S―asymptotically ω―periodic Solutions of R―L Fractional Derivative―Integral Equa

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【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

【Rreference】

[1]D.Araya.C.Lizama，Almost automorphic mild solutions to fractional differential equations[J].Nonlinear Anal.， 2007，10.004.

[2]C.Cuevas， E，Hernandez， psedo-almost periodic solutions for abstract partial functional differential equations[J].Appl. Math. Lett.，（2008）doi：10.1016.

[3]C.Cuevas，M.pinto，Existence and uniqueness of pseudo almost periodic solutions of semilinear Cauchy problems with non dense domain， Nonlinear Anal.，45（2001）73-83.

[4]Diagana.G..M.Almostautomorphic solutions to semilinear evolution equations， [J].Funct.differ.Equ.，13（2） （2006）195-206.

[5]T.diagana.G..M.，N.vanminh， Almost automorphic solutions of evolutions equations[J] Prov.Amer. Math.Soc.，132（2004）3289-3298.

[6]Claudio Cuevas， Julio Cesar de souza. S-asymptotically ω-periodic solutions of semilinear fractional integral-differential equations[J].Applied Mathematics Letters， 22 （2009）： 865-870.