Journal of Function Spaces

Volume 2014 (2014), Article ID 192542, 9 pages

http://dx.doi.org/10.1155/2014/192542

## Asymptotically Stable Solutions of a Generalized Fractional Quadratic Functional-Integral Equation of Erdélyi-Kober Type

^{1}Department of Mathematics, Sciences Faculty for Girls, King Abdulaziz University, Jeddah, Saudi Arabia^{2}Department of Mathematics, Faculty of Science, Damanhour University, Damanhour, Egypt^{3}Department of Mathematics, Rzeszów University of Technology, Al. Powstańców Warszawy 8, 35-959 Rzeszów, Poland

Received 7 April 2014; Accepted 22 May 2014; Published 16 June 2014

Academic Editor: Józef Banaś

Copyright © 2014 Mohamed Abdalla Darwish and Beata Rzepka. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We study a generalized fractional quadratic functional-integral equation of Erdélyi-Kober type in the Banach space . We show that this equation has at least one asymptotically stable solution.

#### 1. Introduction

Quadratic integral equations with nonsingular kernels have received a lot of attention because of their useful applications in describing numerous events and problems of the real world. For example, quadratic integral equations are often applicable in kinetic theory of gases, in the theory of neutron transport, and in the traffic theory; see [1–8]. The existence of solutions for several classes of nonlinear quadratic integral equations with nonsingular kernels has been studied by several authors, for example, Argyros [9], Banaś et al. [10–12], Benchohra and Darwish [13, 14], Caballero et al. [15–17], Darwish et al. [18, 19], Leggett [20], and Stuart [21]. There is a great interest in studying singular quadratic integral equations by many authors, after the appearance of Darwish’s paper [22], for example, Banaś and O’Regan [23], Banaś and Rzepka [24, 25], Darwish [26, 27], Darwish and Sadarangani [28], Darwish and Ntouyas [29], Darwish et al. [30], and Wang et al. [31, 32].

In this paper, we will study the quadratic functional-integral equation of fractional order where and .

If and , we obtain a quadratic Urysohn-Volterra integral equation of fractional order studied by Banas’ and O’Regan in [23] while in the case where , , and , we get a fractional quadratic integral equation of Hammerstein-Volterra type studied by Darwish in [22]. Moreover, in the case where , we obtain the quadratic functional-integral equation of fractional order studied by Darwish and Sadarangani in [28].

The aim of this paper is to prove the existence of solutions of (1) in the space of real functions, defined, continuous, and bounded on an unbounded interval. Moreover, we will obtain some asymptotic characterization of solutions of (1). Our proof depends on suitable combination of the technique of measures of noncompactness and the Schauder fixed point principle.

#### 2. Notation and Auxiliary Facts

This section is devoted to collecting some definitions and results which will be needed further on. First, we recall from [33–35] that the Erdélyi-Kober fractional integral of a continuous function is defined as When , we obtain Riemann-Liouville fractional integral; that is,

Now, let be an infinite dimensional Banach space with zero element . Let denote the closed ball centered at with radius . The symbol stands for the ball .

If is a subset of , then and denote the closure and convex closure of , respectively. Moreover, we denote by the family of all nonempty and bounded subsets of and by its subfamily consisting of all relatively compact subsets.

Next we give the definition of the concept of a measure of noncompactness [36].

*Definition 1. *A mapping is said to be a measure of noncompactness in if it satisfies the following conditions. (1)The family is nonempty and .(2).(3).(4) for .(5)If , , for and , then .

The family described above is called the kernel of the measure of noncompactness . Let us observe that the intersection set from belongs to . In fact, since for every, then we have that .

In what follows we will work in the Banach space consisting of all real functions defined, bounded, and continuous on . This space is equipped with the standard norm

Next, we give the construction of the measure of noncompactness in which will be used as main tool of the proof of our main result; see [37, 38] and references therein.

Let us fix a nonempty and bounded subset of and numbers and . For arbitrary function let us denote by the modulus of continuity of the function on the interval ; that is, Further, let us put Moreover, for a fixed number let us define Let us mention that the kernel consists of all nonempty and bounded sets such that functions belonging to are locally equicontinuous on . On the other hand, the kernel is the family containing all nonempty and bounded sets in the space such that the thickness of the bundle formed by the graphs of functions belonging to tends to zero at infinity.

Finally, with the help of the above quantities we can define a measure of noncompactness as The function is a measure of noncompactness in the space [36, 37].

In the end of this section, we recall the definition of the asymptotic stability solutions which will be used in the proof of our main result. To this end we assume that is a nonempty subset of the space . Let be a given operator. We consider the following operator equation:

*Definition 2. *One says that solutions of (9) are asymptotically stable if there exists a ball such that and such that for each there exists such that for arbitrary solutions , of this equation belonging to the inequality is satisfied for any .

#### 3. The Existence and Asymptotic Stability of Solutions

In this section we will study (1) assuming that the following hypotheses are satisfied. is a continuous and bounded function on . is continuous and the function is bounded on with . Moreover, there exists a continuous function such that for all and for any . is continuous and there exists a continuous function such that for all and for any . is a continuous function. Moreover, there exist a function being continuous on and a function being continuous and nondecreasing on with such that for all such that and for all . For further purpose let us define the function by . The functions defined by , , , and are bounded on and the functions and vanish at infinity; that is, . There exists a positive solution of the inequality and , where , , , and .

Now, we are in a position to state and prove our main result.

Theorem 3. *Let the hypotheses be satisfied. Then (1) has at least one solution and all solutions of this equation belonging to the ball are asymptotically stable.*

*Proof. *Denote by the operator associated with the right-hand side of (1). Then, (1) takes the form
where
Here, and are the superposition operators, generated by the functions and involved in (1), defined by
respectively, where is an arbitrary function defined on (see [39]).

Solving (1) is equivalent to finding a fixed point of the operator defined on the space .

For convenience, we divide the proof into several steps.*Step 1* ( is continuous on ). To prove the continuity of the function on it suffices to show that if , then is continuous function on , thanks to , , and . For this purpose, take an arbitrary and fix and . Assume that are such that . Without loss of generality we can assume that . Then we get
Let us denote
then we obtain
Thus
where

In view of the uniform continuity of the function on we have that as . From the above inequality we infer that the function is continuous on the interval for any . This yields the continuity of on and, consequently, the function is continuous on .*Step 2* ( is bounded on ). In view of our hypotheses for arbitrary and for a fixed we have
Hence, is bounded on , thanks to hypothesis . *Step 3* ( maps the ball into itself). Steps and allow us to conclude that the operator transforms into itself. Moreover, from the last estimate we have
From the last estimate with hypothesis we deduce that there exists such that the operator maps into itself.*Step 4* (an estimate of with respect to the quantity ). Let us take a nonempty set . Then, for arbitrary and for a fixed , we obtain
Hence, we can easily deduce the following inequality:
Now, taking into account hypothesis we obtain
where . Obviously, in view of hypothesis we have that .*Step 5* (an estimate of with respect to the modulus of continuity ). Take arbitrary numbers and . Choose a function and take such that . Without loss of generality we can assume that . Then, taking into account our hypotheses and (21), we have
In the last estimates, we have denoted by
Hence,
Since the function is uniformly continuous on the set , the function is uniformly continuous on the set and the function is uniformly continuous on the set , where
we have
It is easy to see that because is bounded on , is bounded on , and .

Therefore, from the last estimate we derive the following one:

Hence we have
*Step 6* ( is contraction with respect to the measure of noncompactness ). From (27) and (34) and the definition of the measure of noncompactness given by formula (8), we obtain
*Step 7.* We construct a nonempty, bounded, closed, and convex set on which we will apply a fixed point theorem. In the sequel let us put , , and so on. In this way we have constructed a decreasing sequence of nonempty, bounded, closed, and convex subsets of such that for . Hence, in view of (35) we obtain
This implies that . Hence, taking into account Definition 1 we infer that the set is nonempty, bounded, closed, and convex subset of . Moreover, . Also, the operator maps into itself.*Step 8* ( is continuous on the set ). Let us fix a number and take arbitrary functions such that . Using the fact that and keeping in mind the structure of sets belonging to we can find a number such that for each and we have that . Since maps into itself, we have that . Thus, for we get
On the other hand, let us assume . Then we obtain
Now, taking into account (37) and (38) and hypothesis we conclude that the operator is continuous on the set .*Step 9* (application of Schauder fixed point principle). Linking all above-obtained facts about the set and the operator and using the classical Schauder fixed point principle we deduce that the operator has at least one fixed point in the set . Obviously the function is a solution of the quadratic integral equation (1). Moreover, since we have that all solutions of (1) belonging to are asymptotically stable in the sense of Definition 2. This completes the proof.

#### 4. Example

In this section, we present an example as an application of Theorem 3.

Consider the following integral equation of fractional order: Equation (39) is a special case of (1), where , , is a positive constant, and

It is easy to check that the assumptions of Theorem 3 are satisfied. In fact we have that the function is continuous and bounded on and .

The function satisfies assumption with and , being .

Moreover, the function satisfies assumption with .

The function satisfies assumption with , , , and .

Next, we are going to check that assumption is satisfied. The functions , , , and appearing in assumption take the form It is easy to see that .

Moreover we have , , and .

Therefore the inequality in assumption has the form We can easily check that the number is a solution of the inequality (43) for . Now, by Theorem 3, we infer that our equation has a solution in and all solutions of (39) which belongs to are asymptotically stable in the sense of the Definition 2.

#### Conflict of Interests

The authors declare that there is no conflict of interests in the submitted paper.

#### References

- V. C. Boffi and G. Spiga, “An equation of Hammerstein type arising in particle transport theory,”
*Journal of Mathematical Physics*, vol. 24, no. 6, pp. 1625–1629, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. T. Kelley, “Approximation of solutions of some quadratic integral equations in transport theory,”
*Journal of Integral Equations*, vol. 4, no. 3, pp. 221–237, 1982. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. C. Boffi and G. Spiga, “Nonlinear removal effects in time-dependent particle transport theory,”
*Zeitschrift für Angewandte Mathematik und Physik*, vol. 34, no. 3, pp. 347–357, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Spiga, R. L. Bowden, and V. C. Boffi, “On the solutions to a class of nonlinear integral equations arising in transport theory,”
*Journal of Mathematical Physics*, vol. 25, no. 12, pp. 3444–3450, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. M. Case and P. F. Zweifel,
*Linear Transport Theory*, Addison-Wesley, Reading, Mass, USA, 1967. View at MathSciNet - S. Chandrasekhar,
*Radiative Transfer*, Oxford University Press, London, UK, 1950. View at MathSciNet - K. Deimling,
*Nonlinear Functional Analysis*, Springer, Berlin, Germany, 1985. View at Publisher · View at Google Scholar · View at MathSciNet - S. Hu, M. Khavanin, and W. Zhuang, “Integral equations arising in the kinetic theory of gases,”
*Applicable Analysis*, vol. 34, no. 3-4, pp. 261–266, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. K. Argyros, “Quadratic equations and applications to Chandrasekhar's and related equations,”
*Bulletin of the Australian Mathematical Society*, vol. 32, no. 2, pp. 275–292, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Banas, J. Rocha, and K. B. Sadarangani, “Solvability of a nonlinear integral equation of Volterra type,”
*Journal of Computational and Applied Mathematics*, vol. 157, no. 1, pp. 31–48, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Banaś, J. Caballero, J. Rocha, and K. Sadarangani, “Monotonic solutions of a class of quadratic integral equations of Volterra type,”
*Computers & Mathematics with Applications*, vol. 49, no. 5-6, pp. 943–952, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Banaś and A. Martinon, “Monotonic solutions of a quadratic integral equation of Volterra type,”
*Computers & Mathematics with Applications*, vol. 47, no. 2-3, pp. 271–279, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Benchohra and M. A. Darwish, “On unique solvability of quadratic integral equations with linear modification of the argument,”
*Miskolc Mathematical Notes*, vol. 10, no. 1, pp. 3–10, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Benchohra and M. A. Darwish, “On quadratic integral equations of Urysohn type in Fréchet spaces,”
*Acta Mathematica Universitatis Comenianae. New Series*, vol. 79, no. 1, pp. 105–110, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Caballero, D. O'Regan, and K. Sadarangani, “On solutions of an integral equation related to traffic flow on unbounded domains,”
*Archiv der Mathematik*, vol. 82, no. 6, pp. 551–563, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Caballero, B. López, and K. Sadarangani, “Existence of nondecreasing and continuous solutions for a nonlinear integral equation with supremum in the kernel,”
*Zeitschrift für Analysis und ihre Anwendungen*, vol. 26, no. 2, pp. 195–205, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Caballero, J. Rocha, and K. Sadarangani, “On monotonic solutions of an integral equation of Volterra type,”
*Journal of Computational and Applied Mathematics*, vol. 174, no. 1, pp. 119–133, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. Darwish, “On solvability of some quadratic functional-integral equation in Banach algebra,”
*Communications in Applied Analysis*, vol. 11, no. 3-4, pp. 441–450, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. Darwish, J. Banaś, and E. O. Alzahrani, “The existence and attractivity of solutions of an Urysohn integral equation on an unbounded interval,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 147409, 9 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - R. W. Leggett, “A new approach to the H-equation of Chandrasekhar,”
*SIAM Journal on Mathematical Analysis*, vol. 7, no. 4, pp. 542–550, 1976. View at Publisher · View at Google Scholar · View at MathSciNet - C. A. Stuart, “Existence theorems for a class of non-linear integral equations,”
*Mathematische Zeitschrift*, vol. 137, pp. 49–66, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. Darwish, “On quadratic integral equation of fractional orders,”
*Journal of Mathematical Analysis and Applications*, vol. 311, no. 1, pp. 112–119, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Banaś and D. O'Regan, “On existence and local attractivity of solutions of a quadratic Volterra integral equation of fractional order,”
*Journal of Mathematical Analysis and Applications*, vol. 345, no. 1, pp. 573–582, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Banaś and B. Rzepka, “On local attractivity and asymptotic stability of solutions of a quadratic Volterra integral equation,”
*Applied Mathematics and Computation*, vol. 213, no. 1, pp. 102–111, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Banaś and B. Rzepka, “Nondecreasing solutions of a quadratic singular Volterra integral equation,”
*Mathematical and Computer Modelling*, vol. 49, no. 3-4, pp. 488–496, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. Darwish, “On monotonic solutions of a singular quadratic integral equation with supremum,”
*Dynamic Systems and Applications*, vol. 17, no. 3-4, pp. 539–549, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. Darwish, “On existence and asymptotic behaviour of solutions of a fractional integral equation,”
*Applicable Analysis*, vol. 88, no. 2, pp. 169–181, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. Darwish and K. Sadarangani, “On existence and asymptotic stability of solutions of a functional-integral equation of fractional order,”
*Journal of Convex Analysis*, vol. 17, no. 2, pp. 413–426, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. Darwish and S. K. Ntouyas, “On a quadratic fractional Hammerstein-Volterra integral equation with linear modification of the argument,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 74, no. 11, pp. 3510–3517, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. Darwish, J. Henderson, and D. O'Regan, “Existence and asymptotic stability of solutions of a perturbed fractional functional-integral equation with linear modification of the argument,”
*Bulletin of the Korean Mathematical Society*, vol. 48, no. 3, pp. 539–553, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Wang, C. Zhu, and M. Fečkan, “Solvability of fully nonlinear functional equations involving Erdélyi-Kober fractional integrals on the unbounded interval,”
*Optimization*, 2014. View at Publisher · View at Google Scholar - J. Wang, C. Zhu, and M. Fečkan, “Analysis of Abel-type nonlinear integral equations with weakly singular kernels,”
*Boundary Value Problems*, vol. 2014, article 20, 2014. View at Publisher · View at Google Scholar - J. A. Alamo and J. Rodríguez, “Operational calculus for modified Erdélyi-Kober operators,”
*Serdica*, vol. 20, no. 3-4, pp. 351–363, 1994. View at Google Scholar · View at MathSciNet - M. A. Darwish and K. Sadarangani, “On Erdélyi—Kober type quadratic integral equation with linear modification of the argument,”
*Applied Mathematics and Computation*, vol. 238, pp. 30–42, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - V. S. Kiryakova and B. N. Al-Saqabi, “Transmutation method for solving Erdélyi-Kober fractional differintegral equations,”
*Journal of Mathematical Analysis and Applications*, vol. 211, no. 1, pp. 347–364, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Banaś and K. Goebel,
*Measures of Noncompactness in Banach Spaces*, vol. 60 of*Lecture Notes in Pure and Applied Mathematics*, Marcel Dekker, New York, NY, USA, 1980. View at MathSciNet - J. Appell, J. Banaś, and N. Merentes, “Measures of noncompactness in the study of asymptotically stable and ultimately nondecreasing solutions of integral equations,”
*Zeitschrift für Analysis und ihre Anwendungen*, vol. 29, no. 3, pp. 251–273, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Banaś, “Measures of noncompactness in the study of solutions of nonlinear differential and integral equations,”
*Central European Journal of Mathematics*, vol. 10, no. 6, pp. 2003–2011, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Appell and P. P. Zabrejko,
*Nonlinear Superposition Operators*, vol. 95 of*Cambridge Tracts in Mathematics*, Cambridge University Press, Cambridge, UK, 1990. View at Publisher · View at Google Scholar · View at MathSciNet