Some remarks concerning a condition for the non-oscillation of linear differential equations

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E. Moldovan (Popoviciu)
Institutul de Calcul

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E. Moldovan (Popoviciu), Some remarks concerning a condition for the non-oscillation of linear differential equations. (Russian) Mathematica (Cluj) 1 (24)1959 45–48.

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Mathematica Cluj

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SOME REMARKS ON ONE NON-OSCILLATORY SIGN FOR LINEAR DIFFERENTIAL EQUATIONS Elena Moldovan Cluj

    • In the work of V. A. KONDRATIEV [1] a necessary and sufficient condition for the oscillation*) in a given interval of integrals of the equation is given
(1) L ( y ) = y ′ ′ + p ( x ) y ′ + q ( x ) y = 0 (1) L ( y ) = y ′ ′ + p ( x ) y ′ + q ( x ) y = 0 {:(1)L(y)=y^('')+p(x)y^(')+q(x)y=0:}(1)L(y)=y′′+p(x)y′+q(x)y=0
Assuming that p ( x ) p ( x ) p(x)p(x)And q ( x ) q ( x ) q(x)q(x)are continuous functions on the interval [ a , b ] [ a , b ] [a,b][a,b], Kondratiev's theorem states:
in order for the integrals of equation (1) to be non-oscillatory in [ a , b ] [ a , b ] [a,b][a,b], it is necessary and sufficient that there exists a complex and twice differentiable function u(x) satisfying the condition
(2) L ( u ) ≦ 0 (2) L ( u ) ≦ 0 {:(2)L(u) <= 0:}(2)L(u)≦0
The sufficiency of the condition is given in the work of de la VallÉE Poussin, [6]. In the proposed note, we give an interpretation of this theorem, from which follows one of its generalizations for linear equations of order higher than two.
2. - Assume that the integrals of equation (1) are non-oscillating in [ a , b ] [ a , b ] [a,b][a,b]Then the set M 2 M 2 M_(2)M2integrals of equation (1) are a set of the type I 2 [ a , b ] I 2 [ a , b ] I_(2)[a,b]I2[a,b], i.e. whatever the different points may be x 1 , x 2 ∈ [ a , b ] x 1 , x 2 ∈ [ a , b ] x_(1),x_(2)in[a,b]x1,x2∈[a,b], and whatever the numbers may be y 1 , y 2 y 1 , y 2 y_(1),y_(2)y1,y2set M 2 M 2 M_(2)M2contains one, and only one, function y ( x ) y ( x ) y(x)y(x)so what y ( x i ) = y i , i = 1 , 2 y ( x i ) = y i , i = 1 , 2 y(x_(i))=y_(i),i=1,2y(xi)=yi,i=1,2Under this assumption, the following property is known: if f ( x ) f ( x ) f(x)f(x)there is a twice differentiable function in [ a , b ] [ a , b ] [a,b][a,b]and if there are three points x 1 < x 2 < x 3 x 1 < x 2 < x 3 x_(1) < x_(2) < x_(3)x1<x2<x3V [ a , b ] [ a , b ] [a,b][a,b]such,
that some integral of equation (1) is equal to or less than the function at these points f ( x ) f ( x ) f(x)f(x), then there is a point ξ ∈ ( x 1 , x 3 ) ξ ∈ ( x 1 , x 3 ) xi in(x_(1),x_(3))ξ∈(x1,x3), such that f ′ ′ ( ξ ) + ϕ ( ξ ) f ′ ( ξ ) + + q ( ξ ) f ( ξ ) = 0 f ′ ′ ( ξ ) + ϕ ( ξ ) f ′ ( ξ ) + + q ( ξ ) f ( ξ ) = 0 f^('')(xi)+phi(xi)f^(')(xi)++q(xi)f(xi)=0f′′(ξ)+ϕ(ξ)f′(ξ)++q(ξ)f(ξ)=0. This property implies the following remark: if the function f ( x ) f ( x ) f(x)f(x)satisfies one of the inconsistencies f ′ ′ ( x ) + p ( x ) f ′ ( x ) ÷ q ( x ) f ( x ) < 0 f ′ ′ ( x ) + p ( x ) f ′ ( x ) ÷ q ( x ) f ( x ) < 0 f^('')(x)+p(x)f^(')(x)-:q(x)f(x) < 0f′′(x)+p(x)f′(x)÷q(x)f(x)<0or > 0 > 0 > 0>0, tagda f ( x ) f ( x ) f(x)f(x)cannot coincide at more than two points in the interval [ a , b ] [ a , b ] [a,b][a,b]with no integrals of equation (1).
Definition I. - Function f ( x ) f ( x ) f(x)f(x), defined on the interval [ a , b ] [ a , b ] [a,b][a,b], is called two-valued relative to the set M 2 M 2 M_(2)M2, if at any y ( x ) ∈ M 2 y ( x ) ∈ M 2 y(x)inM_(2)y(x)∈M2, equality y ( x ) = f ( x ) y ( x ) = f ( x ) y(x)=f(x)y(x)=f(x)cannot occur at more than two points in the interval [ a , b ] [ a , b ] [a,b][a,b].
This definition is a special case of one definition given in [3] for the case of a set of type I n [ a , b ] I n [ a , b ] I_(n)[a,b]In[a,b]From Theorem 13 of [3] it follows that if f ( x ) f ( x ) f(x)f(x)continuous in [ a , b ] [ a , b ] [a,b][a,b]and two-valued with respect to the set M 2 M 2 M_(2)M2then whatever the points are x 1 < x 2 < x 3 ∈ [ a , b ] x 1 < x 2 < x 3 ∈ [ a , b ] x_(1) < x_(2) < x_(3)in[a,b]x1<x2<x3∈[a,b], from y ( x ) ∈ M 2 y ( x ) ∈ M 2 y(x)inM_(2)y(x)∈M2And f ( x i ) = y ( x i ) , i = 1 , 2 f ( x i ) = y ( x i ) , i = 1 , 2 f(x_(i))=y(x_(i)),i=1,2f(xi)=y(xi),i=1,2it always follows that f ( x 3 ) - y ( x 3 ) > 0 f ( x 3 ) - y ( x 3 ) > 0 f(x_(3))-y(x_(3)) > 0f(x3)-y(x3)>0or f ( x 3 ) - y ( x 3 ) < 0 f ( x 3 ) - y ( x 3 ) < 0 f(x_(3))-y(x_(3)) < 0f(x3)-y(x3)<0.
Definition 2. - If on any system of three points x 1 < x 2 < x 3 x 1 < x 2 < x 3 x_(1) < x_(2) < x_(3)x1<x2<x3from [ a , b ] [ a , b ] [a,b][a,b], from y ( x ) ∈ M 2 u y ( x i ) = f ( x i ) ; i = 1 , 2 y ( x ) ∈ M 2 u y ( x i ) = f ( x i ) ; i = 1 , 2 y(x)inM_(2)uy(x_(i))=f(x_(i));i=1,2y(x)∈M2uy(xi)=f(xi);i=1,2one of the inequalities follows for all f ( x 3 ) - y ( x 3 ) > , ≧ , ≦ f ( x 3 ) - y ( x 3 ) > , ≧ , ≦ f(x_(3))-y(x_(3)) > , >= , <=f(x3)-y(x3)>,≧,≦or < 0 < 0 < 0<0, then the function f ( x ) f ( x ) f(x)f(x)is called convex, non-concave, non-convex respectively with respect to the set M 2 M 2 M_(2)M2on the interval [ a , b ] [ a , b ] [a,b][a,b].
Here we do not dwell on the fact that a similar definition can be given by replacing the interval [ a , b ] [ a , b ] [a,b][a,b]through some of its sub-intervals.
If f ( x ) f ( x ) f(x)f(x)is a twice differentiable function, then it can be proved [2] that from the inequality f ′ ′ ( x ) + p ( x ) f ′ ( x ) + q ( x ) f ( x ) > 0 , x ∈ [ a , b ] f ′ ′ ( x ) + p ( x ) f ′ ( x ) + q ( x ) f ( x ) > 0 , x ∈ [ a , b ] f^('')(x)+p(x)f^(')(x)+q(x)f(x) > 0,x in[a,b]f′′(x)+p(x)f′(x)+q(x)f(x)>0,x∈[a,b]it follows that f ( x ) f ( x ) f(x)f(x)is a convex function relative to the set M 2 M 2 M_(2)M2, and from the inequality f ′ ′ ( x ) + p ( x ) f ′ ( x ) + q ( x ) f ( x ) < 0 , x ∈ [ a , b ] f ′ ′ ( x ) + p ( x ) f ′ ( x ) + q ( x ) f ( x ) < 0 , x ∈ [ a , b ] f^('')(x)+p(x)f^(')(x)+q(x)f(x) < 0,x in[a,b]f′′(x)+p(x)f′(x)+q(x)f(x)<0,x∈[a,b]it follows that f ( x ) f ( x ) f(x)f(x)is a concave function with respect to the set M 2 M 2 M_(2)M2Both of these properties apply here to the entire interval. [ a , b ] [ a , b ] [a,b][a,b]. If the inequality L ( f ) > 0 L ( f ) > 0 L(f) > 0L(f)>0or < 0 < 0 < 0<0имет место на некстором нітервале J ⊂ [ a . b ] J ⊂ [ a . b ] J sub[ab]J⊂[a.b]it entails convexity, respectively concavity on the interval J J JJIf so, L ( f ) ≧ 0 L ( f ) ≧ 0 L(f) >= 0L(f)≧0or ≦ 0 ≦ 0 <= 0≦0, then the function f ( x ) f ( x ) f(x)f(x)is non-concave, respectively non-convex.
Using these concepts, we can prove Kondratiev's theorem. Let us assume that the condition of the theorem is satisfied. Let V ( x ) V ( x ) V(x)V(x)wavering on [ a , b ] [ a , b ] [a,b][a,b]integral of equation (1). Let x 1 < x 2 x 1 < x 2 x_(1) < x_(2)x1<x2two consecutive points of a set of points from [ a , b ] [ a , b ] [a,b][a,b]on which V ( x ) V ( x ) V(x)V(x)vanishes. Let's assume that V ( x ) > 0 V ( x ) > 0 V(x) > 0V(x)>0on ( x 1 , x 2 x 1 , x 2 x_(1),x_(2)x1,x2). Then, by virtue of Sturm's theorem, the integrals of equation (1) are non-oscillatory on the interval ( x 1 , x 2 x 1 , x 2 x_(1),x_(2)x1,x2). Therefore, from L ( u ) ≦ 0 L ( u ) ≦ 0 L(u) <= 0L(u)≦0it follows that u ( x ) u ( x ) u(x)u(x)is non-convex with respect to M 2 M 2 M_(2)M2on the interval ( x 1 , x 2 x 1 , x 2 x_(1),x_(2)x1,x2). But, since u ( x ) u ( x ) u(x)u(x)complex function, there is a constant c c cc, such that c . V ( x ) c . V ( x ) cV(x)c.V(x)coincides in at least two points x 1 ′ < x 2 ′ x 1 ′ < x 2 ′ x_(1)^(') < x_(2)^(')x1′<x2′from ( x 1 , x 2 ) ( x 1 , x 2 ) (x_(1),x_(2))(x1,x2)With u ( x ) u ( x ) u(x)u(x), A u ( x 2 - ε ) - c V ( x 2 - ε ) > 0 u ( x 2 - ε ) - c V ( x 2 - ε ) > 0 u(x_(2)-epsi)-cV(x_(2)-epsi) > 0u(x2-ε)-cV(x2-ε)>0at a sufficiently small ε > 0 ε > 0 epsi > 0ε>0, which contradicts the property of non-convexity of the function u ( x ) u ( x ) u(x)u(x). Thus, equation (1) cannot have any oscillating integral on the interval [ a , b ] [ a , b ] [a,b][a,b]The sufficiency of the condition is proven.
To prove the necessity it is sufficient to note that if the integrals of equation (1) are non-oscillatory on the interval [ a , b ] [ a , b ] [a,b][a,b]then the integral y ( x ) y ( x ) y(x)y(x)satisfying the conditions y ( a ) = y ( b ) = α > 0 y ( a ) = y ( b ) = α > 0 y(a)=y(b)=alpha > 0y(a)=y(b)=α>0, is positive throughout the interval [ a , b ] [ a , b ] [a,b][a,b].
3. - The application of the concept of a convex function in the above sense leads us to a more general theorem of comparison, containing Kondratiev's theorem as a special case.
Let a differential equation be given
(3) L n ( y ) = y ( n ) + a 1 ( x ) y ( n - 1 ) + ⋯ + a n ( x ) y = 0 (3) L n ( y ) = y ( n ) + a 1 ( x ) y ( n - 1 ) + ⋯ + a n ( x ) y = 0 {:(3)L_(n)(y)=y^((n))+a_(1)(x)y^((n-1))+cdots+a_(n)(x)y=0:}(3)Ln(y)=y(n)+a1(x)y(n-1)+⋯+an(x)y=0
Where a i ( x ) , i = 1 , 2 , … , n a i ( x ) , i = 1 , 2 , … , n a_(i)(x),i=1,2,dots,nai(x),i=1,2,…,ncontinuous functions in a finite and closed interval [ a , b ] [ a , b ] [a,b][a,b]Let us denote by M 2 M 2 M_(2)M2set of integrals of equation (3). Many M n M n M_(n)Mn- by analogy with the definitions given in work [3] - we will call it a set of TNPA I n [ a , b ] I n [ a , b ] I_(n)[a,b]In[a,b]if for any system of n different points x 1 , x 2 , … , x n x 1 , x 2 , … , x n x_(1),x_(2),dots,x_(n)x1,x2,…,xnfrom [ a , b ] [ a , b ] [a,b][a,b]and for any system of arbitrary numbers y 1 , y 2 , … , y n y 1 , y 2 , … , y n y_(1),y_(2),dots,y_(n)y1,y2,…,ynset M n M n M_(n)Mncontains one, and only one, integral passing through the points M i ( x i , y i ) , i = 1 , 2 , … , n M i ( x i , y i ) , i = 1 , 2 , … , n M_(i)(x_(i),y_(i)),i=1,2,dots,nMi(xi,yi),i=1,2,…,n. Keeping the notation from [3], the function f ( x ) f ( x ) f(x)f(x)defined on the interval J ⊆ [ a , b ] J ⊆ [ a , b ] J sube[a,b]J⊆[a,b], we will call convex, non-concave, non-convex or concave with respect to the set M n M n M_(n)Mnif for any system n + 1 n + 1 n+1n+1points x 1 < x 2 < ⋯ ⋯ < x n < x n + 1 x 1 < x 2 < ⋯ ⋯ < x n < x n + 1 x_(1) < x_(2) < cdots cdots < x_(n) < x_(n+1)x1<x2<⋯⋯<xn<xn+1, belonging to the interval J J JJ, from y ( x ) ∈ M n y ( x ) ∈ M n y(x)inM_(n)y(x)∈MnAnd y ( x i ) = f ( x i ) ' i = 1 , 2 , … , n y ( x i ) = f ( x i ) ' i = 1 , 2 , … , n y(x_(i))=f(x_(i))^("'")i=1,2,dots,ny(xi)=f(xi)'i=1,2,…,nalways flows out
f ( x n + 1 ) - y ( x n + 1 ) > , ≧ , ≦ , resp. < 0 f ( x n + 1 ) - y ( x n + 1 ) > , ≧ , ≦ , resp. < 0 f(x_(n+1))-y(x_(n+1)) > , >= , <= ,"corresponding" < 0f(xn+1)-y(xn+1)>,≧,≦,resp.<0
Coraacho's mean value theorem, due to G. Polya [5], allows us to state that if a set M n M n M_(n)Mnis a set of tynas I n [ a , b ] I n [ a , b ] I_(n)[a,b]In[a,b], then the inequalities
(4) L n ( f ) > , ≧ , ≦ , < 0 For x ∈ I ⊆ [ a , b ] (4) L n ( f ) > , ≧ , ≦ , < 0 For x ∈ I ⊆ [ a , b ] {:(4)L_(n)(f) > "," >= "," <= "," < 0quad"for"quad x in I sube[a","b]:}(4)Ln(f)>,≧,≦,<0Forx∈I⊆[a,b]
imply convexity, non-concaveness, non-convexity, and concavity of a function f ( x ) f ( x ) f(x)f(x)on the interval I I II.
If instead of the differential operator L n ( y ) L n ( y ) L_(n)(y)Ln(y), we will consider the operator G n ( y ) = G n ( x , y , y ′ , … , y ( n ) ) G n ( y ) = G n ( x , y , y ′ , … , y ( n ) ) G_(n)(y)=G_(n)(x,y,y^('),dots,y^((n)))Gn(y)=Gn(x,y,y′,…,y(n)), where the function G n ( u 1 , u 2 , … , u n + 1 ) G n ( u 1 , u 2 , … , u n + 1 ) G_(n)(u_(1),u_(2),dots,u_(n+1))Gn(u1,u2,…,un+1), defined in the area
(D) a ≦ u 1 ≦ b , - ∞ < u k < + ∞ , k = 2 , 3 , … , n + 1 (D) a ≦ u 1 ≦ b , - ∞ < u k < + ∞ , k = 2 , 3 , … , n + 1 {:(D)a <= u_(1) <= b","quad-oo < u_(k) < +oo","quad k=2","3","dots","n+1:}(D)a≦u1≦b,-∞<uk<+∞,k=2,3,…,n+1
and is continuous with respect to the set of its variables in D D DD, then the above concepts (a set of type I n [ a , b ] I n [ a , b ] I_(n)[a,b]In[a,b]and the concept of convexity) can be extended to a set G n G n G_(n)Gnintegrals of a differential equation
(5) G n ( x , y , y ′ , … , y ( n ) ) = 0 . (5) G n ( x , y , y ′ , … , y ( n ) ) = 0 . {:(5)G_(n)(x","y","y^(')","dots","y^((n)))=0.:}(5)Gn(x,y,y′,…,y(n))=0.
Assuming that G n G n G_(n)Gnis a set of type I n [ a , b ] I n [ a , b ] I_(n)[a,b]In[a,b]Polya's theorem for equation (5), given by us in [2], holds. From this theorem it follows that the inequalities
G n ( x , f ( x ) , f ′ ( x ) , … , f ( n ) ( x ) ) >≧ , ≦ or < 0 , x ∈ I ⊆ [ a , b ] G n ( x , f ( x ) , f ′ ( x ) , … , f ( n ) ( x ) ) >≧ , ≦ or < 0 , x ∈ I ⊆ [ a , b ] G_(n)(x,f(x),f^(')(x),dots,f^((n))(x))>≧, <= "or" < 0,quad x in I sube[a,b]Gn(x,f(x),f′(x),…,f(n)(x))>≧,≦or<0,x∈I⊆[a,b]
imply convexity, non-concaveness, non-convexity, respectively concavity of the function f ( x ) f ( x ) f(x)f(x)on the interval I I II.
Let us also recall that the property of the set M n M n M_(n)Mnbe a set of type I n [ a , b ] I n [ a , b ] I_(n)[a,b]In[a,b]is equivalent to the following property: no function from M h M h M_(h)Mhdoes not vanish n n nnonce a [ a , b ] [ a , b ] [a,b][a,b]This follows from the linearity of the Set M n M n M_(n)Mn. Therefore, in case n = 2 n = 2 n=2n=2, this property means the non-oscillation of the integrals of equation (1)
TEOPEMA - In order for many M ,, integrands of the equation M ,, integrands of the equation M_(",, integrals of the equation")M,, integrands of the equation(3) was a set of type I n [ a , b ] , n ≧ 2 I n [ a , b ] , n ≧ 2 I_(n)[a,b],n >= 2In[a,b],n≧2, it is necessary and sufficient that such an equation (5) exists so that the set G n G n G_(n)Gnwas a set of type I n [ a I n [ a I_(n)[aIn[a, in] and the following conditions were met:
1 ∘ 1 ∘ 1^(@)1∘If n = 2 , G n n = 2 , G n n=2,G_(n)n=2,Gncontains a positive function in [ a , b ] [ a , b ] [a,b][a,b], If n > 2 , G n n > 2 , G n n > 2,G_(n)n>2,Gncontains a function y ( x ) y ( x ) y(x)y(x)such that y ( a ) = 0 y ( a ) = 0 y(a)=0y(a)=0And y ( x ) > 0 y ( x ) > 0 y(x) > 0y(x)>0For x ∈ [ a , b ] x ∈ [ a , b ] x in[a,b]x∈[a,b];
2 ∘ . L n ( y ) ≦ G n ( x , y , y ′ , … , y ( n ) ) , x ∈ [ a , b ] 2 ∘ . L n ( y ) ≦ G n ( x , y , y ′ , … , y ( n ) ) , x ∈ [ a , b ] 2^(@).L_(n)(y) <= G_(n)(x,y,y^('),dots,y^((n))),x in[a,b]2∘.Ln(y)≦Gn(x,y,y′,…,y(n)),x∈[a,b]for each integral of equation (3) in [ a , b ] [ a , b ] [a,b][a,b].
The proof is straightforward. The necessity of the theorem's condition follows from the fact that if M n M n M_(n)Mnis a type of interethnicity I n [ a , b ] I n [ a , b ] I_(n)[a,b]In[a,b], Then M n M n M_(n)Mncontains one function y ( x ) y ( x ) y(x)y(x), satisfying the condition 1 ∘ 1 ∘ 1^(@)1∘, and conditional 2 ∘ 2 ∘ 2^(@)2∘also takes place.
To prove it, let us assume that the conditions of the theorem are satisfied. Inequality 2 ∘ 2 ∘ 2^(@)2∘implies the non-convexity of all integrals of equation (3) with respect to the set G 7 G 7 G_(7)G7If equation (3) had an integral V ( x ) V ( x ) V(x)V(x)vanishing n n nnonce, then the function c V ( x ) c V ( x ) cV(x)cV(x)Where c c cc- a constant chosen appropriately -, coincides with the fact that c y ( x ) c y ( x ) c quad y(x)cy(x)V n n nnpoints x 1 < x 2 < … < x n x 1 < x 2 < … < x n x_(1) < x_(2) < dots < x_(n)x1<x2<…<xnand there would be a point x n + 1 x n + 1 x_(n+1)xn+1, such that x n < x n + 1 ≦ b x n < x n + 1 ≦ b x_(n) < x_(n+1) <= bxn<xn+1≦bAnd c V ( x n + 1 ) - y ( x n + 1 ) < 0 c V ( x n + 1 ) - y ( x n + 1 ) < 0 cV(x_(n+1))-y(x_(n+1)) < 0cV(xn+1)-y(xn+1)<0, which contradicts the property of the integrals of equation (3) to be non-concave with respect to the set G n G n G_(n)Gn.

Literature

i. V. A. Kondrat'ev, An elementary derivation of a necessary and sufficient condition for the non-oscillation of the solution of a linear differential equation, Uspekhi Mat. Nauk. Vol. 12, 3(75), 159-160, 1957.
2. E. Moldovan, Asupra unor teorem de medie. Com. Acad. RPRT 6, 7-12, 1954.
3. E. Moldovan, Sur une généralisation des fonctions convexes Mathematica, 1(24), 1959 .
4. H. Poincaré, L'intermediaire des Mathematiciens. Vol. 1, 141-144, 1894.
5. G. Polya, On the mean-value theorem coresponding to a given homogenous differential equation. Trans. Am. Math. Soc. T. 24, 312-324, 1922.
6. De La Vallée Ponssin, Sur l'équation differentielle linéaire du second ordre Journ. de Math pure et appl. (9), 8, 125-144, 1929.
Received by the editor on December 1, 1958.
  1. *) The integral of equation (1) is called non-oscillating in interval I if it vanishes at no more than one point in this interval. Otherwise, the integral is called oscillating.
1959

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