S-asymptotically -periodic Solutions of R-L Fractional Derivative-Integral Equation

WANG Bing

(School of science, Tianjin, University of Technology and Education, Tianjin 300222, China)

【Abstract】The aim of this paper is to study the S-asymptotically ω-periodic solutions of R-L fractional derivative-integral equation:

where １＜α＜２, Ａ:Ｄ(Ａ)Ｘ→Ｘ is a linear densely defined operator of sectorial type on a completed Banach space Ｘ, f is a continuous function satisfying a suitable Lipschitz type condition. We will use the contraction mapping theory to prove problem (1) and (2) has a unique S-asymptotically ω-periodic solution if the function f satisfies Lipshcitz condition.

【Key words】S-asymptotically ω-periodic mild solutions; R-L Fractional Derivative-Integral; Equation

0　Introduction

Fractional order calculus is the theory of arbitrary order differential and integral, it is unified with the integer order differential and integral calculus, is a generalization of the fractional calculus. When the proposed integer order differential and integral calculus, fractional order calculus is also the inevitable is put forward. The score is not only a rational number, also can be irrational fraction, to some extent, the fractional order calculus can be calculated into integer order differential and integral calculus. Fractional order differential equations with deep physical background and rich theoretical connotation, refers to the fractional order differential equation with fractional order derivative or fractional integral equation. The fractional order derivative and fractional integrals in the physical, biological, chemical, and other disciplines has been widely used, such as dynamic systems with chaotic behavior, chaotic dynamic system, a complex material or the dynamics of porous media, such as randomwalk with memory[1-5].

1　Preliminaries

To state our main result, we need some elimmentary deffinitions.

Definition 1.1　([6]) Let A be a closed and linear operator. If there exist 0＜θ＜π/2，M＞0，μ∈R such that its resolvent exist outside the sector

On the other hand, it is easy to derive that

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