Fractional nonlocal impulsive quasilinear multi-delay integro-differential systems

Amar Debbouche

doi:10.1186/1687-1847-2011-5

Abstract

In this article, sufficient conditions for the existence result of quasilinear multi-delay integro-differential equations of fractional orders with nonlocal impulsive conditions in Banach spaces have been presented using fractional calculus, resolvent operators, and Banach fixed point theorem. As an application that illustrates the abstract results, a nonlocal impulsive quasilinear multi-delay integro-partial differential system of fractional order is given. AMS Subject Classifications. 34K05, 34G20, 26A33, 35A05.

Highlights

Many fractional models can be represented by the following system dα u(t) dtα +A(t, u(t))u(t) = f (t, u(t), u(β(t)))t g(t, s, u(s), u(γ (s))) ds, (1:1)u(0) + h(u) = u0, (1:2)u(ti) = Ii(u(ti)), (1:3)in a Banach space X, where 0
The existence results to evolution equations with nonlocal conditions in Banach space was studied first by Byszewski [13,14], subsequently, many authors were pointed in the same field, see reference therein
Much attention has been paid to existence results for the impulsive differential and integrodifferential equations of fractional order in abstract spaces, see Benchohra et al [2,24]

Summary

Introduction

The existence results to evolution equations with nonlocal conditions in Banach space was studied first by Byszewski [13,14], subsequently, many authors were pointed in the same field, see reference therein. The existence of solutions of fractional abstract differential equations with nonlocal initial condition was investigated by N’Guérékata [22] and Li [23].

Results

Conclusion