Nonoscillatory bounded solutions of neutral differential systems

doi:10.1016/j.na.2007.01.014

Nonlinear Analysis: Theory, Methods & Applications

Volume 68, Issue 7, 1 April 2008, Pages 1816-1824

https://doi.org/10.1016/j.na.2007.01.014 Get rights and content

Abstract

The article contains some sufficient conditions for the existence of a nonoscillatory bounded solution of the nonlinear neutral differential systems. The main results are presented in several theorems.

Introduction

We consider nonlinear neutral differential systems $\frac{d^{n}}{d t^{n}} [y_{1} (t) - a_{1} (t) y_{1} (t - τ_{1})] = p_{1} (t) g_{1} (y_{2} (t - σ_{2})) + f_{1} (t),$ $\frac{d^{n}}{d t^{n}} [y_{2} (t) - a_{2} (t) y_{2} (t - τ_{2})] = p_{2} (t) g_{2} (y_{1} (t - σ_{1})) + f_{2} (t),$ which we rewrite in the form $\frac{d^{n}}{d t^{n}} [y_{i} (t) - a_{i} (t) y_{i} (t - τ_{i})] = p_{i} (t) g_{i} (y_{3 - i} (t - σ_{3 - i})) + f_{i} (t), t \geq t_{0},$ where $n \geq 1$ is integer, $τ_{i}, σ_{i} \in [0, \infty), a_{i}, p_{i}, f_{i} \in C ([t_{0}, \infty), R), g_{i} \in C (R, R)$ , $i = 1, 2$ .

Let $r = max {τ_{1}, τ_{2}, σ_{1}, σ_{2}}$ . By a solution of the system (1) we mean a function $y = (y_{1}, y_{2}) \in C ([t_{1} - r, \infty), R^{2})$ for some $t_{1} \geq t_{0}$ such that $y_{i} (t) - a_{i} (t) y_{i} (t - τ_{i}), i = 1, 2$ are continuously differentiable on $[t_{1}, \infty)$ and such that the system (1) is satisfied for $t \geq t_{1}$ .

By using Krasnoselskii’s and Schauder’s fixed point theorems and new techniques we obtained some sufficient conditions for the existence of a nonoscillatory bounded solution of the systems (1). The oscillatory case of the functions $p_{i} (t)$ and $f_{i} (t), i = 1, 2$ , is also possible. The existence of unbounded nonoscillatory solutions of neutral differential systems is treated in [3] and the existence of nonoscillatory solutions of scalar neutral equations in [4], [5] and in papers cited therein.

We will consider the following cases:

$0 < a_{i} (t) \leq a_{i} < 1$ , $1 < a_{i} \equiv a_{i} (t) < \infty$ , $a_{i} (t) \equiv 1$ , $i = 1, 2$ , and their combination.

The following fixed point theorems will be used.

Lemma 1.1 [1], Krasnoselskii’s Fixed Point Theorem

Let $X$ be a Banach space, let $Ω$ be a bounded closed convex subset of $X$ and let $Q, S$ be maps of $Ω$ into $X$ such that $Q x + S y \in Ω$ for every pair $x, y \in Ω$ . If $Q$ is a contractive and $S$ is completely continuous, then the equation $Q x + S x = x$ has a solution in $Ω$ .

Lemma 1.2 [1], [2], Schauder’s Fixed Point Theorem

Let $Ω$ be a closed, convex and nonempty subset of a Banach space $X$ . Let $S : Ω \to Ω$ be a continuous mapping such that $S Ω$ is a relatively compact subset of $X$ . Then $S$ has at least one fixed point in $Ω$ . That is there exists an $x \in Ω$ such that $S x = x$ .

Section snippets

Main results

Theorem 2.1

Suppose that $0 < a_{i} (t) \leq a_{i} < 1$ , $\int_{t_{0}}^{\infty} t^{n - 1} | p_{i} (t) | d t < \infty,$ and $| \int_{t_{0}}^{\infty} t^{n - 1} f_{i} (t) d t | < \infty, i = 1, 2 .$ Then(1)has a nonoscillatory bounded solution.

Proof

With regard to (2), (3) we choose a $T > t_{0}$ sufficiently large that $\frac{1}{(n - 1)!} \int_{T}^{\infty} s^{n - 1} | p_{i} (s) | d s \leq \frac{1 - a_{i}}{5 M_{i}}, \frac{1}{(n - 1)!} | \int_{T}^{\infty} s^{n - 1} f_{i} (s) d s | \leq \frac{1 - a_{i}}{10},$ $i = 1, 2$ , where $M_{i} = {max}_{(2 - 2 a_{i}) / 5 \leq y_{3 - i} \leq 1} {| g_{i} (y_{3 - i}) |}, i = 1, 2$ .

Let $C ([t_{0}, \infty), R^{2})$ be the set of all continuous vector functions $y (t) = (y_{1} (t), y_{2} (t))$ with the norm $‖ y ‖ = {sup}_{t \geq t_{0}} {| y_{1} (t) |, | y_{2} (t) |} < \infty$ . Then $C ([t_{0}, \infty), R^{2})$ is a Banach space. We define a closed, bounded and

Acknowledgements

The research was supported by the grant 1/3238/06 from the Scientific Grant Agency of the Ministry of Education of the Slovak Republic and the Slovak Academy of Sciences.

References (5)

L.H. Erbe et al.
Oscillation Theory for Functional Differential Equations
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I. Györi et al.
Oscillation Theory of Delay Differential Equations with Applications
(1991)

There are more references available in the full text version of this article.

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