Džurina, Jozef

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Institute of Mathematics AS CR, v. v. i.

Mathematica SlovacaJozef DžurinaProperty (A) of third order differential equations with deviating argumentMathematica Slovaca, Vol. 45 (1995), No. 4, 395--402Persistent URL: http://dml.cz/dmlcz/136656Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 1995Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czMathematica Slovaca ©1995 , . . _ . . _ /-^.-r-x », . o r ._ . .~~ Mathematical Institute Math. Slovaca, 45 (1995), NO. 4, 395-402 Slovák Academy of Sciences PROPERTY (A) OF THE THIRD ORDER DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENT J O Z E F DZURINA (Communicated by Milan Medved') ABSTRACT. The aim of this paper is to derive sufficient conditions for property (A) of delay and advanced differential equations of the form i(iS))Ь*иo)=» We consider the differential equations of the form {MM^)))+m^(,))=o- (i) We always assume that (i) r{(t), 0 ^ i ^ 2, r(t) and p(t) are continuous on [to,oo), rj(t) > 0, p(t) > 0, and r(t) —> oo as t —> oo. t (ü) m=ІЉ i(s) We say that the operator co as ť —> co, ѓ = l , 2 . to is in the canonical form if (ii) holds. It is well known that any differential operator of the form (2) can always be represented in a canonical form in an essentially unique way (see T r e n c h [9]). A M S S u b j e c t C l a s s i f i c a t i o n (1991): Pr imary 34C10. K e y w o r d s : functional differential equation, property (A), third-order. 395 JOZEF DlURINA For convenience, we introduce the following functions: Liu(t) = ^ r - ^ i f _ i w ( t ) , i = 1, 2 , Lsu(t) = ftL2u(t). The domain V(L^) of L3 is defined to be the set of all functions u: [TU1 co) —> R such that Liu(t), 0 ^ i ^ 3, exist and are continuous on [Tu ,oo). A nontrivial solution of (1) is called oscillatory if it has arbitrarily large zeros; otherwise it is called nonosdilatory. The asymptotic behavior of the nonoscillatory solutions of (1) is described in the following lemma, which is a generalization of a lemma of K i g u r a d z e [4; Lemma 3]. LEMMA 1. Let u(t) be a nonoscillatory solution of (1). Then there exist an integer £, £ G {0, 2} . and t\ ^ to such that u(t)Liu(t)>0, O^i^l, . 0 for all t ^ ti. (3) (-iy-£u(t)Liu(t)>0, £ ^ i < 3 , A function u(t) satisfying (3) is said to be a function of degree £. The set of all nonoscillatory solutions of degree £ of (1) is denoted by Mi. If we denote by JV the set of all nonoscillatory solutions of (1), then, by Lemma 1, J\f = A/Q U A/2 . Following K u s a n o and N a i t o [5], we say that equation (1) has prop-erty (A) if Af = Af0. In the recent papers (see, e.g., [1], [2], [5], [6] and [8]), various kinds of suf-ficient conditions for equation (1) to have property (A) appear. In this paper, we investigate delay and advanced equations of the form (1), and wre present some criteria for property (A) of (1) which generalize and extend those in [1], [5] and [6]. Define the function h{tyto\r1,r2) = I r i (s) / r2(x) dxds. : • / . . ( . ) / , THEOREM 1. Assume that r(t) < t and r(t) is increasing. Let g(t) be a continuous function satisfying 9(t)>t, T(g(t))^t. (4) 396 PROPERTY (A) OF THE THIRD ORDER EQUATIONS ... Let 9(t) limmf / p ( s ) r 0 ( r ( s ) ) / 2 ( r ( s ) , r 0 ; r i , r 2 ) ds > 1 . (5) t Then equation (1) has property (A) . P r o o f . For the sake of contradiction, we assume that (1) has a positive nonoscillatory solution u(t) such that u E JV2. Then (3) reduces to L0u(t) > 0, L\u(t) > 0 , L2u(t) > 0 and Lsu(t) < 0, for t^ tx. (6) An integration of (1) from t ( ^ t\) to oo yields oo L2u(t) > IP(S)U(T(S)) ds, t>tx. (7) t On the other hand, integrating twice the identity L2u(t) = L2u(t) from t\ to t , we have in view of (6) t s L0u(t) ^ / r i (s) / r2(x)L2u(x) dxds ^ L2^(t)/2(t,^i;ri,r2). (8) t l t ! Combining (7) with (8) and taking (i) and (6) into, account we get oo L2u(t)^ / p ( s ) r 0 ( r ( s ) )L 2 i x (T(s ) ) / 2 ( r ( s ) , t i ; r i , r 2 ) ds *(*) / L>(s)r0(T(s))L27i(r(s))/2(T(s),ti;ri,r2) ds t 9(t) ^ L2u(T(g(t))) / p ( s ) r 0 ( r ( s ) ) / 2 ( r ( s ) , t i ; r i , r 2 ) ds , t ^ *2 , t g(t) where t2 ^ t\ is large enough. Since T(g(t)) ^ £ and L2u(t) is decreasing, the above inequalities yield g{t) 1 ^ / p(s)r0(r(s))/2(r(s),ri;ri,r2) d s , r ^ r2 , which contradicts (5) . The proof is complete. • For a special choice of the function g(t) we have the following corollary: 397 JOZEF DZURINA COROLLARY 1. Assume that r(t) < t and r(i) is increasing. Let T~~ (t) be the inverse function to r(t). Let liminf / p(s)ro(r(s))J2(T(s), ^o; rur2) ds > 1. t—VOO J \ / \ / t Then equation (1) has property (A) . P r o o f . Since the function g(t) = T - 1 ( £ ) satisfies (4), the assertion of this corollary follows immediately from Theorem 1. • For a special case of (1) we have the following criteria for property (A) (see [5] and [1]): THEOREM A. Let p(t) > 0 . r(t) ^ t and r'(t) > 0 . Let oo liminf r(i) / (r(s) - r(t))p(s) ds > - . t Then the delay equation y"'(t)+p(t)y(T(t))=0 (9) has property (A) . T H E O R E M B. Let p(t) > 0, r(t) < t and Tf(t) > 0 . Let oo 1 liminf т2(t) / p(s) ds > t^°° x ' J * ч ' 3^3 Tb,en equation (9) /ms property (A). In the following illustrative examples, we compare criteria from Theorems A and B with that in Corollary 1. E x a m p l e 1. Let us consider the third order delay equation y'"(t) + \y(\Tt) = 0 , o > 0 a n d t ^ l . (10) By Corollary 1, equation (10) has property (A) for all a > 0. On the other hand, by Theorem A (or Theorem B), equation (10) has property (A) if a > -r (or a > —-=-•). E x a m p l e 2. Let us consider the third order delay equation y'"V+ (0.01)4ioo ^(°-01«) = °. ' ^ - ("> By Corollary 1, equation (11) has property (A) . On the other hand, by the above-mentioned result of K u s a n o and N a i t o , Theorem A fails for (11). Now we turn to advanced equations of the form (1). 398 PROPERTY (A) OF THE THIRD ORDER EQUATIONS ... THEOREM 2. Assume that r(t) ^ t. Let oo liminf R2(t) fp(s)ro(r(s))(R1(r(s)) - R±(t)) ds > 1. (12) t Then equation (1) has property (A). P r o o f . For the sake of contradiction, we assume that u(t) is a positive nonoscillatory solution of (1) such that u G Af2. Then (6) and (7) hold for t ^ t \ . By integrating the identity L\u(t) = L\u(t) from t\ to t, we get, in view of (6), t L0u(t) > / r±(s)Liu(s) ds. (13) t i Collecting (7) and (13) and using that L\u(t) is increasing we obtain oo r(s) L2u(t) ^ p(s)r0(r(s)) / r1(x)L1u(x) dxds t Ѓi oo r(s) ^ / p(s)ro{T(s)) / ri(x)L\u(x) dxds \^) t t oo ^ Lxu(t) Jp(s)r0(r(s)) (iti(r(s)) - R±(t)) ds , t ^ t2 , t where t2 ^ t i is large enough. E 1 i a s in [3] has generalized the third lemma of K i g u r a d z e . From his result (see also [7]), we have the following relation between L\u(t) and L2u(t): LlU(t) ^ L2u(t)R2(t), t ^ t i . (15) Then (14) and (15) yield oo 1 ^ R2(t) fp(s)ro(r(s))(R1(r(s)) - R^t)) ds , t ^ t2 , t which contradicts (12). The proof is complete. • The following result for advanced equation (9) is due to O 1 a h [6]. 399 JOZEF D2URINA THEOREM C. Let p(t) and r(t) satisfy ( i ) . Assume that r(t) ^ t. Let r(t) f(s~t)2p(s) liminf / (s-tYp(s) d s > 2. t — Ю O t Then equation (9) has property (A) . E x a m p l e 3. Let us consider the third order advanced equation ym(i) + -^y(\t) = 0 , a > 0 , A ^ l and t ^ l , (16) By Theorem 2, equation (16) has property (A) if a(X — 0.5) > 1. On the other hand, Theorem C guarantees property (A) of (16) if a stronger condition holds, namely a ( l n A + ! - 2 3 ? - ! ) > 2 -R e m a r k 1. In the above examples, we have compared our results concern­ing (9) with those in [1], [5] and [6]. Note that Theorems 1 and 2 can also be applied to more general equation (1). Let us consider the mixed differential equation L*u(t) +p1(t)u(r1(t)) +p2(t)u(r2(t)) = 0 , (17) where L3 is defined as for (1) and (hi) Pi(t) and Ti(t) are continuous on [to, 00), Pi(t) > 0, 1 ^ i ^ 2, Ti(£) is increasing, Ti(£) < t, r2(t) ^ t , and Ti(£) —• 00 as t —+ 00. As Lemma 1 holds also for (17), we can seek criteria for property (A) of (17). THEOREM 3. Assume that (hi) holds. Let g(t) be a continuous function sat-isfying 9(t)>t, TX(g(t)) ^ t . Let if lirninf< / p1(s)r0(r(s))I2(Ti(s),r0; r1, r2) ds 00 \ +Ri(t) f P2(s)r0(r(s))(R1(T2(s)) - Ri(t)) ds \ > 1. (18) 400 PROPERTY (A) OF THE THIRD ORDER EQUATIONS . . . Then equation (17) has property (A) . P r o o f . Assume that (17) has a positive solution u(t) G Af2. Proceeding exactly as in the proofs of Theorems 1 and 2 we can verify that f f 1 > \ / Pi(s)r0(r(s))l2(r1(s),to]r1,r2) ds oo ^ +R2(t) f p2(s)ro(r(s))(R1(r2(s)) - Rx(t)) ds\ , which contradicts (18). • E x a m p l e 4. Let us consider the third order mixed equation y'"(t) + jt[y(xt) + y(j)} = 0> a > 0 > o < A < i a n d t ^ i . (19) We put g(t) = -y. Then, by Theorem 3, equation (19) has property (A) if a ( x - i - f l n A ) > 1 -REFERENCES [1] DZURINA, J . : Comparison theorems for nonlinear ODE's, Math . Slovaca 42 (1992), 299-315. [2] DZURINA, J . : Asymptotic properties of third-order differential equations with deviating argument, Czechoslovak Math. J. (To appear) . [3] ELIAS, U. : Generalizations of an inequality of Kiguradze, J. Math . Anal. Appl. 97 (1983), 277-290. [4] KIGURADZE, I. T. : On the oscillation of solutions of the equation drnu/dtrn + a ( t ) | u | n s i g n u = 0 . (Russian), Mat. Sb. 65 (1964), 172-187. [5] KUSANO, T.—NAITO, M. : Comparison theorems for functional differential equations with deviating arguments, J. Math . Soc. Japan 3 (1981), 509-532. [6] OLAH, R. : jVote on the oscillation of differential equation with advanced argument, Math. Slovaca 3 3 (1981), 241-248. [7] SEDA, V. : Nonosdilatory solutions of differential equations with deviating argument, Czechoslovak Math. J. 36 (1986), 93-107. [8] SKERLIK, A . : Oscillation theorems for third order nonlinear differential equations, Math. Slovaca 42 (1992), 471-484. 401 JOZEF DZURINA [9] TRENCH, W. F. : Canonical forms and principal systems for general disconjugate equa-tions, Trans. Amer. Math . Soc 189 (1974), 319-327. Received June 24, 1993 Department of Mathematical Analysis Revised November 15, 1993 Faculty of Science Safdrik University Jesennd 5 SK-04I 54 Košice SLOVAKIA E-mail: dzurina@turing.upjs.sk 402 

Property $(A)$ of third order differential equations with deviating argument

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Property $(A)$ of third order differential equations with deviating argument

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Property $(A)$ of third order differential equations with deviating argument