S-asymptoticallyω-periodic Solutionsof R-L Fractional Derivative-Integral Equation

WANG Bing

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

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，Mffgt;0，μ∈R such that its resolvent exist outside the sector

Then we call A is a sectorial of typeμ.

Definition 1.2 ([6])Let A be a closed and linear operator with domain D（A）defined on a Banach space X.If there existμ∈R and a strongly continuous function Sα（t）:R+→B（X）such that

Itispointed out herethat Sα（t）:R+→B（X）satisfiestheestimatewhich will be used in the proof of Theorem 1.

Definition 1.3 ([6])Let A generatesan integrablesolution operator Sα（t）.A function u（t）∈SAPω（X）is said to be a mild solutions of equation(1)and(2)if u（t）is a S-asymptoticallyω-periodic function and satisfies

Definition 1.4 ([6])A Continuous bounded function f is called Sasymptoticallyω-periodic,if there exists a constantωffgt;0 such that

In this paper the notation SAPω（x）stands for the subspace of Cb（[0，+∞），X）consisting of the S-asymptotically ω-periodic functions,where the notation Cb（[0，+∞），X）stands for the space consisting of bounded continuous function.

2 Main results

Theorem 1 Let A is sectorial of typeμ＜0,f be a continuous function and integrable function on[0，+∞）,there exist constant L such that

||f（x）-f（y）||≤L||x-y||，

if 2L|μ|-1/απ＜αsin（π/α）,then the equation(1)and(2)has a unique S-asymptoticallyω-periodic mild solution.

Proof.Suppose that u（t）∈SAPω（X）,define the operator Γαon space SAPω（X）:

On the other hand,it is easy to derive that||Γα（u1（t））-Γα（u2（t））||≤2L｜μ|-1/απαsin（π/α）||u1-u2||∞

ObviouslyΓαis a contraction map since 2L｜μ|-1/απ＜αsin（π/α）.Therefore,Γαhas a fixed point which solves(1)and(2).

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