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BOUNDED OSCILLATION FOR SECOND-ORDER NONLINEAR DELAY DIFFERENTIAL EQUATIONS†
BOUNDED OSCILLATION FOR SECOND-ORDER NONLINEAR DELAY DIFFERENTIAL EQUATIONS†
Journal of Applied Mathematics & Informatics. 2014. Sep, 32(3_4): 447-454
Copyright © 2014, Korean Society of Computational and Applied Mathematics
  • Received : February 01, 2013
  • Accepted : January 21, 2014
  • Published : September 28, 2014
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XIA SONG
QUANXIN ZHANG

Abstract
Two necessary and sufficient conditions for the oscillation of the bounded solutions of the second-order nonlinear delay differential equation are obtained by constructing the sequence of functions and using inequality technique. AMS Mathematics Subject Classification: 34C10.
Keywords
1. Introduction
Consider the second-order nonlinear delay differential equation
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The paper assumes the following conditions hold:
  • (H1)a(t) ∈C1([t0, ∞), (0, ∞)),q(t) ∈C([t0, ∞), [0, ∞)), and witht→ ∞,
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  • (H2)ƒ(x) ∈ C(R,R) is non-decreasing function , andwithx≠ 0;
  • (H3)τ(t) ∈ C([t0, ∞),R),τ(t) ≤t, and
We call that x ( t ) ∈ C 1 ([ Tx , ∞), R) ( Tx t 0 ) is the solution of equation (1.1) if a ( t ) x ′( t ) ∈ C 1 ([ Tx , ∞). R) and x ( t ) satisfy (1.1) for t ∈ [ Tx , ∞). We suppose that every solution of (1.1) can be extended in [ t 0 , +∞). In any infinite interval [ T , +∞), we call x ( t ) is a regular solution of (1.1) if x ( t ) is not the eventually identically zero. The regular solution of (1.1) is said to be oscillatory in case it has arbitrarily large zero point; otherwise, the solution is said to be nonoscillatory.
For the equation (1.1), if a ( t ) ≡ 1, the equation (1.1) becomes
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For the equation (1.2), if ƒ ( x ) = x , τ ( t ) = t , and q ( t ) = c ( t ), the equation (1.2) is simplified to be the second-order linear differential equation
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There are some oscillation criteria for the equation (1.3), and one of the most important criteria is given by Wintner [1] as follows: If
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the equation (1.3) is oscillatory. In 1978, Kamenev [2] improved the result of Wintner. He proved that if
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where λ is a constant and λ > 1, the equation (1.3) is oscillatory.
In recent years, the oscillation theory and its application of differential equations have been greatly concerned. For example, you can see the recent monographs [3 - 5] . In particular, the result on oscillation criteria of second-order differential equation is very rich (see [6 - 16] ), but the most results obtained establish the sufficient condition of the oscillation for differential equations. Generally, the necessary and sufficient condition is difficult to obtain. This article discusses the oscillation for the bounded solution of the equation (1.1), and establishes two necessary and sufficient conditions of oscillation for the bounded solution of (1.1) by constructing a sequence of functions and using inequality technique.
2. Main results
Theorem 2.1. Suppose that ( H 1 )−( H 3 ) hold. Then the necessary and sufficient condition of the oscillation for every bounded solution of the equation (1.1) is
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Proof . Sufficiency. Suppose that there is nonoscillatory bounded solution x ( t ) of the equation (1.1). Without loss of generality we assume that x ( t ) is eventually positive, then there exists t 1 ( t 1 t 0 ) such that as t t 1 ,
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From the equation (1.1), we get
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Thus, we can determine that
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Actually, if there exists t 2 ( t 2 t 1 ) such that
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noticing a ( t ) x ′( t ) is nonincreasing, we can obtain
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as t t 2 , i.e.
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Integrating the above formula from t 2 to t , by ( H 1 ), we get that
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This contradicts with that x ( t ) is eventually positive, so (2.2) holds. Thus, for t t 1 , we have
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Then there exists t 3 ( t 3 t 2 ) and l ( l > 0) such that when t t 3 ,
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Substituting (2.3) into equation (1.1), we get
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Multiplying the both ends of the above formula by A ( t ) , we can get
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Because
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we can obtain
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Substituting the above into (2.5), we get
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Integrating the above formula from t 3 to t , we obtain
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Because x ( t ) is increasing and bounded, there exists C ( C > 0) such that
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This contradicts with the condition (2.1). The proof of sufficient section is completed.
Necessity. If
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then there is T t 0 such that for t T , we have
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Constructing the sequence of functions such that
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for k = 1, 2, · · · , Then
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Suppose that
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Noticing ƒ ( u ) is non-decreasing, then
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Thus, by mathematical induction, for any positive integer k , we get
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Therefore, the limit of the sequence of functions { xk ( t )} exists, i.e.
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and 1 ≤ x ( t ) ≤ 2, t τ ( T ): Applying Lebesgue control convergence theorem to (2.6), we can get
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Derivation of the both sides of the above formula and multiplying them by a ( t ), we get that for t > T
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Continually after derivation of the both sides of the above formula, we get
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It is easy to see that x ( t ) is an eventually positive bounded solution of (1.1), which is contradictory to that every bounded solution of the equation (1.1) is oscillatory. The proof is completed.
Remark 2.1. For linear case of (1.1) (i.e. ƒ ( x ) = x ), oscillation criterion relative to Theorem 2.1 has been obtained in Corollary 2 in [17] .
By Theorem 2.1, we can obtain for equation (1.2)
Corollary 2.2. Suppose that ( H 1 )−( H 3 ) hold. Then every bounded solution of equation (1.2) is oscillatory if and only if
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Corollary 2.3. Assume that ( H 1 )−( H 3 ) hold. Then every bounded solution of (1.1) is oscillatory if and only if
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Proof . Sufficiency. Suppose that there is nonoscillatory bounded solution x ( t ) of the equation (1.1). Without loss of generality, we assume that x ( t ) is eventually positive, then there exists t 1 ( t 1 t 0 ) such that
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for t t 1 . Using arguments similar to ones in the proof of Theorem 2.1, we can get (2.4), i.e.
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Integrating the above from t to t + v , we get
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Noticing (2.2) and letting v → ∞, we get that
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Integrating the above from T ( T t 2 ) to t ( t T ), it follows
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Let t → ∞ to acquire the limits of both sides of the above. Because x ( t ) is bounded and increasing, it is easy to get
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This is contradictory to the condition (2.8). The proof of sufficiency is completed.
Necessity. Suppose that
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and there exists T ( T t 0 ) such that
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for t T . Construct the sequence of functions and let
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k = 1, 2, · · · . Similarly to the proof of Theorem 2.1, by the mathematical induction for any positive integer k , we have
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So the limit of { xk ( t )} exists, i.e.
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and 1 ≤ x ( t ) ≤ 2, t τ ( T ): By Lebesgue control convergence theorem to (2.9), it follows that
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Derivation of the both sides of the above and multiplying them by a ( t ), we get that for t > T
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Continuing to take the derivatives of the both sides of the above, we can get
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Thus x ( t ) is an eventually positive bounded solution of (1.1), which is contradictory to that every bounded solution of the equation (1.1) is oscillatory. The proof is completed.
By Theorem 2.3, we can get for equation (1.2).
Corollary 2.4. Suppose that ( H 1 )−(< H 3 ) hold. Then every bounded solution of (1.2) is oscillatory if and only if
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Example 2.5. Consider second-order linear differential equation
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Here
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The conditions (H 1 )-(H 3 ), (2.1) and (2.8) are clearly satisfied. Altogether, by Theorems 2.1 and 2.3, every bounded solution of the equation (2.11) is oscillatory. In fact, x ( t ) = cos ln t is a bounded oscillatory solution of the equation (2.11).
BIO
Xia Song received M.Sc. from Shandong University, China. She is now a lecturer of Department of Mathematics in Binzhou University, Shandong, China. Her current research interests include differential equations and dynamical systems.
Department of Mathematics, Binzhou University, Shandong 256603, P.R. China.
e-mail: songxia119@163.com
Quanxin Zhang was made professor in 1997. He is the incumbent Dean of the Department of Mathematics in Binzhou University, he was selected as the “ExcellentWell-known Teacher of Shandong Province”. His current research interests include differential equations and dynamical systems.
Department of Mathematics, Binzhou University, Shandong 256603, P.R. China.
e-mail: 3314744@163.com
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