On generalized boundary value problems for a class of fractional differential inclusions

Irene Benedetti,Valentina Taddei,Valeri Obukhovskii

doi:10.1515/fca-2017-0075

Abstract

Abstract We prove existence of mild solutions to a class of semilinear fractional differential inclusions with non local conditions in a reflexive Banach space. We are able to avoid any kind of compactness assumptions both on the nonlinear term and on the semigroup generated by the linear part. We apply the obtained theoretical results to two diffusion models described by parabolic partial integro-differential inclusions.

Highlights

Due to the more flexibility given by the non-integer derivatives, fractional calculus is an excellent tool for the description of memory and hereditary properties of various materials and processes
Riemann-Liouville fractional derivative, the Caputo derivative of a constant is zero and it allows a physical interpretation of the initial conditions as well as of boundary conditions
We extend to semilinear differential inclusions a recent result obtained in [7] given for fully nonlinear inclusions

Summary

Introduction

Due to the more flexibility given by the non-integer derivatives, fractional calculus is an excellent tool for the description of memory and hereditary properties of various materials and processes. In all quoted results and usually in literature in order to solve fractional differential problems of type (1.3) in an infinite dimensional framework some compactness assumptions are required on the semigroup generated by the nonlinear part, or on the nonlinear term. A regularity assumption in terms of measures of non compactness is required on the non linear term or the linear part is assumed to generate a compact semigroup (or a compact evolution operator) Unlike all those results, by means of a technique based on weak topology and developed in [6], we are able to prove the existence of at least a solution of problem (1.3) avoiding any kind of compactness hypotheses both on the nonlinear term F and on semigroup generated by the linear part

Preliminaries

Existence result