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 ![](https://www.scirp.org/html/7-7401568\306ef420-cfdd-425d-8e10-a163f8d1f7ce.jpg)
•
(
is the identity operator in
)•
for ![](https://www.scirp.org/html/7-7401568\aa87bea2-fc49-4daf-8b95-52f3418a4948.jpg)
• 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
![](https://www.scirp.org/html/7-7401568\4bfd8120-1f63-4c25-9884-bc59e3976bef.jpg)
The space
is furnished with the following standard norm
![](https://www.scirp.org/html/7-7401568\a1c22a1b-070b-40dd-a83a-eb014c6dc785.jpg)
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 ![](https://www.scirp.org/html/7-7401568\48814202-4d9c-4e6a-ab75-d73cf77758a1.jpg)
• 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
![](https://www.scirp.org/html/7-7401568\cd9c5394-3be0-42af-88c2-665facc35d35.jpg)
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
• ![](https://www.scirp.org/html/7-7401568\ce0d4173-a86b-46a7-ba2c-704c99f28d54.jpg)
• (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
![](https://www.scirp.org/html/7-7401568\a2ee4d13-5c75-4b0f-a759-70c626be7236.jpg)
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 ![](https://www.scirp.org/html/7-7401568\1d2dcd5e-4f3e-49e8-915b-139bd2225a5a.jpg)
Definition 2 [14] A continuous function
is said to be a mild solution of (0.1) if
satisfies to
![](https://www.scirp.org/html/7-7401568\c277e5ee-4839-4a4c-9dd4-42459abc36eb.jpg)
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 ![](https://www.scirp.org/html/7-7401568\a4a7cbde-536a-447b-805c-14a2e9ab0b5e.jpg)
Proof. Consider the operator
defined by the formula
(4)
for
and
. Let
![](https://www.scirp.org/html/7-7401568\6842fb17-4cc8-4cb5-acc1-f9570b3b84fb.jpg)
where
![](https://www.scirp.org/html/7-7401568\04712f47-a459-4ce9-98c8-f8bab9f27b47.jpg)
and
![](https://www.scirp.org/html/7-7401568\a9c85627-d71e-4e29-a15e-350a0a79a8e1.jpg)
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
![](https://www.scirp.org/html/7-7401568\24297b07-3933-4393-8ee7-463a42598750.jpg)
Then, keeping in mind that
, we obtain, that there exists
so big that
(8)
Next, for
, denote
the operator defined by
![](https://www.scirp.org/html/7-7401568\aa46ff74-7f05-488b-8d45-7941bc0c9f8c.jpg)
For
, we have,
![](https://www.scirp.org/html/7-7401568\c715bb3f-8cba-4bd6-bf1c-4752aef27b36.jpg)
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,
![](https://www.scirp.org/html/7-7401568\8d2b132b-e6ad-473f-892d-bebf0fad338f.jpg)
Thus,
(9)
Observe that for any
there exists
such that
for all
,
and
,
such that ![](https://www.scirp.org/html/7-7401568\4134c583-fac0-412a-916c-d15f414e30d0.jpg)
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 ![](https://www.scirp.org/html/7-7401568\7fe85679-8eb4-40a3-9ece-cf7325612ba5.jpg)
Proof. Define the operator
by:
(13)
for
Kipping in mind the result of lemma (0.2), we get, for
.
(14)
Put
and
![](https://www.scirp.org/html/7-7401568\860b1ccd-ce40-4fbd-89f4-7444ade680d0.jpg)
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
![](https://www.scirp.org/html/7-7401568\b4cdbefc-595a-42f4-b164-81fa9b09c2e2.jpg)
Consider the operator
![](https://www.scirp.org/html/7-7401568\4c8f8c08-a27e-4b83-b8ec-e27e70ef4b6f.jpg)
defined by
![](https://www.scirp.org/html/7-7401568\1bb4551d-2a64-4873-b7c6-207489c769a2.jpg)
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
![](https://www.scirp.org/html/7-7401568\2025c6b0-f2af-49d5-b0b4-0df128674ac7.jpg)
Bay using the Jensen’s inequality it is not difficult to see that
![](https://www.scirp.org/html/7-7401568\8bd2c03f-e20c-40ac-b00b-1e25ac888422.jpg)
To check conditions
and
it is enough to take
![](https://www.scirp.org/html/7-7401568\89a2103a-3686-4e9c-86c4-e3b1c7303e72.jpg)
Let be a positive function
defined on
.
![](https://www.scirp.org/html/7-7401568\749145c9-74a0-4b99-9491-72da8be35d9a.jpg)
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.