EXISTENCE OF SOLUTIONS FOR IMPULSIVE NEUTRAL FUNCTIONAL INTEGRO DIFFERENTIAL EQUATIONS WITH NONLOCAL CONDITIONS

In this paper, by using fractional power of operators and Sadovskii’s fixed point theorem, we study the existence of mild solution for a certain class of impulsive neutral functional integrodifferential equations with nonlocal conditions. The results we obtained are a generalization and continuation of the recent results on this issue.


INTRODUCTION
Impulsive differential equations, that are differential equations involving impulsive effect, appear as a natural description of several real world problems. Many evolution process that have a sudden change in their states such as mechanical systems with impact, biological systems such as heart beats, blood flows, population dynamics, theoretical physics, radiophysics, pharmacokinetics, mathematical economy, chemical technology, electric technology, metallurgy, ecology, industrial robotics, biotechnology process, chemistry, engineering, control theory, medicine and so on. Adequate mathematical models of such processes are systems of differential equations with impulses, see the monographs of Bainov and Simeonov (13), Bainov, Lakshmikantham and Simeonov (14), the papers (10,15) and the references therein.
wave speeds, studied in (19) can be modeled in the form integrodifferential equation of neutral type. For more details on this theory and on its applications we refer to the monographs of Lakshmikantham et al. (14), and Samoilenko and Perestyuk (4) for the case of ordinary impulsive system and for partial differential and for partial functional differential equations with impulses.
The starting point of this paper is the work in papers (11,12). Especially, authors in (12) investigated the existence of solutions for the system. arising from many fields such as electronics, fluid dynamics, biological models, and chemical kinetics. Most of these phenomena cannot be described through classical differential equations. That is Why in recent years they have attracted more and more attention of several mathematicians, physicists and engineers. Impulsive integrodifferential equations has undergone rapid development over the years and played very important role in modern applied mathematical models of real process. Recently, several authors (3,9,11) have investigated the impulsive integrodifferential equations in abstract spaces. We refer to the papers Wang and Wei (20) and Guo (21) and the references cited therein. Particularly, neutral (integro) differential equations arise in many areas of applied mathematics. For instance, the system of heat conduction with finite by using fractional powers of operators and Sadovskii's fixed point theorem. And in (11), authors studied the following impulsive functional integrodifferential equation with nonlocal conditions of the form by using Schaefer's fixed point theorem. Motivated by above mentioned works (11,12), the main purpose of this paper is to prove the existence of mild solutions for the following impulsive neutral functional integrodifferential equations in a Banach space .
(H4) There exist positive constants 1 and 2 such that ∶  is completely continuous and there exist continuous non decreasing functions : + → + such that for each ∈ . = on is called an evolution system if the following conditions hold: For our convenience let us take 0, 1 0 , … , 0 , 0 = 0.
Moreover, there exists constant > 0 such that for a convex, closed and bounded set of , then has fixed point in (where . denotes Kuratowski's measure of noncompactness).

EXISTENCE RESULTS
In order to define the solution of the problem 6 − (8), we consider the following space For each positive integer , let Dividing on both sides by and taking the lower = ∈  ∶ ║ ║ ≤ , 0 ≤ ≤ . limit as → , we get Then for each , is clearly a bounded closed convex set in  . Since by (9) and (H1) the following relation holds: This contradicts (12). Hence for some positive integer ,  .
= 0 0≤ ≤ ║ 1 − 2 ║ Thus, Next we will show that the operator has a fixed point on , which implies eq 6 − (8) has a To this end, we decompose as = 1 + 2 , where the operator 1 , 2 are defined on respectively, by 1 = − , So by assumption 0 < 0 < 1, we see that 1 is contraction.
To prove that 2 is compact, firstly we prove that 2 is continuous on . Let for 0 ≤ ≤ , and we will verify that 1 is a → in , then by H2(i) and H5 , = 1,2, … , is continuous By the dominated convergence theorem, we have contraction while 2 is a compact operator.
It remains to prove that = ( 2 ) ∶ ∈ is relatively compact in . (0) is relatively compact in . Let 0 < ≤ be fixed and 0 < < . For ∈ , we define Then from the compactness of ( , ) for − > 0, we obtain = ( 2, ) ∶ ∈ is relatively compact in for every , 0 < < . Moreover, for every ∈ , we have  Therefore, there ara relatively compact sets arbitrarily close to the set . Hence the set is also relatively compact in .
Thus, by Arzela-Ascoli theorem, 2 is a compact operator. Those arguments enable us to conclude that = 1 + 2 is a condensing map on , and by the fixed point theorem of Sadovskii there exists a fixed point (. ) for on . Therefore, the nonlocal Cauchy problem 6 − (8) has a mild solution and the proof is completed.