Controllability for a new class of fractional neutral integro-differential evolution equations with infinite delay and nonlocal conditions

Jun Du,Wei Jiang,Azmat Ullah Khan Niazi,Denghao Pang

doi:10.1186/s13662-017-1182-6

Abstract

In this paper, we apply the fractional calculus and a suitable fixed point theorem with the measure of noncompactness to give the sufficient conditions of the controllability for a new class of fractional neutral integro-differential evolution systems with infinite delay and nonlocal conditions. The results are obtained here under some weakly noncompactness conditions. Thus they improve and generalize many well-known results. At the end of this paper, two examples are given to explain our abstract conclusions.

Highlights

1 Introduction In the last two decades, the theory of fractional differential equations have become an active area of investigation due to their applications in many fields such as viscoelasticity, electrochemistry, control, porous media, electromagnetic, etc
Liang and Yang [ ] presented weakly controllable conditions for the fractional evolution system with nonlocal initial conditions. Inspired by these facts and [, ], in this manuscript we consider the controllability for a new class of fractional neutral integro-differential evolution systems with infinite delay and nonlocal initial conditions, CDq[x(t) – g(t, xt)] + A[x(t) – g(t, xt)] = f (t, xt, x(t)) + Bu(t), t ∈ [, a], x( ) =
In Section, by the Mönch fixed point theorem, we prove the exact controllability of fractional neutral integro-differential evolution equations with nonlocal conditions and infinite delay

Summary

Introduction

In the last two decades, the theory of fractional differential equations have become an active area of investigation due to their applications in many fields such as viscoelasticity, electrochemistry, control, porous media, electromagnetic, etc. (see [ – ]). Ravichandran and Baleanu [ ] investigated the controllability of fractional functional integro-differential systems with an infinite delay in Banach spaces by means of fixed point theorem and phase space theory.

Results

Conclusion