Approximate controllability result for nonlinear impulsive neutral fuzzy stochastic differential equations with nonlocal conditions

S Narayanamoorthy,S Sowmiya

doi:10.1186/s13662-015-0454-2

Abstract

This paper deals with the approximate controllability of impulsive neutral fuzzy stochastic differential equations with nonlocal conditions in a Banach space by using the concept of fuzzy numbers whose values are normal, convex, upper semicontinuous and compact. The hypotheses are obtained by Schauder’s and Banach fixed point theorems. The results are obtained by the evolution operator.

Highlights

In, Zadeh initiated the development of the modified set theory known as fuzzy set theory, which is a tool that makes possible the description of vague notions and manipulations with them
The fuzzy stochastic differential equations could be applicable in the investigation of numerous engineering and economic problems where the phenomena are subjected to randomness and fuzziness simultaneously
Motivated by the above considerations, in this paper we investigate the approximate controllability for nonlinear impulsive neutral fuzzy stochastic differential equations with nonlocal conditions described by d x(t) – h t, x(t) = A x(t) – h t, x(t) + Bu(t) dt + f t, x(t) dt

Summary

Introduction

In , Zadeh initiated the development of the modified set theory known as fuzzy set theory, which is a tool that makes possible the description of vague notions and manipulations with them. Impulsive differential equations have been developed in modeling impulsive problems in physics, population dynamics, ecology, biological systems, biotechnology, industrial robotics, pharmacokinetics, optimal control, and so forth. Associated with this development, a theory of impulsive differential equations has been given extensive attention. Zang and Li [ ] discussed the concept of approximate controllability of fraction impulsive neutral stochastic differential equations with nonlocal conditions. Motivated by the above considerations, in this paper we investigate the approximate controllability for nonlinear impulsive neutral fuzzy stochastic differential equations with nonlocal conditions described by d x(t) – h t, x(t) = A x(t) – h t, x(t) + Bu(t) dt + f t, x(t) dt.

Preliminaries

Conclusion