Existence of mild solutions for Sobolev-type Hilfer fractional evolution equations with boundary conditions

Haide Gou,Baolin Li

doi:10.1186/s13661-018-0965-3

Abstract

This paper is concerned with the fractional differential equations of Sobolev type with boundary conditions in a Banach space. With the help of the properties of Hilfer fractional calculus, the theory of propagation families as well as the theory of the measure of noncompactness and fixed point methods, we obtain the existence results of mild solutions for Sobolev-type fractional evolution differential equations involving the Hilfer fractional derivative. Finally, an example is presented to illustrate the main result.

Highlights

In the last decades, fractional calculus and fractional differential equations have attracted much attention; we refer to [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17] and the references therein
Fractional differential equations have been applied to various fields successfully, for example, physics, engineering, chemistry, aerodynamics, electrodynamics of a complex medium, polymer rheology, and they have been emerging as an important area of investigation in the last few decades; see [1, 18, 21,22,23]
It is a development in the theory and application of fractional differential equations with the Riemann–Liouville fractional derivative or the Caputo fractional derivative, see [24,25,26,27,28,29,30,31] and the references therein

Summary

Introduction

Fractional calculus and fractional differential equations have attracted much attention; we refer to [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17] and the references therein. The existence result of mild solutions of fractional integrodifferential equations of Sobolev type with nonlocal condition in a separable Banach space was studied by using the theory of propagation families as well as the theory of the measures of noncompactness and the condensing maps [6]. Motivated by the above discussion, in this paper, we use fixed point theorems combined with the theory of propagation families to discuss the existence of mild solutions for existence results for Hilfer fractional evolution differential equations of Sobolev type with boundary conditions of the form. We say the pair (A, B) is the generator of an μ-resolvent family, if there exist a ≥ 0 and a strongly continuous function Kμ : [0, ∞) → B(E) such that Kμ(t) is exponentially bounded, {λμ : Re λ > a} ⊂ ρ(A), and for all x ∈ E, λμB – A –1Bu = e–λtKμ(t)u dt, Re λ > a. Lemma 2.12 ([34]) Assume that {W (t)}t≥0 is a norm continuous family for t > 0 and W (t) ≤ M, {Kμ(t)}t>0 and {Sν,μ(t)}t>0 are strongly continuous for t > 0

Main results

Conclusions