Analytical solutions to fractional evolution equations with almost sectorial operators

Xiao-Li Ding,Bashir Ahmad

doi:10.1186/s13662-016-0927-y

Abstract

In this paper, with the aid of functional analysis, for almost sectorial operators and some fixed point theorems, we study the existence and uniqueness of mild solutions to fractional neutral evolution equations with almost sectorial operators. We also show that mild solutions can become strong and classical solutions under appropriate assumptions. Finally, we present an example to illustrate the applicability of our results.

Highlights

Throughout this paper, by (X, · ) we denote a Banach space
Several elliptic differential operators considered in the spaces of continuous functions or Lebesgue spaces belong to the class of sectorial operators
In Section, we prove the existence and uniqueness of mild solutions to fractional neutral evolution equations (FNEEs) with almost sectorial operators

Summary

Introduction

Throughout this paper, by (X, · ) we denote a Banach space. As usual, for a linear operator A, D(A), R(A), and σ (A) stand for the domain, range, and spectrum of A, respectively. A sectorial operator is a linear operator A in a Banach space whose spectrum lies in a closed sector Sω = {z ∈ C\{ } | | arg z| ≤ μ} ∪ { } for some ≤ ω < π and whose resolvent (z – A)– satisfies the estimate (z – A)– ≤ M|z|– for all z ∈/ Sω. In [ ], the author discussed mild solutions for abstract fractional differential equations with almost sectorial operators and infinite delay. In Section , we introduce some notation, definitions, and basic properties about fractional derivatives and functional analysis associated with almost sectorial operators. The following properties of the Wright-type function (cf [ ]) are useful in establishing the definition of mild solutions to FNEEs with almost sectorial operators.

Complex powers

Conclusions