- Research Article
- Open Access
On the Existence of Locally Attractive Solutions of a Nonlinear Quadratic Volterra Integral Equation of Fractional Order
© Mohamed I. Abbas. 2010
- Received: 19 May 2010
- Accepted: 25 November 2010
- Published: 6 December 2010
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 .
- Banach Space
- Fractional Order
- Existence Result
- Operator Equation
- Attractive Solution
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.
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.
where denotes the gamma function.
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.
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 ).
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 employ a hybrid fixed point theorem of Dhage  for proving the existence result.
Theorem 2.4 (Dhage ).
is Lipschitz with the Lipschitz constant ,
is completely continuous,
for all , and
has a solution and the set of all solutions is compact in .
We consider the following set of hypotheses in the sequel.
for all and .
(H3) The function is continuous and .
for all such that and for all .
Obviously the function is continuous on .
In what follows we will assume additionally that the following conditions are satisfied.
are bounded on and vanish at infinity, that is, .
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 .
Set . Consider the closed ball in centered at origin 0 and of radius , where .
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 .
for all . This shows that is a Lipschitz on with the Lipschitz constant .
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.
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 .
From the uniform continuity of the function on and the function in , we get as .
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.
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 .
for all . Taking the supremum over , we obtain for all . Hence hypothesis of Theorem 2.4 holds.
and therefore . Now we apply Theorem 2.4 to conclude that (1.1) has a solution on
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.
In this section we provide an example illustrating the main existence result contained in Theorem 3.2.
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. ), 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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