Existence Theorem for a Nonlinear Functional Integral Equation and an Initial Value Problem of Fractional Order in L1(R+) ()
1. Introduction
The class of functional integral equations of various types plays very important role in numerous mathematical research areas. An interesting feature of functional integral equations is its role in the study of many problems of functional differential Equations [1-4].
In this work we study the solvability of the following initial value problem
(1)
where
denotes the fractional derivative of order
of
with
. Such initial value problem of arbitrary order (1) was investigated in [5-7]. To achieve this goal, let us consider the integral equation
(2)
which is different from that studied in [2].
Section 2 contains some basic results. Our main result will be given in Section 3. Solvability of the considered initial value problem will be discussed in Section 4.
2. Basic Concepts
This section is devoted to recall some notations and known results that will be needed in the sequel.
If
is a Lebesgue measurable subset of the set of real numbers
then we use the symbol
to denote the Lebesgue measure of
. Let
be the space of all real functions defined and Lebesgue measurable on the set
. If
then the norm of
is defined as:
![](https://www.scirp.org/html/20-7401268\6c211b77-52fa-4a38-827c-e0a256d84f75.jpg)
when
we will write
instead of ![](https://www.scirp.org/html/20-7401268\9ccf6bf1-0273-49ff-88db-f8bb79b1d7f4.jpg)
2.1. The Superposition Operator
An important operator called the superposition operator can be investigated in the theories of differential integral and functional equations [4,8-10]. It can be defined as follows:
Definition 1. Assume that
satisfies Carathéodory conditions, that is it is measurable in
for any
and continuous in
for almost all
where
. Then for every measurable function
on the interval
we assign the function:
![](https://www.scirp.org/html/20-7401268\4c67a139-3005-432c-ada3-8e8c141aa209.jpg)
The operator F defined in this way is called the superposition operator generated by the function
.
Carathéodory [11] gave the first contribution to the theory of the superposition operator and proved its measurability according to the measurability of
.
We state the following result giving the necessary and sufficient condition so that the superposition operator
generated by
will map continuously
into itself [12].
Theorem 2. Let
satisfy the conditions in Definition 1. The superposition operator
generated by the function
maps continuously the space
into itself if and only if:
![](https://www.scirp.org/html/20-7401268\8b9c4eb5-a1aa-4d79-bf18-835e71f23da4.jpg)
for all
and
, where
is a function that belongs to
and
is a nonnegative constant.
It is known that a real valued continuous function is measurable and that the converse is not necessarily true. However, for the converse we have the following results due to Dragoni [13].
Theorem 3. Let
be a bounded interval and
be a function satisfying Caratheodory conditions. Then for each
there exists a closed subset
of the interval I such that
and
is continuous.
2.2. Volterra Integral Operator
We proceed by recalling some basic facts concerning the linear Volterra integral operator in the Lebesgue space
Suppose
is a given function which is measurable with respect to both variables where
![](https://www.scirp.org/html/20-7401268\7d098ed8-c720-4e1a-beac-28e55c7334a0.jpg)
For an arbitrary function
define Volterra integral operator as follows:
![](https://www.scirp.org/html/20-7401268\7c85c13c-755b-4274-8298-49ef396dbe1d.jpg)
It is well known that if
then it is continuous [4,9].
In general, it is rather difficult to find necessary and sufficient conditions for the function
guaranteeing that the integral operator
transforms the space
into itself. Some special cases of this problem were discussed in [4,14]. In this direction we state the next result [15]:
Theorem 4. Let
be measurable on
and such that
![](https://www.scirp.org/html/20-7401268\d281adf7-bed5-4dc8-8d28-e89b75d1c718.jpg)
Then the Volterra integral operator
generated by
maps (continuously) the space
into itself and the norm
of this operator is majorized by the number
.
Observe that if
is a nonempty and measurable subset of
then we can also consider the linear Volterra integral operator associated with the Lebesgue space ![](https://www.scirp.org/html/20-7401268\969f6ffa-4b78-4ee0-980b-12232776d2c3.jpg)
NamelyIf
where
is a nonempty and measurable subset of
then we extend
to the whole half axis
by putting
for
. Then we can treat
in the usual way. When the operator
transforms
into itself its norm will be denoted by
.
2.3. Measures of Weak Noncompactness
Let us assume that
is an infinite dimensional Banach space with the norm
and the zero element
. Denote by
the family of all nonempty and bounded subsets of
and by
its subfamily consisting of all relatively weakly compact sets. The symbol
stands for the weak closure of a set
and the symbol
will denote the convex closed hull (with respect to the norm topology) of a set
. We denote by
the ball centered at
and of radius
. We write
instead of
In what follows we accept the following definition [16]
Definition 5. A function
is said to be a measure of weak noncompactness if it satisfies the following conditions: The Family 1) The family
is nonempty and is nonempty and ker ![](https://www.scirp.org/html/20-7401268\f08484c8-246b-4db6-a722-8dfb519379ca.jpg)
2) ![](https://www.scirp.org/html/20-7401268\a86d3b9c-5ccd-4b0d-8e69-34696be79217.jpg)
3) ![](https://www.scirp.org/html/20-7401268\93bca2bc-05b4-48e0-ab76-19b62db3f93b.jpg)
4)
for
.
5) ![](https://www.scirp.org/html/20-7401268\1bb8d06a-0dc9-4f7d-864f-d64d1fc9669a.jpg)
And if
then the intersection is nonempty ![](https://www.scirp.org/html/20-7401268\4a1edd19-6f5b-4786-9765-06c6be516b13.jpg)
The family
is said to be the kernel of the measure of weak noncompactness
. Let us observe that the intersection set
from 5) belongs to
. Indeed, since
for every
then we have that
.
We can construct a useful measure of weak noncompactness in the space
that based on the following criterion for weak noncompactness due to Dieudonné [17,18].
Theorem 6. A bounded set
is relatively weakly compact in
if and only if the following two conditions are satisfied:
a) for any
there exists
such that if meas.
Then
for all,
.
b) for any
there is
such that
for any ,
.
Now, for a nonempty and bounded subset
of the space
let us define:
(3)
where
![](https://www.scirp.org/html/20-7401268\7b2747c4-1986-4c1a-a339-288a1fb9e4d9.jpg)
and
![](https://www.scirp.org/html/20-7401268\eacabe42-03a3-4885-93d2-06cdf81f5809.jpg)
It can be shown [17] that the function
is a measure of weak noncompactness in the space
such that
, for any
, where
denotes the De Blasi measure of weak noncompactness in
. Moreover,
.
In our approach we will need the following fixed point theorem due to Schauder.
Theorem 7. Let
be a nonempty, convex, closed, and bounded subset of a Banach space
. Let
be a completely continuous mapping. Then
has at least one fixed point in
.
2.4. Fractional Calculus
The definitions of both differential operator and the integral operator of fractional order are stated as follows [19,20].
Definition 8. Let
The RiemmanLiouville (R-L) fractional integral of the function
of order
is defined as
![](https://www.scirp.org/html/20-7401268\b0e4a244-5c9d-4640-9225-1ef22425e6a9.jpg)
Definition 9. Let
be an absolutely continuous function on
. Then the fractional derivative of order
of
is defined as
![](https://www.scirp.org/html/20-7401268\a77875cb-cedd-40df-a751-6765a3619f3f.jpg)
We state here some results concerning the above mentioned operators:
1) Let
, then i) ![](https://www.scirp.org/html/20-7401268\1d6462a0-f8a0-40ed-a863-672221fcfa27.jpg)
ii) ![](https://www.scirp.org/html/20-7401268\fcc4e9f8-2337-41ff-bdb9-7c76fb5090c3.jpg)
2) The operator
maps
into itself continuously.
3. Existence Theorem
Consider the integral Equation (2) and let
denotes the operator determined by the right hand side of this equation, i.e.,
(4)
where
In fact the operator
can be written as the product
of the linear Volterra operator
![](https://www.scirp.org/html/20-7401268\67910aec-3217-48e5-b3a5-23a04d4e0c1a.jpg)
and the superposition operator
![](https://www.scirp.org/html/20-7401268\ab51c82c-6694-4f46-8208-48487dc68d32.jpg)
Therefore Equation (4) can be written as:
(5)
To establish our main result concerning existence of an integrable solution of Equation (2) we impose suitable conditions on the functions involved in that equation. Namely we assume 1) The functions
satisfy the Caratheodory conditions and there exist functions
and constants
such that
![](https://www.scirp.org/html/20-7401268\d8984f7f-3118-4e4a-856e-b2dc92cd97bf.jpg)
holds for all ![](https://www.scirp.org/html/20-7401268\f7eeffd9-b238-4933-bec1-7eb6136fb77b.jpg)
2) The functions
satisfy the Caratheodory conditions and the linear Volterra operators
associated with
map
into itself.
3)
is increasing, absolutely continuous and there exists a constant
such that
a.e. on
.
4)
.
Now we can state our main result in the next theorem.
Theorem 10. Under the above assumptions the Equation (2) has at least one solution ![](https://www.scirp.org/html/20-7401268\290e2f79-94cf-4efd-9c31-6063426100ea.jpg)
Proof. Since
is a nonlinear operator defined by Equation (5), then based on assumptions i) and ii) if
, then
Moreover, from Equation (5), and noting that
according to our assumptions are indeed bounded, we have
![](https://www.scirp.org/html/20-7401268\4e2bb68f-c99c-4ebc-bdf5-d0af6a36f07b.jpg)
The above estimate shows that the operator
maps
into itself, where
![](https://www.scirp.org/html/20-7401268\8ba0238d-84c5-40c4-9d97-5ed3c6bdf56a.jpg)
Moreover, according to Theorem 2, we deduce that the operator
is continuous on the space
.
Next, to prove that
is a contraction, let
be a nonempty subset of
Fix
and take a measurable subset
such that
. Then for any
, we get
![](https://www.scirp.org/html/20-7401268\bf38d275-310b-42bc-adc7-649c2d2c6a17.jpg)
where the symbol
denotes the operator norm acting from the space
into itself. Also in the above calculation we used the fact that
for
. From the absolute continuity of the function
and the obvious equality
.
and using Theorem 6 we obtain
(6)
Furthermore, fixing
we can deduce that
![](https://www.scirp.org/html/20-7401268\7946c8e1-5252-4ede-ab20-4dbdfdd51d05.jpg)
where the symbol
denotes the operator norm acting from the space
into itself. Now according to the fact that the set consisting of one element is weakly compact, by using Theorem 6 and the formula
![](https://www.scirp.org/html/20-7401268\c343f0d8-6576-4ce7-bb41-6d4f0df3fe04.jpg)
and since
we get
(7)
According to Equation (3), combining (6) and (7), we get
(8)
Put
. Clearly, according to assumption iv)
. Consider the sequence of sets
, where
and so on. Obviously this sequence is decreasing i.e.
for
Moreover,
. Apart from this, all sets belonging to this sequence are closed and convex, so weakly closed. On the other hand in view of inequality (8) we have
![](https://www.scirp.org/html/20-7401268\e7ee95d2-be10-429f-b954-a8299cd4f41a.jpg)
which yields that ![](https://www.scirp.org/html/20-7401268\c9e3bf0e-09dc-4ee0-a3ef-c322e2ae2abb.jpg)
Consequently, by axiom 5) of Definition 5 we infer that the set
![](https://www.scirp.org/html/20-7401268\98213210-30eb-4989-8a82-82197c50a4dc.jpg)
is nonempty, closed, convex and weakly compact (in view of
). Moreover,
.
In the sequel we show that the set
is relatively compact in the set
.
To do this let us take an arbitrary sequence
and fix arbitrarily a number
. Since
is weakly compact, in view of Theorem 6 we deduce that there exists
such that for any natural number
the following inequality is satisfied
(9)
To apply the classical Schauder fixed point theorem, we need to prove that the set
is relatively compact in
. For this aim let us consider the functions
on the set
and the functions
on the set
.
In view of Theorem 3 we can find a closed subset
of the interval
such that
(where
) and such that the functions ![](https://www.scirp.org/html/20-7401268\7f2fed88-053f-4578-b37a-28f5cff1fe5e.jpg)
and
are continuous. Hence we infer that
are uniformly continuous.
In what follows we show that
is an equicontinuous on
, for that let us take arbitrarily
. Without loss of generality we can assume that
. Then, keeping in mind our assumptions, for an arbitrary fixed
we obtain:
![](https://www.scirp.org/html/20-7401268\aa57581c-2819-4bf7-8653-96bc8daf6266.jpg)
where
denotes the modulus of continuity of the function
on the set
and
![](https://www.scirp.org/html/20-7401268\88e8e5cd-7f42-4a84-a144-cf0923bf33ed.jpg)
By rearranging the order of double integrations, we get
![](https://www.scirp.org/html/20-7401268\d8b9f242-3e7e-4bb1-ac59-2dce9651936d.jpg)
From the above estimate and the consideration of the fact that
we obtain
![](https://www.scirp.org/html/20-7401268\64949260-3e93-4462-b1c3-9d9a55002b1e.jpg)
Now, utilizing the fact that the sequence
is weakly compact and taking into account Theorem 6 we can show that the number
![](https://www.scirp.org/html/20-7401268\ebb0aca4-e13d-4bd3-b31a-caff04e32843.jpg)
is arbitrarily small provided the number
is taken to be sufficiently small (it is a consequence of the fact that a one element set is weakly compact in
).
Furthermore,
![](https://www.scirp.org/html/20-7401268\dc14bb53-8bd4-4957-b752-135e6bf392a9.jpg)
Hence
![](https://www.scirp.org/html/20-7401268\5096adb5-bf9f-4d76-8edc-67fed8535240.jpg)
Hence consequently the sequence
is a sequence of uniformly bounded and equicontinuous functions on
. Hence, in view of Ascoli-Arzela theorem we deduce that the sequence
is relatively compact subset in the space
.
Further observe that the above reasoning does not depend on the choice of
. Thus we can construct a sequence
of closed subsets of the interval
such that
as
and such that the sequence
is relatively compact in every space
. Passing to subsequences if necessary we can assume that
is a Cauchy sequence in each space
, for ![](https://www.scirp.org/html/20-7401268\852cf1bf-b167-4a0b-af86-6fc03c8c92b7.jpg)
In what follows, utilizing the fact that the set
is weakly compact, let us choose a number
such that for each closed subset
of the interval
such that
we have
(10)
for any
.
Keeping in mind the fact that the sequence
is a Cauchy sequence in each space
we can choose a natural number
such that ![](https://www.scirp.org/html/20-7401268\31a3bf3a-93dc-46e9-93b8-69aa6da9ac53.jpg)
and for arbitrary natural numbers
the following inequality holds
![](https://www.scirp.org/html/20-7401268\8c9c54e6-acd7-47eb-9557-27f30d27ef20.jpg)
for any
. Obviously without loss of generality we can assume that
.
Now, using the above facts and (10) we obtain
(11)
Finally, from (10) and (11) we get
![](https://www.scirp.org/html/20-7401268\9bec7ed5-2d70-4cb5-82e0-5b2bdf6787af.jpg)
which means that
is a Cauchy sequence in the space
Hence we conclude that the set
is relativelycompact in this space.
In the last step of the proof let us consider the set
In view of the Mazur theorem we infer that the set
is compact in the space
. Moreover, we have that the operator
transforms continuously the set
into itself. Thus the classical Schauder fixed point principle gives that
has at least one fixed point. This proves that there exists at least one
that solves Equation (4).
4. Nonlinear Equation of Convolution Type
Assume that
is an integrable function. For an arbitrary function
set
![](https://www.scirp.org/html/20-7401268\1421f9c6-631a-4e2b-bbd5-c1223c7ed38b.jpg)
This operator
is a linear integral operator of convolution type and maps
into itself continuously.
Now, consider the following condition
![](https://www.scirp.org/html/20-7401268\cef2bc08-f5fb-4a1c-af7c-81f67489abee.jpg)
Then we have the following Corollary
Corollary 11. Let the hypotheses i)-v) are satisfied. Then a nonlinear equation of convolution type
(12)
has at least one integrable solution
.
In the next subsection, we prove an existence theorem for integral equation of fractional order as a special form of Equation (12).
Initial Value Problems of Fractional Order
As a special case of Equation (14), we consider
(13)
where
![](https://www.scirp.org/html/20-7401268\c1a59b03-7f2b-40a3-84a5-f7b21cd08496.jpg)
and
. Equation (13) is an integral equation of fractional order that can be written in the form
(14)
Obviously, Equation (14) has at least one integrable solution
.
Definition 12. By a solution of the initial value problem (1) we mean an absolutely continuous function x satisfies the initial value problem (1).
Theorem 13. Let
and
.
If assumptions i)-iii) and v) are satisfied, then the initial value problem (1) has at least one solution
.
Proof. Let
be a solution of the integral Equation (14). Putting
![](https://www.scirp.org/html/20-7401268\0f805011-b82a-4ddf-b19c-0d6adc854a3a.jpg)
Since
is integrable, then
![](https://www.scirp.org/html/20-7401268\3e8b11c9-97ec-4ded-acc8-ceac662d3379.jpg)
where
. Moreover, the integral
of integrable function
is absolutely continuous then
![](https://www.scirp.org/html/20-7401268\72049aba-ef41-4094-b8b3-80841487524d.jpg)
Then we have,
![](https://www.scirp.org/html/20-7401268\7ce040f4-4096-43f5-8780-cfb6fae13761.jpg)
Furthermore, we obtain
![](https://www.scirp.org/html/20-7401268\d2954107-baa4-4e31-9ab1-907c524fe3c0.jpg)
Consequently, Equation (14) gives
![](https://www.scirp.org/html/20-7401268\13b64d6a-3c94-457b-9dab-7ba9e7cb5968.jpg)
Since
is integrable and absolutely continuous, then
![](https://www.scirp.org/html/20-7401268\a0b6e5c7-3d89-4f18-98b7-aae8bdcb22bc.jpg)
Clearly,
. Hence we deduce that
is an absolutely continuous function satisfies the initial value problem (1). Hence the proof is complete.
5. Conclusion
The existence theorem of functional integrable equation in the space of Lebesgue integrable functions on unbounded interval
is presented and proved. As an application of this theorem, we investigated the existence of solution of the suggested initial value problems of fractional order.