Second Order Impulsive Neutral Functional Differential Inclusions

Abstract

Abstract. In this paper, we investigate the existence of solutions of second order im-pulsive neutral functional differential inclusions which the nonlinearity F admits convexand non-convex values. Some results under weaker conditions are presented. Our resultsextend previous ones. The methods rely on a fixed point theorem for condensing mul-tivalued maps and Schaefer’s fixed point theorem combined with lower semi-continuousmultivalued operators with decomposable values. 1. IntroductionIn this paper, we consider the existence results of solutions for the followingsecond-order neutral functional differential inclusions with the form(1.1)p(t)u 0 (t)−Z tt−τ q(s)u(s)ds 0 ∈ F(t,u t ), a.e.t∈ [0,T]\{t 1 ,t 2 ,··· ,t m },(1.2) ∆u| t=t k = I k (u(t −k )), ∆u 0 | t=t k = J k (u(t −k )), k= 1,2,··· ,m,(1.3) u(t) = a(t),t∈ [−τ,0],u 0 (0) = η,where F: [0,T] × C→ P(R n ) is a multi-valued map, C= {ϕ : [−τ,0] → R ; ϕis continuous everywhere except for a finite number of points ˜tat which ϕ(˜t − ) andϕ(˜t