Existence results for Riemann-Liouville fractional neutral evolution equations

Yi-Liang Liu,Jing-Yun Lv

doi:10.1186/1687-1847-2014-83

Abstract

In this paper, by using the fractional power of operators and the theory of measure of noncompactness, we discuss a class of fractional neutral evolution equations with Riemann-Liouville fractional derivative. We establish sufficient conditions for the existence of mild solutions for fractional neutral evolution equations in the cases C 0 semigroup is compact or noncompact. We give an example to illustrate the applications of the abstract results.

Highlights

1 Introduction A strong motivation for studying fractional differential equations comes from the fact that fractional order derivatives and integrals have extensive applications in viscoelasticity, analytical chemistry, electromagnetic, neuron modeling, and biological sciences, and the theory of fractional calculus has attracted great interest from the mathematical science research community
Heymans and Podlubny [ ] demonstrated that it is possible to attribute physical meaning to initial conditions expressed in terms of Riemann-Liouville fractional derivatives or integrals on the field of the viscoelasticity, and such initial conditions are more appropriate than physically interpretable initial conditions
Fractional evolution equations with the Caputo fractional derivative with difference conditions were studied by many authors, but much less is known about the fractional evolution equations with Riemann-Liouville fractional derivative; see [ – ]

Summary

Introduction

A strong motivation for studying fractional differential equations comes from the fact that fractional order derivatives and integrals have extensive applications in viscoelasticity, analytical chemistry, electromagnetic, neuron modeling, and biological sciences, and the theory of fractional calculus has attracted great interest from the mathematical science research community. Heymans and Podlubny [ ] demonstrated that it is possible to attribute physical meaning to initial conditions expressed in terms of Riemann-Liouville fractional derivatives or integrals on the field of the viscoelasticity, and such initial conditions are more appropriate than physically interpretable initial conditions. LDq +x(t) = Ax(t) + (Fx)(t), t ∈ J := ( , a], < q < , (I –+qx)( ) + g(x) = x , where LDα + is the Riemann-Liouville fractional derivative of order q, I –+q is the RiemannLiouville integral of order – q, A is the infinitesimal generator of a C -semigroup {T(t), t ≥ } on a Banach space X. Motivated by the above work, the aim of this paper is to study the existence of solutions for fractional neutral evolution equations with a Riemann-Liouville fractional derivative of the form.

Preliminaries

Existence of mild solutions

Conclusions