Existence of positive S-asymptotically periodic solutions of the fractional evolution equations in ordered Banach spaces

Qiang Li,Lishan Liu,Mei Wei

doi:10.15388/namc.2021.26.24176

Abstract

In this paper, we discuss the asymptotically periodic problem for the abstract fractional evolution equation under order conditions and growth conditions. Without assuming the existence of upper and lower solutions, some new results on the existence of the positive S-asymptotically ω-periodic mild solutions are obtained by using monotone iterative method and fixed point theorem. It is worth noting that Lipschitz condition is no longer needed, which makes our results more widely applicable.

Highlights

Let (E, · ) be an ordered Banach space, whose positive cone K := {x ∈ E: x θ} is a normal cone with normal constant N, θ is the zero element of E
The S-asymptotically periodic functions have been widely studied in fractional evolution equations, and the existence and uniqueness of S-asymptotically ω-periodic solutions have been well studied
In [35], Shu studied a class of semilinear neutral fractional evolution equations with delay and obtained the existence and uniqueness of the positive S-asymptotically ω-periodic mild solutions by using contraction mapping principle in positive cone

Summary

Introduction

Let (E, · ) be an ordered Banach space, whose positive cone K := {x ∈ E: x θ} is a normal cone with normal constant N , θ is the zero element of E. In [7, 8], by means of the monotone iterative method Chen presented the existence and uniqueness of the positive mild solutions for the abstract fractional evolution equations under certain initial conditions. In [35], Shu studied a class of semilinear neutral fractional evolution equations with delay and obtained the existence and uniqueness of the positive S-asymptotically ω-periodic mild solutions by using contraction mapping principle in positive cone. Without assuming the existence of upper and lower solutions, some new results on the existence of the positive S-asymptotically ω-periodic mild solution are obtained by using monotone iterative method and fixed point theorem.

Preliminaries

Main results

Example