Posts by Diana Otrocol

Abstract

In this paper we study some properties of the solutions of a second order system of functional-differential equations with maxima, of mixed type, with “boundary” conditions. We use the weakly Picard operator technique.

Authors

Diana Otrocol
Technical University of Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

Keywords

weakly Picard operator; equations of mixed type; equations with maxima

References

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Paper coordinates

D. Otrocol, Qualitative properties of solutions for mixed type functional-differential equations with maxima, Miskolc Mathematical Notes, 20 (2019) no. 2, pp. 1119–1128,
DOI: 10.18514/MMN.2019.1946

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About this paper

Journal

Miskolc Mathematical Notes

Publisher Name

DOI

10.18514/MMN.2019.1946

Print ISSN

1787-2405

Online ISSN

1787-2413

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[1] D. D. Bainov and S. G. Hristova, Differential equations with maxima. New York: Chapman & Hall/CRC Pure and Applied Mathematics, 2011.
[2] D. D. Bainov and D. Mishev, Oscillation theory of operator-differential equations. Singapore: World Scientific, 1995.
[3] L. Georgiev and V. G. Angelov, “On the existence and uniqueness of solutions for maximum equations.” Glas. Mat., vol. 37, no. 2, pp. 275–281, 2002.
[4] D. Otrocol, “Systems of functional differential equations with maxima, of mixed type.” Electron. J. Qual. Theory Differ. Equ., vol. 2014, no. 5, pp. 1–9, 2014, doi: https://doi.org/10.14232/ejqtde.2014.1.5.
[5] D. Otrocol and I. A. Rus, “Functional-differential equations with maxima of mixed type argument.” Fixed Point Theory, vol. 9, no. 1, pp. 207–220, 2008.
[6] D. Otrocol and I. A. Rus, “Functional-differential equations with “maxima” via weakly Picard operators theory.” Bull. Math. Soc. Sci. Math. Roumanie, vol. 51(99), no. 3, pp. 253–261, 2008.
[7] I. A. Rus, Generalized contractions and applications. Cluj-Napoca: Cluj University Press, 2001.
[8] I. A. Rus, “Functional differential equations of mixed type, via weakly Picard operators.” Seminar on Fixed Point Theory Cluj-Napoca, vol. 3, pp. 335–346, 2002.
[9] I. A. Rus, “Picard operators and applications.” Sci. Math. Jpn., vol. 58, no. 1, pp. 191–219, 2003.
[10] I. A. Rus, A. Petrus¸el, and M. A. S¸ erban, “Weakly Picard operators: equivalent definitions, applications and open problems.” Fixed Point Theory, vol. 7, no. 1, pp. 3–22, 2006.
[11] B. Zhang and G. Zhang, “Qualitative properties of functional equations with “maxima”.” Rocky Mt. J. Math., vol. 29, no. 1, pp. 357–367, 1999

QUALITATIVE PROPERTIES OF SOLUTIONS FOR MIXED TYPE FUNCTIONAL-DIFFERENTIAL EQUATIONS WITH MAXIMA

Diana Otrocol Department of Mathematics
Technical University of Cluj-Napoca
Memorandumului Street 28
400114 Cluj-Napoca
Romania Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
P.O.Box. 68-1
400110, Cluj-Napoca
Romania dotrocol@ictp.acad.ro, Diana.Otrocol@math.utcluj.ro

Abstract.

Key words and phrases:

Weakly Picard operator, equations of mixed type, equations with maxima

2010 Mathematics Subject Classification:

Primary 34K10; Secondary 47H10

1. Introduction

Differential equations with maxima are often met in the applications, for instance in the theory of automatic control. Numerous results on existence and uniqueness, asymptotic stability as well as numerical solutions have been obtained. To name a few, we refer the reader to [BainovHristova, Bainov, Georgiev], [Zhang] and the references therein.

The main goal of the presented paper is to study a second order functional-differential equations with maxima, of mixed type, using the theory of weakly Picard operators. The theory of Picard operators was introduced by I. A. Rus (see [Rus93] and [Rus]) to study problems related to fixed point theory. This abstract approach is used by many mathematicians and it seemed to be a very useful and powerful method in the study of integral equations and inequalities, ordinary and partial differential equations (existence, uniqueness, differentiability of the solutions), etc.

We consider the following functional-differential equation

-x^{\prime\prime}(t)=f(t,x(t),\underset{t-h_{1}\leq\xi\leq t+h_{2}}{\max}x(\xi)),\ t\in[a,b]

(1.1)

with the ”boundary” conditions

\left\{\begin{array}[c]{l}x(t)=\varphi(t),\ t\in[a-h_{1},a],\\ x(t)=\psi(t),\ t\in[b,b+h_{2}].\end{array}\right.

(1.2)

The novelty of this paper lies in the fact that ’maxima’ is taken on interval $[t-h_{1},t+h_{2}]$ , where $h_{1},\ h_{2}>0$ . Our results extend and improve corresponding theorems in the existing literature (see, e.g. [Rus02], [OtrocolEJQTDE, OtrocolRus, OtrocolRusFPT]). Also, in the end an extremal principle for the solution is given.

We suppose that:

(C₁)

$h_{1},h_{2},\ a\$ and $b\in\mathbb{R},\ a<b,\ h_{1}>0,\ h_{2}>0;$
(C₂)

$f\in C([a,b]\times\mathbb{R}^{2},\mathbb{R})$ ;
(C₃)

there exists $L_{f}>0$ such that

$\left|f(t,u_{1},u_{2})-f(t,v_{1},v_{2})\right|\leq L_{f}\underset{i=1,2}{\max}\left|u_{i}-vi\right|,$

for all $t\in[a,b]$ and $u_{i},v_{i}\in\mathbb{R},i=1,2;$
(C₄)

$\varphi\in C([a-h_{1},a],\mathbb{R})$ and $\psi\in C([b,b+h_{2}],\mathbb{R}).$

Let $G$ be the Green function of the following problem

-x^{\prime\prime}=\chi,\ x(a)=0,\ x(b)=0,\ \chi\in C[a,b].

The problem (1.1)–(1.2), $x\in C[a-h_{1},b+h_{2}]\cap C^{2}[a,b]$ is equivalent with the following fixed point equation

x(t)=\left\{\begin{array}[c]{l}\varphi(t),\ t\in[a-h_{1},a],\\ w(\varphi,\psi)(t)+\\ \quad+\int_{a}^{b}G(t,s)f(s,x(s),\underset{s-h_{1}\leq\xi\leq s+h_{2}}{\max}x(\xi))ds,\ t\in[a,b],\\ \psi(t),\ t\in[b,b+h_{2}],\end{array}\right.

(1.3)

$x\in C[a-h_{1},b+h_{2}],$ where

w(\varphi,\psi)(t):=\frac{t-a}{b-a}\psi(b)+\frac{b-t}{b-a}\varphi(a).

The equation (1.1) is equivalent with

x(t)=\left\{\begin{array}[c]{l}x(t),\ t\in[a-h_{1},a],\\ w(x|_{[a-h_{1},a]},x|_{[b,b+h_{2}]})(t)+\\ \quad+\int_{a}^{b}G(t,s)f(s,x(s),\underset{s-h_{1}\leq\xi\leq s+h_{2}}{\max}x(\xi))ds,\ t\in[a,b],\\ x(t),\ t\in[b,b+h_{2}],\end{array}\right.

(1.4)

$x\in C[a-h_{1},b+h_{2}].$

In what follow we consider the operators $B_{f},E_{f}:C[a-h_{1},b+h_{2}]\rightarrow C[a-h_{1},b+h_{2}]$ defined by

B_{f}(x)(t):=\left\{\begin{array}[c]{l}\varphi(t),\ t\in[a-h_{1},a],\\ w(\varphi,\psi)(t)+\\ \quad+\int_{a}^{b}G(t,s)f(s,x(s),\underset{s-h_{1}\leq\xi\leq s+h_{2}}{\max}x(\xi))ds,\ t\in[a,b],\\ \psi(t),\ t\in[b,b+h_{2}],\end{array}\right.

and

E_{f}(x)(t):=\left\{\begin{array}[c]{l}x(t),\ t\in[a-h_{1},a],\\ w(x|_{[a-h_{1},a]},x|_{[b,b+h_{2}]})(t)+\\ \quad+\int_{a}^{b}G(t,s)f(s,x(s),\underset{s-h_{1}\leq\xi\leq s+h_{2}}{\max}x(\xi))ds,\ t\in[a,b],\\ x(t),\ t\in[b,b+h_{2}].\end{array}\right.

Let $X:=C[a-h_{1},b+h_{2}]$ and $X_{\varphi,\psi}:=\{x\in X|\ x|_{[a-h_{1},a]}=\varphi,\ x|_{[b,b+h_{2}]}=\psi\}$ . It is clear that

X=\underset{\begin{array}[c]{c}\varphi\in C[a-h_{1},a]\\ \psi\in C[b,b+h_{2}]\end{array}}{\cup}X_{\varphi,\psi}

is a partition of $X.\$ We have

Lemma 1.1.

We suppose that the conditions $(C_{1}),\ (C_{2})$ and $(C_{4})$ are satisfied. Then

(a)

$B_{f}(X)\subset X_{\varphi,\psi}$ and $B_{f}(X_{\varphi,\psi})\subset X_{\varphi,\psi};$
(b)

$B_{f}|_{X_{\varphi,\psi}}=E_{f}|_{X_{\varphi,\psi}}.$

In this paper we shall prove that, if $L_{f}$ is small enough, then the operator $E_{f}$ is weakly Picard operator and we shall study the equation (1.1) in the terms of this operator.

2. Picard and Weakly Picard operators

In this paper we use the terminologies and notations from [Rus, Rus02, Rus93]. Let us recall now some essential definitions and fundamental results.

Let $(X,d)$ be a metric space and $A:X\rightarrow X$ an operator. We denote by $\;A^{0}=1_{X},\;A^{1}=A,\;A^{n+1}:=A\circ A^{n},n\in\mathbb{N}$ the iterates of the operator $A$ ;

We also use the following notations: $F_{A}:=\{x\in X\mid A(x)=x\}$ - the fixed point set of $A$ ;

$I(A):=\{Y\subset X\mid A(Y)\subset Y,Y\neq\emptyset\}$ - the family of the nonempty invariant subset of $A$ ;

We begin with the definitions of a Picard and weakly Picard operator.

Definition 2.1.

Let $(X,d)$ be a metric space. An operator $A\colon X\rightarrow X$ is a Picard operator (PO) if there exists $x^{\ast}\in X$ such that:

(i)

$F_{A}=\{x^{\ast}\};$
(ii)

the sequence $(A^{n}(x_{0}))_{n\in\mathbb{N}}$ converges to $x^{\ast}$ for all $x_{0}\in X$ .

Definition 2.2.

Let $(X,d)$ be a metric space. An operator $A\colon X\rightarrow X$ is a weakly Picard operator (WPO) if the sequence $(A^{n}(x))_{n\in\mathbb{N}}$ converges for all $x\in X$ , and its limit (which may depend on $x$ ) is a fixed point of $A$ .

Definition 2.3.

If $A$ is weakly Picard operator then we consider the operator $A^{\infty}$ defined by

A^{\infty}\colon X\rightarrow X,\ A^{\infty}(x):=\underset{n\rightarrow\infty}{\lim}A^{n}(x).

Remark 2.1.

It is clear that $A^{\infty}(X)=F_{A}=\{x\in X\mid A(x)=x\}.$

The following results are very useful in the sequel.

Lemma 2.1.

Let $(X,d,\leq)$ be an ordered metric space and $A\colon X\rightarrow X$ an operator. We suppose that $A$ is WPO and increasing. Then, the operator $A^{\infty}$ is increasing.

Lemma 2.2.

(Abstract Gronwall lemma) Let $(X,d,\leq)$ be an ordered metric space and $A\colon X\rightarrow X$ an operator. We suppose that $A$ is WPO and increasing. Then:

(a)

$x\leq A(x)\Rightarrow x\leq A^{\infty}(x);$
(b)

$x\geq A(x)\Rightarrow x\geq A^{\infty}(x).$

Lemma 2.3.

(Abstract comparison lemma) Let $(X,d,\leq)$ an ordered metric space and $A,B,C\colon X\rightarrow X$ be such that:

(i)

the operators $A,B,C$ are WPOs;
(ii)

$A\leq B\leq C;$
(iii)

the operator $B$ is increasing.

Then $x\leq y\leq z$ implies that $A^{\infty}(x)\leq B^{\infty}(y)\leq C^{\infty}(z).$

Theorem 2.1.

Let $(X,d)$ be a complete metric space and $A,B\colon X\rightarrow X$ two operators. We suppose that

(i)

the operator $A$ is an $\alpha$ -contraction;
(ii)

$F_{B}\neq\emptyset;$
(iii)

there exists $\eta>0$ such that

$d(A(x),B(x))\leq\eta,\ \forall x\in X.$

Then, if $F_{A}=\{x_{A}^{\ast}\}$ and $x_{B}^{\ast}\in F_{B},$ we have

d(x_{A}^{\ast},x_{B}^{\ast})\leq\frac{\eta}{1-\alpha}.

Another important notion is the following

Definition 2.4.

Let $A$ be a weakly Picard operator and $c>0.$ The operator $A\$ is $c$ -weakly Picard operator if

d(x,A^{\infty}(x))\leq cd(x,A(x)),\ \forall x\in X.

For the $c$ -PO_s and $c$ -WPO_s we have the following lemma.

Theorem 2.2.

Let $(X,d)$ be a metric space and $A_{i}\colon X\rightarrow X,\ i=1,2.$ Suppose that

(i)

the operator $A_{i}$ is $c_{i}$ -WPO_s, $i\!=\!1,2;$
(ii)

there exists $\eta>0$ such that

$d(A_{1}(x),A_{2}(x))\leq\eta,\ \forall x\in X.$

Then $H(F_{A_{1}},F_{A_{2}})\leq\eta\max(c_{1},c_{2}).$

Example 2.1.

Let $(X,d)$ be a complete metric space and $A\colon X\rightarrow X$ an $\alpha$ -contraction. Then $A$ is $\frac{1}{1-\alpha}$ -PO.

Example 2.2.

Let $(X,d)$ be a complete metric space and $A:X\rightarrow X$ continuous and $\alpha$ -graphic contraction. Then $A$ is $\frac{1}{1-\alpha}$ -WPO.

For more details on WPOs theory see [Rus], [Rus93] and [RusPetrusel].

3. Existence and uniqueness

Our first result is the following

Theorem 3.1.

We suppose that:

(a)

the conditions $(C_{1})$ – $(C_{4})$ are satisfied;
(C₅)

$\frac{L_{f}}{8}(b-a)^{2}<1$ .

Then the problem (1.1)–(1.2) has a unique solution which is the uniform limit of the successive approximations.

Proof.

Consider the Banach space $(C[a-h_{1},b+h_{2}],\left\|\cdot\right\|)$ where $\left\|\cdot\right\|$ is the Chebyshev norm, $\left\|\cdot\right\|:=\underset{a-h_{1}\leq t\leq b+h_{2}}{\max}\left|x(t)\right|.$

The problem (1.1)–(1.2) is equivalent with the fixed point equation

B_{f}(x)=x,\ x\in C[a-h_{1},b+h_{2}].

From the condition $(C_{3})$ we have, for $t\in[a,b]$

	$\displaystyle\left\|B_{f}(x)(t)-B_{f}(y)(t)\right\|\leq$
	$\displaystyle\leq L_{f}\int_{a}^{b}G(t,s)\left[\max\left\|x(s)-y(s)\right\|,\bigg\|\underset{a-h_{1}\leq\xi\leq b+h_{2}}{\max}x(\xi)-\underset{a-h_{1}\leq\xi\leq b+h_{2}}{\max y(\xi)}\bigg\|\right]ds$
	$\displaystyle\leq L_{f}\int_{a}^{b}G(t,s)\underset{a-h_{1}\leq\xi\leq b+h_{2}}{\max}\left\|x(s)-y(s)\right\|ds\leq$
	$\displaystyle\leq\frac{L_{f}}{8}(b-a)^{2}\left\\|x-y\right\\|.$

This implies that $B_{f}$ is an $\alpha$ -contraction, with $\alpha=\frac{L_{f}}{8}(b-a)^{2}.$ The proof follows from the contraction principle. ∎

Remark 3.1.

From the proof of Theorem 3.1, it follows that the operator $B_{f}$ is PO. Since

B_{f}|_{X_{\varphi,\psi}}=E_{f}|_{X_{\varphi,\psi}}

and

X:=C[a-h_{1},b+h_{2}]=\underset{\varphi,\psi}{\cup}X_{\varphi,\psi},\ E_{f}(X_{\varphi,\psi})\subset X_{\varphi,\psi}

hence, the operator $E_{f}$ is WPO and

F_{E_{f}}\cap X_{\varphi,\psi}=\{x_{\varphi,\psi}^{\ast}\},\forall\varphi\in C[a-h_{1},a],\ \forall\psi\in C[b,b+h_{2}],

where $x_{\varphi,\psi}^{\ast}$ is the unique solution of the problem (1.1)–(1.2).

Remark 3.2.

$E_{f}$ is $\alpha$ -graphic contraction, i.e.

\left\|E_{f}^{2}(x)-E_{f}(x)\right\|\leq\alpha\left\|x-E_{f}(x)\right\|,\ \forall x\in C[a-h_{1},b+h_{2}].

4. Inequalities of Čaplygin type

Now we consider the operators $E_{f}$ and $B_{f}$ on the ordered Banach space $(C[a-h_{1},b+h_{2}],\left\|\cdot\right\|,\leq).\$ We have

Theorem 4.1.

We suppose that:

(a)

the conditions $(C_{1})-(C_{4})$ are satisfied;
(b)

$\frac{L_{f}}{8}(b-a)^{2}<1$ ;
(c)

$f(t,\cdot,\cdot):\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ is increasing, $\forall t\in[a,b].$

Let $x$ be a solution of equation (1.1) and $y$ a solution of the inequality

-y^{\prime\prime}(t)\leq f(t,y(t),\underset{t-h_{1}\leq\xi\leq t+h_{2}}{\max}y(\xi)),\ t\in[a,b].

Then if

y(t)\leq x(t),\forall t\in[a-h_{1},a]\cup[b,b+h_{2}]\text{ we obtain that }y\leq x.

Proof.

Let us consider the operator $\widetilde{w}:C[a-h_{1},b+h_{2}]\rightarrow C[a-h_{1},b+h_{2}]$ defined by

\widetilde{w}(z)(t):=\left\{\begin{array}[c]{l}z(t),\ t\in[a-h_{1},a],\\ w(z|_{[a-h_{1},a]},z|_{[b,b+h_{2}]})(t),\ t\in[a,b],\\ z(t),\ t\in[b,b+h_{2}].\end{array}\right.

First of all we remark that

w(y|_{[a-h_{1},a]},y|_{[b,b+h_{2}]})\leq w(x|_{[a-h_{1},a]},x|_{[b,b+h_{2}]})

and

\widetilde{w}(y)\leq\widetilde{w}(x).

In the terms of the operator $E_{f},$ we have

x=E_{f}(x)\text{ and }y\leq E_{f}(y).

From the conditions $(C_{1}),(C_{2})$ and $(C_{3})$ follows that the operator $E_{f}$ is WPO. Also, from condition (c) we have that $E_{f}$ is an increasing operator. Applying Lemma 2.1, we have that the operator $E_{f}^{\infty}$ is increasing. From Theorem 3.1 we have that $E_{f}(X_{\varphi,\psi})\subset X_{\varphi,\psi},\ \forall\varphi,\psi\in\mathbb{R}.\ E_{f}|_{X_{\varphi,\psi}}$ is a contraction and since $\widetilde{w}(z)\in X_{\varphi,\psi}$ then

E_{f}^{\infty}(\widetilde{w})=E_{f}^{\infty}(y),\ \forall y\in X_{\varphi,\psi}.

Let $y\leq E_{f}(y)$ , since $E_{f}$ is increasing, from the Gronwall lemma (Lemma 2.2) we get $y\leq E_{f}^{\infty}(y).$ Also $y,\widetilde{w}(y)\in X_{w(y)}$ , so $E_{f}^{\infty}(y)=E_{f}^{\infty}(\widetilde{w}(y))$ . But $w(y)\leq w(x),\ E_{f}^{\infty}$ is increasing and $E_{f}^{\infty}(\widetilde{w}(x))=E_{f}^{\infty}(x)=x.$ So

y\leq E_{f}^{\infty}(y)=E_{f}^{\infty}(\widetilde{w}(y))\leq E_{f}^{\infty}(\widetilde{w}(x))=E_{f}^{\infty}(x)=x.

∎

5. Data dependence: monotony

In this section we study the monotony of the solution of the problem (1.1)–(1.2) with respect to $\varphi,\ \psi$ and $f$ .

Theorem 5.1.

Let $f_{i}\in C([a,b]\times\mathbb{R}^{2},\mathbb{R}),i=1,2,3,$ be as in Theorem 3.1. We suppose that:

(i)

$f_{1}\leq f_{2}\leq f_{3};$
(ii)

$f_{2}(t,\cdot,\cdot,\cdot):\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ is monotone increasing;

Let $x_{i}$ be a solution of the equation

-x_{i}^{\prime\prime}(t)=f_{i}(t,x(t),\underset{t-h_{1}\leq\xi\leq t+h_{2}}{\max}x(\xi)),\ t\in[a,b]\text{ and }i=1,2,3.

Then, $x_{1}(t)\leq x_{2}(t)\leq x_{3}(t),\ \forall t\in[a-h_{1},a]\cup[b,b+h_{2}]$ , implies that $x_{1}\leq x_{2}\leq x_{3}$ , i.e. the unique solution of the problem (1.1)–(1.2) is increasing with respect to $f,\ \varphi$ and $\psi.$

Proof.

From Theorem 3.1, the operators $E_{f_{i}},i=1,2,3,\$ are WPOs. From the condition (ii) the operator $E_{f_{2}}$ is monotone increasing. From the condition (i) it follows that

E_{f_{1}}\leq E_{f_{2}}\leq E_{f_{3}}.

On the other hand we remark that

\widetilde{w}(x_{1})\leq\widetilde{w}(x_{2})\leq\widetilde{w}(x_{3})

and

x_{i}=E_{f_{i}}^{\infty}(\widetilde{w}(x_{i})),\ i=1,2,3.

So, the proof follows from Lemma 2.3. ∎

6. Data dependence: continuity

Consider the boundary value problem (1.1)–(1.2) and suppose the conditions of the Theorem 3.1 are satisfied. Denote by $x^{\ast}(\cdot;\varphi,\psi,f),\$ the solution of this problem. We state the following result:

Theorem 6.1.

Let $\varphi_{i},\psi_{i},f_{i},i=1,2$ be as in the Theorem 3.1. Furthermore, we suppose that there exists $\eta_{i}>0,i=1,2\$ such that

(i)

$\left|\varphi_{1}(t)-\varphi_{2}(t)\right|\leq\eta_{1},\ \forall t\in[a-h_{1},a]$ and $\left|\psi_{1}(t)-\psi_{2}(t)\right|\leq\eta_{1},\ \forall t\in[b,b+h_{2}]$ ;
(ii)

$\left|f_{1}(t,u_{1},u_{2})-f_{2}(t,u_{1},u_{2})\right|\leq\eta_{2},\forall t\in C[a,b],u_{1},u_{2}\in\mathbb{R}.$

Then

\left\|x_{1}^{\ast}(t;\varphi_{1},\psi_{1},f_{1})-x_{2}^{\ast}(t;\varphi_{2},\psi_{2},f_{2})\right\|\leq\frac{8\eta_{1}+(b-a)^{2}\eta_{2}}{8-L_{f}(b-a)^{2}},

where $x_{i}^{\ast}(t;\varphi_{i},\psi_{i},f_{i})$ are the solution of the problem (1.1)–(1.2) with respect to $\varphi_{i},\psi_{i},f_{i},$ $i=1,2,\$ and $L_{f}=\max(L_{f_{1}},L_{f_{2}}).$

Proof.

Consider the operators $B_{\varphi_{i},\psi_{i},f_{i}},i=1,2.$ From Theorem 3.1 these operators are contractions.

Additionally

\left\|B_{\varphi_{1},\psi_{1},f_{1}}(x)-B_{\varphi_{2},\psi_{2},f_{2}}(x)\!\right\|\leq\eta_{1}+\eta_{2}\frac{(b-a)^{2}}{8},

$\forall x\in C[a-h_{1},b+h_{2}].$

Now the proof follows from Theorem 2.1, with $A:=B_{\varphi_{1},\psi_{1},f_{1}},B=B_{\varphi_{2},\psi_{2},f_{2}},\eta=\eta_{1}+\eta_{2}\frac{(b-a)^{2}}{8}$ and $\alpha:=\frac{L_{f}}{8}(b-a)^{2}$ where $L_{f}=\max(L_{f_{1}},L_{f_{2}}).$ ∎

In what follow we shall use the $c$ -WPOs techniques to give some data dependence results using Theorem 2.2.

Theorem 6.2.

Let $f_{1}$ and $f_{2}$ be as in the Theorem 3.1. Let $S_{E_{f_{1}}},S_{E_{f_{2}}}$ be the solution sets of system (1.1) corresponding to $f_{1}$ and $f_{2}$ . Suppose that there exists $\eta>0,$ such that

\left|f_{1}(t,u_{1},u_{2})-f_{2}(t,u_{1},u_{2})\right|\leq\eta

(6.1)

for all $t\in[a,b],u_{1},u_{2}\in\mathbb{R}.$

Then

H_{\left\|\cdot\right\|_{C}}(S_{E_{f_{1}}},S_{E_{f_{2}}})\leq\frac{(b-a)^{2}\eta}{8-L_{f}(b-a)^{2}},

where $L_{f}=\max(L_{f_{1}},L_{f_{2}})$ and $H_{\left\|\cdot\right\|_{C}}$ denotes the Pompeiu-Housdorff functional with respect to $\left\|\cdot\right\|_{C}$ on $C[a,b].$

Proof.

In the condition of Theorem 3.1, the operators $E_{f_{1}}$ and $E_{f_{2}}$ are $c_{1}$ -WPO and $c_{2}$ -weakly Picard operators.

Let

X_{\varphi,\psi}:=\{x\in X|\ x|_{[a-h_{1},a]}=\varphi,\ x|_{[b,b+h_{2}]}=\psi\}.

It is clear that $E_{f_{1}}|_{X_{\varphi,\psi}}=B_{f_{1}},\ E_{f_{2}}|_{X_{\varphi,\psi}}=B_{f_{2}}.$ Therefore,

\left|E_{f_{1}}^{2}(x)-E_{f_{1}}(x)\right|\leq\tfrac{1}{8}L_{f_{1}}(b-a)^{2}\left|E_{f_{1}}(x)-x\right|,

\left|E_{f_{2}}^{2}(x)-E_{f_{2}}(x)\right|\leq\tfrac{1}{8}L_{f_{2}}(b-a)^{2}\left|E_{f_{2}}(x)-x\right|,

for all $x\in C[a-h_{1},b+h_{2}].$

Now, choosing

\alpha_{i}=\tfrac{1}{8}L_{f_{i}}(b-a)^{2},i=1,2,

we get that $E_{f_{1}}$ and $E_{f_{2}}$ are $c_{1}$ -weakly Picard operators and $c_{2}$ -weakly Picard operators with $c_{1}=(1-\alpha_{1})^{-1}\$ and $\ c_{2}=(1-\alpha_{2})^{-1}$ . From (6.1) we obtain that

\left\|E_{f_{1}}(x)-E_{f_{2}}(x)\right\|_{C}\leq(b-a)^{2}\eta,

$\forall x\in C[a-h_{1},b+h_{2}].$ Applying Theorem 2.2 we have that

H_{\left\|\cdot\right\|_{C}}(S_{E_{f_{1}}},S_{E_{f_{2}}})\leq\frac{(b-a)^{2}\eta}{8-L_{f}(b-a)^{2}},

where $L_{f}=\max(L_{f_{1}},L_{f_{2}})$ and $H_{\left\|\cdot\right\|_{C}}$ is the Pompeiu-Housdorff functional with respect to $\left\|\cdot\right\|_{C}$ on $C[a-h_{1},b+h_{2}].$ ∎

7. Extremal principle

We consider the following differential equation with maxima

x^{\prime\prime}(t)+p(t)x(t)+q(t)\underset{t-h_{1}\leq\xi\leq t+h_{2}}{\max}x(\xi)=0

(7.1)

where $p,q\in C([a,b],\mathbb{R})$ .

Remark 7.1.

The function $x(t)$ is an eventually negative solution of equation (7.1) if and only if $y(t)=-x(t)$ is an eventually positive solution of the equation

y^{\prime\prime}(t)+p(t)y(t)+q(t)\underset{t-h_{1}\leq\xi\leq t+h_{2}}{\min}y(\xi)=0.

(7.2)

From the above remark we see that the positive and the negative solution of equation (7.1) need to be discussed separately.

Theorem 7.1.

(Extremal principle) Let $p(t),q(t)<0,$ for all $t\in]a,b[$ and $x$ be a solution of (7.1). Then

(a)

if $\max\{x(t)|\ t\in[a,b]\}=x(t_{0})$ and $x(t_{0})>0$ then $t_{0}\in\{a,b\};$
(b)

if $min\{x(t)|\ t\in[a,b]\}=x(t_{0})$ and $x(t_{0})<0$ then $t_{0}\in\{a,b\}.$

Proof.

(a)

Let $t_{0}\in]a,b[$ be such that, $x(t_{0})>0$ is the maximum value of $t_{0}$ on $[a,b]$ . Since $x\in C^{2}[a,b]$ we have that $x(t_{0})>0,x^{\prime}(t_{0})=0,x^{\prime\prime}(t_{0})\leq 0$ . From (7.1) we have

0=x^{\prime\prime}(t_{0})+p(t_{0})x(t_{0})+q(t_{0})\underset{t_{0}-h_{1}\leq\xi\leq t_{0}+h_{2}}{\max}x(\xi)<0\text{.}

This is a contradiction, and therefore our assumption is wrong. So $t_{0}\in\{a,b\}.$

(b)

Let $t_{0}\in]a,b[$ be such that, $x(t_{0})<0$ is the minimum value of $t_{0}$ on $[a,b]$ . Since $x\in C^{2}[a,b]$ we have that $x(t_{0})<0,x^{\prime}(t_{0})=0,x^{\prime\prime}(t_{0})\geq 0$ . From (7.1) we have

0=x^{\prime\prime}(t_{0})+p(t_{0})x(t_{0})+q(t_{0})\underset{t_{0}-h_{1}\leq\xi\leq t_{0}+h_{2}}{\max}x(\xi)>0\text{.}

This is a contradiction, and therefore our assumption is wrong. So $t_{0}\in\{a,b\}.$

∎

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	$\displaystyle\left\|B_{f}(x)(t)-B_{f}(y)(t)\right\|\leq$
	$\displaystyle\leq L_{f}\int_{a}^{b}G(t,s)\left[\max\left\|x(s)-y(s)\right\|,\bigg\|\underset{a-h_{1}\leq\xi\leq b+h_{2}}{\max}x(\xi)-\underset{a-h_{1}\leq\xi\leq b+h_{2}}{\max y(\xi)}\bigg\|\right]ds$
	$\displaystyle\leq L_{f}\int_{a}^{b}G(t,s)\underset{a-h_{1}\leq\xi\leq b+h_{2}}{\max}\left\|x(s)-y(s)\right\|ds\leq$
	$\displaystyle\leq\frac{L_{f}}{8}(b-a)^{2}\left\\|x-y\right\\|.$

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QUALITATIVE PROPERTIES OF SOLUTIONS FOR MIXED TYPE FUNCTIONAL-DIFFERENTIAL EQUATIONS WITH MAXIMA

Abstract.

Key words and phrases:

2010 Mathematics Subject Classification:

1. Introduction

Lemma 1.1.

2. Picard and Weakly Picard operators

Definition 2.1.

Definition 2.2.

Definition 2.3.

Remark 2.1.

Lemma 2.1.

Lemma 2.2.

Lemma 2.3.

Theorem 2.1.

Definition 2.4.

Theorem 2.2.

Example 2.1.

Example 2.2.

3. Existence and uniqueness

Theorem 3.1.

Proof.

Remark 3.1.

Remark 3.2.

4. Inequalities of Čaplygin type

Theorem 4.1.

Proof.

5. Data dependence: monotony

Theorem 5.1.

Proof.

6. Data dependence: continuity

Theorem 6.1.

Proof.

Theorem 6.2.

Proof.

7. Extremal principle

Remark 7.1.

Theorem 7.1.

Proof.

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