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On the Existence of Locally Attractive Solutions of a Nonlinear Quadratic Volterra Integral Equation of Fractional Order
Advances in Difference Equations volume 2010, Article number: 127093 (2010)
Abstract
The authors employs a hybrid fixed point theorem involving the multiplication of two operators for proving an existence result of locally attractive solutions of a nonlinear quadratic Volterra integral equation of fractional (arbitrary) order. Investigations will be carried out in the Banach space of real functions which are defined, continuous, and bounded on the real half axis .
1. Introduction
The theory of differential and integral equations of fractional order has recently received a lot of attention and now constitutes a significant branch of nonlinear analysis. Numerous research papers and monographs have appeared devoted to differential and integral equations of fractional order (cf., e.g., [1–6]). These papers contain various types of existence results for equations of fractional order.
In this paper, we study the existence of locally attractive solutions of the following nonlinear quadratic Volterra integral equation of fractional order:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ1_HTML.gif)
for all and
, in the space of real functions defined, continuous, and bounded on an unbounded interval.
It is worthwhile mentioning that up to now integral equations of fractional order have only been studied in the space of real functions defined on a bounded interval. The result obtained in this paper generalizes several ones obtained earlier by many authors.
In fact, our result in this paper is motivated by the extension of the work of Hu and Yan [7]. Also, We proceed and generalize the results obtained in the papers [8, 9].
2. Notations, Definitions, and Auxiliary Facts
Denote by the space of Lebesgue integrable functions on the interval
, which is equipped with the standard norm. Let
and let
be a fixed number. The Riemann-Liouville fractional integral of order
of the function
is defined by the formula:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ2_HTML.gif)
where denotes the gamma function.
It may be shown that the fractional integral operator, transforms the space
into itself and has some other properties (see [10–12]).
Let be the space of continuous and bounded real-valued functions on
and let
be a subset of
. Let
be an operator and consider the following operator equation in
, namely,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ3_HTML.gif)
for all . Below we give different characterizations of the solutions for the operator equation (2.2) on
. We need the following definitions in the sequel.
Definition 2.1.
We say that solutions of (2.2) are locally attractive if there exists an and an
such that for all solutions
and
of (2.2) belonging to
we have that:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ4_HTML.gif)
Definition 2.2.
An operator is called Lipschitz if there exists a constant
such that
for all
. The constant
is called the Lipschitz constant of
on
.
Definition 2.3 (Dugundji and Granas [13]).
An operator on a Banach space
into itself is called compact if for any bounded subset
of
,
is a relatively compact subset of
. If
is continuous and compact, then it is called completely continuous on
.
We seek the solutions of (1.1) in the space of continuous and bounded real-valued functions defined on
. Define a standard supremum norm
and a multiplication "
" in
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ5_HTML.gif)
Clearly, becomes a Banach space with respect to the above norm and the multiplication in it. By
we denote the space of Lebesgue integrable functions on
with the norm
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ6_HTML.gif)
We employ a hybrid fixed point theorem of Dhage [14] for proving the existence result.
Theorem 2.4 (Dhage [14]).
Let be a closed-convex and bounded subset of the Banach space
and let
be two operators satisfying:
-
(a)
is Lipschitz with the Lipschitz constant
,
-
(b)
is completely continuous,
-
(c)
for all
, and
-
(d)
where
.
Then the operator equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ7_HTML.gif)
has a solution and the set of all solutions is compact in .
3. Existence Result
We consider the following set of hypotheses in the sequel.
(H1) The function is continuous, and there exists a bounded function
with bound
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ8_HTML.gif)
for all and
.
(H2) The function defined by
is bounded with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ9_HTML.gif)
(H3) The function is continuous and
.
(H4) The function is continuous. Moreover, there exist a function
being continuous on
and a function
being continuous on
with
and such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ10_HTML.gif)
for all such that
and for all
.
For further purposes let us define the function by putting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ11_HTML.gif)
Obviously the function is continuous on
.
In what follows we will assume additionally that the following conditions are satisfied.
(H5) The functions defined by the formulas
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ12_HTML.gif)
are bounded on and vanish at infinity, that is,
.
Remark 3.1.
Note that if the hypotheses and
hold, then there exist constants
and
such that:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ13_HTML.gif)
Theorem 3.2.
Assume that the hypotheses hold. Furthermore, if
, where
and
are defined in Remark 3.1, then (1.1) has at least one solution in the space
. Moreover, solutions of (1.1) are locally attractive on
.
Proof.
Set . Consider the closed ball
in
centered at origin 0 and of radius
, where
.
Let us define two operators and
on
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ14_HTML.gif)
for all .
According to the hypothesis , the operator
is well defined and the function
is continuous and bounded on
. Also, since the function
is continuous on
, the function
is continuous and bounded in view of hypothesis
. Therefore
and
define the operators
. We will show that
and
satisfy the requirements of Theorem 2.4 on
.
The operator is a Lipschitz operator on
. In fact, let
be arbitrary. Then by hypothesis
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ15_HTML.gif)
for all . Taking the supremum over
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ16_HTML.gif)
for all . This shows that
is a Lipschitz on
with the Lipschitz constant
.
Next, we show that is a continuous and compact operator on
. First we show that
is continuous on
. To do this, let us fix arbitrary
and take
such that
. Then we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ17_HTML.gif)
Since is continuous on
, then it is bounded on
, and there exists a nonnegative constant, say
, such that
. Hence, in view of hypothesis
, we infer that there exists
such that
for
. Thus, for
we derive that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ18_HTML.gif)
Furthermore, let us assume that . Then, evaluating similarly to the above we obtain the following estimate:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ19_HTML.gif)
where ,  
,  
.
Therefore, from the uniform continuity of the function on the set
we derive that
as
. Hence, from the above-established facts we conclude that the operator
maps the ball
continuously into itself.
Now, we show that is compact on
. It is enough to show that every sequence
in
has a Cauchy subsequence. In view of hypotheses
and
, we infer that:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ20_HTML.gif)
for all . Taking the supremum over
, we obtain
for all
. This shows that
is a uniformly bounded sequence in
. We show that it is also equicontinuous. Let
be given. Since
, there is constant
such that
for all
.
Let be arbitrary. If
, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ21_HTML.gif)
From the uniform continuity of the function on
and the function
in
, we get
as
.
If , then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ22_HTML.gif)
as .
Similarly, if with
, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ23_HTML.gif)
Note that if , then
and
. Therefore from the above obtained estimates, it follows that:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ24_HTML.gif)
As a result, as
. Hence
is an equicontinuous sequence of functions in
. Now an application of the Arzelá-Ascoli theorem yields that
has a uniformly convergent subsequence on the compact subset
of
. Without loss of generality, call the subsequence of the sequence itself.
We show that is Cauchy sequence in
. Now
as
for all
. Then for given
there exists an
such that for
, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ25_HTML.gif)
This shows that is Cauchy. Since
is complete, then
converges to a point in
. As
is closed,
converges to a point in
. Hence,
is relatively compact and consequently
is a continuous and compact operator on
.
Next, we show that for all
. Let
be arbitrary, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ26_HTML.gif)
for all . Taking the supremum over
, we obtain
for all
. Hence hypothesis
of Theorem 2.4 holds.
Also we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ27_HTML.gif)
and therefore . Now we apply Theorem 2.4 to conclude that (1.1) has a solution on
Finally, we show the local attractivity of the solutions for (1.1). Let and
be any two solutions of (1.1) in
defined on
, then we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ28_HTML.gif)
for all . Since
,
and
, for
, there are real numbers
,
and
such that
for
,
for all
and
for all
. If we choose
, then from the above inequality it follows that
for
, where
. This completes the proof.
4. An Example
In this section we provide an example illustrating the main existence result contained in Theorem 3.2.
Example 4.1.
Consider the following quadratic Volterra integral equation of fractional order:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ29_HTML.gif)
where .
Observe that the above equation is a special case of (1.1). Indeed, if we put and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ30_HTML.gif)
Then we can easily check that the assumptions of Theorem 3.2 are satisfied. In fact, we have that the function is continuous and satisfies assumption
, where
and
as in assumption
. We have that the function
is continuous and it is easily seen that
as
, thus assumption
is satisfied. Next, let us notice that the function
satisfies assumption
, where
,
and
. Thus
. To check that assumption
is satisfied let us observe that the functions
appearing in that assumption take the form:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F127093/MediaObjects/13662_2010_Article_1244_Equ31_HTML.gif)
Thus it is easily seen that as
. Finally, let us note that in Remark 3.1 there are two constants
such that
. It is also easy to check that
,
and
. Then
. Hence, taking into account that
(cf. [4]), all the assumptions of Theorem 3.2 are satisfied and (4.1) has a solution in the space
. Moreover, solutions of (4.1) are uniformly locally attractive in the sense of Definition 2.1.
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Abbas, M. On the Existence of Locally Attractive Solutions of a Nonlinear Quadratic Volterra Integral Equation of Fractional Order. Adv Differ Equ 2010, 127093 (2010). https://doi.org/10.1155/2010/127093
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DOI: https://doi.org/10.1155/2010/127093