Mild Solutions of Fractional Semilinear Integro-Differential Equations on an Unbounded Interval

In this paper, we study the existence of mild solutions for fractional semilinear integro-differential equations in an arbitrary Banach space associated with operators generating compact semigroup on the Banach space. The arguments are based on the Schauder fixed point theorem.


Introduction
The purpose of the present paper is to present an alternative approach to the existence of solution of fractional semilinear integro-differential equations in an arbitrary Banach space X of the form is a given function.Differential equations of fractional order have recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering.This equations also serve as an tool for the description of hereditary properties of various materials and processes.For details, see [1][2][3][4][5].The most important problem examined up to now is that concerning the existence of solutions of considered equations.In order to solve (1), many different methods have been applied in the literature.Most of these methods use the notion of a measure of noncompactness in Banach spaces, see [6][7][8][9][10].Such a method can be to apply in this work.The method we are going to use is to reduce the existence of mild solutions of fractional semilinear integro-differential equations of type (0.1) to searching a fixed points of a suitable map on the space tempered by an arbitrary positive real continuous function defined on

 
p t   .In order to prove the existence of fixed points, we shall rely on the Schauder theorem.Moreover, an application to fractional differential equations is provided to illustrate the results of this work.

Preliminary Tools
In what follows, X will represent a Banach space with norm . .Denote by   is a given function defined and continuous on the interval   with real positive values.Denote by the Banach space consisting of all functions

 
x x t  defined and continuous on   with values in the Banach space X such that Let us recall two facts:  The convergence in

 
, is the uniform convergence in the compact intervals, i.e.

C X
is relatively compact if and only if the restrictions to   0,T of all functions from X form an equicontinuous set for each and ( ) > 0 T M t is relatively compact in X for each t   ,where said to be bounded if the there is a function : It may be shown that the fractional integral operator into itself and has some other properties (see [6][7][8], for example).More generally, we can consider the operator I  on the function space  consisting of real functions being locally integrable over .
  The following result is well known, one can see Michalski [12]  Lemma 1 For all 0   and Our consideration is based on following Schauder fixed point theorem.

Existence of Mild Solutions
The following hypotheses well be needed in the sequel.

 (A)  
A t is a bounded linear operator on E for each 0 t  and generates a uniformly continuous evolution system   , ,  satisfies the Caratheodory type conditions, i.e.

 
,.,. f t continuous for a.e.0 t  , (ii) there exists a continuous positive function where where Our main result is given by the following theorem.Theorem 2 If the Banach space X is separable.
Assume that the hypotheses and satisfied.Then for each 0 x X  , the problem (0.1) has at least one mild solution x in , for Consider the operator F defined by the formula The estimate (0.3) guarantee the convergence of the integral   G t .In the other hand, observe that if : is nondecreasing function, then the func-Obviously, the function is continuous, positive and decreasing.In the space let us consider the set is also nondecreasing on   .There- fore, the function is will defined and nondecreasing on Next, put From the estimate (7), that F transforms


. Applying the properties of  and F we get Then, keeping in mind that , we Observe that for any there exists 0 Then, by the m f onotonicity o  and for all and  we have , note that implies that .Accordin e above results, , Then, we have that

Example
In str tain Th ow l d this section, we illu ate the main result con ed in eorem 2 by the foll ing quadratic fractiona ifferential equation : , and Clearly A is dens ined in X ely def and is the infinitesimal generator of a strongly continuous semigroup Let be a positive function  defined on   .
itself.In what follows we show that      is continuous.To do this, let us fix x


, the right-hand side of the above inequality tends to zero as 2 1  is relatively compact, because are similar to that in Theorem 2.
the basis of Theorem 2, we conclude that Equation (4.1) has at least one mild solution in the space [4] L. Byszewski, "Theorems about the Existence and Uniqueness of Solutions of a Semilinear Evolution Nonlocal Cauchy Problem," Journal of Mathematical Analysis and d d , Thus F is completely continuous on