Boundedness and stability in nonlinear delay difference equations employing fixed point theory

In this paper we study stability and boundedness of the nonlinear difference equation x(t+1)=a(t)x(t)+c(t)Δx(t−g(t))+q(x(t),x(t−g(t))). In particular we study equi-boundedness of solutions and the stability of the zero solution of this equation. Fixed point theorems are used in the analysis

Stability in nonlinear neutral differential equations with variable delays using fixed point theory

The purpose of this paper is to use a fixed point approach to obtain asymptotic stability results of a nonlinear neutral differential equation with variable delays. An asymptotic stability theorem with a necessary and sufficient condition is proved. In our consideration we allow the coefficient functions to change sign and do not require bounded delays. The obtained results improve and generalize those due to Burton, Zhang and Raffoul. We end by giving three examples to illustrate our work

Almost Automorphic Solutions of Delayed Neutral Dynamic Systems on Hybrid Domains

We study the existence of almost automorphic solutions of the delayed neutral dynamic system on hybrid domains that are additively periodic. We use exponential dichotomy and prove uniqueness of projector of exponential dichotomy to obtain some limit results leading to sufficient conditions for existence of almost automorphic solutions to neutral system. Unlike the existing literature we prove our existence results without assuming boundedness of the coefficient matrices in the system. Hence, we significantly improve the results in the existing literature. Finally, we also provide an existence result for an almost periodic solutions of the system

A STUDY OF THE STABILITY IN NEUTRAL NONLINEAR DIFFERENTIAL EQUATIONS WITH FUNCTIONAL DELAY VIA FIXED POINTS

In this paper, we use a modification of Krasnoselskii's fixed point theorem introduced by Burton (see b0 Theorem 3) to obtain stability results of the zero solution of the totally nonlinear neutral differential equations with functional delayx′(t)=-a(t)h(x(t-τ(t)))+c(t)x′(t-τ(t))+G(t,x(t),x(t-τ(t))).The stability of the zero solution of this eqution provided that h(0)=G(t,0,0)=0. The Caratheodory condition is used for the function G

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

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