Existence of Asymptotically Almost Automorphic Mild Solutions of Semilinear Fractional Differential Equations

Junfei Cao,Zaitang Huang,Gaston M N’Guérékata

doi:10.1155/2018/8243180

Abstract

This paper is concerned with the existence of asymptotically almost automorphic mild solutions to a class of abstract semilinear fractional differential equations Dtαxt=Axt+Dtα-1Ft,xt,Bxt, t∈R, where 1<α<2, A is a linear densely defined operator of sectorial type on a complex Banach space X and B is a bounded linear operator defined on X, F is an appropriate function defined on phase space, and the fractional derivative is understood in the Riemann-Liouville sense. Combining the fixed point theorem due to Krasnoselskii and a decomposition technique, we prove the existence of asymptotically almost automorphic mild solutions to such problems. Our results generalize and improve some previous results since the (locally) Lipschitz continuity on the nonlinearity F is not required. The results obtained are utilized to study the existence of asymptotically almost automorphic mild solutions to a fractional relaxation-oscillation equation.

Highlights

Combining the fixed point theorem due to Krasnoselskii and a decomposition technique, we prove the existence of asymptotically almost automorphic mild solutions to such problems
The almost periodic function introduced seminally by Bohr in 1925 plays an important role in describing the phenomena that are similar to the periodic oscillations which can be observed frequently in many fields, such as celestial mechanics, nonlinear vibration, electromagnetic theory, plasma physics, engineering, and ecosphere
In [5, 6], the authors gave an important overview about the theory of almost automorphic functions and their applications to differential equations

Summary

Introduction

The almost periodic function introduced seminally by Bohr in 1925 plays an important role in describing the phenomena that are similar to the periodic oscillations which can be observed frequently in many fields, such as celestial mechanics, nonlinear vibration, electromagnetic theory, plasma physics, engineering, and ecosphere. A is a linear densely defined sectorial operator on a complex Banach space X, F : R × X × X 󳨀→ X, and (t, x, y) 󳨀→ F(t, x, y) is a weighted pseudo almost automorphic function in t ∈ R for each x, y ∈ X satisfying suitable conditions. Mophou [63] investigated the existence and uniqueness of weighted pseudo almost automorphic mild solution to the fractional differential equation: Dαt x (t) = Ax (t) + Dαt −1F (t, x (t) , Bx (t)) ,. As application and to illustrate our main results, we will examine some sufficient conditions for the existence of asymptotically almost automorphic mild solutions to the fractional relaxation-oscillation equation given by.

Preliminaries

Asymptotically Almost Automorphic Mild Solutions

Applications