Existence of mild solutions for fractional nonautonomous evolution equations of Sobolev type with delay

Haide Gou,Baolin Li

doi:10.1186/s13660-017-1526-5

Abstract

In this paper, we deal with a class of nonlinear fractional nonautonomous evolution equations with delay by using Hilfer fractional derivative, which generalizes the famous Riemann-Liouville fractional derivative. The definition of mild solutions for the studied problem was given based on an operator family generated by the operator pair (A,B) and probability density function. Combining the techniques of fractional calculus, measure of noncompactness, and fixed point theorem with respect to k-set-contractive, we obtain a new existence result of mild solutions. The results obtained improve and extend some related conclusions on this topic. At last, we present an application that illustrates the abstract results.

Highlights

Fractional differential equations have been successfully applied to various fields, for example, physics, engineering, chemistry, aerodynamics, electrodynamics of complex medium, and polymer rheology, and they have been emerging as an important area of investigation in the last few decades; see [ – ]
5 Conclusions In this paper, we deal with a class of nonlinear fractional nonautonomous evolution equations with delay by using the Hilfer fractional derivative, which generalizes the famous Riemann-Liouville fractional derivative
Combining the techniques of fractional calculus, measure of noncompactness, and fixed point theorem with respect to a k-set-contractive operator, we obtain a new result on the existence of mild solutions with the assumption that the nonlinear term satisfies some growth condition and noncompactness measure condition

Summary

Introduction

Fractional differential equations have been successfully applied to various fields, for example, physics, engineering, chemistry, aerodynamics, electrodynamics of complex medium, and polymer rheology, and they have been emerging as an important area of investigation in the last few decades; see [ – ]. ) A strongly continuous operator family {W (t)}t≥ of D(B) to a Banach space E such that {W (t)}t≥ is exponentially bounded, which means that, for any u ∈ D(B), there exist a > and M > such that We introduce the following assumptions: (H ) {W (t)}t≥ is a norm-continuous family for t > and uniformly bounded, that is, there exists M > such that W (t) ≤ M.

Results

Conclusion