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 
of the form
where 
and 
generates an evolution system
, satisfying:
• 
, where 
denotes the Banach space of bounded linear operators from 
into 
• 
(
is the identity operator in
)• 
for 
• the mapping 
is strongly continuous in 
and 
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 [1-5]. 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 [6-10]. 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
. 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.
will represent a Banach space with norm
. Denote by 
the space of continuous functions
. Now, let us assume thet 
is a given function defined and continuous on the interval 
with real positive values. Denote by
the Banach space consisting of all functions 
defined and continuous on 
with values in the Banach space 
such that
is furnished with the following standard norm
is the uniform convergence in the compact intervals, i.e. 
converge to 
in 
if and only if 
is uniformly convergent to 
on compact subsets of 
is relatively compact if and only if the restrictions to 
of all functions from X form an equicontinuous set for each 
and 
is relatively compact in 
for each 
,where
, See [
is said to be bounded if the there is a function 
such that 
for each 
and
.
the space of real functions defined and Lebesgue integrable on 
and equiped with the standard norm. For 
and for a fixed number 
we define the Riemann-Liouville fractional integral of order 
of the function 
by putting
transforms the space 
into itself and has some other properties (see [6-8], for example). More generally, we can consider the operator 
on the function space 
consisting of real functions being locally integrable over
.
and
.
be a closed convex subset of the Banach space
. Suppose 
and 
is compact (i.e., bounded sets in 
are mapped into relatively compact sets). Then, 
has a fixed point in
.
is a bounded linear operator on 
for each 
and generates a uniformly continuous evolution system 
such that
satisfies the Caratheodory type conditions, i.e. 
is measurable for 
and 
continuous for a.e.
, (ii) there exists a continuous positive function 
such that
and all
.
is continuous on
• (ii) 
being continuous such that
where 
is continuous and increasing function with
there exist 
such that
.
is said to be a mild solution of (0.1) if 
satisfies to
is separable.
and 
are satisfied. Then for each
, the problem (0.1) has at least one mild solution 
in
, for 
defined by the formula
and
. Let
. In the other hand, observe that if 
is nondecreasing function, then the function 
is also nondecreasing on
.Therefore, the function 
is will defined and nondecreasing on 
Next, put
is continuous, positive and decreasing. In the space 
let us consider the set
is closed convex of
. Next, let 
Applying assumptions 
(1) and 
(2) we have
transforms 
into itself. In what follows we show that 
is continuous. To do this, let us fix 
and take arbitrary sequence 
such that 
converge to 
in
. Further, let us fix
. Applying the properties of 
and 
we get
, we obtain, that there exists 
so big that
, denote 
the operator defined by
, we have,
we have 
this fact proves that 
is continuous on
.
, we should prove that 
is equicontinuous on 
and 
is relatively compact for each 
and
. For any 
and 
we get,
there exists 
such that 
for all
,
and
, 
such that 
and for all 
, we get
. Keeping in mind the continuity of
, the right-hand side of the above inequality tends to zero as
.
, then we have
, note that 
implies that 
and
. According to the above results, we have
.
, 
is equicontinuous. Meanwhile, 
is relatively compact because that 
is uniformly bounded. Thus 
is completely continuous on
. By Schauder fixed point theorem, we deduce that 
has a fixed point 
in
.
satisfies the Caratheodory type conditions, i.e. 
is measurable for 
and 
continuous for a.e.
• 
for a.e. 
and all
.
is separable. Assume that the hypotheses 
and 
are satisfied. Then for each
, the problem (0.1) has at least one mild solution 
in
, for 
by:
Kipping in mind the result of lemma (0.2), we get, for
.
and
is a self-mapping of
. We omit the proof of continuity 
and 
is relatively compact, because are similar to that in Theorem 2.
Let 
be a complete probability measure space. Let 
the space of
-measurable maps 
with
.
is densely defined in 
and is the infinitesimal generator of a strongly continuous semigroup
in
. Observe that the above equation is a special case of Equation (1) if we put 
and
and 
it is enough to take
defined on
.
.