle\eta\left(\xi_{1}\right)=\eta\left(\xi_{2}\right)=\ldots\ldots\ldots$

conditions

in number of $n-1$ The existence of such a non-identically zero integral results from the hypothesis that $Y_{n}$ has all the properties $I_{n}^{(1)}(a,b),I_{n}^{(2)}(a,b),\ldots,I_{n}^{(k-1)}(a,b)$ which appears in (33) and from the fact that the multiplicity orders $p_{j_{1}}^{**},p_{j_{2}}^{**},\ldots,p_{j_{\beta}}^{**}$ which occur under conditions (38), satisfy the inequality

\max\left\{p_{j_{1}}^{**},p_{j_{2}}^{**},\ldots,p_{j_{m}}^{**}\right\}\leqq k-1

(39)

which results from (35), (36) and (28).
Since by hypothesis the properties are true $P_{n}^{(1)}(a,b)$ , $P_{n}^{(2)}(a,b),\ldots,P_{n}^{(k-1)}(a,b)$ , from relation (39) it follows that the integral $\eta(x)$ , assumed non-identically null, satisfies the relations:

	$\displaystyle\eta^{\left(p_{j_{1}}^{}\right)}\left(x_{j_{1}}\right)\neq 0,\quad\eta^{\left(p_{j_{2}}^{}\right)}\left(x_{j_{2}}\right)\neq 0,\ldots,\eta^{\left(p_{j_{\beta}}^{**}\right)}\left(x_{j_{\beta}}\right)\neq 0$		(40)
	$\displaystyle\eta^{\prime}\left(\xi_{1}\right)\neq 0,\quad\eta^{\prime}\left(\xi_{2}\right)\neq 0,\ldots,\eta^{\prime}\left(\xi_{n-\nu-1}\right)\neq 0$

It also follows that the only distinct roots in $(a,b)$ of the integral $\eta(x)$ there are numbers $\left.x_{j_{1}},x_{j_{2}},\ldots,x_{j_{\beta}};\xi_{1},\xi_{2},\ldots,\xi_{n-\nu-1\cdot}{}^{5}\right)$ Taking into account this last observation, then the relations (38), considered together with (40), as well as the observation made previously that the numbers $p_{1}^{**},p_{2}^{**},\ldots,p_{m}^{**}$

⁰⁰footnotetext: ⁵ ) This statement also results from the hypothesis that the family

Y_{n}

has the properties

I_{n}^{(1)}(a,b),\ldots,I_{n}^{(k-1)}(a,b)

are respectively of the same parity as the numbers $\pi_{1},\pi_{2},\ldots,\pi_{m}$ , it follows that at least one of the integrals $\eta(x)$ or $-\eta(x)$ , will have the same sign as $y_{0}(x)$ , in sufficiently small neighborhoods of the points $x_{1},x_{2},\ldots,x_{m}$ , possibly with the exception of the means of these neighborhoods. We will denote by $\bar{\eta}(x)$ that of the integrals $\eta(x)$ and $-\eta(x)$ , which satisfies this desire.

Next, we will note with $(\Gamma)$ equation curve $y=y_{0}(x)$ and with $\left(\Gamma_{\varepsilon}\right)$ equation curve $y=\varepsilon\bar{\eta}(x)$ , where $\varepsilon$ is a parameter. We propose to examine how the curves ( $\Gamma$ ) and ( $\Gamma_{\varepsilon}$ ), when $\varepsilon$ tends to zero through positive values.

We note from the beginning that whatever $\varepsilon>0$ , integral curves ( $\Gamma$ ) and ( $\Gamma_{\varepsilon}$ ) cannot coincide identically in the interval ( $a,b$ ), since in the additional nodes $\xi_{1},\xi_{2},\ldots,\xi_{n-v-1}$ (whose number is greater than or equal to 1, as shown previously) the integral $\bar{\eta}(x)$ is canceled, while $y_{0}(x)$ takes values other than zero.

Referring first to the numbers $x_{i_{1}},x_{i_{2}},\ldots,x_{i_{\alpha}}$ , we find that they represent for the function $y_{0}(x)$ roots of even order. This statement is immediately justified, taking into account that $p_{i_{1}}^{**}=p_{i_{2}}^{**}=\ldots=p_{i_{\alpha}}^{**}=0$ , as well as (35) and (36). On the contrary, the function $\bar{\eta}(x)$ does not cancel at any of the points $x_{i_{1}},x_{i_{2}},\ldots,x_{i_{\alpha}}$ . By the way the integral was chosen $\bar{\eta}(x)$ between the integrals $\eta(x)$ and - $\eta(x)$ , it follows that in sufficiently small neighborhoods of the points $x_{i_{1}},x_{i_{2}},\ldots,x_{i_{\alpha}}$ , but with the exception of the means of these neighborhoods, the relationship takes place

\operatorname{sgn}\{\bar{\eta}(x)\}=\operatorname{sgn}\left\{y_{0}(x)\right\}.

From these findings it follows that if the parameter $\varepsilon$ is positive and lower than a certain threshold, then in each of the respective neighborhoods of the numbers $x_{i_{1}},x_{i_{2}},\ldots,x_{i_{\alpha}}$ , the curves $(\Gamma)$ and $\left(\Gamma_{\varepsilon}\right)$ will intersect in two distinct points. We will denote the abscissas of these intersection points, respectively, with $\bar{x}_{i_{1}},\bar{x}_{i_{1}},\bar{x}_{i_{2}},\overline{\bar{x}}_{i_{2}},\ldots,\bar{x}_{i_{\alpha}},\overline{\bar{x}}_{i_{\alpha}}$ .

Then, referring to the roots $x_{j_{1}},x_{j_{2}},\ldots,x_{j_{\beta}}$ , we will divide them into two subsets as follows. We will denote by $x_{j_{k_{1}}},x_{j_{k_{2}}},\ldots,x_{j_{k_{\gamma}}}$ those roots $x_{j}$ , for which the number $\pi_{j}$ correspondingly is equal to 1 , and with $x_{j_{l_{1}}},x_{j_{l_{2}}},\ldots,x_{j_{l_{\delta}}}$ those, for which the number $\pi_{j}$ correspondingly is greater than 1 .

We observe, taking into account (35) and (36), that the following equalities hold:

	$\displaystyle\pi_{j_{k_{1}}}=p_{j_{k_{1}}}=p_{t_{k_{1}}}^{}=p_{j_{k_{1}}}^{*}=1$
	$\displaystyle\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots$		(41)
	$\displaystyle\pi_{j_{k_{\gamma}}}=p_{j_{k_{\gamma}}}=p_{j_{k_{\gamma}}}^{}=p_{j_{k_{\gamma}}}^{*}=1$

From here and from (40), it follows that the numbers $x_{j_{k_{1}}},x_{j_{k_{2}}},\ldots,x_{j_{k_{\gamma}}}$ represents for $y_{0}(x)$ and $\bar{\eta}(x)$ simple roots, and therefore that $y_{0}^{\prime}(x)$ and $\bar{\eta}^{\prime}(x)$ take non-zero values for each of them. It follows from this that if the parameter $\varepsilon>0$ is below a certain threshold, then the curves ( $\Gamma$ ) and ( $\Gamma_{\varepsilon}$ ) will intersect at the points $x_{j_{k_{1}}},x_{j_{k_{2}}},\ldots,x_{j_{k_{\delta}}}$ crossing each other at these points.

Finally, referring to the remaining roots $x_{j_{l_{1}}},x_{j_{l_{2}}},\ldots,x_{j_{l_{\delta}}}$ , we observe, taking into account (35) and (36), that for them the relations hold:

0<p_{j_{1}}^{**}\leqq\pi_{j_{l_{1}}}-2;\quad 0<p_{j_{l_{2}}}^{**}\leqq\pi_{j_{j_{2}}}-2;\ldots,0<p_{j_{l_{\delta}}}^{**}\leqq\pi_{j_{l_{\delta}}}-2.

which shows us (taking into account (38)), that each of the numbers $x_{j_{l_{1}}},x_{j_{l_{2}}},\ldots,x_{j_{l_{\delta}}}$ represents a root for both the function $\bar{\eta}(x)$ as well as for $y_{0}(x)$ - and that the order of any one of these roots, with respect to the function $\bar{\eta}(x)$ , is at least two units smaller than the order of the same root relative to the function $y_{0}(x)$ On the other hand, as previously specified, the numbers $p_{j_{1}l_{1}}^{**},p_{j_{2}}^{**},\ldots,p_{j_{l_{\delta}}}^{**}$ are respectively of the same parity as the numbers $\pi_{j_{l_{1}}},\pi_{j_{l_{2}}},\ldots,\pi_{j_{l_{\delta}}}$ From these findings, the following results emerge:
$1^{\circ}$ Whatever the value of the parameter $\varepsilon$ , functions $y_{0}(x)$ and $\varepsilon\bar{\eta}(x)$ take equal values at the points $x_{j_{l_{1}}},x_{j_{l_{2}}},\ldots,x_{j_{l_{\delta}}}$ , the coincidence taking place respectively up to their derivatives of order

\begin{gathered}p_{j_{1}}^{**}-1,p_{j_{2}}^{**}-1,\ldots\\ \ldots,p_{j_{l}}^{**}-1\end{gathered}

So the curves $(\Gamma)$ and $\left(\Gamma_{\varepsilon}\right)$ will present at these points, contacts of tangency and order $p_{j_{1}}^{**}-1,p_{j_{2}}^{**}-1,\ldots,p_{j_{l_{\delta}}}^{**}-1$ .
$2^{\circ}$ If the parameter $\varepsilon>0$ is small enough, then the functions $y_{0}(x)$ and $\varepsilon\bar{\eta}(x)$ will take equal values in more $2\delta$ points, located two by two in sufficiently small neighborhoods of the numbers $x_{j_{1}},x_{j_{l_{2}}},\ldots,x_{j_{l_{\delta}}}$ We will note the abscissas of these intersection points, respectively with $\bar{x}_{j_{l_{1}}},\bar{x}_{j_{l_{1}}},\bar{x}_{j_{l_{2}}},\overline{\bar{x}}_{j_{l_{2}}},\ldots,\bar{x}_{j_{\delta}},\overline{\bar{x}}_{j_{\delta}}$ (fig. 7 or 8). This statement is based on the following lemma:

Le ma 8. If two functions $f(x)$ and $g(x)$ defined in an interval $(a,b)$ , have the following properties:
$1^{\circ}$ . admit in $(a,b)$ continuous derivatives of the order $p$ respectively $p+2k$ , where $k\geqq 1$ ,
$2^{\circ}$ . possesses in the interval ( $a,b$ ) a common root $x_{0}$ , which for $f(x)$ is a multiple of the order $p$ , and for $g(x)$ is a multiple of the order $p+2k$ ,
$3^{\circ}.\operatorname{sgn}\left\{f^{(p)}\left(x_{0}\right)\right\}=\operatorname{sgn}\left\{g^{(p+2k)}\left(x_{0}\right)\right\}\neq 0$ .
Under these conditions, given a neighborhood ( $x_{0}-\delta,x_{0}+\delta$ ), contained in the interval ( $a,b$ ), for this neighborhood there is a threshold $E>0$ , so that if the parameter e satisfies the inequalities $0<\varepsilon<E$ , then the functions $\varepsilon/(x)$ and $g(x)$ take common values in at least 3 distinct points in the vicinity $\left(x_{0}-\delta,x_{0}+\delta\right)$ considered. One of these points is $x_{0}$ , in which the corresponding curves have a tangent contact of the order of $p-1$ , and at the other two intersection points the curves cross each other.

The proof of this lemma is easy, developing the functions $\varepsilon f(x)$ and $g(x)$ with Taylor's formula at the point $x_{0}$ .

Returning to the proof of Lemma 7, let us consider the function $\widetilde{y}_{\mathbf{e}}(x)=y_{0}(x)-$ - $\varepsilon\bar{\eta}(x)$ , which obviously represents an integral of the differential equation (1) and which is not identically zero in the interval ( $a,b$ ), since - as previously shown - the functions $y_{0}(x)$ and $\varepsilon\bar{\eta}(x)$ cannot coincide identically in the interval $(a,b)$ , whatever the value the parameter takes $\varepsilon$ . T, Taking into account the previous findings regarding the behavior of the curves $(\Gamma)$ and ( $\Gamma_{\varepsilon}$ ) between them, we arrive at the following conclusion based on Lemma 8. If the parameter $\varepsilon$ takes positive values, below a certain threshold $E$ , then the integral $\widetilde{y}_{\varepsilon}(x)$ it cancels out for the following distinct values ⁶ ) in the interval $(a,b)$ :

	$\displaystyle\bar{x}_{i_{1}},\overline{\bar{x}}_{i_{1}},\bar{x}_{i_{2}},\overline{\bar{x}}_{i_{2}},\ldots\ldots,\bar{x}_{i_{a}},\bar{x}_{i_{\alpha}}$
	$\displaystyle x_{j_{k_{1}}},x_{j_{k_{2}}},\ldots,x_{j_{k_{\gamma}}}$		(42)
	$\displaystyle\bar{x}_{j_{l_{1}}},\bar{x}_{j_{l_{1}}},\bar{x}_{j_{l_{2}}},\bar{x}_{j_{l_{2}}},\ldots,\bar{x}_{j_{l_{\delta}}},\bar{x}_{j_{l_{\delta}}}$
	$\displaystyle x_{j_{l_{1}}},x_{j_{l_{2}}},\ldots,x_{l_{l_{\delta}}}$

The roots that appear in the last row of (42) have relative to the function $\tilde{y}_{\varepsilon}(x)$ , respectively the orders $p_{j_{l_{1}}}^{**},p_{j_{l_{2}}}^{**},\ldots,p_{j_{l_{\delta}}}^{**}$ . T, taking into account (39), it follows that these orders are at most equal to the number $k-1$ .

It is then found that the total number of conditions for the integral to be cancelled is $\tilde{y}_{k}(x)$ , corresponding to the roots of (42), is greater than or at least equal to the sum:

N=2\alpha+\gamma+2\delta+p_{j_{l_{1}}}^{**}+p_{j_{l_{2}}}^{**}+\ldots+p_{j_{l_{\delta}}}^{**}

(43)

⁰⁰footnotetext: ⁶ ) The values written in the first and third lines depend on

\varepsilon

We will show that this sum is greater than or at least equal to n. Indeed, taking into account the fact that the numbers $p_{i_{1}}^{**},p_{i_{2}}^{**},\ldots,p_{i_{\alpha}}^{**}$ are all equal to zero (by hypothesis), it follows from (36) that the numbers $p_{i_{1}}^{*},p_{i_{2}}^{*},\ldots,p_{i_{\alpha}}^{*}$ are all equal to 2, and then from (35) - that the numbers $p_{i_{1}},p_{i_{2}},\ldots,p_{i_{\alpha}}$ are less than or at most equal to 2. From this last statement the inequality results

2\alpha\geqq p_{i_{1}}+p_{i_{2}}+\ldots+p_{i_{\alpha}}.

(44)

Next, from (41) the equality results:

\gamma=p_{j_{k_{1}}}+p_{j_{k_{2}}}+\ldots+p_{j_{k_{\gamma}}}

(45)

and finally, from (36) the inequalities result:

p_{j_{l_{1}}}^{**}\geqq p_{j_{l_{1}}}^{*}-2,\quad p_{j_{l_{2}}}^{**}\geqq p_{j_{l_{2}}}^{*}-2,\ldots,p_{j_{l_{\delta}}}^{**}\geqq p_{j_{l_{\delta}}}^{*}-2.

(46)

Adding the inequalities (46) together, we obtain:

2\delta+p_{j_{l_{1}}}^{**}+p_{j_{l_{2}}}^{**}+\ldots+p_{j_{l_{\delta}}}^{**}\geqq p_{j_{l_{1}}}^{*}+p_{j_{l_{2}}}^{*}+\ldots+p_{j_{l_{\delta}}}^{*}

(47)

Then, taking into account the inequalities $p_{i}^{*}\geqq p_{i}(i=1,2,\ldots,m)$ , which results from (35), the inequality is deduced from (47):

2\delta+p_{j_{1}}^{**}+p_{j_{1}}^{**}+\ldots+p_{j_{l}}^{**}\geqq p_{j_{l_{1}}}+p_{j_{l_{2}}}+\ldots+p_{j_{l_{\delta}}}.

(48)

Adding the inequalities (44), (45), (48) member by member, we obtain the inequality:

	$\displaystyle N\geqq\left(p_{i_{1}}+p_{i_{2}}+\cdots+p_{i_{\alpha}}\right)+\left(p_{j_{k_{1}}}+p_{j_{k_{2}}}+\ldots+p_{j_{k_{\gamma}}}\right)+$
	$\displaystyle+\left(p_{j_{l_{1}}}+p_{j_{l_{2}}}+\ldots+p_{j_{l_{\delta}}}\right).$		(49)

But how among the clues $i_{1},i_{2},\ldots,i_{\alpha};j_{k_{1}},j_{k_{2}},\ldots,j_{k_{\gamma}};j_{l_{1}},j_{l_{2}},\ldots,j_{l_{\delta}}$ , each of the clues appears $1,2,\ldots,m$ , once and only once, it follows that the expression in the second member of inequality (49) represents the sum of the numbers $p_{i}$ corresponding to all roots $x_{1},x_{2},\ldots,x_{m}$ considered, of the integral $y_{0}(x)$ .

Thus inequality (49) is transcribed:

N\geqq p_{1}+p_{2}+\ldots+p_{m}.

(

\prime

)

However, according to the first relation in (28), it follows that the sum on the right-hand side of the inequality ( $49^{\prime}$ ) is equal to the number $n$ , so that inequality (49) becomes $N\equiv n$ , or, taking into account (43):

N=2\alpha+\gamma+2\delta+p_{j_{l_{1}}}^{**}+p_{j_{l_{2}}}^{**}+\ldots+p_{j_{\delta}}^{**}\geqq n.

(50)

Ultimately, the following result is obtained:
The integral is not identically zero $\widetilde{y}(x)$ of the differential equation (1), vanishes in the interval ( $a,b$ ), at least for the values indicated in table (42) - all
these values being distinct. The roots $x_{j_{l_{1}}},x_{j_{l_{2}}},\ldots,x_{j_{l_{\delta}}}$ which appear in the last row of (42), have relative to the integral $\widetilde{y}_{\mathrm{e}}(x)$ , the multiplicity orders, respectively $p_{j_{l_{1}}}^{**},p_{j_{l_{2}}}^{**},\ldots,p_{j_{l_{\delta}}}^{**}$ , satisfying the inequality

\max\left\{p_{j_{l_{1}}}^{**},p_{j_{l_{2}}}^{**},\ldots,p_{j_{l_{\delta}}}^{**}\right\}\leqq k-1

which results from (39). The other roots of the integral $\widetilde{y}_{\varepsilon}(x)$ , which appear in table (42), have orders greater than or at least equal to 1. Inequality (50) also occurs.

Taking into account Lemma 1, these results obtained regarding the integral $\tilde{y}_{e}(x)$ are in contradiction with the hypothesis that the family $Y_{n}$ own the properties $I_{n}^{(1)}(a,b),I_{n}^{(2)}(a,b),\ldots,I_{n}^{(k-1)}(a,b)$ .

It ultimately follows that if the family $Y_{n}$ has the properties (33), then $Y_{n}$ also has the property $I_{n}^{(k)}(a,b)$ .

Next, we will prove the following lemma:
Le ma 9. If the family $Y_{n}$ has all the properties indicated in (33), then that family also has the property $P_{n}^{(k)}(a,b)$ .

Demonstration. Let $y_{0}(x)$ a non-identical integral zero in $(a,b)$ , of the differential equation (1), which in $m$ distinct points $x_{1},x_{2},\ldots,x_{m}$ from the interval ( $a,b$ ), satisfies conditions (29) and (30). Let us denote by $\pi_{1},\pi_{2},\ldots,\pi_{m}$ multiplicity orders of the roots $x_{1},x_{2},\ldots,x_{m}$ of the integral $y_{0}(x)$ We want to prove that under the assumption that $Y_{n}$ has the properties (33), the relations (31) hold, i.e. the equalities

\pi_{1}=p_{1},\pi_{2}=p_{2},\ldots,\pi_{m}=p_{m}

We note from the beginning that the integral $y_{0}(x)$ considered, cannot have in the interval ( $a,b$ ) other roots, besides $x_{1},x_{2},\ldots,x_{m}$ , since otherwise the statement of Lemma 7 would be contradicted. We will assume in the following that $x_{1}<x_{2}<\ldots<x_{m}$ .

With these clarifications, we will first demonstrate that:

\left.\begin{array}[]{l}\text{ In ipotezele adoptate, numerele }\pi_{1},\pi_{2},\ldots,\pi_{m}\quad\text{ şi }\\ p_{1},p_{2},\ldots,p_{m}\text{ sînt respectiv de aceeaşi paritate. }\end{array}\right\}

For this purpose, we consider the integral $\widetilde{y}_{\varepsilon}(x)=y_{0}(x)-\varepsilon\bar{\eta}(x)$ , where $y_{0}(x)$ is the integral chosen previously, and $\bar{\eta}(x)$ is also an integral of the differential equation (1), constructed relative to the roots $x_{1},x_{2},\ldots,x_{m}$ of the integral $y_{0}(x)$ , following the procedure used previously in the proof of Lemma 7. We will, however, take into account the following differences regarding the integral $y_{0}(x)$ , which occur in the case of the present lemma.

First, it is worth noting that instead of the first equality in (28), in the present case the equality is considered $p_{1}+p_{2}+\ldots+p_{m}=n-1$ , as indicated
in (30). Then, from the second relation (30) it follows that at least one of the numbers $p_{1},p_{2},\ldots,p_{m}$ is equal to the number $k$ Here, unlike the case of Lemma 7, we will also consider the case when only one of the numbers $p_{1},p_{2},\ldots,p_{m}$ is equal to the number $k$ , all others being smaller than $k$ .

Since by hypothesis we have $k>2$ , it follows, taking into account (35) and (36), that for at least one of the indices $i$ , the strict inequality takes place $p_{i}^{**}<p_{i}$ . Hence, taking into account the inequalities $p_{i}^{**}\leqq p_{i},(i=1,2,\ldots,m)$ , which also results from (35) and (36), and then from the first relation in (30), we deduce the inequality $\sum_{i=1}^{m}p_{i}^{**}\leqq n-2$ , from which it results in $\bmod$ obviously the inequality:

v=p_{j_{1}}^{**}+p_{j_{2}}^{**}+\ldots+p_{j_{\beta}}^{**}\leqq n-2

(52)

This inequality is analogous to the inequality ( $37^{\prime}$ ), established during the proof of Lemma 7. Just as there, one chooses at will $n-v-1$ distinct nodes $\xi_{1}<\xi_{2}<\ldots<\xi_{n-v-1}$ from the interval ( $x_{m},b$ ). Taking into account (52), the number of these additional nodes satisfies the inequality $n-v-1\geqq 1$ For what follows, it is useful to point out that the integral $y_{0}(x)$ , cannot be canceled for any of the values $\xi_{1},\xi_{2},\ldots,\xi_{n-\nu-1}$ chosen, since - as shown previously - this integral does not have in the interval ( $a,b$ ) other roots besides $x_{1},x_{2},\ldots,x_{m}$ .

Next, we will consider a non-zero non-identical integral $\eta(x)$ , which satisfies the conditions (38) regarding the roots $x_{1},x_{2},\ldots,x_{m}$ of the integral $y_{0}(x)$ , and relating to the nodes $\xi_{1},\xi_{2},\ldots,\xi_{n-\nu-1}$ As in the case of Lemma 7, it is shown that under the adopted assumptions, at least one of the integrals $\eta(x)$ or $-\eta(x)$ will have the same sign as $y_{0}(x)$ in sufficiently small neighborhoods of the points $x_{1},x_{2},\ldots,x_{m}$ , possibly with the exception of the means of these neighborhoods. Noting with $\bar{\eta}(x)$ the one that satisfies this desire, we will consider $\tilde{y}_{\varepsilon}(x)=y_{0}(x)-\varepsilon\bar{\eta}(x)$ It is also found that whatever the value of the parameter $\varepsilon$ , functions $y_{0}(x)$ and $\varepsilon\bar{\eta}(x)$ cannot be identically equal in the interval ( $a,b$ ), and therefore the integral $\widetilde{y}_{\varepsilon}(x)$ cannot be identically zero in this interval. As in the proof of Lemma 7, it is shown that if the parameter $\varepsilon$ is positive and below a certain threshold, then the integral $\tilde{y}_{e}(x)$ will cancel out at some distinct points in the interval ( $a,b$ ), indicated in table (42). The roots appearing in the last row of (42), have relative to the integral $\tilde{y}_{\mathrm{e}}(x)$ , the orders of multiplicity $p_{j_{1}}^{**},p_{j_{1}}^{**},\ldots,p_{j_{1}}^{**}$ As follows from (39), these orders satisfy the relation

\max\left\{p_{j_{l_{1}}}^{**},p_{j_{l_{2}}}^{**},\ldots,p_{j_{l_{\delta}}}^{**}\right\}\leqq k-1

(53)

It is found that the number of conditions for the integral to be cancelled is $\widetilde{y}_{8}(x)$ , corresponding to the roots written in table 1 (42), is at least equal to the sum

N=2\alpha+\gamma+2\delta+p_{j_{1}}^{**}+p_{j_{1}}^{**}+\ldots+p_{j_{l}}^{**}

(54)

3 - Mathematics studies and research

It is shown, as in the proof of Lemma 7, that the inequalities (44), (45), (46), (47), (48), (49) hold, as well as the inequality

N\equiv p_{1}+p_{2}+\ldots+p_{m}

(55)

which is analogous to the inequality ( $49^{\prime}$ ). From this last inequality, taking into account the first relation in ( 30 ), we obtain the inequality

N\geqq n-1

(56)

We now proceed to the actual proof of property (51). We assume by the absurd converse that among the roots $x_{1},x_{2},\ldots,x_{m}$ of the integral $y_{0}^{\prime}(x)$ , there would be at least one root $x_{\tau}$ for which numbers $p_{\tau}$ and $\pi_{\tau}$ corresponding would be of different parity.

Taking into account the definition relation (35), it follows that

p_{\mathrm{r}}^{*}=p_{\mathrm{r}}+1

(57)

and since $p_{\tau}\geqq 1$ , it follows from (57) that $p_{\tau}^{*}\geqq 2$ , and then from (36) that

p_{\tau}^{**}=p_{\tau}^{*}-2=p_{\tau}-1

(58)

The following two cases can be presented, as follows: $p_{\tau}^{**}=0$ or $p_{\tau}^{**}>0$
Case 1 : $p_{\mathrm{r}}^{**}=0$ In this case, from (58) we deduce the equality

p_{\tau}=1

(59)

On the other hand, since I assumed that $p_{\tau}^{**}=0$ , it follows that the index $\tau$ represents one of the indicators $i_{1},i_{2},\ldots,i_{a}$ Then, taking into account the fact that the numbers $p_{i_{1}}^{**},p_{i_{2}}^{**},\ldots,p_{i_{\alpha}}^{**}$ are all equal to zero, it follows from (36) that the numbers $p_{i_{1}}^{*},p_{i_{2}}^{*},\ldots,p_{i_{\alpha}}$ are all equal to 2, and from (35) - that the numbers $p_{i_{1}},p_{i_{2}},\ldots$ …, $p_{i_{\alpha}}$ are less than or at least equal to 2 . From here, taking into account also the equality (59), the strict inequality results

2\alpha>p_{i_{1}}+p_{i_{2}}+\ldots+p_{i_{a}}

(60)

analogous to inequality (44), with the difference that instead of the sign $\geqq$ , in this case the sign of strict inequality appears.

Following the reasoning that led us from equality (43) to inequality (49'), we come to the conclusion that in the present case, due to strict inequality (60), in relation (49') the sign of strict inequality will appear, instead of the sign $\geqq$ . Thus, the strict inequality will finally be obtained $N>p_{1}+p_{2}+\ldots+p_{m}$ , from which, taking into account the first relation in (30), the inequality will result

N>n-1

(61)

which shows us that the number of cancellation conditions that the non-identical integral satisfies is zero $\tilde{y}_{\varepsilon}(x)$ , referring to the roots indicated in table (42), is greater than or at least equal to $n$ . At the same time, however, the relation (53) also holds
. According to Lemma 1, this situation is in contradiction with the properties $I_{n}^{(1)}(a,b),I_{n}^{(2)}(a,b),\ldots,I_{n}^{(k-1)}(a,b)$ of the family $Y_{n}$ , assumed true by hypothesis.

Case 2: $p_{r}^{**}\geqq 1$ In this case, from (58) we deduce that $p_{r}\geqq 2$ and hence, a fortiori, the inequality $\pi_{\tau}\geqq 2$ According to the previous statements, it follows that the index $\tau$ represents one of the indicators $j_{l_{1}},j_{l_{2}},\ldots,j_{l_{8}}$ . Then, in another order of ideas, taking into account the equality (57), as well as the inequalities $p_{i}^{*}\geqq p_{i}(i=1,2,\ldots,m)$ , which results from (35), we deduce by adding the inequalities (46) member by member, the following strict inequality:

2\delta+p_{j_{l_{1}}}^{**}+p_{j_{l_{2}}}^{**}+\ldots+p_{j_{l_{\delta}}}^{**}>p_{j_{l_{1}}}+p_{j_{l_{2}}}+\ldots+p_{j_{l_{\delta}}}

(62)

analogous to inequality (48).
Following the reasoning that led us from equality (43) to inequality (49'), we arrive at the conclusion that due to strict inequality (62), the strict inequality will take place instead of inequality (49'). $N>p_{1}+p_{2}++\ldots+p_{m}$ , and based on the first relation in (30)-the strict inequality $N>n-1$ This inequality shows us that the number of cancellation conditions that the non-identically zero integral satisfies is $\tilde{y}_{e}(x)$ , regarding the roots indicated in table (42), is at least equal to $n$ . But at the same time, the relation (53) also occurs. This circumstance is in contradiction with the properties $I_{n}^{(1)}(a,b)$ , $I_{n}^{(2)}(a,b),\ldots,I_{n}^{(k-1)}(a,b)$ of the family $Y_{n}$ , assumed true by hypothesis.

In conclusion, since the two cases exhaust all possible cases relative to $p_{\tau}^{**}$ , it follows from their examination that property (51), qed

Next, we consider a non-zero non-identical integral $y_{0}(x)$ , of equation (1), which in $m$ distinct points $x_{1},x_{2},\ldots,x_{m}$ from the interval ( $a,b$ ), satisfies conditions (29) and (30). We denote respectively by $\pi_{1},\pi_{2},\ldots,\pi_{m}$ multiplicity orders of the roots $x_{1},x_{2},\ldots,x_{m}$ of the integral $y_{0}(x)$ We will prove the following property:

If the family $Y_{n}$ has the properties (33), then under the assumption that the integral $y_{0}(x)$ satisfies conditions (29) and (30), it follows that this integral satisfies relations (31).

To demonstrate this property, let us assume absurdly that $y_{0}(x)$ would not satisfy the relations (31). Then, at least for one of the roots $x_{i}(i=1,2,\ldots,m)$ , the strict inequality will hold $\pi_{i}>p_{i}$ For the sake of clarity, we will assume that this root corresponds to index 1, i.e.

\pi_{1}>p_{1}

(68)

In these hypotheses, we note from the beginning that if $p_{1}<k$ , then, taking into account (68), we can consider with respect to the root $x_{1}$ , instead of the number $p_{1}$ , the number $p_{1}+1$ (which is less than or at most equal to $k$ ),
and by this substitution, the sum appearing in the first relation in (30) will be equal to $n$ Thus, according to Lemma 1, the property would be contradicted $I_{n}^{(k)}(a,b)$ of the family $Y_{n}$ , property ensured by Lemma 7.

The inequality ultimately results $p_{1}\geqq k$ , and since by hypothesis the second relation in (30) takes place, the equality results

p_{1}=k.

(69)

Taking into account relations (68) and (69), the strict inequality results $\pi_{1}>k$ and then, taking into account the property (51) established previously, the inequality will result

\pi_{1}\geqq k+2.

(70)

Regarding the number $k$ , we observe that the inequality must necessarily occur

k\leqq n-3,

(71)

whereas, otherwise, that is, in the case of $k>n-3$ , it would follow from (70) that $\pi_{1}>n-1$ and so that $x_{1}$ is a multiple root of order at least equal to $n$ for the integral $y_{0}(x)$ This result would, however, be incompatible with the hypothesis that $y_{0}(x)$ is not identically null in the interval ( $a,b$ ).

Regarding orders $\pi_{1},\pi_{2},\ldots,\pi_{m}$ , we will distinguish the following two cases.

\text{ Cazul }1:\pi_{1}\geqq k+2;\quad\pi_{2}=\pi_{3}=\ldots=\pi_{m}=1,\quad(2<k\leqq n-3).

In this case, at each of the points $x_{2},x_{3},\ldots,x_{m}$ equation curve $y=y_{0}(x)$ will cross the axis $Ox$ , since the function $y_{0}(x)$ changes sign at these points. Let $\varepsilon$ a nonzero real number and let $\mu_{\varepsilon}(x)$ the integral of equation (1), which satisfies at the point $x_{1}$ the following Cauchy conditions:

	$\displaystyle\mu_{\varepsilon}\left(x_{1}\right)=y_{0}\left(x_{1}\right)=0;\quad\mu_{\varepsilon}\left(x_{1}\right)=y_{0}^{\prime}\left(x_{1}\right)=0;\ldots$
	$\displaystyle\ldots,\mu_{\varepsilon}^{(k)}\left(x_{1}\right)=y_{0}^{(k)}\left(x_{1}\right)=0;\quad\mu_{\varepsilon}^{(k+1)}\left(x_{1}\right)=\varepsilon;$		(72)
	$\displaystyle\mu_{\varepsilon}^{(k+2)}\left(x_{1}\right)=y_{0}^{(k+2)}\left(x_{1}\right);\ldots,\quad\mu_{\varepsilon}^{(n-1)}\left(x_{1}\right)=y_{0}^{(n-1)}\left(x_{1}\right).$

Put into words, the integral $\mu_{\varepsilon}(x)$ satisfy at point $x_{1}$ the same Cauchy conditions as $y_{0}(x)$ , except for the derivative of order $k+1$ , which takes the value $\varepsilon\neq 0$ in the case of the function $\mu_{\varepsilon}(x)$ , while in the case of the integral $y_{0}(x)$ this derivative takes the value zero.

If the parameter $\varepsilon$ is small in absolute value, then the Cauchy conditions that the integral satisfies $\mu_{\varepsilon}(x)$ are close to the Cauchy conditions, which are satisfied by $y_{0}(x)$ , and if $\varepsilon$ tends to zero then $\mu_{\varepsilon}(x)$ will tend uniformly towards $y_{0}(x)$ in any closed interval $\left[a_{1},b_{1}\right]$ , contained in ( $a,b$ ).

Since the equation curve $y=y(x)$ crosses the axis $Ox$ in the points $x_{2},x_{3},\ldots,x_{m}$ , it follows that no matter how small the neighborhoods of these roots are chosen, there will be a threshold for them $E$ , so that whatever the number $\varepsilon$
satisfying inequalities $0<\varepsilon<E$ , integral curve $y=\mu_{\varepsilon}(x)$ corresponding to the number $\varepsilon$ thus chosen, to cross the axis $Ox$ in each of the chosen neighborhoods at one point. We will note the abscissas of these intersection points with $\bar{x}_{2},\bar{x}_{3},\ldots,\bar{x}_{m}$ In addition, as can be seen from relations (72), the integral $\mu_{\varepsilon}(x)$ still admits the root $x_{1}$ , with the order of multiplicity $k+1$ It follows therefore that the integral $\mu_{\varepsilon}(x)$ has in the interval ( $a,b$ ) root $\bar{x}_{1}=x_{1}$ multiple of the order $k+1$ and others $m-1$ distinct roots $\bar{x}_{2},\bar{x}_{3},\ldots,\bar{x}_{m}$ , different from $x_{1}$ and having odd orders of multiplicity. For these $m$ roots $\bar{x}_{1},\bar{x}_{2},\ldots,\bar{x}_{m}$ of the integral $\mu_{\varepsilon}(x)$ , conditions (29) and (30) are satisfied, relative to the integral $\mu_{\varepsilon}(x)$ with the numbers $\bar{p}_{1}=k,\bar{p}_{2}=\bar{p}_{3}=\ldots=\bar{p}_{m}=1$ , this is because the equalities occur $\bar{p}_{1}=p_{1},\bar{p}_{2}=p_{2},\ldots,\bar{p}_{m}=p_{m}$ , which results from (69), as well as from the hypothesis $\pi_{2}=\pi_{3}=\ldots=\pi_{m}=1$ , specific to the considered case. However, the multiplicity order of the root $x_{1}$ relative to the integral $\mu_{\varepsilon}(x)$ it is by hypothesis $\bar{\pi}_{1}=k+1$ , as shown by equality (72). It follows therefore that for the root $x_{1}$ of the integral $\mu_{\varepsilon}(x)$ , the numbers $\bar{p}_{1}=k$ and $\pi_{1}=k+1$ are of different parities. This result, however, contradicts the property (51) established previously. The contradiction comes from the false hypothesis (68). By removing it, the statement (67) results, qed

case $2:\pi_{1}\geqq k+2;\quad\max\left\{\pi_{2},\pi_{3},\ldots,\pi_{m}\right\}\geqq 2;(2<k<n-2)$ .
Just as above, we will assume by absurdity that for at least one of the roots $x_{i},(i=1,2,\ldots,m)$ , the inequality occurs $\pi_{i}>p_{i}$ For the sake of clarity, we will assume that this root is $x_{1}$ , that is, inequality (68) occurs.

We will associate numbers $p_{2},p_{3},\ldots,p_{m}$ (except for $p_{1}=k$ ), respectively the numbers $p_{2}^{**},p_{3}^{**},\ldots,p_{m}^{**}$ defined by relations (35) and (36). From these relations and from the previously established property (51), it follows:

p_{i}^{**}=\left\{\begin{array}[]{ll}p_{i}-2,&\text{ dacă }p_{i}>1\\ p_{i},&\text{ dacă }p_{i}=1\end{array}\quad(i=2,3,\ldots,m).\right.

Let us show that in the adopted hypotheses, at least for one of the indices $i=2,3,\ldots,m$ equality occurs $p_{i}^{**}=p_{i}-2$ Indeed, taking into account (73), it will suffice to show that at least one of the numbers $p_{2},p_{3},\ldots,p_{m}$ is greater than the number 1, that is

\max\left\{p_{2},p_{3},\ldots,p_{m}\right\}\geqq 2.

(

\prime

)

Let us suppose absurdly that $p_{2}=p_{3}=\ldots=p_{m}=1$ . Hence, according to the property $I_{n}^{(k)}(a,b)$ of the family $Y_{n}$ (property established by Lemma 7), the equality necessarily results, $\pi_{2}=\pi_{3}=\ldots=\pi_{m}=1$ , which contradicts the hypothesis $\max\left\{\pi_{2},\pi_{3},\ldots,\pi_{m}\right\}\geqq 2$ , specific to the considered case. It follows therefore that for at least one of the indices $i=2,3,\ldots,m$ equality occurs $p_{i}^{**}=p_{i}-2$ From here and from (73) the inequality follows

\nu=p_{1}+\sum_{i=2}^{m}p_{i}^{**}\leqq p_{1}+\left(\sum_{i=2}^{m}p_{i}\right)-2.

(74)

Taking into account equality $p_{1}+p_{2}+\ldots+p_{m}=n-1$ from (30), it follows from (74) that the inequality

v=p_{1}+\sum_{i=2}^{m}p_{i}^{**}\leqq n-3.

(75)

Assuming that the roots $x_{1},x_{2},\ldots,x_{m}$ of the integral $y_{0}(x)$ considered, satisfy the inequalities

x_{1}<x_{2}<\ldots\ldots<x_{m}

(76)

we will choose in the interval ( $x_{m},b$ ), $n-v-1$ distinct nodes

\xi_{1},\xi_{2},\ldots\ldots,\xi_{n-v-1}.

(77.)

Taking into account (75), the inequality results $n-v-1\geqq 2$ , which shows us that the set of additional nodes (77) is not empty.

Be it now $\eta(x)$ a non-zero non-identical integral of equation (1), satisfying the following conditions:

	$\displaystyle\eta\left(x_{1}\right)=\eta^{\prime}\left(x_{1}\right)=\ldots\ldots=\eta^{(k-1)}\left(x_{1}\right)=0$
	$\displaystyle\eta\left(x_{2}\right)=\ldots\ldots=\eta^{\left(p_{2}^{**}-1\right)}\left(x_{2}\right)=0$
	$\displaystyle\eta\left(x_{3}\right)=\ldots\ldots=\eta^{\left(p_{3}^{**}-1\right)}\left(x_{3}\right)=0$		(78)
	$\displaystyle\eta\left(x_{m}\right)=\ldots\ldots=\eta^{\left(p_{m}^{**}-1\right)}\left(x_{m}\right)=0$
	$\displaystyle\eta\left(\xi_{1}\right)=\eta\left(\xi_{2}\right)=\ldots\ldots=\eta\left(\xi_{n-\nu-1}\right)=0.$

Of these conditions, those that correspond to the numbers $x_{i}$ for which $p_{i}^{**}=0$ , do not make sense and consequently we will ignore them. Taking into account (69) and (74), it is found that the number of cancellation conditions in (78) is equal to

k+\left(\sum_{i=2}^{m}p_{i}^{**}\right)+n-v-1=v+(n-v-1)=n-1.

(79)

We also observe that the highest order of derivation occurring in (78) is equal to the number $k-1$ , which follows from (73) and the second relation in (30). According to the property $I_{n}^{(k)}(a,b)$ of the family $Y_{n}$ (established in Lemma 7), it follows that there is such a non-identically zero integral $\eta(x)$ , which satisfies the conditions (78).

We will divide the set of roots $x_{2},x_{3},\ldots,x_{m}$ into two categories, as follows: We denote by $x_{i_{1}},x_{i_{2}},\ldots,x_{i_{\alpha}}$ those roots $x_{i}$ , for which the number $p_{i}$ correspondingly satisfies the equality $p_{i}=1^{7}$ ), and with $x_{j_{1}},x_{j_{2}}$ , $\ldots,x_{j_{\beta}}$ those roots $x_{j}$ , for which the corresponding number $p_{j}$ satisfy

⁰⁰footnotetext: ⁷ ) The set of these roots could be empty.

inequality, $p_{j}>1$ Obviously that $\alpha+\beta=m-1$ , since the root was not taken into account $x_{1}$ Taking into account formulas (73), we obtain the equalities

	$\displaystyle p_{i_{1}}^{}=p_{i_{1}}=1;\quad p_{i_{2}}^{}=p_{i_{2}}=1;\ldots;\quad p_{i_{a}}^{**}=p_{i_{a}}=1$
	$\displaystyle p_{j_{1}}^{}=p_{j_{1}}-2;\quad p_{j_{2}}^{}=p_{j_{2}}-2;\ldots;\quad p_{j_{\beta}}^{**}=p_{j_{\beta}}-2.$		(80)

From these equalities, taking into account the second relation in (30), the relation results

\max\left\{p_{i_{1}}^{**},p_{i_{2}}^{**},\ldots,p_{i_{\alpha}}^{**};\quad p_{j_{1}}^{**},p_{j_{2}}^{**},\ldots,p_{j_{\beta}}^{**}\right\}\leqq k-2.

(81)

We will now show that the relations hold:

	$\displaystyle\eta^{\left(p_{i_{1}}^{}\right)}\left(x_{i_{1}}\right)\neq 0,\quad\eta^{\left(p_{i_{2}}^{}\right)}\left(x_{i_{2}}\right)\neq 0,\ldots,\eta^{\left(p_{i_{\alpha}}^{**}\right)}\left(x_{i_{\alpha}}\right)\neq 0$		(82)
	$\displaystyle\eta^{\left(p_{j_{1}}^{}\right)}\left(x_{j_{1}}\right)\neq 0,\quad\eta^{\left(p_{j_{2}}^{}\right)}\left(x_{j_{2}}\right)\neq 0,\ldots,\eta^{\left(p_{j_{\beta}}^{*}\right)}\left(x_{j_{\beta}}\right)\neq 0.$

Indeed, the failure to satisfy any relation in (82) would contradict the property $I_{n}^{(k)}(a,b)$ of the family $Y_{n}$ (true by Lemma 7), because the number of cancellation conditions in (78), which the non-identical integral verifies, is zero $\eta(x)$ , is equal to $n-1$ , and since the relation (81) holds. Taking into account (80), the relations (82) are transcribed:

	$\displaystyle\eta^{\prime}\left(x_{i_{1}}\right)\neq 0,\quad\eta^{\prime}\left(x_{i_{2}}\right)\neq 0,\ldots,\quad\eta^{\prime}\left(x_{i_{\alpha}}\right)\neq 0$
	$\displaystyle\eta^{\left(p_{j_{1}}-2\right)}\left(x_{j_{1}}\right)\neq 0,\quad\eta^{\left(p_{j_{2}}-2\right)}\left(x_{j_{2}}\right)\neq 0,\ldots,\eta^{\left(p_{j_{\beta}}-2\right)}\left(x_{j_{\beta}}\right)\neq 0.$		(83)

These relations together with equalities (78) show us that, regarding the integral $\eta(x)$ , the roots $x_{i_{1}},x_{i_{2}},\ldots,x_{i_{\alpha}}$ are simple, and $x_{j_{1}},x_{j_{2}},\ldots,x_{j_{\beta}}$ have the orders respectively $p_{j_{1}}-2,p_{j_{2}}-2,\ldots,p_{j_{\beta}}-2.{}^{8)}$

Referring to the 1st integral $y_{0}(x)$ and taking into account equalities $p_{i_{1}}=p_{i_{2}}=\ldots=p_{j_{\alpha}}=1$ , we deduce for the same reasons as above, the relations:

y_{0}^{\prime}\left(x_{i_{1}}\right)\neq 0,\quad y_{0}^{\prime}\left(x_{i_{2}}\right)\neq 0,\ldots,y_{0}^{\prime}\left(x_{i_{\alpha}}\right)\neq 0,

which shows us that the roots $x_{i_{1}},x_{i_{2}},\ldots,x_{i_{\alpha}}$ are simple relative to the integral $y_{0}(x)$ .

Then, based on the previously established property (51), it follows that the multiplicity orders $\pi_{j_{1}},\pi_{j_{2}},\ldots,\pi_{j_{\beta}}$ of the roots $x_{j_{1}},x_{j_{2}},\ldots,x_{j_{\beta}}$ relative to the integral $y_{0}(x)$ , are of the same parity as the numbers $p_{j_{1}},p_{j_{2}},\ldots,p_{j_{\beta}}$ and taking into account relations (78), (80) and (83), it follows that they are of the same parity with the orders of the same roots $x_{j_{1}},x_{j_{2}},\ldots,x_{j_{\beta}}$ , relative to the integral $\eta(x)$ .

Also based on property (51), it is found that the multiplicity order of the root $x_{1}$ , regarding the integral $y_{0}(x)$ , is of the same parity as the number

⁰⁰footnotetext: ⁸ ) Numbers

x_{j}

for which

p_{j}^{**}=0

there are no roots for

\eta(x)

NUMBER $p_{1}$ , which based on equality (69) is equal to $k$ . Also, the multiplicity order of the same root $x_{1}$ , regarding the integral $\eta(x)$ is of the same parity as the number $p$ corresponding, which, as the first series of equalities in (78) shows us, is equal to $k$ In conclusion, it follows that the orders of multiplicity of the root $x_{1}$ , relative to the two integrals $y_{0}(x)$ and $\eta(x)$ , are of the same parity with each other.

We finally obtained the following result:
The multiplicity orders of the roots $x_{1},x_{2},\ldots,x_{m}$ , relatively 1st integral $\eta(x)$ , are respectively of the same parity with the orders of the same roots, relative to the integral $y_{0}(x)$ .

In another context, we find that the non-identical integral is zero $y_{0}(x)$ cannot have in the range ( $a,b$ ) roots other than $x_{1},x_{2},\ldots,x_{m}$ , since otherwise, taking into account relations (29) and (30), the property would be contradicted $I_{n}^{k(}a,b$ ) of the family $Y_{n}$ , true property by Lemma 7. For the same reasons, the non-identical integral is zero $\eta(x)$ cannot have in the range $(a,b)$ roots other than $x_{1},x_{2},\ldots,x_{m},\xi_{1},\xi_{2},\ldots,\xi_{n-\nu-1}$ .

From all these conclusions, still taking into account the fact that all the roots $\xi_{1},\xi_{2},\ldots,\xi_{n-\nu-1}$ of the integral $\eta(x)$ are located outside the range $\left[x_{1},x_{m}\right]$ , it follows that if we consider the integrals $\eta(x)$ and - $\eta(x)$ , then one of them will preserve in sufficiently small neighborhoods of the roots $x_{1},x_{2},\ldots,x_{m}$ same sign as the integral $y_{0}(x)$ (except, possibly, for the means of these neighborhoods). The integral that satisfies this requirement will be denoted by $\bar{\eta}(x)$ . So $\bar{\eta}(x)$ is non-identically null, satisfies conditions (78) and in addition the equality

\operatorname{sgn}\{\bar{\eta}(x)\}=\operatorname{sgn}\left\{y_{0}(x)\right\}

valid in sufficiently small neighborhoods of the points $x_{1},x_{2},\ldots,x_{m}$ , with the possible exception of the means of these neighborhoods.

We now consider the integral curve $(\Gamma)$ of equation $y_{0}(x)$ , and the integral curve ( $\Gamma_{e}$ ) of equation $y=\varepsilon\bar{\eta}(x)$ , where $\varepsilon$ is a parameter, taking positive values. We will now examine how the curves are situated relative to each other. $(\Gamma)$ and $\left(\Gamma_{\varepsilon}\right)$ , when $\varepsilon\rightarrow 0^{+}$ .

Referring first to the numbers $x_{i_{1}},x_{i_{2}},\ldots,x_{i_{\alpha}}$ and taking into account the previously established fact that these numbers are simple roots for both $\bar{\eta}(x)$ as well as for $y_{0}(x)$ , it follows that if the parameter $\varepsilon$ takes positive values, below a certain threshold $E_{1}$ , then the curves $(\Gamma)$ and $\left(\Gamma_{e}\right)$ they will cross at the points $x_{i_{1}},x_{i_{2}},\ldots,x_{i_{\alpha}}$ (without being tangent to each other at these points).

Next, referring to the roots $x_{j_{1}},x_{j_{2}},\ldots,x_{j_{\beta}}$ , and taking into account relations (29), (78) and (83), we reach the following conclusions:
$1^{\circ}$ In the points $x_{j_{1}},x_{j_{2}},\ldots,x_{j_{\beta}}$ , the curves $(\Gamma)$ and $\left(\Gamma_{\varepsilon}\right)$ presents tangency and order contacts, respectively $p_{j_{1}}-3,p_{j_{2}}-3,\ldots,p_{j_{\beta}}-3$ , that is, the functions $y_{0}(x)$ and $\bar{\eta}(x)$ coincide at these points, respectively up to their derivatives of the order $p_{j_{1}}-3,p_{j_{2}}-3,\ldots,p_{j_{\beta}}-3$ , inclusive. ⁹ )

⁰⁰footnotetext: ⁹ ) By hypothesis the numbers

p_{j_{1}},p_{j_{2}},\ldots,p_{j_{\beta}}

are greater than or equal to 2. In the case when any of these numbers, for example

p_{j_{1}}

, is equal to 2 , the above result should be interpreted as follows: The curve

(\Gamma)

has in the corresponding point

x_{j_{1}}

a contact order

$2^{\circ}$ .If the parameter $\varepsilon$ is sufficiently small, then, taking into account that the orders of multiplicity $\pi_{j_{1}},\pi_{j_{2}},\ldots,\pi_{j_{\beta}}$ of the roots $x_{j_{1}},x_{j_{2}},\ldots,x_{j_{\beta}}$ , relative to the integral $y_{0}(x)$ , are of the same parity as the orders $p_{j_{1}}-2,p_{j_{2}}-2,\ldots,p_{j_{\beta}}-2$ of the same roots relative to the integral $\bar{\eta}(x)$ and also taking into account the strict inequalities

\pi_{j_{1}}>p_{j_{1}}-2,\quad\pi_{j_{2}}>p_{j_{2}}-2,\ \ldots,\ \pi_{j_{\beta}}>p_{j_{\beta}}-2,

it follows, based on Lemma 8, that, if the parameter $\varepsilon$ is positive and below a certain threshold $E_{2}$ , then, in given, sufficiently small, neighborhoods of the points $x_{j_{1}},x_{j_{2}},\ldots,x_{j_{\beta}}$ , the curves $(\Gamma)$ and $(\Gamma_{\varepsilon})$ intersect at the points

\bar{x}_{j_{1}},\ \overline{\bar{x}}_{j_{1}},\ \bar{x}_{j_{2}},\ \overline{\bar{x}}_{j_{2}},\ \ldots,\ \bar{x}_{j_{\beta}},\ \overline{\bar{x}}_{j_{\beta}},

distinct from each other and different from $x_{j_{1}},x_{j_{2}},\ldots,x_{j_{\beta}}$ , crossing each other at these points.

Finally, referring to the root $x_{1}$ , from (68), (69) and (78) we deduce that whatever the value of the parameter $\varepsilon$ , the curves $(\Gamma)$ and $\left(\Gamma_{\varepsilon}\right)$ will present at this point a contact of order at least equal to $k-1$ .

In conclusion, taking into account the above findings, we reach the following result:

If the parameter $\varepsilon>0$ is small enough, then the integral $\tilde{y}_{e}(x)==y_{0}(x)-\varepsilon\bar{\eta}(x)$ of the differential equation (1), will admit in the interval ( $a,b$ ), apart from the root $x_{1}$ , multiple of order $\geqq k$ , and the following distinct roots:

	$\displaystyle x_{i_{1}},\quad x_{i_{2}},\ldots\ldots,x_{i_{\alpha}}$
	$\displaystyle x_{j_{1}}\quad x_{j_{2}},\ldots\ldots,x_{j_{\beta}}$		(84)
	$\displaystyle\bar{x}_{j_{1}},\quad\overline{\bar{x}}_{j_{1}},\quad\bar{x}_{j_{2}}\overline{\bar{x}}_{j_{2}},\ldots\ldots,\bar{x}_{j_{\beta}},\overline{\bar{x}}_{j_{\beta}}$

having orders of multiplicity respectively greater than or at least equal to the numbers

	$\displaystyle 1,1,\ldots\ldots,1$
	$\displaystyle p_{j_{1}}-2,p_{j_{2}}-2,\ldots\ldots,p_{j_{\beta}}-2$		(85)
	$\displaystyle 1,1,1,1,\ldots\ldots,1,1$

The sum of the numbers in (85) is equal to

s=\alpha+p_{j_{1}}+p_{j_{2}}+\ldots+p_{j_{\beta}}

and taking into account the first relations in (80), then the first relation in (30) as well as the equality (69), it results for the sum $s$ equal

	$\displaystyle s=\left(p_{i_{1}}+p_{i_{2}}+\cdots+p_{i_{\alpha}}\right)+\left(p_{j_{1}}+p_{j_{2}}+\ldots+p_{j_{\beta}}\right)=$		(86)
	$\displaystyle=n-1-p_{1}=n-1-k$

Referring to the root $x_{1}$ of the integral $\bar{\eta}(x)$ and taking into account the conditions (78) that this integral satisfies, we distinguish the following two subcases as follows: $\bar{\eta}^{(k)}(x)$ is it canceled or not for the value $x=x_{1}$ .

⁰⁰footnotetext:

\pi_{j_{1}}-1

with the axis

Ox

, where

\pi_{j_{1}}

is an even number greater than or at least equal to 2, and the curve

\left(\Gamma_{\varepsilon}^{1}\right)

does not intersect the axis

Ox

in the vicinity of this point

x_{j_{1}}

, located relative to this axis on the side where the curve (

\Gamma

Subcase 1: $\bar{\eta}\left(x_{1}\right)=\bar{\eta}^{\prime}\left(x_{1}\right)=\ldots=\bar{\eta}^{(k-1)}\left(x_{1}\right)=0;\bar{\eta}^{(k)}\left(x_{1}\right)\neq 0$ In this
subcase, if we take into account the inequality $\pi_{1}\geq k+2$ from (70), as well as the hypothesis specific to the case considered, it is found that the curves $(\Gamma)$ and $(\Gamma_{\varepsilon})$ presents in the point $x_{1}$ a tangential contact of the order $k-1$ , that is, the coincidence of the functions $y_{0}(x)$ and $\bar{\eta}(x)$ at the point $x=x_{1}$ is carried out up to their derivatives of the order $k-1$ , inclusive.

Then, taking into account the previously established fact that the numbers $\pi_{1}$ (with $\pi_{1}\geq k+2$ ) and $k$ , which represent the orders of multiplicity of the root $x_{1}$ relative to integrals $y_{0}(x)$ and $\bar{\eta}(x)$ , are of the same parity, it follows, based on Lemma 8, that, if the parameter $\varepsilon$ takes positive values below a certain threshold $E_{2}$ , then, in the vicinity of the point $x_{1}$ , the curves $(\Gamma)$ and $(\Gamma_{\varepsilon})$ will intersect at two more distinct points $\bar{x}_{1}$ and $\overline{\bar{x}}_{1}$ from the range $(a,b)$ , different from $x_{1}$ and the points in (84).

Taking these into account, it ultimately results that, if the parameter $\varepsilon$ takes positive and sufficiently small values, then the integral

\widetilde{y}_{\varepsilon}(x)=y_{0}(x)-\varepsilon\bar{\eta}(x)

satisfies, in relation to the points $x_{1},\bar{x}_{1},\overline{\bar{x}}_{1}$ and with the points in (84), a number of $s+k+2$ cancellation conditions, that is, taking into account also the relation (86), a number of

(n-1-k)+k+2=n+1

cancellation conditions.

However, since $\widetilde{y}_{\varepsilon}(x)$ cannot be identically null in the range $(a,b)$ , whatever it is $\varepsilon\neq 0$ (which results from the fact that $y_{0}(\xi_{1})\neq 0$ and $\bar{\eta}(\xi_{1})=0$ ), it would follow from the above that the family $Y_{n}$ would not have the property $I_{n}^{(k)}(a,b)$ This result contradicts the statement of Lemma 7.

We thus conclude that, under the assumptions of Lemma 9, subcase 1 cannot occur. Subcase 2: $\bar{\eta}\left(x_{1}\right)=\bar{\eta}^{\prime}\left(x_{1}\right)=\ldots=\bar{\eta}^{(k-1)}\left(x_{1}\right)=\bar{\eta}^{(k)}\left(x_{1}\right)=0$ We will
first show that in this subcase, assuming that the family $Y_{n}$ has all the properties (33), the equality also holds $\bar{\eta}^{(k+1)}\left(x_{1}\right)=0$ , that is, the multiplicity order of the root $x_{1}$ , relative to the integral $\bar{\eta}(x)$ , is greater than or at least equal to $k+2$ Indeed, we first observe that the cancellation conditions (78), which are satisfied by the non-identically zero integral $\bar{\eta}(x)$ , have the form of conditions (29), (30), which verify $y_{0}(x)$ , in the sense that the total number of conditions written in (78) is equal to $n-1$ (as shown by equality (79)), and that the number of cancellation conditions in (78) in the case of any node among the nodes $x_{1},x_{2},\ldots,x_{m},\xi_{1},\xi_{2},\ldots,\xi_{n-\nu-1}$ , does not exceed the number $k$ .

Let's note with $p\left(x_{1};\bar{\eta}\right)$ NUMBER $p$ from (78), regarding the integral $\bar{\eta}(x)$ and regarding the root $x_{1}$ , and with $\pi\left(x_{1};\bar{\eta}\right)$ order of the same root $x_{1}$ regarding the integral $\bar{\eta}(x)$ As the first relation in (78) shows, we have $p\left(x_{1},\bar{\eta}\right)=k$ , and from the specific hypothesis of the considered case we have $\pi\left(x_{1},\bar{\eta}\right)\geqq\geqq k+1$ We remember the relationships:

p\left(x_{1},\bar{\eta}\right)=k,\quad\pi\left(x_{1},\bar{\eta}\right)\geqq k+1.

According to property (51), which we state regarding the integral $\bar{\eta}(x)$ , the numbers $p\left(x_{1};\bar{\eta}\right)$ and $\pi\left(x_{1};\bar{\eta}\right)$ must be of the same parity. From here, taking into account the previous relations, it follows

\pi\left(x_{1};\bar{\eta}\right)\geqq k+2,

(87)

and thus the statement made previously is demonstrated.

Next, referring to the other roots of the integral $\bar{\eta}(x)$ , which appears in (78), we observe that the strict inequality takes place

	$\displaystyle\max\left\{p\left(x_{2};\bar{\eta}\right),\right.$	$\displaystyle\left.p\left(x_{3};\bar{\eta}\right),\ldots,p\left(x_{m};\bar{\eta}\right);p\left(\xi_{1};\bar{\eta}\right),\ldots,p\left(\xi_{n-\nu-1},\bar{\eta}\right)\right\}<$
	$\displaystyle<$	$\displaystyle\max\left\{p\left(x_{2};y_{0}\right),p\left(x_{3};y_{0}\right),\ldots,p\left(x_{m};y_{0}\right)\right\}$		(88)

where $p\left(x_{i};\bar{\eta}\right),\quad(i=2,3,\ldots,m)$ and $p\left(\xi_{j};\bar{\eta}\right),\quad(j=1,2,\ldots,n-v-1)$ , represent the numbers respectively $p$ from (78), regarding the roots $x_{i}$ and $\xi_{j}$ of the integral $\bar{\eta}(x)$ , and $p\left(x_{i};y_{0}\right),(i=2,3,\ldots,m)$ represents the numbers $p$ from (29), regarding the roots $x_{i}$ of the integral $y_{0}(x)$ .

Inequality (88) is deduced taking into account relations (78), (29), (80), as well as property (73').

We now consider instead of the integral $y_{0}(x)$ , the non-identical integral is zero $\bar{\eta}(x)$ Taking into account relations (78), (79) and (87), we can present for $\bar{\eta}(x)$ , one of the two cases mentioned above for the integral $y_{0}(x)$ As previously demonstrated, the specific conditions of case 1 are inconsistent with the hypothesis that the family $Y_{n}$ has the properties indicated in (33). It follows that the integral $\bar{\eta}(x)$ must satisfy the conditions of case 2, and at the same time the strict inequality (88) will hold for it, which plays a reduction role. Repeating the reasoning used in case 2, but relative to the integral $\bar{\eta}(x)$ , we will be led to consider an integral $\bar{\eta}_{1}(x)$ , non-identical null in the interval ( $a,b$ ), satisfying conditions analogous to conditions (78), (79), (87), (88). It is found, as above, that $\bar{\eta}_{1}(x)$ must satisfy the conditions of case 2. Next, starting from the integral $\bar{\eta}_{1}(x)$ is obtained in the same $\bmod$ an integral $\bar{\eta}_{2}(x)$ , non-identical null in the interval $(a,b)$ , also satisfying conditions analogous to conditions (78), (79), (87), (88). Thus, taking into account the fact that each time the strict inequality (88) occurs, which plays a role of reduction, we will eventually arrive at an integral $\bar{\eta}_{N}(x)$ , non-identical null in the interval ( $a,b$ ) and which will satisfy the conditions of case 1 considered previously. However, as previously demonstrated, the specific hypotheses of this case are in contradiction with the properties of (33). We finally arrive at a contradiction, which comes from the absurd hypothesis (68). Thus the property $P_{n}(a,b)$ of the family $Y_{n}$ is established.

Returning to the proof of Theorem 1, from the statements of Lemmas 7 and 9 it follows that if the family $Y_{n}$ has all the properties indicated in (33), then that family also has the properties $I_{n}^{(k)}(a,b)$ and $P_{n}^{(k)}(a,b)$ According to the principle of induction, the statement of theorem 1 follows.

Next, we will assume that the differential equation (1) has continuous coefficients in the semi-closed interval1 $[a,b)$ , and therefore that the family $Y_{n}$ of the integrals of this equation consists of functions defined in the interval $[a,b)$ Definitions $1,2,3$ previous data relative to an open range $(a,b)$ , extend to the case of a semi-closed interval $[a,b)$ We will denote that by
$I_{n}[a,b),I_{p_{1},p_{2},\ldots,p_{m}}[a,b),I_{n}^{*}[a,b)$ family properties $Y_{n}$ , highlighted by these definitions. It is easily seen that the lemmas $1,2,3$ established on the occasion of Theorem 1, in the case of an open interval ( $a,b$ ), are also extended to the case of a semi-closed interval $[a,b)$ We will now establish the following theorem:
theorem 2. If the differential equation (1) has continuous coefficients in the semiclosed interval $[a,b)$ , and if the family $Y_{n}$ of the integrals of this equation has the property $I_{n}^{*}(a,b)$ , then the family $Y_{n}$ also has the property $I_{n}^{*}[a,b)$ .

To prove this theorem, we will first establish the following theorem:

Lemma 10. We assume that the differential equation (1) has continuous coefficients in the open interval ( $a,b$ ) and that the family $Y_{n}$ of the integrals of this equation has the property $I_{n}^{*}(a,b)$ . Either $n-1$ NUMBERS $x_{1}<x_{2}<\ldots<x_{n-1}$ , chosen arbitrarily from the interval $(a,b)$ and be $h_{1}(x),h_{2}(x),\ldots,h_{n-1}(x)$ non-identical zero integrals of the differential equation (1), satisfying conditions ¹⁰ ):

	$\displaystyle h_{1}\left(x_{1}\right)=h_{1}\left(x_{2}\right)=\ldots\ldots\ldots\ldots..=h_{1}\left(x_{n-1}\right)=0$
	$\displaystyle h_{2}\left(x_{1}\right)=h_{2}\left(x_{2}\right)=\ldots=h_{2}\left(x_{n-2}\right)=0,\quad h_{2}\left(x_{n-1}\right)=1$
	$\displaystyle h_{n-1}\left(x_{1}\right)=0,h_{n-1}\left(x_{2}\right)=1.$		(89)

In these hypotheses, the relationships take place

h_{1}(x)\neq 0,\quad w\left(h_{1},h_{2}\right)\neq 0,\ldots,w\left(h_{1},h_{2},\ldots,h_{n-1}\right)\neq 0

(90)

valid in the intervals $\left(a,x_{1}\right)$ and $\left(x_{n-1},b\right)$ . Here $w\left(h_{1},h_{2},\ldots,h_{i}\right)$ represents the Wronskian of the functions $h_{1}(x),h_{2}(x),\ldots,h_{i}(x)$ .

The proof we give below is somewhat analogous to the proof of Theorem IV established by G. Pó1y a in [23] ¹¹⁾ . Thus, since by hypothesis the integral is not identically zero $h_{1}(x)$ is cancelled for $n-1$ distinct values $x_{1},x_{2},\ldots,x_{n-1}$ from the interval ( $a,b$ ), it follows that this integral cannot be cancelled for any other value in this interval, since otherwise the property would be contradicted $I_{n}^{*}(a,b)$ of the family $Y_{n}$ (according to Lemma 1). Thus the first relation in (90) is established. Next, we adopt the notations $h_{1}(x)=w_{1}$ and $w\left(h_{1},h_{2},\ldots,h_{k}\right)=w_{k},\quad(k=2,3,\ldots,n-1)$ .

Using the method of induction, let us assume that in both intervals $\left(a,x_{1}\right)$ and $\left(x_{n-1},b\right)$ relationships take place

w_{1}\neq 0,\quad w_{2}\neq 0,\ldots,w_{k}\neq 0

(91)

⁰⁰footnotetext: ¹⁰ ) The existence of such non-identically zero integrals is ensured by the property

I_{n}^{*}(a,b)

.
¹¹ ) In the statement of Theorem IV in the work [23] by G. Póly it is assumed that the family

Y_{n}

has the property

I_{n}^{*}[a,b)

, that is, it is an interpolator of the order

n

in the half-closed interval

[a,b)

, and the nodes

x_{1},x_{2},\ldots,x_{n-1}

, which occur under conditions (89), are all considered to be confused at the point

x=a

These hypotheses show the validity of relations (90) in the open interval

(a,b)

See THEOREM A, stated below.

NUMBER $k$ satisfying the inequality $1\leqq k<n-1$ In these hypotheses we will demonstrate that the relationship also holds $w_{k+1}\neq 0$ in the intervals $\left(a,x_{1}\right)$ and $\left(x_{n-1},b\right)$ For this purpose, we consider the linear combination

h(x)=C_{1}h_{1}(x)+C_{2}h_{2}(x)+\ldots+C_{k}h_{k}(x)+h_{k+1}(x),

(92)

where $C_{1},C_{2},\ldots,C_{k}$ represent arbitrary constants for the moment. We note from the outset that whatever values these constants take, the corresponding function $h(x)$ is cancelled at the points $x_{1},x_{2},\ldots,x_{n-k-1}$ , this is because each of the functions $h_{1}(x),h_{2}(x),\ldots,h_{k}(x),h_{k+1}(x)$ , which intervene in the expression of the function $h(x)$ , cancels out at these points. Now let $x_{0}$ an arbitrary point in the open interval $\left(a,x_{1}\right)$ , or from the range $\left(x_{n-1},b\right)$ . We determine the constants $C_{1},C_{2},\ldots,C_{k}$ from (92) so that $h(x)$ to cancel by $k$ times at the point $x_{0}$ , that is, to admit the value $x_{0}$ as a multiple root of order $\geqq k$ Such a determination is possible, and this in only one way, since the determinant of the $k$ equations that are formed by writing these conditions, is $w\left(h_{1},h_{2},\ldots,h_{k}\right)\neq 0$ This determinant is nonzero according to hypothesis (91). Let $\bar{C}_{1},\bar{C}_{2},\ldots,\bar{C}_{k}$ the values of the constants thus determined $C_{1},C_{2},\ldots,C_{k}$ , and let $\bar{h}(x)$ the corresponding function, obtained using formula (92). This function will therefore have as roots the numbers $x_{0},x_{1},\ldots,x_{n-k-1}$ , the root $x_{0}$ having the order of multiplicity greater than or at least equal to $k$ It follows from this that $\bar{h}(x)$ is cancelled in total by $(n-k-1)+k=n-1$ times in the interval ( $a,b$ ). We also observe that the integral $\bar{h}(x)$ is not identically null in the interval $(a,b)$ , since $\bar{h}\left(x_{n-k}\right)=h_{k+1}\left(x_{n-k}\right)=1$ .

Since by hypothesis, the family $Y_{n}$ of the integrals of the differential equation (1) has the property $I_{n}^{*}(a,b)$ , it follows from Lemma 3 that the roots $x_{1},x_{2},\ldots,x_{n-k}$ of the integral $\bar{h}(x)$ are simple, and the root $x_{0}$ cannot have a multiplicity order greater than $k$ .

We now use the following identity established by G. Pó1y a in [23]:

w\left(h_{1},h_{2},\ldots,h_{k},y\right)\equiv\frac{w_{k}^{2}}{w_{k-1}}\cdot\frac{d}{dx}\frac{w_{k-1}^{2}}{w_{k-2}w_{k}}\cdots\frac{d}{dx}\frac{w_{2}^{2}}{w_{1}w_{3}}\cdot\frac{d}{dx}\frac{w_{1}^{2}}{w_{0}w_{2}}\cdot\frac{d}{dx}\frac{y}{w_{1}}

(93)

valid in any range of the axis $Ox$ , in which the functions $w_{1}=h_{1}(x),w_{2}==w\left(h_{1},h_{2}\right),\ldots,w_{k}=w\left(h_{1},h_{2},\ldots,h_{k}\right)$ does not cancel, and for any function $y(x)$ having continuous derivatives up to the order $k$ inclusively in that interval. In formula (93) it was noted $w_{0}\equiv 1$ .

We now assume that the functions $h_{1}(x),h_{2}(x),\ldots,h_{k}(x)$ which occur in (93), are those considered in the statement of the present lemma, that is, those that satisfy conditions (89). In the case of these functions, taking into account hypothesis (91), it follows that identity (93) is valid in the open intervals ( $a,x_{1}$ ) and $\left(x_{n-1},b\right)$ , whatever the function $y(x)$ , having continuous derivatives up to the order $k$ inclusive. In particular, replacing in (93) by $y(x)$ with the function $\bar{h}(x)$ , we obtain the identity

w\left(h_{1},h_{2},\ldots,h_{k},\bar{h}\right)=\frac{w_{k}^{2}}{w_{k-1}}\cdot\frac{d}{dx}\frac{w_{k-1}^{2}}{w_{k-2}w_{k}}\cdots\frac{d}{dx}\frac{w_{2}^{2}}{w_{1}w_{3}}\cdot\frac{d}{dx}\frac{w_{1}^{2}}{w_{0}w_{2}}\cdot\frac{d}{dx}\frac{\bar{h}}{w_{1}}

(94)

Taking into account the relation (92) which defines the function $h(x)$ , it is observed that

w\left(h_{1},h_{2},\ldots,h_{k},\bar{h}\right)=w\left(h_{1},h_{2},\ldots,h_{k},h_{k+1}\right)\equiv w_{k+1}.

(95)

On the other hand, taking into account the previously established fact that the multiplicity order of the root $x_{0}$ relative to the function $\bar{h}(x)$ cannot be greater than $k$ , and taking into account also the relations (91), assumed valid in both intervals $\left(a,x_{1}\right)$ and $\left(x_{n-1},b\right)$ , it is obtained by successively applying Rolle's theorem that the second member of the identity (94) does not vanish at the point $x_{0}$ . From here, taking into account the identity (95), it follows that the function $\varpi_{k+1}$ does not cancel at the point $x_{0}$ But how? $x_{0}$ was arbitrarily chosen in the range $\left(a,x_{1}\right)$ or ( $x_{n-1},b$ ), the relationship results $w_{k+1}\neq 0$ , valid in each of these intervals, qed

Based on the principle of induction, Lemma 10 follows.
Remark. Assuming that the differential equation (1) has the coefficients $a_{i}(x),\quad(i=1,2,\ldots,n)$ , continuous in the semiclosed interval $[a,b)$ and choosing the nodes $x_{1},x_{2},\ldots,x_{n-1}$ which occur in conditions (89), so that they all coincide at the point $x=a$ , the above reasoning leads to the following theorem, established by G. Pó1y a in the paper [23]:
theorem a. ¹² ) We assume that the differential equation (1) has the coefficients $a_{i}(x),(i=1,2,\ldots,n)$ , continuous in the semi-closed interval $[a,b)$ . Either $h_{1}(x),h_{2}(x),\ldots,h_{n-1}(x)$ a system of integrals of equation (1) satisfying at the point $x=a$ conditions:

	$\displaystyle h_{1}(a)=h_{1}^{\prime}(a)=\ldots\ldots\ldots=h_{1}^{(n-2)}(a)=0,\quad h_{1}^{(n-1)}(a)=1$
	$\displaystyle h_{2}(a)=h_{2}^{\prime}(a)=\ldots=h_{2}^{(n-3)}(a)=0,\quad h_{2}^{(n-2)}(a)=1$
	………………………………………..		(96)
	$\displaystyle h_{n-2}(a)=h_{n-2}(a)=0,\quad h_{n-2}^{n}(a)=1$
	$\displaystyle h_{n-1}(a)=0,\quad h_{n-1}^{\prime}(a)=1.$

Then, assuming that the family $Y_{n}$ of the integrals of equation (1) has the property $I_{n}^{*}[a,b)$ , it follows that the integrals $h_{1}(x),h_{2}(x),\ldots,h_{n-1}(x)$ considered above, I am satisfied in the range ( $a,b$ ) relationships:

h_{1}(x)\neq 0,w\left(h_{1},h_{2}\right)\neq 0,\ldots,w\left(h_{1},h_{2},\ldots,h_{n-1}\right)\neq 0

(97)

We will prove Theorem 2 by induction relative to the natural number $n$ which represents the order of the differential equation. For $n=1$ , the property expressed by Theorem 2 is obvious. We will assume that this property is true for the natural number $n-1$ and we will prove its validity for the number $n$ For this purpose, we will assume that the family $Y_{n}$ of the integrals of the differential equation (1) has the property $I_{n}^{*}(a,b)$ . (This hypothesis intervenes in the statement of theorem 2). From this hypothesis, based on lemma 3, it follows that the differential equation (1) does not admit any non-identically zero integral, which has $n$ roots (distinct or not) in the open interval ( $a,b$ ). To prove the statement of Theorem 2, it will be sufficient (also based on Lemma 3)

⁰⁰footnotetext: ¹² ) In the paper [23], this theorem bears the order number IV.

let us show that the differential equation (1) does not admit any non-identically zero integral, which has $n$ roots (distinct or not) in the semiclosed interval $[a,b)$ Let us suppose by absurdity that there exists a non-zero non-identical integral $y_{0}(x)\in Y_{n}$ , which should have $n$ roots in the interval $[a,b)$ . Then, necessarily one of the roots of the integral $y_{0}(x)$ must coincide with the extremity $x=a$ of the interval $[a,b)$ , because otherwise, if all the $n$ roots would be inside the interval ( $a,b$ ), then according to Lemma 1 the property would be contradicted $I_{n}(a,b)$ assumed true by hypothesis. In what follows we will denote by $\xi_{1}=a,\xi_{2},\ldots,\xi_{m}$ the distinct roots of $[a,b)$ of the integral $y_{0}(x)$ considered, and with $\pi_{1},\pi_{2},\ldots,\pi_{m}$ their orders of multiplicity. Obviously, $\pi_{1}+\pi_{2}+\ldots+\pi_{m}\geqq n$ We will assume that the inequalities hold $\xi_{1}<\xi_{2}<\ldots<\xi_{m}$ .

Either now $x_{1}<x_{2}<\ldots<x_{n-1}$ values taken arbitrarily from the range $\left(\xi_{m},b\right)$ We consider $n-1$ FULL $h_{1}(x),h_{2}(x),\ldots,h_{n-1}(x)$ satisfying the conditions ( 89 ), but in which $x_{1},x_{2},\ldots,x_{n-1}$ represents the previously chosen nodes, from the interval $\left(\xi_{m},b\right)$ Such integrals exist, since by hypothesis it was assumed that the family $Y_{n}$ has the property $I_{n}^{*}(a,b)$ According to Lemma 10, it follows that in the interval ( $a,x_{1}$ ) the relationships take place:

	$\displaystyle h_{1}(x)\neq 0,\quad w\left(h_{1}h_{2}\right)\neq 0,\ldots,w\left(h_{1},h_{2},\ldots,h_{n-1}\right)\neq 0$		(98)
	$\displaystyle\text{ pentru }x\in\left(a,x_{1}\right).$

In addition, the integral $h_{1}(x)$ canceling out in $n-1$ points, $x_{1},x_{2},\ldots,x_{n-1}$ from the range $(a,b)$ , it follows that it cannot be cancelled in the interval $(a,b)$ , since otherwise the property would be contradicted $I_{n}^{*}(a,b)$ of the family $Y_{n}$ (according to Lemma 1). Moreover, the integral $h_{1}(x)$ cannot be canceled even at the point $x=a$ Indeed, assuming absurdly that $h_{1}(a)=0$ , let us consider the integral $y_{e}(x)$ of the differential equation (1), which satisfies the conditions

y_{\varepsilon}(a+\varepsilon)=h_{1}(a),\quad y_{\varepsilon}^{\prime}(a+\varepsilon)=h_{1}^{\prime}(a),\ldots,y_{\varepsilon}^{(n-1)}(a+\varepsilon)=h_{1}^{(n-1)}(a),

where $\varepsilon$ represents a parameter that satisfies the inequalities

0<\varepsilon<\xi_{m}-a.

If the parameter $\varepsilon$ tends to zero through positive values, then the integral $y_{\varepsilon}(x)$ tends uniformly towards $h_{1}(x)$ on any closed interval $[a,b_{1}]$ , contained in the interval $[a,b)$ .

Hence, taking into account the fact that the roots $x_{1},x_{2},\ldots,x_{n-1}$ of the integral $h_{1}(x)$ are simple (which follows from the statement of Lemma 4), it follows that, for $\varepsilon>0$ small enough, integral $y_{\varepsilon}(x)$ appropriate will have $n-1$ distinct roots

\bar{x}_{1},\bar{x}_{2},\ldots,\bar{x}_{n-1},

located in the range $(\xi_{m},b)$ and will also cancel at the point $x=a+\varepsilon$ , belonging to the interval $(a,\xi_{m})$ .

This would result in a non-zero integral $y_{\varepsilon}(x)$ of the differential equation (1), which cancels out in at least $n$ points in the range $(a,b)$ This situation, however, contradicts the property $I_{n}^{*}(a,b)$ of the family $Y_{n}$ .

In conclusion, the inequality occurs

h_{1}(a)\neq 0.

Next, let's perform the change of function on the differential equation (1)

y(x)=h_{1}(x)z

(99)

Then, from (1) we obtain the differential equation

\bar{L}_{n}(z)=h_{1}(x)\left[z^{(n)}+\bar{a}(x)z^{(n-1)}+\ldots+\bar{a}_{n-1}(x)z^{\prime}\right]=0

having the coefficients $\bar{a}_{1}(x),\ldots,\bar{a}_{n-1}(x)$ continue in the interval [ $a,x_{1}$ ), in which the function $h_{1}(x)$ does not cancel. Let us note

z^{\prime}=u.

(100)

Taking into account the fact that $h_{1}(x)$ is nonzero in the interval [ $a,x_{1}$ ), the previous equation reduces in this interval to the 1st equation

\bar{L}_{n-1}(u)=u^{(n-1)}+\bar{a}_{1}(x)u^{(n-2)}+\ldots+\bar{a}_{n-1}(x)u=0,

(101)

having continuous coefficients in the interval $\left[a,x_{1}\right)$ .
The integral $y_{0}(x)$ of the differential equation (1), the integral will correspond to $u_{0}(x)=\frac{d}{dx}\left(\frac{y_{0}(x)}{h_{1}(x)}\right)$ of equation (101). Since by hypothesis the function $y_{0}(x)$ is canceled at least by $n$ times in the interval $\left[a,x_{1}\right)$ , namely at the points $\xi_{1}=a,\xi_{2},\ldots,\xi_{m}$ , with order of multiplicity respectively $\pi_{1},\pi_{2},\ldots,\pi_{m}$ , obviously satisfying the inequality $\pi_{1}+\pi_{2}+\ldots+\pi_{m}\geqq n$ , it follows that the integral $u_{0}(x)$ of equation (101) will cancel at least by $n-1$ times in the interval $\left[a,x_{1}\right)$ On the other hand, from the fact that the integral $y_{0}(x)$ is not identically null in ( $a,b$ ), it follows that and $u_{0}(x)$ is not identically null in the interval $\left[a,x_{1}\right)$ Indeed, assuming the opposite, that is, $u_{0}(x)\equiv 0$ in the interval $\left[a,x_{1}\right)$ , it would result, taking into account the changes of variables (99) and (100), that the identity takes place $y_{0}(x)\equiv Ch_{1}(x)$ in the interval $\left[a,x_{1}\right)$ and therefore in the entire interval $[a,b)$ In this identity, $C$ represents a constant. Since $y_{0}(x)$ is by hypothesis non-identically null in the interval ( $a,b$ ), would result from the identity $y_{0}(x)\equiv Ch_{1}(x)$ that $C\neq 0$ . At the same time, it would also follow that the integral $b_{1}(x)$ is cancelled in the interval [ $a,x_{1}$ ) for all values in this interval, for which it also cancels out $y_{0}(x)$ . But the function $y_{0}(x)$ is cancelled by $n$ times in the interval $\left[a,x_{1}\right)$ in the points $\xi_{1}=a,\xi_{2},\ldots,\xi_{m}$ with the respective multiplicity orders $\pi_{1},\pi_{2},\ldots,\pi_{m}\left(\pi_{1}+\pi_{2}+\ldots+\pi_{m}\geqq n\right)$ . However, since the integral $y_{0}(x)$ 'is not identically null in the range ( $a,b$ ), it follows that the number $m$ of these points satisfies the inequality $m\geqq 2$ and therefore that the function $y_{0}(x)$ vanishes at least at one point inside the interval ( $a,x_{1}$ ). On the other hand, as follows from (89), the integral $h_{1}(x)$ is also canceled by $n-1$ times in the interval $\left[x_{1},b\right)$ . One would ultimately come to the conclusion that the integral is not identically zero $h_{1}(x)$ of equation (1) is canceled at least by $n$ times in the open interval ( $a,b$ ), which would contradict the property $I_{n}^{*}(a,b)$ of the family $Y_{n}$ , property assumed to be true by hypothesis.

We finally obtain the following result: The integral $u_{0}(x)$ equation (101) is not identically zero in the interval [ $a,x_{1}$ ) and is cancelled at least by $n-1$ times in this interval.

Next, the integrals $h_{2}(x),\ldots,h_{n-1}(x)$ of equation (1) will correspond to the integrals $u_{2}(x),\ldots,u_{n-1}(x)$ of equation (101), having the expressions:

u_{i}(x)=\frac{d}{dx}\left(\frac{h_{i}(x)}{h_{1}(x)}\right),\quad i=2,\ldots,n-1

(102)

These integrals are defined in the interval $\left[a,x_{1}\right)$ , in which $h_{1}(x)$ does not cancel. Using the identity

w\left[\left(\frac{h_{2}}{h_{1}}\right)^{\prime},\left(\frac{h_{3}}{h_{1}}\right)^{\prime},\ldots,\left(\frac{h_{i}}{h_{1}}\right)^{\prime}\right]=\frac{1}{h_{1}^{i}}w\left(h_{1},h_{2},\ldots,h_{i}\right).

established by G. Pó1y a in [23], and taking into account formulas (102), it is found that the integrals $u_{2}(x),\ldots,u_{n-1}(x)$ of equation (101) satisfy in the interval ( $a,x_{1}$ ) the following conditions analogous to conditions (98):

	$\displaystyle u_{2}(x)\neq 0,\quad w\left(u_{2},u_{3}\right)\neq 0,\ldots,w\left(u_{2},u_{3},\ldots,u_{n-1}\right)\neq 0$		(103)
	$\displaystyle\text{ pentru }x\in\left(a,x_{1}\right).$

Thus we conclude that the differential equation (101) has continuous coefficients in the semi-closed interval $\left[a,x_{1}\right)$ and satisfies conditions (103) in the open interval $\left(a,x_{1}\right)$ According to Theorem I in [23] by G. Pó1ya (see Theorem $B$ stated below), it follows that the family $U_{n-1}$ of the integrals of the differential equation (101) will have the property $I_{n-1}^{*}\left(a,x_{1}\right)$ . Then, taking into account the hypothesis initially made, namely that the property expressed by Theorem 2 in the present paper is true for the natural number $n-1^{13}$ ), it results that the family $U_{n-1}$ of the integrals of the differential equation (101) also has the property $I_{n-1}^{*}\left[a,x_{1}\right)$ This conclusion is, however, incompatible with the existence of the integral $u_{0}(x)$ of the same equation (101), which as previously shown is not identically zero in the interval [ $a,x_{1}$ ) and also cancels out in this interval $n-1$ times. This contradiction comes from the absurd hypothesis that under the conditions of theorem 2, the differential equation (1) would admit in the interval $[a,b)$ a non-zero non-identical integral $y_{0}(x)$ , which is to be cancelled by $n$ times in this interval. It follows therefore that under the assumptions of theorem 2 in the present paper, the differential equation (1) does not admit any integral $y_{0}(x)$ of this kind, and therefore that the family $Y_{n}$ has the property $I_{n}^{*}[a,b)$ , what

Before formulating consequences of the results obtained above, we will first state the following theorem which is also given by G. Pó1 ya in the paper [23]:
theorem B. ¹⁴⁾ If the differential equation (1) has continuous coefficients in the open interval $(a,b)$ , and if she admits $n-1$ FULL $h_{1}(x)$ , $h_{2}(x),\ldots,h_{n-1}(x)$ , satisfying in this interval the relations:

h_{1}(x)\neq 0,\quad w\left(h_{1},h_{2}\right)\neq 0,\ldots,w\left(h_{1},h_{2},\ldots,h_{n-1}\right)\neq 0

then the family $Y_{n}$ of the integrals of the differential equation (1) has the property $I_{n}^{*}(a,b)$ .

result

From G. Pó 1 ya's theorems A and B, taking into account theorem 2 demonstrated above, it follows:

⁰⁰footnotetext: ¹³ ) This hypothesis was made when applying the principle of induction.
¹⁴ ) This theorem has the order number II in the cited work [23].

theorem 3. Assuming that the differential equation (1) has continuous coefficients in the semiclosed interval [ $a,b$ ), the necessary and sufficient condition that Tamil $Y_{n}$ of the integrals of equation (1) have the property $I_{n}^{*}[a,b)$ , it is like for any system of $n-1$ FULL $h_{1}(x),h_{2}(x),\ldots,h_{n-1}(x)$ of the differential equation (1), satisfying the conditions (96), the relations (97) must take place, in the open interval $(a,b)$ .

Remark. We assume that the differential equation (1) has continuous coefficients in a semi-closed interval [ $a,b$ ) and that it admits a particular system of $n-1$ FULL $h_{1}(x),h_{2}(x),\ldots,h_{n-1}(x)$ , satisfying relations (97) in the interval ( $a,b$ ). Under these assumptions, any system of $n-1$ integrals satisfying the conditions (96), will also verify the relations (97) in the interval ( $a,b$ ).

This property immediately follows from G. Póly's theorems A and B and from theorem 2 of the present paper.

Next, taking into account Theorem 1 established in this paper, as well as Theorem 3 stated above, we obtain:
Theorem 4. Under the assumption that the differential equation (1) has continuous coefficients in the semi-closed interval [ $a,b$ ), the necessary and sufficient condition that the family $Y_{n}$ of the integrals of this equation have the property $I_{n}[a,b)$ , it is like for any system of $n-1$ FULL $h_{1}(x),h_{2}(x),\ldots,h_{n-1}(x)$ of the differential equation (1), satisfying the conditions (96), the relations (97) take place in the open interval ( $a,b$ ).

Applications

Determining the maximum range $[a,b)$ , with the left extremity given, $\hat{in}$ which family $Y_{n}$ of the integrals of the differential equation (1) has the property $I_{n}[a,b)$ and therefore the property $I_{n}^{*}[a,b)$ .

This problem is related to numerous works on polylocal boundary value problems in linear differential equations. Among them we cite the works $[1-3,8,11,13,14,19,20,23,27]$ .

This paper provides a solution to this problem, related to the results obtained in the previous paragraphs.

We assume that a differential equation (1) is given, having the coefficients $a_{i}(x)$ continuous in the interval ( $-\infty,+\infty$ ). Either $x=a$ any given real number. We propose to determine the semi-closed interval $[a,b)$ , of maximum length, in which the family $Y_{n}$ of the integrals of the differential equation (1) has the property $I_{n}[a,b)$ (hence the property $I_{n}^{*}[a,b)$ ). To this end, taking into account theorem 4 as well as the observation made on the occasion of theorem 3, we can proceed as follows:

We will first consider some particular system of $n-1$ FULL $h_{1}(x),h_{2}(x),\ldots,h_{n-1}(x)$ of the differential equation (1), satisfying the conditions
(96), and then we will determine the maximum open interval ( $a,b$ ) in which the relations (97) take place, for the chosen integral system. The semi-closed interval $[a,b)$ will be the searched interval.

Example:

Let the linear and homogeneous differential equation with constant coefficients, of order 3 be

a_{0}y^{\prime\prime\prime}+a_{1}y^{\prime\prime}+a_{2}y^{\prime}+a_{3}y=0

(104)

We assume that the characteristic polynomial associated with this equation has a real root $r$ and two complex roots $\alpha+i\beta$ We aim to determine intervals of maximum length, of the form $[a,b)$ , in which the family $Y_{3}$ of the integrals of this differential equation has the respective property $I_{3}[a,b)$ (hence the property $I_{3}^{*}[a,b)$ ).

As shown in [2], the length of these maximal interpolation intervals does not depend on the number a representing the left extremum of the interval (this is because such a differential equation remains unchanged if any translation is performed on the independent variable). Noting with $l$ the desired length and taking $a=0$ , the problem returns to finding the maximum interval of the form $[0,l)$ in which the family $Y_{3}$ has the property $I_{3}[0,l)$ .

Also in [2] it was shown that using a change of variables of the form

x=\beta^{-1}t,\quad y=e^{\frac{a}{\beta}t}z(t)

the given differential equation is transformed into another differential equation with real constant coefficients, whose characteristic polynomial has as complex roots the numbers $+i$ and $-i$ Noting with $l^{*}$ length of the maximum interval of the form $\left[0,l^{*}\right)$ in which the set of integrals of the transformed equation has the property $I_{3}\left[0,l^{*}\right)$ , it is easily seen that the equality holds $l=|\beta|^{-1}l^{*}$ Thus, without restricting the generality of the problem, it can be assumed that the two complex roots of the characteristic polynomial associated with the given equation (104), are $+i$ and $-i$ We will also denote it with $r$ , the real root of the characteristic polynomial associated with this equation. We also note that we can assume this root to be non-negative, a situation that can always be achieved by performing the change of independent variable $x=-\xi$ .

Therefore, we will assume in the following that the roots of the characteristic polynomial associated with the differential equation (104) are $r\geqq 0,+i$ and $-i$ In this hypothesis, the general integral of equation (104) will be written

y(x)=C_{1}e^{rx}+C_{2}\sin x+C_{3}\cos x

(105)

To find out the number $l$ accordingly, we will use the working method presented previously.

We first determine two integrals $h_{1}(x)$ and $h_{2}(x)$ of the given differential equation, which satisfies the conditions:

	$\displaystyle h_{1}(0)=h_{1}^{\prime}(0)=0,\quad h_{1}^{\prime\prime}(0)=1$
	$\displaystyle h_{1}(0)=0,\quad h_{2}^{\prime}(0)=1,\quad h_{2}^{\prime\prime}(0)=0$		(106)

It is found that

		$\displaystyle h_{1}(x)=\frac{1}{1+r^{2}}\left(e^{rx}-r\sin x-\cos x\right)$
		$\displaystyle h_{2}(x)=\sin x$

Then, we calculate:

		$\displaystyle w_{1}(x)=h_{1}(x)=\frac{1}{1+r^{2}}\left(e^{rx}-r\sin x-\cos x\right)$
		$\displaystyle w_{2}(x)=\frac{1}{1+r^{2}}\left[e^{rx}(\cos x-r\sin x)-1\right]$

We will need to find the maximum open interval ( $0,l$ ) in which the relationship takes place $w_{1}(x)\neq 0$ and $w_{2}(x)\neq 0$ Length $l$ of this interval is given by the formula $l=\min\left\{p_{1},p_{2}\right\}$ , where $p_{1}$ and $p_{2}$ represent the smallest positive roots of the equations $w_{1}(x)=0$ , respectively $w_{2}(x)=0$ We distinguish the following two cases, as $r>0$ or $r=0$ .

Case 1: $r>0$ .

To study the behavior of the function in this case $w_{1}(x)$ in the interval $(0,\infty)$ , we calculate.

\frac{d}{dx}w_{1}(x)=\frac{1}{1+r^{2}}\left(re^{rx}-r\cos x+\sin x\right)

It is directly observed that in the considered case it takes place in the interval ( $0,\pi$ ), the inequality $\frac{d}{dx}w_{1}(x)>0$ . So in this interval, the function $w_{1}(x)$ is, increasing and how $w_{1}(0)=0$ , it follows that $w_{1}(x)$ takes positive values in the interval ( $0,\pi$ ).

On the other hand, we observe that if $x\geqq\pi$ , then the inequalities take place:

\left(1+r^{2}\right)w_{1}(x)=e^{rx}-r\sin x-\cos x>e^{r\pi}-r-1>0.

It follows from this that in the interval [ $\pi,\infty$ ), the function $w_{1}(x)$ does not cancel, always remaining positive. Ultimately, we obtain the result that in the considered case, the function $w_{1}(x)$ has no positive roots.

Regarding the behavior of the function $w_{2}(x)$ , it is obtained as soon as $\frac{d}{dx}w_{2}(x)=-e^{rx}\sin x$ , from which it follows that the function $w_{2}(x)$ is decreasing
in the interval ( $0,\pi$ ) and increasing in the interval ( $\pi,2\pi$ ). We obtain the following variation table:

$x$	0	$\pi$	$2\pi$
$\frac{d}{dx}w_{2}(x)$	$0---0$	+++++0
$w_{2}(x)$	$0\quad\searrow$	$\min\quad\nearrow\frac{1}{1+r^{2}}\left(e^{2\pi r}-1\right)$

From this picture, it is seen that if $r>0$ , then the smallest positive root of the function $w_{2}(x)$ is located in the interval ( $\pi,2\pi$ ).

In conclusion, in the case $r>0$ considered, we have $l=\rho_{2}$
Case 2 : $r=0$ In this case we have

w_{1}(x)=1-\cos x,\quad w_{2}(x)=\cos x-1,

and it is seen that $l=\rho_{1}=\rho_{2}=2\pi$ Finally
, we find the following result, established in another way in [2]:

If the characteristic polynomial associated with the differential equation (104) has a real root $r\geqq 0$ , as well as two complex roots +i and - $i$ , then the number $l$ corresponding to the considered differential equation is equal to the root in the interval ( $\pi,2\pi$ ] of the equation

\left(1+r^{2}\right)w_{2}(x)=e^{rx}(\cos x-r\sin x)-1=0

Computing Institute, RPR Academy - Cluj Branch.

BIBLIOGRAPHY:

1.

O. Ar a mă, The bilocal problem and the theorem of differential inequalities with confused nodes, of SA Ciaplighin, for second-order linear differential equations. Studii și Cerc. de Mat. (Cluj), IX, 7-38 (1958).
2.

O. Aramă, D. Ripianu, On the polylocal problem for linear differential equations with constant coefficients (I). Studii și Cerc. de Mat. (Cluj), VIII, 37-74 (1957).
3 - On the polylocal problem for linear differential equations with constant coefficients (II). Studii și Cerc. de Mat. (Cluj), VIII, 211-265 (1957).
3.
- •
  
  On the polylocal problem with confused nodes for linear differential equations with constant coefficients. Studia Universitatum V. Babeş et Bolyai, III, 3 ser. I, 95-116 (1958).
4.

PR Beesack, Non-oscillation and disconjugacy in the complex domain. (doc. diss. Washington Univ., 1955). Dissertation. Abstrs., 1955, 15, no. 5, 837.
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  Non-oscillation and disconjugacy in the complex domain. Trans. Amer. Math. Soc., 81, no. 1, 211-242 (1956).
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M. Biernacki, On an interpolation problem relative to linear differential equations. Ann. from Soc. Polish. of Math., XX, 169-214 (1947).
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S. Cinquini, Boundary value problems for order differential equations $n$ . Ann. della R. Sc. Norm. Sup. un di Pisa, (2), 9, 61-77 (1940).
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  On the comparison theorem of Kneser-Hille. Math. Scand., 5, 2,255-260 (1957).
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S. Z aidman, Evaluations of the distance between the zeros of the solutions of differential equations. Review of the CI Parhon University and the Bucharest Polytechnic, no. 6-7 (1955).

Comparative results on some polylocal boundary value problems for linear differential equations

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COMPARATIVE RESULTS ON SOME POLYLOCAL LIMIT PROBLEMS FOR LINEAR DIFFERENTIAL EQUATIONS

Case 2: $\alpha\geqq 2,(n\geqq 5)$ .

Proof of Theorem 1.

result

Applications

Example:

Case 1: $r>0$ .

BIBLIOGRAPHY:

Related Posts

Comparative results on some polylocal boundary value problems for linear differential equations

Abstract

Authors

Keywords

Paper coordinates

PDF

About this paper

Journal

Publisher Name

DOI

Print ISSN

Online ISSN

References

Paper (preprint) in HTML form

COMPARATIVE RESULTS ON SOME POLYLOCAL LIMIT PROBLEMS FOR LINEAR DIFFERENTIAL EQUATIONS

Case 2:α≧2,(n≧5)\alpha\geqq 2,(n\geqq 5).

Proof of Theorem 1.

result

Applications

Example:

Case 1:R>0r>0.

BIBLIOGRAPHY:

Related Posts

Case 2: $\alpha\geqq 2,(n\geqq 5)$ .

Case 1: $r>0$ .