Eighth ACIS International Conference on Software Engineering, Artificial Intelligence, Networking, and Parallel/Distributed Computing
Existence of Nonoscillatory Solution for Second Order Nonlinear Neutral Delay Differential Equation ∗ Jinzhu Zhang Department of Mathematics North University of China College Attached to North China University Taiyuan 030008, P.R. China Abstract
obtained some sufficient conditions of the existence of nonoscillatory solution for Eq.(1.1). In this paper, motivated by recent paper [2,3], we shall extend the results of [1] to the case P (t) = 1, give some sufficient conditions for the existence of nonoscillatory solution of Eq.(1.1), and indicate that the condition (H3) is redundant. Let ϕ ∈ C([t0 − σ, ∞), R), where σ = max{τ, σ1 , σ2 }, be a given function and let x 0 be a given constant. By the method of steps, it is easy to know that Eq.(1.1) has a unique solution x ∈ C([t 0 , ∞), R) in the sense that x(t) + P (t)x(t − τ ) is twice continuously differential for t ≥ t0 , x(t) satisfies Eq.(1.1) and
This paper has considered the existence of nonoscillatory solutions for second-order nonlinear neutral delay differential equation with positive and negative coefficients. Some sufficient conditions for existence of nonoscillatory solutions are obtained.
1. Introduction Recently, Yu and Wang [1] studied the existence of nonoscillatory solutions for the second-order neutral delay differential equation:
x(s) = ϕ(s) for s ∈ [t0 − σ, t0 ],
[r(t)(x(t) + P (t)x(t − τ )) ] + Q1 (t)f (x(t − σ1 )) −Q2 (t)g(x(t − σ2 )) = 0, t ≥ t0 , (1.1)
[x(t) + P (t)x(t − τ ))]t=t0 = x0 . As is customary, a solution of Eq.(1.1) is called oscillatory if it has arbitrarily large zeros, and otherwise it is nonoscillatory. Eq.(1.1) is oscillatory if all its solutions are oscillatory.
where τ > 0, σ1 , σ2 ≥ 0, P, Q1 , Q2 , r ∈ C([t0 , ∞), R) and f, g ∈ C(R, R). (H1) f and g satisfy local Lipschitz condition and xf (x) > 0, xg(x) > 0, for x = 0. (H2) r(t) > 0, Qi ≥ 0, and t 1 1, 2, where R(t) = t0 r(t) ds.
∞
2. Main Result
R(t)Qi (t)dt < ∞, i =
Theorem. Suppose that conditions (H1) and (H2) hold. If P (t) satisfies one of the following ranges:
(H3) aQ1 (t) − Q2 (t) is eventually nonnegative for every a > 0.
(i) P (t) ≡ 1,
(ii)
0 < P (t) ≤ p1 < 1,
(iii) 1 < p2 ≤ P (t) ≤ p1 , (iv) − 1 < −p2 ≤ P (t) < 0,
In [1], Yu proved that Eq.(1.1) (P (t) = ±1) has a nonoscillatory solution under conditions (H1) − (H3) and
(v)
∗ This work is supported by the National Sciences Foundation of China(10471040), Science Foundations of Shanxi Province(2006011009), and Supported by Program for New Century Excellent Talents in University(NCET050271)
0-7695-2909-7/07 $25.00 © 2007 IEEE DOI 10.1109/SNPD.2007.211
Huimin Zhang Department of Mathematics North University of China Taiyuan 030051, P. R. China
Zhen Jin Department of Mathematics North University of China Taiyuan 030051, P. R. China
[email protected]
− p2 ≤ P (t) ≤ −p1 < −1.
then Eq.(1.1) has a nonoscillatory solution. Proof. (i) P (t) ≡ 1. Let L f (A), Lg (A) denote Lipschitz constants of functions f, g on the set A, respectively,
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L = max{Lf (A), Lg (A)}, α1 = maxx∈A {f (x)}, α2 = maxx∈A {g(x)}, and M = max{L, α1 , α2 }. Choose a t1 > t0 + σ sufficiently large such that ∞ 1 (2.1) (R(u) − R(t))Qi (u)du < , i = 1, 2 M 3 t
≤
≥2−M
∞ t
∞ s
L(Q1 (u) + Q2 (u))|x1 − x2 |duds ∞ t
(R(u) − R(t))(Q1 (u) + Q2 (u))du
t1
∞
t1
α2 R(s)Q2 (s)ds ≤ 1 − p1 N1 − M1 ,
(2.4)
where M1 and N1 are positive constants such that 1 < N1
t0 + σ sufficiently large such that ∞ 1 − p1 , (2.2) R(s)[Q1 (s) + Q2 (s)]ds < 2L t1 ∞ α1 R(s)Q1 (s)ds ≤ N1 − 1, (2.3)
Clearly, T x is continuous. For every x ∈ A and t ≥ t 1 , using (2.1), we get (T x)(t) ∞ ∞ t+2jτ 1 Q1 (u)f (x(u − σ1 ))duds ≥2− t+(2j−1)τ r(s) s j=1 ∞ t+(2j−1)τ 1 Q1 (u)f (x(u − σ1 ))duds + t+(2j−2)τ r(s) s 1 r(s)
∞
where we used sup norm.
Define a mapping T : A → X as follows (T x)(t) ⎧ ∞ ∞ t+2jτ 1 2 − j=1 t+(2j−1)τ r(s) ⎪ s [Q1 (u)f (x(u − σ1 )) ⎪ ⎪ ⎪ ⎨ = −Q2 (u)g(x(u − σ2 ))]duds, t ≥ t1 ⎪ ⎪ ⎪ ⎪ ⎩ (T x)(t1 ), t0 ≤ t ≤ t1 .
t
1 r(s)
≤ 23 x1 − x2 .
A = {x ∈ X : 1 ≤ x(t) ≤ 3, t ≥ t0 }.
∞
t
≤ M x1 − x2
hold, for t ≥ t 1 . Let X be the set of all continuous and bounded functions on [t0 , ∞) with the sup norm. Set
=2−
∞
∞ 1 Q2 (u)g(x(u − σ2 ))duds t+(2j−1)τ r(s) s j=1 ∞ t+(2j−1)τ 1 Q2 (u)g(x(u − σ2 ))duds + t+(2j−2)τ r(s) s t+2jτ
Clearly, T x is continuous. For every x ∈ A and t ≥ t 1 , using (2.3), we get ∞ (T x)(t) ≤ 1 + R(s)Q1 (s)f (x(s − σ1 ))ds
∞ 1 ∞ = 2 + t r(s) Q2 (u)g(x(u − σ2 ))duds ∞ s ≤ 2 + M t [R(u) − R(t)]Q2 (u)du ≤ 3. Thus we prove that T A ⊂ A. To apply the contraction principle, we need to prove that T is a contraction mapping on A since A is a bounded, closed, and convex subset of X. Now, for x1 , x2 ∈ A and t ≥ t1 we have
t1
∞
≤ 1 + t1 α1 R(s)Q1 (s)ds ≤ N1 Furthermore, from (2.4), we have ∞ (T x)(t) ≥ 1 − p1 N1 − R(s)Q2 (s)g(x(s − σ2 ))ds
|(T x1 )(t) − (T x2 )(t)| ∞ ∞ t+2jτ 1 L(Q1 (u)|x1 − x2 | ≤ t+(2j−1)τ r(s) s j=1 +Q2 (u)|x1 − x2 |)duds
t1
≥ 1 − p1 N 1 −
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∞ t1
α2 R(s)Q2 (s)ds ≥ M1 .
Thus we prove that T A ⊂ A. To apply the contraction principle, we need to prove that T is a contraction mapping on A since A is a bounded, closed, and convex subset of X. Now, for x1 , x2 ∈ A and t ≥ t1 we have
≥ M2 . Thus we prove that T A ⊂ A. To apply the contraction principle, we need to prove that T is a contraction mapping on A since A is a bounded, closed, and convex subset of X. Now, for x1 , x2 ∈ A and t ≥ t1 , we have |(T x1 )(t) − (T x2 )(t)|
|(T x1 )(t) − (T x2 )(t)| ∞ R(s)[Q1 (s) + Q2 (s)]ds} ≤ x1 − x2 {p1 + L
≤
t1
= q1 x1 − x2 where we used sup norm. From (2.2), we obtain q 1 < 1.
t1
p1 N2 α2 R(s)Q2 (s)ds ≤ 1 − − p1 M 2 , p2
(2.7)
∞
α2 R(s)Q2 (s)ds ≤ 1 − M3 ,
(2.10)
M3 < 1 < N3 (1 − p2 ). Let X be the set of all continuous and bounded functions on [t0 , ∞) with the sup norm. Set
Define a mapping T : A → X as follows (T x)(t) ⎧ ) ) ∞ 1 ⎪ − Px(t+τ + PR(t+τ ⎪ P (t+τ ) (t+τ ) (t+τ ) t+τ [Q1 (s)f (x(s − σ1 )) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −Q2 (s)g(x(s − σ2 ))]ds ⎪ ⎪ ⎪ ⎪ ⎨ t+τ 1 = R(s)[Q1 (s)f (x(s − σ1 )) + P (t+τ ) t1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −Q2 (s)g(x(s − σ2 ))]ds, t ≥ t1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (T x)(t1 ), t0 ≤ t ≤ t1 .
A = {x ∈ X : M3 ≤ x(t) ≤ N3 , t ≥ t0 }. Define a mapping T : A → X as follows (T x)(t) ∞ ⎧ 1 − P (t)x(t − τ ) + R(t) t1 [Q1 (s)f (x(s − σ1 )) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −Q2 (s)g(x(s − σ2 ))]ds ⎪ ⎪ ⎪ ⎪ ⎨ t = + t1 R(s)[Q1 (s)f (x(s − σ1 )) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −Q2 (s)g(x(s − σ2 ))]ds, t ≥ t1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (T x)(t1 ), t0 ≤ t ≤ t1 .
Clearly, T x is continuous. For every x ∈ A and t ≥ t 1 , using (2.6), we get
t1
(2.9)
where M3 and N3 are positive constants such that
A = {x ∈ X : M2 ≤ x(t) ≤ N2 , t ≥ t0 }.
∞
R(s)[Q1 (s) + Q2 (s)]ds}
α1 R(s)Q1 (s)ds ≤ N3 (1 − p2 ) − 1,
t1
Let X be the set of all continuous and bounded functions on [t0 , ∞) with the sup norm. Set
∞ t1
1 < p1 N2 < p2 (1 − p1 M2 ).
1 1 + p2 p2
t1
(iv) −1 < −p2 ≤ P (t) < 0. Choose a t1 > t0 + σ sufficiently large such that ∞ 1 − p2 , (2.8) R(s)[Q1 (s) + Q2 (s)]ds < 2L t1
where M2 and N2 are positive constants such that
(T x)(t) ≤
∞
where we used sup norm. From (2.5), we obtain q 2 < 1.
t1
∞
− x2 {1 + L
= q2 x1 − x2 .
(iii) 1 < p2 ≤ P (t) ≤ p1 < +∞. Choose a t1 > t0 +σ sufficiently large such that ∞ p2 − 1 , (2.5) R(s)[Q1 (s) + Q2 (s)]ds < 2L t1 ∞ α1 R(s)Q1 (s)ds ≤ p2 N2 − 1, (2.6)
1 p2 x1
Clearly, T x is continuous. For every x ∈ A and t ≥ t 1 , using (2.9), we get ∞ (T x)(t) ≤ 1 + p2 N3 + R(s)Q1 (s)f (x(s − σ1 ))ds
R(s)Q1 (s)f (x(s − σ1 ))ds ≤ N2 .
Furthermore, from (2.7), we have ∞ 1 N2 1 (T x)(t) ≥ − − R(s)Q2 (s)g(x(s − σ2 ))ds p1 p2 p1 t1
t1
≤ N3
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Furthermore, from (2.10), we have ∞ R(s)Q2 (s)g(x(s − σ2 ))ds ≥ M3 . (T x)(t) ≥ 1 −
Furthermore, from (2.12), we have (T x)(t)
t1
M4 1 1 + − ≥ p2 p2 p1
Thus we prove that T A ⊂ A. To apply the contraction principle, we need to prove that T is a contraction mapping on A since A is a bounded, closed, and convex subset of X. Now, for x1 , x2 ∈ A and t ≥ t1 , we have
∞ t1
3. Conclusions In this paper, we considered the existence of a class of nonoscillatory solutions for second-order nonlinear neutral delay differential equation with positive and negative coefficients. Some sufficient conditions for existence of nonoscillatory solutions are obtained, the results obtained generalize and improve noticeably the known results in previous literatures.
t1
where M4 and N4 are positive constants such that N4 (p1 − 1) > 1 > (p2 − 1)M4 .
References
Let X be the set of all continuous and bounded functions on [t0 , ∞) with the sup norm. Set A = {x ∈ X : M4 ≤ x(t) ≤ N4 , t ≥ t0 }. Define a mapping T : A → X as follows (T x)(t) ⎧ ) ) ∞ 1 ⎪ − P (t+τ − Px(t+τ + R(t+τ ⎪ ) (t+τ ) P (t+τ ) t+τ [Q1 (s)f (x(s − σ1 )) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −Q2 (s)g(x(s − σ2 ))]ds ⎪ ⎪ ⎪ ⎪ ⎨ t+τ 1 = R(s)[Q1 (s)f (x(s − σ1 )) + P (t+τ ) t1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −Q2 (s)g(x(s − σ2 ))]ds, t ≥ t1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (T x)(t1 ), t0 ≤ t ≤ t1 Clearly, T x is continuous. For every x ∈ A and t ≥ t 1 , using (2.13), we get
≤
1 N4 1 + + p1 p1 p1
∞
t1
R(s)Q1 (s)f (x(s − σ1 ))ds ≥ M4 .
= q4 x1 − x2 . where we used sup norm. From (2.11), we obtain q 4 < 1. This completes the proof.
(v) −∞ < −p2 ≤ P (t) ≤ −p1 < −1. Choose a t1 > t0 + σ sufficiently large such that ∞ p1 − 1 , (2.11) R(s)[Q1 (s) + Q2 (s)]ds < 2L t1 ∞ p1 α1 R(s)Q1 (s)ds ≤ [1 − (p2 − 1)M4 ], (2.12) p2 t1 ∞ α2 R(s)Q2 (s)ds ≤ N4 (p1 − 1) − 1, (2.13)
t1
|(T x1 )(t) − (T x2 )(t)| ∞ 1 s[Q1 (s) + Q2 (s)]ds} ≤ x1 − x2 {1 + L p1 t1
R(s)[Q1 (s) + Q2 (s)]ds}
where we used sup norm. From (2.8), we obtain q 3 < 1.
(T x)(t)
∞
Thus we prove that T A ⊂ A. To apply the contraction principle, we need to prove that T is a contraction mapping on A since A is a bounded, closed, and convex subset of X. Now, for x1 , x2 ∈ A and t ≥ t1 , we have
|(T x1 )(t) − (T x2 )(t)| ≤ x1 − x2 {p2 + L = q3 x1 − x2 .
[1] Y. H. Yu, H.Z. Wang, Nonoscillatory solutions of second-order nonlinear neutral delay equations, J. Math. Anal. Appl. 311 (2005) 445–456. [2] X. Y. Lin, Oscillation of second-order nonlinear neutral differential equations, J. Math. Anal. Appl. 311 (2005) 445–456. [3] J. F. Cheng, Z. Annie, Existence of nonoscillation criteria for linear neutral delay equation, J. Sys. Sci. Math. Sci. 24(2004) 389–397 [4] M. R. S. Kulenovi´c, S. Hadˇziomerpahi´c, Existence of nonosillataory solution of second-order liner neutral delay equation, J. Math. Anal. Appl. 228 (1998) 436–448. [5] A. Elbert, Oscillation and nonoscillation criteria for liner second-order differential equation, J. Math. Anal. Appl. 266 (1998), 207–219. [6] W. T. Li, Positive solutions of second-order nonlinear differential equation, J. Math. Anal. Appl. 221 (1998) 326–337.
R(s)Q2 (s)g(x(s − σ2 ))ds ≤ N4 .
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