Mild Solutions of Fractional Semilinear Integro-Differential Equations on an Unbounded Interval ()
1. 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 
of the form
(1)
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.
2. Preliminary Tools
In what follows, 
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
The space 
is furnished with the following standard norm
Let us recall two facts:
• The convergence in 
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 
• A subset 
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 [11].
Definition 1 A nonempty subset 
is said to be bounded if the there is a function 
such that 
for each 
and
.
Namely, denote by 
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
It may be shown that the fractional integral operator 
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
.
The following result is well known, one can see Michalski [12]
Lemma 1 For all 
and
.
(2)
Our consideration is based on following Schauder fixed point theorem.
Theorem 1 [13] Let 
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
.
3. Existence of Mild Solutions
The following hypotheses well be needed in the sequel.
• (A)
is a bounded linear operator on 
for each 
and generates a uniformly continuous evolution system 
such that
• 
• (Cf) (i)
satisfies the Caratheodory type conditions, i.e. 
is measurable for 
and 
continuous for a.e.
, (ii) there exists a continuous positive function 
such that
for a.e. 
and all
.
• (Cu) (i) 
is continuous on
• (ii) 
being continuous such that
where 
is continuous and increasing function with
(3)
• (iii) For all positive function 
there exist 
such that
.
where 
Definition 2 [14] A continuous function 
is said to be a mild solution of (0.1) if 
satisfies to
Our main result is given by the following theorem.
Theorem 2 If the Banach space 
is separable.
Assume that the hypotheses 
and 
are satisfied. Then for each
, the problem (0.1) has at least one mild solution 
in
, for 
Proof. Consider the operator 
defined by the formula
(4)
for 
and
. Let
where
and
The estimate (0.3) guarantee the convergence of the integral
. 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
(5)
Obviously, the function 
is continuous, positive and decreasing. In the space 
let us consider the set
(6)
Clearly 
is closed convex of
. Next, let 
Applying assumptions 
(1) and 
(2) we have
(7)
From the estimate (7), we deduce that 
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
Then, keeping in mind that
, we obtain, that there exists 
so big that
(8)
Next, for
, denote 
the operator defined by
For
, we have,
Next, by the Lebesgue dominated convergence theorem and (0.8) we derive that for suitable large 
we have 
this fact proves that 
is continuous on
.
Next, from (**) we see that to prove the compactness of
, we should prove that 
is equicontinuous on 
and 
is relatively compact for each 
and
. For any 
and 
we get,
Thus,
(9)
Observe that for any 
there exists 
such that 
for all
,
and
, 
such that 
Then, by the monotonicity of 
and for all 
, we get
(10)
where
. Keeping in mind the continuity of
, the right-hand side of the above inequality tends to zero as
.
• If
, then we have
(11)
• If
, note that 
implies that 
and
. According to the above results, we have
(12)
converging to 0 as
.
So for
, 
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
.
The last result in this article is to prove the existence of solutions to (0.1) but with the following conditions.
• 
satisfies the Caratheodory type conditions, i.e. 
is measurable for 
and 
continuous for a.e.
• 
for a.e. 
and all
.
Theorem 3 If the Banach space 
is separable. Assume that the hypotheses 
and 
are satisfied. Then for each
, the problem (0.1) has at least one mild solution 
in
, for 
Proof. Define the operator 
by:
(13)
for 
Kipping in mind the result of lemma (0.2), we get, for
.
(14)
Put 
and
Next, define the set
(15)
Then, we have that 
is a self-mapping of
. We omit the proof of continuity 
and 
is relatively compact, because are similar to that in Theorem 2.
4. Example
In this section, we illustrate the main result contained in Theorem 2 by the following quadratic fractional differential equation
(16)
for 
Let 
be a complete probability measure space. Let 
the space of
-measurable maps 
with
Consider the operator
defined by
Put
.
Clearly 
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
Bay using the Jensen’s inequality it is not difficult to see that
To check conditions 
and 
it is enough to take
Let be a positive function 
defined on
.
Thus, on the basis of Theorem 2, we conclude that Equation (4.1) has at least one mild solution in the space
.
5. Acknowledgements
I thank the referee for their invaluable advices, comments and suggestions.