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Existence of nonoscillatory solutions for system of higher-order neutral differential equations with distributed coefficients and delays
Advances in Difference Equations volume 2017, Article number: 41 (2017)
Abstract
In this paper we consider the existence of nonoscillatory solutions for a system of higher-order neutral differential equations with distributed coefficients and delays. 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 coefficients and delays
-
(1)
where n is a positive integer, \(n\geq1, 0< a< b, 0< c< d, 0< e< f\);
-
(2)
\(r\in C ([t_{0}, \infty), R^{+} ), r(t)>0, p\in C ([t_{0}, \infty)\times[a, b], R ), \mathbf{h}\in C ([t_{0}, \infty ), \mathbf{R} )\),
-
(3)
\(\mathbf{x}\in\mathbf{R}^{n}, Q_{i}\) is continuous \(n\times n\) matrix on \([t_{0}, \infty), i=1, 2\).
Recently there have been a lot of activities concerning the existence of nonoscillatory solutions for neutral differential equations with positive and negative coefficients. In 2005, the existence of nonoscillatory solutions of the first-order linear neutral delay differential equations
was investigated by Zhang et al. [1]. In 2012, Candan [2] studied the higher-order nonlinear differential equation
In 2013, Candan [3] has investigated the existence of nonoscillatory solutions for the system of higher-order nonlinear neutral differential equations
In the same year, Liu et al. [4] has obtained the existence of nonoscillatory solutions for the system of higher-order neutral differential equations
As can be seen from the development process of the above equations, the delay of neutral part in the discussed differential equations were all constant delays. However, the case for distributed deviating arguments is rather rare; see [5, 6]. In 2015, Candan and Gecgel [6] studied the systems of higher-order neutral differential equations with distributed delay
the discussion only covered the condition for the coefficient being \(0<\int_{a_{3}}^{b_{3}}\tilde{P}(t,\xi)x(t-\xi)\,d\xi<\frac{1}{2}\) and \(-\frac{1}{2}<\int_{a_{3}}^{b_{3}}\tilde{P}(t,\xi)x(t-\xi)\,d\xi<0\). However, in this paper, the difficulty in establishing a feasible operator was settled by skillful use of \(r(t)\), and the coefficients \(\int_{a}^{b}p_{2}(t,\xi)\,d\xi\) in the neutral part were all discussed in four cases, that is, \((-\infty,-1),(-1,0), (0,1),(1,+\infty)\). Thus, in view of the above, this paper may have theoretical value as well as practical application value. For related work, we refer the reader to [7–10].
A solution of the system of equations (1) is a continuous vector function \(\mathbf{x}(t)\) defined on \(([t_{1}-\mu, \infty), \mathbf{R}^{n})\), for some \(t_{1}>t_{0}\), such that \(r(t)\mathbf{x}(t)-\int_{a}^{b}p(t,\theta)\mathbf{x}(t-\theta)\,d\theta\) is n times continuously differentiable and the system of equations (1) holds for all \(n\geq1\). Here, \(\mu=\max\{b, \tau, \sigma\}\).
2 The main results
Theorem 1
Assume that \(0\leq\int _{a}^{b}p(t,\theta)\,d\theta\leq p_{1}<1\) and
Then equation (1) has a bounded nonoscillatory solution.
Proof
Let Λ be the set of all continuous and bounded vector functions on \([t_{0}, \infty)\) with the sup norm. Set \(A=\{x\in\Lambda, M_{1}\leq \Vert \mathbf{x}(t)\Vert \leq M_{2}, t\geq t_{0}\}\), where \(M_{1}, M_{2}\) are two positive constants and c is a constant vector, such that \(p_{1}M_{2}+\frac{M_{1}}{p_{1}}<\Vert \mathbf{c}\Vert <M_{2}, 1\leq r(t)\leq\frac{1}{p_{1}}\). From (3), one can choose a \(t_{1}\geq t_{0}\), \(t_{1}\geq t_{0}+\mu\), sufficiently large, \(t\geq t_{1}\), such that
and one defines an operator T on A as follows:
It is easy to see that T x is continuous, for \(t\geq t_{1}, \mathbf{x}\in A\), by using (4), we have
and taking (5) into account, we have
These show that \(TA\subset A\), since A is a bounded, closed, and convex subset of Λ, in order to apply the contraction principle we have to show that T is a contraction mapping on A. For \(\forall \mathbf{x}_{1}, \mathbf{x}_{2}\in A\), and \(t\geq t_{1}\),
Using (6),
which shows that T is a contraction mapping on A and therefore there exists a unique solution, obviously a bounded positive solution of (1) \(\mathbf{x}\in A\), such that \(T\mathbf{x}=\mathbf{x}\). The proof is complete. □
Theorem 2
Assume that \(1< p_{3}\leq\int _{a}^{b}p(t,\theta)\,d\theta\leq p_{2}<2p_{3}<+\infty\), and that (3) holds.
Then equation (1) has a bounded nonoscillatory solution.
Proof
Let Λ be the set of all continuous and bounded vector functions on \([t_{0}, \infty)\) with the sup norm. Set \(A=\{x\in\Lambda, M_{3}\leq \Vert \mathbf{x}(t)\Vert \leq M_{4}, t\geq t_{0}\}\), where \(M_{3}, M_{4}\) are two positive constants such that \(p_{2}M_{4}+2p_{2}M_{3}<\Vert \mathbf{c}\Vert <2 p_{3}M_{4}, 2p_{3}\leq r(t)\leq2p_{2}\). From (3), one can choose a \(t_{1}\geq t_{0}+b\), sufficiently large \(t\geq t_{1}\), such that
and one defines an operator T on A as follows:
It is easy to see that T is continuous, for \(t\geq t_{1}, \mathbf {x}\in A\). By using (7), we have
and taking (8) into account, we have
These show that \(TA\subset A\), since A is a bounded, closed, and convex subset of Λ, in order to apply the contraction principle, we have to show that T is a contraction mapping on A. For \(\forall \mathbf{x}_{1}, \mathbf{x}_{2}\in A\), and \(t\geq t_{1}\),
using (9),
which shows that T is a contraction mapping on A and therefore there exists a unique solution, obviously a bounded positive solution of (1) \(\mathbf{x}\in A\), such that \(T\mathbf{x}=\mathbf{x}\). The proof is complete. □
Theorem 3
Assume that \(-1< p_{4}\leq\int _{a}^{b}p(t,\theta)\,d\theta\leq0\) and that (2) holds.
Then equation (1) has a bounded nonoscillatory solution.
Proof
Let Λ be the set of all continuous and bounded functions on \([t_{0},\infty)\) with the sup norm. Set \(A=\{x\in\Lambda, M_{5}\leq \Vert \mathbf{x}(t)\Vert \leq M_{6}, t\geq t_{0}\}\), where \(M_{5}, M_{6}\) are two positive constants such that \(\frac{M_{5}}{-p_{4}}<\Vert \mathbf{c}\Vert <(1+p_{4})M_{6}, 1\leq r(t)\leq \frac{1}{-p_{4}}\). From (3), one can choose a \(t_{1}\geq t_{0}+b\), sufficiently large \(t\geq t_{1}\) such that
and one defines an operator T on A as follows:
It is easy to see that T is continuous, for \(t\geq t_{1}, \mathbf {x}\in A\). By using (10), we have
and taking (11) into account, we have
These show that \(TA\subset A\), since A is a bounded, closed, and convex subset of Λ, in order to apply the contraction principle, we have to show that T is a contraction mapping on A. For \(\forall \mathbf{x}_{1}, \mathbf{x}_{2}\in A\), and \(t\geq t_{1}\),
Or using (12),
which shows that T is a contraction mapping on A and therefore there exists a unique solution, obviously a bounded positive solution of (1) \(\mathbf{x}\in A\), such that \(T\mathbf{x}=\mathbf{x}\). The proof is complete. □
Theorem 4
Assume that \(-\infty<2p_{5}<p_{6}\leq \int_{a}^{b}p(t,\theta)\,d\theta\leq p_{5}<-1\) and that (3) holds.
Then equation (1) has a bounded nonoscillatory solution.
Proof
Let Λ be the set of all continuous and bounded functions on \([t_{0}, \infty)\) with the sup norm. Set \(A=\{\mathbf{x}\in\Lambda, M_{7}\leq \Vert \mathbf{x}(t)\Vert \leq M_{8}, t\geq t_{0}\}\), where \(M_{7}, M_{8}\) are two positive constants such that \(-2p_{6}M_{7}<\Vert \mathbf{c}\Vert <(-2p_{5}+p_{6})M_{8}\), \(-2p_{5}< r(t)<-2p_{6}\). From (3), one can choose a \(t_{1}\geq t_{0}+b\), sufficiently large \(t\geq t_{1}\), such that
and one defines an operator T on A as follows:
It is easy to see that T is continuous, for \(t\geq t_{1}, \mathbf {x}\in A\). By using (13), we have
and taking (14) into account, we have
These show that \(TA\subset A\), since A is a bounded, closed, and convex subset of Λ, in order to apply the contraction principle, we have to show that T is a contraction mapping on A. For \(\forall \mathbf{x}_{1}, \mathbf{x}_{2}\in A\), and \(t\geq t_{1}\),
Or using (15),
which shows that T is a contraction mapping on A and therefore there exists a unique solution, obviously a bounded positive solution of (1) \(\mathbf{x}\in A\), such that \(T\mathbf{x}=\mathbf{x}\). The proof is complete. □
3 Example
Consider the higher-order neutral differential equation with distributed coefficients and delays
Here, \(n=3\),
It is easy to see that \(1\leq r(t)\leq2\),
thus Theorem 2 holds.
In fact, \(\mathbf{x}(t)=\binom{2+\sin t}{ 2+\sin t}\) is a nonoscillatory solution of equation (16).
References
Zhang, W, Feng, W, Yan, J, Song, J: Existence of nonoscillatory solutions of first-order linear neutral delay differential equations. Comput. Math. Appl. 49, 1021-1027 (2005)
Candan, T: The existence of nonoscillatory solutions of higher order nonlinear neutral differential equations. Appl. Math. Lett. 25, 412-416 (2012)
Candan, T: Existence of nonoscillatory solutions for system of higher order neutral differential equations. Math. Comput. Model. 57, 375-381 (2013)
Liu, Y, Zhang, J, Yan, J: Existence of nonoscillatory solutions for system of higher-order neutral differential equations with distributed deviating arguments. Discrete Dyn. Nat. Soc. 2013, 1-8 (2013)
Candan, T: Existence of nonoscillatory solutions of first-order nonlinear neutral differential equations. Appl. Math. Lett. 26, 1182-1186 (2013)
Candan, T, Gecgel, AM: Existence of nonoscillatory solutions for system of higher order neutral differential equations with distributed delay. J. Comput. Anal. Appl. 18, 266-276 (2015)
Candan, T, Dahiya, RS: Existence of nonoscillatory solutions of first and second order neutral differential equations with distributed deviating arguments. J. Franklin Inst. 347, 1309-1316 (2010)
Liu, Y, Zhang, J, Yan, J: Existence of nonoscillatory solutions of higher order neutral differential equations with distributed deviating arguments. Acta Math. Appl. Sin. 38(2), 235-243 (Chinese) (2015)
Zhou, Y, Zhang, BG: Existence of nonoscillatory solutions of higher-order neutral differential equations with positive and negative coefficients. Appl. Math. Lett. 15, 867-874 (2002)
Györi, I, Ladas, G: Oscillation Theory of Delay Differential Equations with Applications. Clarendon Presss, Oxford (1991)
Acknowledgements
This research is supported by Scientific Research Project Shanxi Datong University (No. 2011K3).
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Liu, Y., Zhao, H. & Yan, J. Existence of nonoscillatory solutions for system of higher-order neutral differential equations with distributed coefficients and delays. Adv Differ Equ 2017, 41 (2017). https://doi.org/10.1186/s13662-016-1061-6
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DOI: https://doi.org/10.1186/s13662-016-1061-6
Keywords
- system
- higher-order
- distributed coefficients and delays
- nonoscillatory solutions
- Banach contraction principle