Oscillation criteria for nonlinear neutral hyperbolic equations with functional arguments

This paper is devoted to the study of oscillatory behavior of solutions to nonlinear neutral hyperbolic equations with functional arguments by using the integral averaging method and generalized Riccati techniques. First, we establish oscillation results for nonlinear neutral hyperbolic equations by...

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Дата:	2011
Автори:	Shoukaku, Y., Stavroulakis, I.P., Yoshida, N.
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Мова:	English
Опубліковано:	Інститут математики НАН України 2011
Назва видання:	Нелінійні коливання
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Цитувати:	Oscillation criteria for nonlinear neutral hyperbolic equations with functional arguments / Y. Shoukaku, I.P. Stavroulakis, N. Yoshida // Нелінійні коливання. — 2011. — Т. 14, № 1. — С. 130-144. — Бібліогр.: 16 назв. — англ.

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spelling	irk-123456789-1753072021-02-01T01:27:28Z Oscillation criteria for nonlinear neutral hyperbolic equations with functional arguments Shoukaku, Y. Stavroulakis, I.P. Yoshida, N. This paper is devoted to the study of oscillatory behavior of solutions to nonlinear neutral hyperbolic equations with functional arguments by using the integral averaging method and generalized Riccati techniques. First, we establish oscillation results for nonlinear neutral hyperbolic equations by reducing the multi-dimensional oscillation problems to one-dimensional oscillation problems for functional differential inequalities. Secondly, we present oscillation results for nonlinear neutral hyperbolic equations by utilizing Riccati techniques. Вивчається коливна поведiнка розв’язкiв нелiнiйних гiперболiчних рiвнянь нейтрального типу з функцiональними аргументами з допомогою методу iнтегрального усереднення та узагальненої технiки Рiккатi. По-перше, отримано результати про коливання для нелiнiйних гiперболiчних рiвнянь нейтрального типу шляхом зведення багатовимiрних задач про коливання до одновимiрних задач про коливання для функцiонально-диференцiальних нерiвностей. По-друге, отримано результати про коливання для нелiнiйних гiперболiчних рiвнянь нейтрального типу з використанням технiки Рiккатi. 2011 Article Oscillation criteria for nonlinear neutral hyperbolic equations with functional arguments / Y. Shoukaku, I.P. Stavroulakis, N. Yoshida // Нелінійні коливання. — 2011. — Т. 14, № 1. — С. 130-144. — Бібліогр.: 16 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/175307 517.9 en Нелінійні коливання Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	This paper is devoted to the study of oscillatory behavior of solutions to nonlinear neutral hyperbolic equations with functional arguments by using the integral averaging method and generalized Riccati techniques. First, we establish oscillation results for nonlinear neutral hyperbolic equations by reducing the multi-dimensional oscillation problems to one-dimensional oscillation problems for functional differential inequalities. Secondly, we present oscillation results for nonlinear neutral hyperbolic equations by utilizing Riccati techniques.
format	Article
author	Shoukaku, Y. Stavroulakis, I.P. Yoshida, N.
spellingShingle	Shoukaku, Y. Stavroulakis, I.P. Yoshida, N. Oscillation criteria for nonlinear neutral hyperbolic equations with functional arguments Нелінійні коливання
author_facet	Shoukaku, Y. Stavroulakis, I.P. Yoshida, N.
author_sort	Shoukaku, Y.
title	Oscillation criteria for nonlinear neutral hyperbolic equations with functional arguments
title_short	Oscillation criteria for nonlinear neutral hyperbolic equations with functional arguments
title_full	Oscillation criteria for nonlinear neutral hyperbolic equations with functional arguments
title_fullStr	Oscillation criteria for nonlinear neutral hyperbolic equations with functional arguments
title_full_unstemmed	Oscillation criteria for nonlinear neutral hyperbolic equations with functional arguments
title_sort	oscillation criteria for nonlinear neutral hyperbolic equations with functional arguments
publisher	Інститут математики НАН України
publishDate	2011
url	http://dspace.nbuv.gov.ua/handle/123456789/175307
citation_txt	Oscillation criteria for nonlinear neutral hyperbolic equations with functional arguments / Y. Shoukaku, I.P. Stavroulakis, N. Yoshida // Нелінійні коливання. — 2011. — Т. 14, № 1. — С. 130-144. — Бібліогр.: 16 назв. — англ.
series	Нелінійні коливання
work_keys_str_mv	AT shoukakuy oscillationcriteriafornonlinearneutralhyperbolicequationswithfunctionalarguments AT stavroulakisip oscillationcriteriafornonlinearneutralhyperbolicequationswithfunctionalarguments AT yoshidan oscillationcriteriafornonlinearneutralhyperbolicequationswithfunctionalarguments
first_indexed	2025-07-15T12:33:26Z
last_indexed	2025-07-15T12:33:26Z
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fulltext	UDC 517 . 9 OSCILLATION CRITERIA FOR NONLINEAR NEUTRAL HYPERBOLIC EQUATIONS WITH FUNCTIONAL ARGUMENTS КРИТЕРIЇ КОЛИВАНЬ ДЛЯ НЕЛIНIЙНИХ ГIПЕРБОЛIЧНИХ РIВНЯНЬ НЕЙТРАЛЬНОГО ТИПУ З ФУНКЦIОНАЛЬНИМИ АРГУМЕНТАМИ Y. Shoukaku Kanazawa Univ. Kanazawa 920-1192, Japan e-mail: shoukaku@t.kanazawa-u.ac.jp I. P. Stavroulakis Univ. Ioannina 451-10, Ioannina, Greece e-mail: ipstav@uoi.gr N. Yoshida Univ. Toyama Toyama 930-8555, Japan e-mail: nori@sci.u-toyama.ac.jp This paper is devoted to the study of oscillatory behavior of solutions to nonlinear neutral hyperbolic equations with functional arguments by using the integral averaging method and generalized Riccati techni- ques. First, we establish oscillation results for nonlinear neutral hyperbolic equations by reducing the multi-dimensional oscillation problems to one-dimensional oscillation problems for functional differential inequalities. Secondly, we present oscillation results for nonlinear neutral hyperbolic equations by utilizing Riccati techniques. Вивчається коливна поведiнка розв’язкiв нелiнiйних гiперболiчних рiвнянь нейтрального типу з функцiональними аргументами з допомогою методу iнтегрального усереднення та узагаль- неної технiки Рiккатi. По-перше, отримано результати про коливання для нелiнiйних гiпербо- лiчних рiвнянь нейтрального типу шляхом зведення багатовимiрних задач про коливання до одновимiрних задач про коливання для функцiонально-диференцiальних нерiвностей. По-друге, отримано результати про коливання для нелiнiйних гiперболiчних рiвнянь нейтрального типу з використанням технiки Рiккатi. 1. Introduction. Consider the hyperbolic equation with functional arguments (E) ∂ ∂t ( r(t) ∂ ∂t ( u(x, t) + l∑ i=1 hi(t)u(x, ρi(t)) )) − − a(t)∆u(x, t)− k∑ i=1 bi(t)∆u(x, τi(t))+ c© Y. Shoukaku, I. P. Stavroulakis, N. Yoshida, 2011 130 ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 1 OSCILLATION CRITERIA FOR NONLINEAR NEUTRAL HYPERBOLIC EQUATIONS . . . 131 + m∑ i=1 qi(x, t)ϕi(u(x, σi(t))) = 0, (x, t) ∈ Ω ≡ G× (0,∞), where ∆ is the Laplacian in Rn and G is a bounded domain of Rn with piecewise smooth boundary ∂G, and the following Dirichlet and Robin (cf. [10]) boundary conditions: (B1) u = 0 on ∂G× [0,∞), (B2) ∂u ∂ν + µu = 0 on ∂G× [0,∞), where ν denotes the unit exterior normal vector to ∂G and µ ∈ C(∂G× [0,∞); [0,∞)). Throughout this paper we assume that: (A1) r(t) ∈ C1([0,∞); (0,∞)), hi(t) ∈ C2([0,∞); [0,∞)), i = 1, 2, . . . , l, a(t), bi(t) ∈ C([0,∞); [0,∞)), i = 1, 2, . . . , k, qi(x, t) ∈ C(Ω; [0,∞)), i = 1, 2, . . . ,m; (A2) ρi(t) ∈ C2([0,∞)R), limt→∞ ρi(t) = ∞, i = 1, 2, . . . , l, τi(t) ∈ C([0,∞);R), limt→∞ τi(t) = ∞, i = 1, 2, . . . , k, σi(t) ∈ C([0,∞);R), limt→∞ σi(t) = ∞, i = 1, 2, . . . ,m; (A3) ϕi(s) ∈ C1(R;R), i = 1, 2, . . . ,m, are convex in (0,∞) and ϕi(−s) = −ϕi(s) for s ≥ 0. Definition 1. By a solution of Eq.(E)we mean a function u ∈ C2(G × [t−1,∞)) ∩ C(G × ×[t̃−1,∞)) which satisfies (E), where t−1 = min { 0, min 1≤i≤l { inf t≥0 ρi(t) } , min 1≤i≤k { inf t≥0 τi(t) }} , t̃−1 = min { 0, min 1≤i≤m { inf t≥0 σi(t) }} . Definition 2. A solution u of Eq. (E) is said to be oscillatory in Ω if u has a zero in G× (t,∞) for any t > 0. Definition 3. We say that the functions (H1, H2) belong to a function class H, denoted by (H1, H2) ∈ H, if (H1, H2) ∈ C(D; [0,∞)) satisfy Hi(t, t) = 0, Hi(t, s) > 0, i = 1, 2, for t > s, where D = {(t, s) : 0 < s ≤ t < ∞}, and the partial derivatives ∂H1/∂t and ∂H2/∂s exist on D such that ∂H1 ∂t (s, t) = h1(s, t)H1(s, t) and ∂H2 ∂s (t, s) = −h2(t, s)H2(t, s), for some functions h1, h2 ∈ Cloc(D;R). ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 1 132 Y. SHOUKAKU, I. P. STAVROULAKIS, N. YOSHIDA In recent years there has been much research activity concerning the oscillation theory of nonlinear hyperbolic equations with functional arguments by employing Riccati techniques. Ri- ccati techniques were used to obtain various oscillation results (cf. Mařı́k [9], Yoshida [15]). For example, we note that Kamenev-type oscillation criteria for hyperbolic equations have been obtained in [3, 6, 12, 14]. On the other hand, interval oscillation criteria for second order di- fferential equation has been investigated by many authors [1, 3, 5, 6, 8, 12, 13]. In particular, Wang, Meng and Liu [12, 13] applied interval oscillation criteria to linear hyperbolic equations with functional arguments. Recently, Cui and Xu [1] presented oscillation criteria for hyperbolic equations which are not of neutral type. It seems that there are no known oscillation results for hyperbolic equations of neutral type, which are obtained by Riccati techniques. The objective of this paper is to establish oscillation ceireria for the nonlinear neutral hyperbolic equation with functional arguments (E) by employing the Riccati method. In Section 2 we reduce our problems to one-dimensional problems for functional differenti- al inequalities, and second order functional differential inequalities are investigated in Secti- on 3 via Riccati inequalities. We present oscillation results for (E) in Section 4 by combining the results of Sections 2 and 3. Two examples which illustrate our main theorems are given in Section 5. 2. Reduction to one-dimensional problems. In this section we reduce the multi-dimensional oscillation problems for (E) to one-dimensional oscillation problems. It is known that the first eigenvalue λ1 of the eigenvalue problem −∆w = λw in G, w = 0 on ∂G is positive, and the corresponding eigenfunction Φ(x) can be chosen so that Φ(x) > 0 in G. Now we let qi(t) = min x∈G qi(x, t). With each solution u(x, t) of the problem (E), (B1) or (E), (B2) we associate functions U(t) and Ũ(t) respectively, defined by U(t) = KΦ ∫ G u(x, t)Φ(x)dx, Ũ(t) = 1 \|G\| ∫ G u(x, t)dx, where KΦ = (∫ G Φ(x) dx )−1 and \|G\| = ∫ G dx. Theorem 1. If the functional differential inequality d dt ( r(t) d dt ( y(t) + l∑ i=1 hi(t)y(ρi(t)) )) + m∑ i=1 qi(t)ϕi(y(σi(t))) ≤ 0 (1) ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 1 OSCILLATION CRITERIA FOR NONLINEAR NEUTRAL HYPERBOLIC EQUATIONS . . . 133 has no eventually positive solutions, then every solution u(x, t) of the problem (E), (B1) is osci- llatory in Ω. Proof. Suppose to the contrary that there exists a nonoscillatory solution u of the problem (E), (B1). Without loss of generality we may assume that u(x, t) > 0 in G × [t0,∞) for some t0 > 0. (The case where u(x, t) < 0 can be treated similarly.) Since (A2) holds, we see that u(x, ρi(t)) > 0, i = 1, 2, . . . , l, u(x, τi(t)) > 0, i = 1, 2, . . . , k, and u(x, σi(t)) > 0, i = = 1, 2, . . . ,m, in G× [t1,∞) for some t1 ≥ t0. Multiplying (E) by KΦΦ(x) and integrating over G, we obtain d dt ( r(t) d dt ( U(t) + l∑ i=1 hi(t)U(ρi(t)) )) − − a(t)KΦ ∫ G ∆u(x, t)Φ(x)dx− m∑ i=1 bi(t)KΦ ∫ G ∆u(x, τi(t))Φ(x) dx + + m∑ i=1 KΦ ∫ G qi(x, t)ϕi(u(x, σi(t)))Φ(x) dx = 0, t ≥ t1. (2) From Green’s formula it follows that KΦ ∫ G ∆u(x, t)Φ(x)dx = −λ1U(t) ≤ 0, t ≥ t1, (3) KΦ ∫ G ∆u(x, τi(t))Φ(x)dx = −λ1U(τi(t)) ≤ 0, t ≥ t1. (4) Using the Jensen’s inequality we observe that m∑ i=1 KΦ ∫ G qi(x, t)ϕi(u(x, σi(t)))Φ(x)dx ≥ m∑ i=1 qi(t)ϕi(U(σi(t))), t ≥ t1, (5) and combining (2) – (5), it follows that d dt ( r(t) d dt ( U(t) + l∑ i=1 hi(t)U(ρi(t)) )) + m∑ i=1 qi(t)ϕi(U(σi(t))) ≤ 0, t ≥ t1. Therefore U(t) is an eventually positive solution of (1). This is a contradiction and the proof is complete. Theorem 2. If the functional differential inequality (1) has no eventually positive solutions, then every solution u(x, t) of the problem (E), (B2) is oscillatory in Ω. Proof. Suppose to the contrary that there exists a nonoscillatory solution u of the problem (E), (B2). Without loss of generality we may assume that u(x, t) > 0 in G × [t0,∞) for some ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 1 134 Y. SHOUKAKU, I. P. STAVROULAKIS, N. YOSHIDA t0 > 0. Since (A2) holds, we see that u(x, ρi(t)) > 0, i = 1, 2, . . . , l, u(x, τi(t)) > 0, i = = 1, 2, . . . , k, and u(x, σi(t)) > 0, i = 1, 2, . . . ,m, in G× [t1,∞) for some t1 ≥ t0. Dividing (E) by \|G\| and integrating over G, we obtain d dt ( r(t) d dt ( Ũ(t) + l∑ i=1 hi(t)Ũ(ρi(t)) )) − − a(t) \|G\| ∫ G ∆u(x, t) dx− k∑ i=1 bi(t) \|G\| ∫ G ∆u(x, τi(t)) dx + + 1 \|G\| m∑ i=1 ∫ G qi(x, t)ϕi(u(x, σi(t)))dx = 0, t ≥ t1. (6) It follows from Green’s formula that∫ G ∆u(x, t)dx = ∫ ∂G ∂u ∂ν (x, t) dS = − ∫ ∂G µ(x, t)u(x, t) dS ≤ 0, t ≥ t1, (7) ∫ G ∆u(x, τi(t))dx = ∫ ∂G ∂u ∂ν (x, τi(t))dS = − ∫ ∂G µ(x, τi(t))u(x, τi(t))dS ≤ 0, t ≥ t1. (8) Using the Jensen’s inequality, we observe that m∑ i=1 KΦ ∫ G qi(x, t)ϕi(u(x, σi(t)))dx ≥ m∑ i=1 qi(t)ϕi(Ũ(σi(t))), t ≥ t1, (9) and combining (6) – (9), it follows that d dt ( r(t) d dt ( Ũ(t) + l∑ i=1 hi(t)Ũ(ρi(t)) )) + m∑ i=1 qi(t)ϕi(Ũ(σi(t))) ≤ 0, t ≥ t1. Therefore Ũ(t) is an eventually positive solution of (1). This is a contradiction and the proof is complete. 3. Second order functional differential inequalities. In this section we establish sufficient conditions for every solution y(t) of the functional differential inequality (1) to have no eventu- ally positive solution. We assume the following hypotheses: (A4) For some j ∈ {1, 2, . . . ,m}, there exists a positive constant σ such that σ′j(t) ≥ σ and σj(t) ≤ t; (A5) ∫ ∞ t0 1 r(t) dt = ∞; ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 1 OSCILLATION CRITERIA FOR NONLINEAR NEUTRAL HYPERBOLIC EQUATIONS . . . 135 (A6) ∑l i=1 hi(t) ≤ 1; (A7) ρi(t) ≤ t, i = 1, 2, . . . , l. Theorem 3. Assume that the hypotheses (A4) – (A7) hold, and moreover assume that (A8) ϕj(s1s2) ≥ ϕj1(s1)ϕj2(s2) for s1 ≥ 0, s2 > 0, where ϕj1(s) ∈ C([0,∞); [0,∞)), ϕj2(s) ∈ C1((0,∞); (0,∞)) and ϕj(s), ϕ′j2(s) are nondecreasing and ϕ′j2(s) > 0 for s > 0. If the Riccati inequality z′(t) + 1 2 1 PK̃(t) z2(t) ≤ −Q(t) (10) for some K̃ > 0 and all large T, has no solution on [T,∞), where PK̃(t) = r(σj(t)) 2K̃σ , (11) Q(t) = qj(t)ϕj1 ( 1− l∑ i=1 hi(σj(t)) ) , (12) then (1) has no eventually positive solutions. Proof. Suppose that y(t) is a positive solution of (1) on [t0,∞) for some t0 > 0. From (1), there exists a j ∈ {1, 2, . . . ,m} such that d dt ( r(t) d dt ( y(t) + l∑ i=1 hi(t)y(ρi(t)) )) + qj(t)ϕj(y(σj(t))) ≤ 0, t ≥ t0. If we define the function z(t) = y(t) + l∑ i=1 hi(t)y(ρi(t)), (13) then we see that (r(t)z′(t))′ ≤ −qj(t)ϕj(y(σj(t))) ≤ 0, t ≥ t0. (14) Since (r(t)z′(t))′ ≤ 0, z(t) > 0 eventually, we observe, using the hypothesis (A5), that z′(t) ≥ 0 (t ≥ t1) for some t1 > t0 (cf. [13], Lemma 2.2). Hence r(t)z′(t) is nonincreasing. Then, we find that z′(t) ≥ 0 or z′(t) < 0 for t ≥ t1 > t0. First we assume that z′(t) < 0 for t ≥ t1. From the well known argument (cf. [13]) we prove that z′(t) ≥ 0 for t ≥ t1. Taking into account (A6) and (A7), from (13) we see that (cf. Yoshida [15]) y(t) ≥ ( 1− l∑ i=1 hi(t) ) z(t), t ≥ t1. (15) ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 1 136 Y. SHOUKAKU, I. P. STAVROULAKIS, N. YOSHIDA In view of (14) and (15), we observe that (r(t)z′(t))′ + qj(t)ϕj1 ( 1− l∑ i=1 hi(σj(t)) ) ϕj2(z(σj(t))) ≤ 0, t ≥ t1. Setting w(t) = r(t)z′(t) ϕj2(z(σj(t))) , we show that w′(t) = (r(t)z′(t))′ ϕj2(z(σj(t))) − r(t)z′(t) ϕ′j2(z(σj(t)))z ′(σj(t))σ ′ j(t) ϕ2 j2(z(σj(t))) . (16) Since z(t) > 0, z′(t) ≥ 0 eventually, it follows that z(σj(t)) ≥ k0 for some k0 > 0. Hence we observe that ϕ′j2(z(σj(t))) ≥ ϕ′j2(k0) ≡ K̃. (17) Substituting (17) into (16), we get w′(t) ≤ −qj(t)ϕj1 ( 1− l∑ i=1 hi(σj(t)) ) − K̃σr(t)z′(t) z′(σj(t)) ϕ2 j2(z(σj(t))) , t ≥ t1. On the other hand, (14) implies that r(σj(t))z ′(σj(t)) ≥ r(t)z′(t), and hence w′(t) + 1 2 ( 2K̃σ r(σj(t)) ) w2(t) ≤ −qj(t)ϕj1 ( 1− l∑ i=1 hi(σj(t)) ) (18) for t ≥ t1. That is, w(t) is a solution of (10) on [t1,∞). This is a contradiction and the proof is complete. Theorem 4. Assume that the hypotheses (A4) – (A8) hold. If for each T > 0 and some K̃ > > 0, there exist (H1, H2) ∈ H, ψ(t) ∈ C1((0,∞); (0,∞)) and a, b, c ∈ R such that T ≤ a < c < < b and 1 H1(c, a) c∫ a H1(s, a) { Q(s)− 1 4 r(σj(s)) K̃σ λ2 1(s, a) } ψ(s) ds + + 1 H2(b, c) b∫ c H2(b, s) { Q(s)− 1 4 r(σj(s)) K̃σ λ2 2(b, s) } ψ(s)ds > 0, (19) ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 1 OSCILLATION CRITERIA FOR NONLINEAR NEUTRAL HYPERBOLIC EQUATIONS . . . 137 where λ1(s, t) = ψ′(s) ψ(s) + h1(s, t), λ2(t, s) = ψ′(s) ψ(s) − h2(t, s). Then (1) has no eventually positive solutions. Proof. Suppose that y(t) is a positive solution of (1) on [t0,∞) for some t0 > 0. At first, we assume that y(t) > 0 on (a, b). Proceeding as in the proof of Theorem 3, we see that there exists a function w(s) which satisfies Q(s)ψ(s) ≤ −w′(s)ψ(s)− K̃σ r(σj(s)) w2(s)ψ(s). (20) Multiplying (20) by H2(t, s) and integrating over [c, t] for t ∈ [c, b), we have t∫ c H2(t, s)Q(s)ψ(s) ds ≤ ≤ − t∫ c H2(t, s)w′(s)ψ(s)ds− t∫ c H2(t, s) K̃σ r(σj(s)) w2(s)ψ(s) ds ≤ ≤ H2(t, c)w(c)ψ(c) + 1 4 t∫ c H2(t, s)λ2 2(t, s) r(σj(s)) K̃σ ψ(s) ds − − t∫ c H2(t, s)  √ K̃σ r(σj(s)) w(s)− 1 2 λ2(t, s) √ r(σj(s)) K̃σ  2 ψ(s) ds, and so 1 H2(t, c) t∫ c H2(t, s) { Q(s)− 1 4 r(σj(s)) K̃σ λ2 2(t, s) } ψ(s)ds ≤ w(c)ψ(c). Letting t → b− in the last inequality, we obtain 1 H2(b, c) b∫ c H2(b, s) { Q(s)− 1 4 r(σj(s)) K̃σ λ2 2(b, s) } ψ(s)ds ≤ w(c)ψ(c). (21) On the other hand, multiplying (20) by H1(s, t) and integrating over [t, c] for t ∈ (a, c], we ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 1 138 Y. SHOUKAKU, I. P. STAVROULAKIS, N. YOSHIDA obtain c∫ t H1(s, t)qj(s)ψ(s) ds ≤ ≤ − c∫ t H1(s, t)w′(s)ψ(s)ds− c∫ t H1(s, t) K̃σ r(σj(s)) w2(s)ψ(s) ds ≤ ≤ −H1(c, t)w(c)ψ(c) + 1 4 c∫ t H1(s, t)λ2 1(s, t) r(σj(s)) K̃σ ψ(s) ds − − c∫ t H1(s, t)  √ K̃σ r(σj(s)) w(s)− 1 2 λ1(s, t) √ r(σj(s)) K̃σ  2 ψ(s) ds, and therefore 1 H1(c, t) c∫ t H1(s, t) { Q(s)− 1 4 r(σj(s)) K̃σ λ2 1(s, t) } ψ(s)ds ≤ −w(c)ψ(c). Letting t → a+ in the last inequality, we obtain 1 H1(c, a) c∫ a H1(s, a) { Q(s)− 1 4 r(σj(s)) K̃σ λ2 1(s, a) } ψ(s)ds ≤ −w(c)ψ(c). (22) Adding (21) and (22), we obtain the following 1 H1(c, a) c∫ a H1(s, a) { Q(s)− 1 4 r(σj(s)) K̃σ λ2 1(s, a) } ψ(s) ds+ + 1 H2(b, c) b∫ c H2(b, s) { Q(s)− 1 4 r(σj(s)) K̃σ λ2 2(b, s) } ψ(s) ds ≤ 0, which contradicts the condition (19). Pick up a sequence {Ti} ⊂ [t0,∞) such that Ti → ∞ as i → ∞. By the assumptions, for each i ∈ N, there exists ai, bi, ci ∈ [0,∞) such that Ti ≤ ai < ci < bi, and (19) holds with a, b, c replaced by ai, bi, ci, respectively. Therefore, every nontrivial solution y(t) of (1) has at least one zero ti ∈ (ai, bi). Noting that ti > ai ≥ Ti, i ∈ N, we see that y(t) is an oscillatory solution of (1). This is a contradiction and the proof is complete. ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 1 OSCILLATION CRITERIA FOR NONLINEAR NEUTRAL HYPERBOLIC EQUATIONS . . . 139 Theorem 5. Assume that the hypotheses (A4) – (A8) hold. If for each T > 0 and some K̃ > 0, there exist functions (H1, H2) ∈ H, ψ(t) ∈ C1((0,∞); (0,∞)), such that lim sup t→∞ t∫ T H1(s, T ) { Q(s)− 1 4 r(σj(s)) K̃σ λ2 1(s, T ) } ψ(s)ds > 0 (23) and lim sup t→∞ t∫ T H2(t, s) { Q(s)− 1 4 r(σj(s)) K̃σ λ2 2(t, s) } ψ(s)ds > 0, (24) then (1) has no eventually positive solutions. Proof. For any T ≥ t0, let a = T and choose T = a in (23). Then there exists c > a such that c∫ a H1(s, a) { Q(s)− 1 4 r(σj(s)) K̃σ λ2 1(s, a) } ψ(s)ds > 0. (25) Next, choose T = c in (24). Then there exists b > c such that b∫ c H2(b, s) { Q(s)− 1 4 r(σj(s)) K̃σ λ2 2(b, s) } ψ(s)ds > 0. (26) Combining (25) and (26), we obtain (19). By the virtue of Theorem 4, the proof is complete. Remark. We give two examples which satisfy the assumption (A8). If ϕj(s) = sγ (γ is the quotient of odd integers), then (A8) is satisfied with ϕj1(s) = ϕj2(s) = sγ . Another example is the case where ϕj(s) = sinh s = es − e−s 2 , ϕj1(s) = min { s, s3, s5, . . . , s2m−1 } , ϕj2(s) = m∑ k=1 s2k−1 (2k − 1)! , where m is a positive integer. In fact, we observe that sinh s1s2 = ∞∑ k=1 (s1s2)2k−1 (2k − 1)! ≥ m∑ k=1 (s1s2)2k−1 (2k − 1)! ≥ min { s1, s 3 1, s 5 1, . . . , s 2m−1 1 } m∑ k=1 s2k−1 2 (2k − 1)! , ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 1 140 Y. SHOUKAKU, I. P. STAVROULAKIS, N. YOSHIDA where we note that min { s1, s 3 1, s 5 1, . . . , s 2m−1 1 } =  s1, s1 ≥ 1, s2m−1 1 , 0 ≤ s1 < 1. 4. Oscillation criteria for Eq. (E). In this section, by combining the results of Sections 2 and 3, we establish sufficient conditions for oscillation of Eq. (E). Using the Riccati inequality, we derive sufficient conditions for every solution of hyperbolic equation (E) to be oscillatory. We are going to use the following lemma which is due to Usa- mi [11]. Lemma 1. If there exists a function ψ(t) ∈ C1([T0,∞); (0,∞)) such that ∞∫ T1 ( p̄(t)\|(ψ′(t))β ψ(t) ) 1 β−1 dt < ∞, ∞∫ T1 1 p̄(t)(ψ(t))β−1 dt = ∞, ∞∫ T1 ψ(t)q̄(t)dt = ∞ for some T1 ≥ T0, then the Riccati inequality x′(t) + 1 β 1 p̄(t) \|x(t)\|β ≤ −q̄(t), where β > 1, p̄(t) ∈ C([T0,∞); (0,∞)) and q̄(t) ∈ C([T0,∞);R), has no solution on [T,∞) for all large T. Combining Theorems 1 – 3 and Lemma 1, we obtain the following theorem. Theorem 6. Assume that the hypotheses (A1) – (A8) hold. If ∞∫ T1 ( PK̃(t)(ψ′(t))β ψ(t) ) dt < ∞, ∞∫ T1 1 PK̃(t)ψ(t) dt = ∞, ∞∫ T1 ψ(t)Q(t)dt = ∞, ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 1 OSCILLATION CRITERIA FOR NONLINEAR NEUTRAL HYPERBOLIC EQUATIONS . . . 141 where PK̃(t) and Q(t) are defined by (11) and (12) for some K̃ > 0, then every solution u(x, t) of (E), (B1) (or (E), (B2)) is oscillatory in Ω. Combining Theorems 1 – 2 and 4, we have the following theorem. Theorem 7. Assume that the hypotheses (A1) – (A8) hold. If for each T > 0 and some K̃ > 0, there exist functions (H1, H2) ∈ H, ψ(t) ∈ C1((0,∞); (0,∞)) and a, b, c ∈ R such that T ≤ a < c < b and (19) hold, then every solution u(x, t) of (E), (B1) (or (E), (B2)) is oscillatory in Ω. Analogously, combining Theorems 1 – 2 and 5 we derive the following. Theorem 8. Assume that the hypotheses (A1) – (A8) hold. If for each T > 0 and some K̃ > 0, there exist functions (H1, H2) ∈ H, ψ(t) ∈ C1((0,∞); (0,∞)) such that (23) and (24) hold, then every solution u(x, t) of (E), (B1) (or (E), (B2)) is oscillatory in Ω. 5. Examples. We present the following examples which illustrate the applicability of our results. Example 1. Consider the problem ∂ ∂t ( e−t ∂ ∂t (u(x, t) + 1 2 u(x, t− π) )) − 1 2 e−t∆u(x, t)− − 1 2 e−t∆u ( x, t+ π 2 ) − e2t∆u(x, t− 2π) + + e2tu(x, t− π) = 0, (x, t) ∈ (0, π)× [1,∞), (27) u(0, t) = u(π, t) = 0. (28) Here n = 1, k = 2, m = 1, r(t) = e−t, h1(t) = 1/2, q1(x, t) = e2t, σ1(t) = t − π and ϕ′12(ξ) = 1 = K̃. It is easy to see that PK̃(t) = 1 2 e−t+π, Q(t) = 1 2 e2t. By choosing ψ(t) = e−2t, H1(s, t) = H2(t, s) = ( et − es )2 , we see that ∞∫ ( 1 2 e −t+π(−2e−2t)2 e−2t ) dt = ∞∫ 2e−3t+π dt < ∞, ∞∫ ( 1 1 2 e −t+π × e−2t ) dt = ∞∫ 2e3t−π dt = ∞, ∞∫ ( e−2t 1 2 e2t ) dt = ∞. ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 1 142 Y. SHOUKAKU, I. P. STAVROULAKIS, N. YOSHIDA Choose now a = 1, b = 2π and c = π and observe that 1 (e− eπ)2 π∫ 1 (e− es)2 { 1 2 e2s − 1 4 e−s+π 4e2s (e− es)2 } e−2s ds + + 1 (e2π − eπ)2 2π∫ π (e2π − es)2 { 1 2 e2s − 1 4 e−s+π 4e4π (e2π − es)2 } e−2sds > 0, that is, the condition (19) is satisfied. Also lim sup t→∞ t∫ T (es − sT )2 { 1 2 e2s − 1 4 e−s+π 4e2s (es − eT )2 } e−2sds = = lim sup t→∞ { 1 4 e2t − et+T + 1 2 ( t− T + 3 2 ) e2T + e−t+π − e−T+π } > 0 and lim sup t→∞ t∫ T (et − ss)2 { 1 2 e2s − 1 4 e−s+π 4e2t (et − es)2 } e−2s ds = = lim sup t→∞ {( 1 2 ( t− T − 3 2 ) − 1 3 eπ−3T ) e2t + et+T + 1 3 e−t+π − 1 4 e2T } > 0, that is, the conditions (23) and (24) hold. Thus, all the conditions of Theorems 6 – 8 are satisfied. Therefore every solution u(x, t) of the problem (27), (28) is oscillatory in (0,∞) × [1,∞). For example, u(x, t) = sinx sin t is such a solution. Example 2. Consider the problem ∂ ∂t ( 1 (t+ π)2 ∂ ∂t (u(x, t) + 1 2 u(x, t− 2π) )) −∆u(x, t) − − 3 2(t+ π)2 ∆u(x, t− 2π)− 3 (t+ π)3 ∆u ( x, t+ π 2 ) + + u(x, t− π) = 0, (x, t) ∈ (0, π)× [1,∞), (29) −ux(0, t) = ux(π, t) = 0. (30) Here n = 1, k = 2, m = 1, r(t) = (t + π)−2, h1(t) = 1/2, q1(x, t) = 1, σ1(t) = t − π and ϕ′12(ξ) = 1 = K̃. It is easy to see that PK̃(t) = 1 2t2 , Q(t) = 1 2 . ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 1 OSCILLATION CRITERIA FOR NONLINEAR NEUTRAL HYPERBOLIC EQUATIONS . . . 143 If we choose ψ(t) = t2, then ∞∫ ( 1 2t2 (2t)2 t2 ) dt = ∞∫ ( 2 t2 ) dt < ∞, ∞∫ ( 1( 1 2t2 ) t2 ) dt = ∞, ∞∫ ( 1 2 t2 ) dt = ∞. Next, choose ψ(t) = 1, H1(s, t) = H2(t, s) = (t − s)2, and a = 1, b = 2π, c = π. It is easy to see that 1 (1− π)2 π∫ 1 (1− s)2 { 1 2 − 1 4s2 4 (1− s)2 } ds+ 1 π2 2π∫ π (2π − s)2 { 1 2 − 1 4s2 4 (2π − s)2 } ds > 0. Moreover, lim sup t→∞ t∫ T (s− T )2 { 1 2 − 1 4 1 s2 4 (s− T )2 } ds = = lim sup t→∞ { 1 6 t3 − 1 2 Tt2 + 1 2 T 2t+ t−1 − 1 6 T 3 − T−1 } > 0 and lim sup t→∞ t∫ T (t− s)2 { 1 2 − 1 4 1 s2 4 (t− s)2 } ds = = lim sup t→∞ { 1 6 t3 − 1 2 Tt2 + 1 2 T 2t+ t−1 − 1 6 T 3 − T−1 } > 0. Thus, all the conditions of Theorems 6 – 8 are satisfied. Therefore, every solution u(x, t) of the problem (29), (30) is oscillatory in (0, π)× [1,∞). One such solution is u(x, t) = cosx sin t. Observe, however, that ∞∫ 1 2 ( 3 2(s+ π)2 + 3 (s+ π)3 ) ds < ∞, and therefore the condition (8) of Theorem 2 given by Deng [2] is not satisfied. Thus, Theorem 2 by Deng [2] can not be applied to this example. ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 1 144 Y. SHOUKAKU, I. P. STAVROULAKIS, N. YOSHIDA 1. Cui S., Xu Z. Interval oscillation theorems for second order nonlinear partial delay differential equations // Different. Equat. and Appl. — 2009. — 1. — P. 379 – 391. 2. Deng L. Oscillation criteria for certain hyperbolic functional differential equations with Robin boundary condition // Indian J. Pure and Appl. Math. — 2002. — 33. — P. 1137 – 1146. 3. Kong Q. Interval criteria for oscillation of second-order linear ordinary differential equations // J. Math. Anal. and Appl. — 1999. — 229. — P. 258 – 270. 4. Li W. N. Oscillation for solutions of partial differential equations with delays // Demonst. Math. — 2000. — 33. — P. 319 – 332. 5. Li W. T. Interval oscillation criteria for second order nonlinear differential equations with damping // Taiwan. J. Math. — 2003. — 7. — P. 461 – 475. 6. Li W. T., Agarwal R. P. Interval oscillation criteria related to integral averaging technique for certain nonlinear differential equations // J. Math. Anal. and Appl. — 2000. — 245. — P. 171 – 188. 7. Li W. N., Cui B. T. Oscillation of solutions of neutral partial functional differential equations // J. Math. Anal. and Appl. — 1999. — 234. — P. 123 – 146. 8. Li W. T., Huo H. F. Interval oscillation criteria for nonlinear second-order differential equations // Indian J. Pure and Appl. Math. — 2001. — 32. — P. 1003 – 1014. 9. Mařı́k R. Oscillation theory of partial differential equations with p-Laplacian // Folia Univ. Agric. Silvic. Mendel. Brun. — Brno, 2008. 10. Stavroulakis I. P., Tersian S. A. Partial differential equations. An introduction with mathematica and MAPLE. — Second Ed. — World Sci. Publ. Co., 2004. 11. Usami H. Some oscillation theorem for a class of quasilinear elliptic equations // Ann. mat. pura ed appl. — 1998. — 175. — P. 277 – 283. 12. Wang J., Meng F., Liu S. Integral average method for oscillation of second order partial differential equations with delays // Appl. Math. and Comput. — 2007. — 187. — P. 815 – 823. 13. Wang J., Meng F., Liu S. Interval oscillation criteria for second order partial differential equations with delays // J. Comput. and Appl. Math. — 2008. — 212. — P. 397 – 405. 14. Yang Q. On the oscillation of certain nonlinear neutral partial differential equations // Appl. Math. Lett. — 2007. — 20. — P. 900 – 907. 15. Yoshida N. Oscillation theory of partial differential equations. — World Sci. Publ. Co. Pte. Ltd., 2008. 16. Zhong Y. H., Yuan Y. H. Oscillation criteria of hyperbolic equations with continuous deviating arguments // Kyungpook Math. J. — 2007. — 47. — P. 347 – 356. Received 27.11.09, after revision — 17.06.10 ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 1

Oscillation criteria for nonlinear neutral hyperbolic equations with functional arguments

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