#### El Hadi Ait Dads, Nadia Drisi, Khalil Ezzinbi, Mohamed Ziat, Exponential dichotomy and (u,v)-pseudo almost automorphic solutions for some ordinary differential equations, 2016 (2016), Article ID 6 (29 May 2016)

Full Text: PDF

Abstract

In this paper, we study $(\mu,\nu)$-pseudo almost automorphic solutions for the nonlinear differential equation $x'(t)=A(t)x(t)+f(t,x(t))$ for $t\in \mathbb{R},$ where $A(t)$ is $n\times n$ matrix and $f:\mathbb{R}\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is an $(\mu,\nu)$-pseudo almost automorphic function satisfying the following condition $|f(t,x)-f(t,y)|\leq l(t)h(|x-y|),$ where $l\in L^{p}(\mathbb{R})\cap L^{\infty}(\mathbb{R})\;(1 and $h:[0,+\infty)\rightarrow [0,+\infty)$ is a nondecreasing function continuous on $[0,+\infty[$, verifying  $h(r) for every $r>0$ and $h(0)=0$. The main tools are exponential dichotomy and a fixed point theorem. This work is motivated by the work [1] where there are a number of inaccuracies in the proof of the main result. An example is given to illustrate the theory.