# On the Existence of Locally Attractive Solutions of a Nonlinear Quadratic Volterra Integral Equation of Fractional Order

- MohamedI Abbas
^{1}Email author

**2010**:127093

https://doi.org/10.1155/2010/127093

© Mohamed I. Abbas. 2010

**Received: **19 May 2010

**Accepted: **25 November 2010

**Published: **6 December 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.

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

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]).

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.

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 employ a hybrid fixed point theorem of Dhage [14] for proving the existence result.

Theorem 2.4 (Dhage [14]).

## 3. Existence Result

We consider the following set of hypotheses in the sequel.

(*H*3) The function
is continuous and
.

*H*4) The function is continuous. Moreover, there exist a function being continuous on and a function being continuous on with and such that

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, .

Remark 3.1.

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 .

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 .

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.

## 4. An Example

In this section we provide an example illustrating the main existence result contained in Theorem 3.2.

Example 4.1.

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.

## Authors’ Affiliations

## References

- Babakhani A, Daftardar-Gejji V:
**Existence of positive solutions of nonlinear fractional differential equations.***Journal of Mathematical Analysis and Applications*2003,**278**(2):434-442. 10.1016/S0022-247X(02)00716-3MATHMathSciNetView ArticleGoogle Scholar - El Borai MM, El-Sayed WG, Abbas MI:
**Monotonic solutions of a class of quadratic singular integral equations of Volterra type.***International Journal of Contemporary Mathematical Sciences*2007,**2**(1–4):89-102.MATHMathSciNetGoogle Scholar - El Borai MM, Abbas MI:
**Solvability of an infinite system of singular integral equations.***Serdica Mathematical Journal*2007,**33**(2-3):241-252.MATHMathSciNetGoogle Scholar - El Borai MM, Abbas MI:
**On some integro-differential equations of fractional orders involving Carathéodory nonlinearities.***International Journal of Modern Mathematics*2007,**2**(1):41-52.MATHMathSciNetGoogle Scholar - Samko SG, Kilbas AA, Marichev OI:
*Fractional Integrals and Derivatives: Theory and Applications*. Gordon and Breach, Yverdon, Switzerland; 1993:xxxvi+976.MATHGoogle Scholar - Srivastava HM, Saxena RK:
**Operators of fractional integration and their applications.***Applied Mathematics and Computation*2001,**118**(1):1-52. 10.1016/S0096-3003(99)00208-8MATHMathSciNetView ArticleGoogle Scholar - Hu X, Yan J:
**The global attractivity and asymptotic stability of solution of a nonlinear integral equation.***Journal of Mathematical Analysis and Applications*2006,**321**(1):147-156. 10.1016/j.jmaa.2005.08.010MATHMathSciNetView ArticleGoogle Scholar - Banaś J, O'Regan D:
**On existence and local attractivity of solutions of a quadratic Volterra integral equation of fractional order.***Journal of Mathematical Analysis and Applications*2008,**345**(1):573-582. 10.1016/j.jmaa.2008.04.050MATHMathSciNetView ArticleGoogle Scholar - Rzepka B:
**On attractivity and asymptotic stability of solutions of a quadratic Volterra integral equation of fractional order.***Topological Methods in Nonlinear Analysis*2008,**32**(1):89-102.MATHMathSciNetGoogle Scholar - Kilbas AA, Trujillo JJ:
**Differential equations of fractional order: methods, results and problems. I.***Applicable Analysis*2001,**78**(1-2):153-192. 10.1080/00036810108840931MATHMathSciNetView ArticleGoogle Scholar - Miller KS, Ross B:
*An Introduction to the Fractional Calculus and Fractional Differential Equations*. John Wiley & Sons, New York, NY, USA; 1993:xvi+366.MATHGoogle Scholar - Podlubny I:
*Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering*.*Volume 198*. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.Google Scholar - Dugundji J, Granas A:
*Fixed Point Theory. I, Monografie Matematyczne*.*Volume 61*. Państwowe Wydawnictwo Naukowe, Warsaw, Poland; 1982:209.Google Scholar - Dhage BC:
**Nonlinear functional boundary value problems in Banach algebras involving Carathéodories.***Kyungpook Mathematical Journal*2006,**46**(4):527-541.MATHMathSciNetGoogle Scholar

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