Existence of solutions for fractional stochastic impulsive neutral functional differential equations with infinite delay

Haibo Bao,Jinde Cao

doi:10.1186/s13662-017-1106-5

Abstract

This paper addresses a class of fractional stochastic impulsive neutral functional differential equations with infinite delay which arise from many practical applications such as viscoelasticity and electrochemistry. Using fractional calculations, fixed point theorems and the stochastic analysis technique, sufficient conditions are derived to ensure the existence of solutions. An example is provided to prove the main result.

Highlights

It is commonly believed that fractional calculus dates back to
Shen and Lam proved that for fractional-order nonlinear system described by Caputo’s or Riemann-Liouville’s definition, any equilibrium cannot be finite-time stable as long as the continuous solution corresponding to the initial value problem globally exists [ ]
In [ ], Liao et al gave the existence theorem of solutions for fractional impulsive neutral functional differential equations with infinite delay by using the Caputo fractional derivative, Hausdorff ’s measure of noncompactness and the theory of Mönch

Summary

Introduction

It is commonly believed that fractional calculus dates back to. Fractional derivatives supply a powerful tool in describing the memory and hereditary properties of many materials and processes [ , ]. Li et al presented a stability theorem for fractionalorder nonlinear dynamic systems [ ] Dynamical behaviors such as existence and stability are basic problems of fractional differential equations [ – ]. In [ ], Liao et al gave the existence theorem of solutions for fractional impulsive neutral functional differential equations with infinite delay by using the Caputo fractional derivative, Hausdorff ’s measure of noncompactness and the theory of Mönch. Since the real environment is influenced by noise inevitably, it is significant to consider the dynamical properties for a fractional stochastic impulsive neutral functional differential equation with infinite delay, especially for the existence of solutions. Motivated by the above discussions, in this paper we aim to study the existence of solutions for fractional stochastic impulsive neutral functional differential equations with infinite delays.

Preliminaries

Conclusions