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 .
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.
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.
2. Notations, Definitions, and Auxiliary Facts
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 2.3 (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 ).
3. Existence Result
We consider the following set of hypotheses in the sequel.
In what follows we will assume additionally that the following conditions are satisfied.
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 . 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 .
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 .
4. An Example
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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