On the integro‐differential equations with reflection

Abstract

In this paper, by developing important properties on the composition of functions with reflection, using some exponential dichotomy properties and an application of the fixed-point theorem, several new sufficient conditions for the existence and the uniqueness of an pseudo almost automorphic solutions with measure for some general-type reflection integro-differential equations. We suppose that the nonlinear part is measure pseudo almost automorphic and in which we distinguish the two constant and variable cases for the Lipschitz coefficients of the functions associated with this part. It is assumed that the linear part of the equation considered admits an exponential dichotomy. Finally, an application is given on the very interesting model of Markus and Yamabe.

Existence Results for Integro-Differential Equations with Reflection

“…Based on composition, completeness, Banach fixed point theorem, and change of variables theorems, we proved two very important results concerning the existence and uniqueness of a single measure paa solution of a new scalar integro-differential system. Compared to previous works, this is first study of oscillations and dynamics of single measure paa solutions for certain integro-differential equations with reflection for the case when bðtÞ 6 ¼ t: Miraoui [30] studied pap solutions with two measures for our equation (3) for the case when K ¼ 0 or h ¼ 0 and bðtÞ ¼ t: Ait Dads et al [9] described equation ( 3) with matrix coefficients for the case when bðtÞ ¼ t: On the other hand, we studied the impact of functions K, f, h and b on the uniqueness of the single measure paa solutions for equation (3). Note that in the special case when b(t) ¼ t, hypotheses (M0) and (h0) are satisfied, therefore the following new results can be deduced from Theorems 1.1 and 1.2.…”