DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi Publishing Corporation 391973 10.1155/2013/391973 391973 Research Article Existence of Nonoscillatory Solutions for System of Higher-Order Neutral Differential Equations with Distributed Deviating Arguments Liu Youjun 1, 2 Zhang Jianwen 1 Yan Jurang 3 Papaschinopoulos Garyfalos 1 Institute of Applied Mechanics and Biomedical Engineering Taiyuan University of Technology Taiyuan, Shanxi 030024 China tyut.edu.cn 2 College of Mathematics and Computer Sciences Shanxi Datong University Datong, Shanxi 037009 China sxdtdx.edu.cn 3 School of Mathematical Sciences Shanxi University Taiyuan, Shanxi 030006 China sxu.edu.cn 2013 19 11 2013 2013 25 07 2013 17 10 2013 2013 Copyright © 2013 Youjun Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, we consider the existence of nonoscillatory solutions for system of variable coefficients higher-order neutral differential equations with distributed deviating arguments. We use the Banach contraction principle to obtain new sufficient conditions for the existence of nonoscillatory solutions.

1. Introduction and Preliminary

In this paper, we consider the system of higher-order neutral differential equations with distributed deviating arguments: (1)[r(t)(x(t)+P(t)x(t-θ))(n-1)]+(-1)ncdQ1(t,τ)x(t-τ)dτ-efQ2(t,σ)x(t-σ)dσ=0,tt0,

where n is a positive integer, n>1, θ>0, 0<c<d, and 0<e<f;

rC([t0,),R+), r(t)>0, and PC([t0,),R);

xRn, Qi is continuous n×n matrix on [t0,), i=1,2, and matrix coefficients system of higher order neutral differential equations with distributed deviating arguments: (2)[r(t)(x(t)+Bx(t-θ))(n-1)]+(-1)ncdQ1(t,τ)x(t-τ)dτ-efQ2(t,σ)x(t-σ)dσ=0,tt0,

where n is a positive integer, n>1, θ>0, 0<c<d, and 0<e<f;

rC([t0,),R+), r(t)>0, and B is nonsingular constant n×n matrix;

xRn, and Qi is continuous n×n matrix on [t0,), i=1,2.

Recently there have been a lot of activities concerning the existence of nonoscillatory solutions for neutral differential equations with positive and negative coefficients. In 2013, Candan  has investigated existence of nonoscillatory solutions for system of higher-order nonlinear neutral differential equations: (3)[x(t)+P(t)x(t-θ)](n)+(-1)n+1[Q1(t)x(t-σ1)-Q2(t)x(t-σ2)]=0 and matrix coefficient system of higher order neutral functional differential equation: (4)[x(t)+Bx(t-θ)](n)+(-1)n+1[Q1(t)x(t-σ1)-Q2(t)x(t-σ2)]=0. In 2012, Candan  studies higher-order nonlinear differential equation: (5)[r(t)[x(t)+P(t)x(t-τ)](n-1)]+(-1)n×[Q1(t)g1(x(t-σ1))kkkkkkkkkkk-Q2(t)g2(x(t-μ))-f(t)]=0. has obtained sufficient conditions for the existence of nonoscillatory solutions. For related work, we refer the reader to the books .

A solution of system of (1) and (2) is a continuous function x(t) defined on ([t1-μ,),Rn), for some t1>t0, such that x(t)-P(t)x(t-θ) and x(t)-Bx(t-θ) are n-1 times continuously differentiable, and r(t)(x(t)-P(t)x(t-θ))(n-1) and r(t)(x(t)-Bx(t-θ))(n-1) are continuously differentiable, and system of (1) and (2) holds for all n>1. Here, μ=max{θ,d,f}.

2. The Main Results Theorem 1.

Assume that 0P(t)p<1/2 and (6)tsn-2r(s)t0scdQi(u,τ)dτduds<,i=1,2. Then, (1) has a bounded nonoscillatory solution.

Proof.

Let Λ be the set of all continuous and bounded vector functions on [t0,) and the sup norm. Set A={xΛ,M1x(t)M2,tt0}, where M1 and M2 are positive constants and b is a constant vector, such that pM2+M1<b<2b<M2+M1. From (6), one can choose a t1t0, t1t0+μ, sufficiently large tt1 such that(7)1(n-2)!t(s-t)n-2r(s)kkkkkkkk×t1s[cdQ1(u,τ)dτ+efQ2(u,σ)dσ]dudsb-pM2-M1M2,t>t1,and define an operator T on A as follows:(8)(Tx)(t)={{t(s-t)n-2r(s)b-P(t)x(t-θ)+1(n-2)!×t(s-t)n-2r(s)×t1s[cdQ1(u,τ)x(u-τ)dτ-efQ2(u,σ)x(u-σ)dσ]duds}tt1,(Tx)(t1)t0tt1.It is easy to see that Tx is continuous, for tt1, xA; by using (7), we have (9)(Tx)(t)=b-P(t)x(t-θ)+1(n-2)!×t(s-t)n-2r(s)kkkkkk×t1s[cdQ1(u,τ)x(u-τ)dτkkkkkkkkkkk-efQ2(u,σ)x(u-σ)dσ]dudsb-px(t-θ)+1(n-2)!×t(s-t)n-2r(s)kkkkk×t1s[cdQ1(u,τ)x(u-τ)dτkkkkkkkkkkk-efQ2(u,σ)dσx(u-σ)]dudsb-px(t-θ)+1(n-2)!×t(s-t)n-2r(s)kkkk×t1s[cdQ1(u,τ)x(u-τ)dτkkkkkkkkk+efQ2(u,σ)x(u-σ)dσ]dudsb-pM2+M2(n-2)!×t(s-t)n-2r(s)kkkkk×t1s[cdQ1(u,τ)dτkkkkkkkkkkk+efQ2(u,σ)dσ]dudsM2; and taking (7) into account, we have (10)(Tx)(t)=b-P(t)x(t-θ)+1(n-2)!×t(s-t)n-2r(s)×t1s[cdQ1(u,τ)x(u-τ)dτ-efQ2(u,σ)x(u-σ)dσ]dudsbpx(t-θ)-1(n-2)!×t(s-t)n-2r(s)×t1s[cdQ1(u,τ)x(u-τ)dτ-efQ2(u,σ)x(u-σ)dσ]dudsb-px(t-θ)-1(n-2)!×t(s-t)n-2r(s)×t1s[cdQ1(u,τ)x(u-τ)dτ+efQ2(u,σ)x(u-σ)dσ]dudsb-pM2-M2(n-2)!×t(s-t)n-2r(s)×t1s[cdQ1(u,τ)dτ+efQ2(u,σ)dσ]dudsM1. These show that TAA. Since A is a bounded, close, convex subset of Λ, in order to apply the contraction principle, we have to show that T is a contraction mapping on A. For all x1,x2A, and tt1,(11)|(Tx1)(t)-(Tx2)(t)|P(t)x1(t-θ)-x2(t-θ)+1(n-2)!×t(s-t)n-2r(s)×t1s[cdQ1(u,τ)x1(u-τ)-x2(u-τ)dτ+efQ2(u,σ)x1(u-σ)-x2(u-σ)dσ]duds.Using (7), (12)|(Tx1)(t)-(Tx2)(t)|x1-x2×(p+1(n-2)!×t(s-t)n-2r(s)t1s[cdQ1(u,τ)dτ+efQ2(u,σ)dσ]duds)<q1x1-x2,(0<q1<1).This implies, with the sup norm, that (13)Tx1-Tx2<q1x1-x2,(0<q1<1), which shows that T is a contraction mapping on A, and therefore there exists a unique solution. Consequently there exists a unique solution of (1) xA of Tx=x. The proof is complete.

Theorem 2.

Assume that 2<p1P(t)p0< and that (6) holds.

Then, (1) has a bounded nonoscillatory solution.

Proof.

Let Λ be the set of all continuous and bounded vector functions on [t0,) and the sup norm. Set A={xΛ,M3x(t)M4,tt0}, where M3 and M4 are positive constants such that p0M3+M4<b<2bp1M4+p0M3. From (6), one can choose a t1t0, t1t0+μ, sufficiently large tt1 such that (14)1(n-2)!t+θ(s-t-θ)n-2r(s)×t1s[cdQ1(u,τ)dτ+efQ2(u,σ)σ]dudsb-M4-p0M3M4,t>t1, and define an operator T on A as follows:(15)(Tx)(t)={1P(t+θ)×{b-x(t+θ)+1(n-2)!×t+θ(s-t-θ)n-2r(s)×t1s[cdQ1(u,τ)x(u-τ)dτ-efQ2(u,σ)x(u-σ)dσ]duds}tt1,(Tx)(t1)t0tt1.It is easy to see that Tx is continuous, for tt1, xA; by using (14), we have (16)(Tx)(t)1p1{-efQ2(u,σ)x(u-σ)dσ]duds}b-x(t-θ)+1(n-2)!×t+θ(s-t-θ)n-2r(s)×t1s[cdQ1(u,τ)x(u-τ)dτ-efQ2(u,σ)x(u-σ)dσ]duds}1p1{-efQ2(u,σ)x(u-σ)dσ]duds}b+x(t-θ)+1(n-2)!×t+θ(s-t-θ)n-2r(s)×t1s[cdQ1(u,τ)x(u-τ)dτ-efQ2(u,σ)x(u-σ)dσ]duds}1p1{-efQ2(u,σ)x(u-σ)dσ]duds}b+x(t-θ)+1(n-2)!×t+θ(s-t-θ)n-2r(s)×t1s[cdQ1(u,τ)x(u-τ)dτ+efQ2(u,σ)x(u-σ)dσ]duds}1p1{b+M4+M4(n-2)!×t+θ(s-t-θ)n-2r(s)×t1s[cdQ1(u,τ)dτ+efQ2(u,σ)dσ]duds}M4; and taking (14) into account, we have (17)(Tx)(t)1p0{-efQ2(u,σ)x(u-σ)dσ]duds}b-x(t-θ)+1(n-2)!×t+θ(s-t-θ)n-2r(s)×t1s[cdQ1(u,τ)x(u-τ)dτ-efQ2(u,σ)x(u-σ)dσ]duds}1p0{-efQ2(u,σ)x(u-σ)dσ]dudsb-x(t-θ)-1(n-2)!×t+θ(s-t-θ)n-2r(s)×t1s[cdQ1(u,τ)x(u-τ)dτ-efQ2(u,σ)x(u-σ)dσ]duds}1p0{-efQ2(u,σ)x(u-σ)dσ]dudsb-x(t-θ)-1(n-2)!×t+θ(s-t-θ)n-2r(s)×t1s[cdQ1(u,τ)x(u-τ)dτ+efQ2(u,σ)x(u-σ)dσ]duds}1p0{b-M4-M4(n-2)!×t+θ(s-t-θ)n-2r(s)×t1s[cdQ1(u,τ)dτ+efQ2(u,σ)dσ]duds}M3. These show that TAA. Since A is a bounded, close, convex subset of Λ, in order to apply the contraction principle, we have to show that T is a contraction mapping on A. For all x1,x2A, and tt1,(18)|(Tx1)(t)-(Tx2)(t)|1p1{efQ2(u,σ)x1(t+θ)-x2(t+θ)+1(n-2)!×t+θ(s-t-θ)n-2r(s)×t1s[cdQ1(u,τ)x1(u-τ)-x2(u-τ)dτ+efQ2(u,σ)x1(u-σ)-x2(u-σ)dσefQ2(u,)]duds},or using (14), (19)|(Tx1)(t)-(Tx2)(t)|x1-x2p1×{1+1(n-2)!×t+θ(s-t-θ)n-2r(s)×t1s[cdQ1(u,τ)dτ+efQ2(u,σ)dσ]duds}<x1-x2. This implies, with the sup norm, that (20)Tx1-Tx2<x1-x2, which shows that T is a contraction mapping on A, and therefore there exists a unique solution. Consequently there exists a unique solution of (1) xA of Tx=x. The proof is complete.

Theorem 3.

Assume that -1/2<p2P(t)0 and that (6) holds.

Then, (1) has a bounded nonoscillatory solution.

Proof.

Let Λ be the set of all continuous and bounded vector functions on [t0,) and the sup norm. Set A={xΛ,M5x(t)M6,tt0}, where M5 and M6 are positive constants such that -p2M6+M5<b<2bM6+M5. From (6), one can choose a t1t0, t1t0+μ, sufficiently large tt1 such that (21)1(n-2)!t(s-t)n-2r(s)×t1s[cdQ1(u,τ)dτ+efQ2(u,σ)dσ]dudsb+p2M6-M5M6,t>t1,       and define an operator T on A as follows(22)(Tx)(t)={b-P(t)x(t-θ)+1(n-2)!×t(s-t)n-2r(s)×t1s[cdQ1(u,τ)x(u-τ)dτ-efQ2(u,σ)x(u-σ)dσ]dudstt1,(Tx)(t1)t0tt1.It is easy to see that Tx is continuous. Since the proof is similar to that of Theorem 1, we omit the remaining part of the proof. The proof is complete.

Theorem 4.

Assume that -<p3P(t)p4<-2 and that (6) holds.

Then, (1) has a bounded nonoscillatory solution.

Proof.

Let Λ be the set of all continuous and bounded vector functions on [t0,) and the sup norm. Set A={xΛ,M7x(t)M8,tt0}, where M7 and M8 are positive constants such that -p3M7+M8<b<2b-p4M8-p3M7. From (6), one can choose a t1t0, t1t0+μ, sufficiently large tt1 such that (23)1(n-2)!t+θ(s-t-θ)n-1r(s)×t1s[cdQ1(u,τ)dτ+efQ2(u,σ)dσ]dudsb+p3M7+M8M8,t>t1,       and define an operator T on A as follows:(24)(Tx)(t)={1P(t+θ)×{+1(n-2)!b-x(t+θ)+1(n-2)!×t+θ(s-t-θ)n-2r(s)×t1s[cdQ1(u,τ)x(u-τ)dτ-efQ2(u,σ)x(u-σ)dσ]duds}tt1,(Tx)(t1)t0tt1.It is easy to see that Tx is continuous. The remaining part of the proof is similar to that of Theorem 2; therefore, it is omitted. The proof is complete.

Theorem 5.

Assume that 0<B<1/2 and that (6) holds.

Then, (2) has a bounded nonoscillatory solution.

Proof.

Let Λ be the set of all continuous and bounded vector functions on [t0,) and the sup norm. Set A={xΛ,N1x(t)N2,tt0}, where N1 and N2 are positive constants such that BN2+N1<b<2bN2+N1. From (6), one can choose a t1t0, t1t0+μ, sufficiently large tt1 such that (25)1(n-2)!t(s-t)n-2r(s)×t1s[cdQ1(u,τ)dτ+efQ2(u,σ)dσ]dudsb-BN2-N1N2,t>t1,       and define an operator T on A as follows:(26)(Tx)(t)={b-Bx(t-θ)+1(n-2)!×t(s-t)n-2r(s)×t1s[cdQ1(u,τ)x(u-τ)dτ-efQ2(u,σ)x(u-σ)dσ]duds  tt1,(Tx)(t1)t0tt1.It is easy to see that Tx is continuous. Since the proof is similar to that of Theorem 1, we omit the remaining part of the proof. The proof is complete.

Theorem 6.

Assume that 0<B-1<1/2 and that (6) holds.

Then, (2) has a bounded nonoscillatory solution.

Proof.

Let Λ be the set of all continuous and bounded vector functions on [t0,) and the sup norm. Set A={xΛ,N3x(t)N4,tt0}, where N3 and N4 are positive constants such that B-1N4+N3<B-1b<2B-1bN4-N3. From (6), one can choose a t1t0, t1t0+μ, sufficiently large tt1 such that (27)1(n-2)!t+θ(s-t-θ)n-2r(s)×t1s[cdQ1(u,τ)dτ+efQ2(u,σ)dσ]dudsB-1b-N3-N4B-1N4B-1,t>t1, and define an operator T on A as follows:(28)(Tx)(t)={B-1{b-x(t+θ)+1(n-2)!×t+θ(s-t-θ)n-2r(s)×t1s[cdQ1(u,τ)x(u-τ)dτ-efQ2(u,σ)x(u-σ)dσ]ds}tt1,(Tx)(t1)  t0tt1.It is easy to see that Tx is continuous, for tt1, xA; by using (27), we have (29)(Tx)(t)B-1b+B-1×{N4+N4(n-2)!×t+θ(s-t-θ)n-2r(s)×t1s[cdQ1(u,τ)dτ-efQ2(u,σ)dσ]duds}N4, and taking (27) into account, we have(30)(Tx)(t)B-1b-B-1×{t+θ(s-t-θ)n-2r(s)N4-N4(n-2)!×t+θ(s-t-θ)n-2r(s)×t1s[cdQ1(u,τ)dτ-efQ2(u,σ)dσ]duds}N3.It is easy to see that Tx is continuous. The remaining part of the proof is similar to that of Theorem 2; therefore, it is omitted. The proof is complete.

3. Example Example 1.

Consider high-order neutral differential equation with distributed deviating arguments: (31)(e-t(x(t)+e-1x(t-1))(3))+2312τe-2t(311272)x(t-τ)dτ-2312σe-2t(2153103)x(t-σ)dσ=0. Here, n=4, c=e=1, d=f=2, θ=1, r(t)=e-t, P(t)=e-1, Q1(t,τ)=(2/3)τe-2t(311/27/2), and Q2(t,σ)=(2/3)σe-2t(215/310/3).

It is easy to see that (32)0<P(t)=e-1<12,tets2t0s1223τe-2u(2153103)dτduds<,tets2t0s1223σe-2u(311272)dσduds<; thus Theorem 1 holds. In fact, x(t)=(e-te-t) is a nonoscillatory solution of (31).

Example 2.

Consider high-order neutral differential equation with distributed deviating arguments: (33)(1|sint|(x(t)+e-π(124353)x(t-π))(2))+3813τ2e-4t(527251)x(t-τ)dτ-3813σ2e-4t(2473113)x(t-σ)dσ=0. Here, n=3, r(t)=1/|sint|, θ=π, c=e=1, d=f=3, B=e-π(124/35/3), Q1(t,τ)=(3/8)τ2e-4t(5/27/251), and Q2(t,σ)=(3/8)σ2e-4t(247/311/3).

It is easy to see that (34)0<B=e-π(124353)<12,t|sins|st0s3813τ2e-4u(527251)dτduds<,t|sins|st0s3813σ2e-4u(2473113)dσduds<; thus Theorem 5 holds. In fact, x(t)=(2+sint2+sint) is a nonoscillatory solution of (33).

Acknowledgments

This research is supported by the Natural Sciences Foundation of China (no. 11172194), the Natural Sciences Foundation of Shanxi Province (no. 2010011008), and the Scientific Research Project Shanxi Datong University (no. 2011K3).

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