Asymptotic Behavior by Krasnoselskii Fixed Point Theorem for Nonlinear Neutral Differential Equations with Variable Delays

Benhadri Mimia

doi:10.5772/intechopen.96040

Abstract

In this paper, we consider a neutral differential equation with two variable delays. We construct new conditions guaranteeing the trivial solution of this neutral differential equation is asymptotic stable. The technique of the proof based on the use of Krasnoselskii’s fixed point Theorem. An asymptotic stability theorem with a necessary and sufficient condition is proved. In particular, this paper improves important and interesting works by Jin and Luo. Moreover, as an application, we also exhibit some special cases of the equation, which have been studied extensively in the literature.

Highlights

For more than one hundred years, Liapunov’s direct method has been very effectively used to investigate the stability problems of a wide variety of ordinary, functional, and partial differential, integro-differential equations
It turns out that the fixed point method is becoming a powerful technique in dealing with stability problems for indeterministic scenes
By the same method of Jin and Luo [14], Ardjouni and Djoudi [6] improved the results of Jin and Luo [14] to the generalized nonlinear neutral differential equation with variable delays of the form x0ðtÞ 1⁄4 ÀaðtÞxðt À τ1ðtÞÞ þ cðtÞx0ðt À τ1ðtÞÞ þ bðtÞGðxσðt À τ2ðtÞÞÞ, t ≥ 0, (8)

Summary

Introduction

For more than one hundred years, Liapunov’s direct method has been very effectively used to investigate the stability problems of a wide variety of ordinary, functional, and partial differential, integro-differential equations. By the same method of Jin and Luo [14], Ardjouni and Djoudi [6] improved the results of Jin and Luo [14] to the generalized nonlinear neutral differential equation with variable delays of the form x0ðtÞ 1⁄4 ÀaðtÞxðt À τ1ðtÞÞ þ cðtÞx0ðt À τ1ðtÞÞ þ bðtÞGðxσðt À τ2ðtÞÞÞ, t ≥ 0, (8). There is a solution xðt, 0, ψÞ of (8) on þ with jxðt, 0, ψÞj ≤ 1: By letting cðtÞ 1⁄4 0 and Gðxσðt À τ2ðtÞÞÞ 1⁄4 xσðt À τ2ðtÞÞ in (8), the present authors [14] have studied, the asymptotic stability and the stability by using Krasnoselskii’s fixed point theorem, under appropriate conditions, of the Eq (4) and improved the results claimed in [9]. An example is provided to illustrate the effectiveness and the merits of the results introduced

Inversion of equation

Example

Conclusion