#### Abstract

Here, we prove the existence of -nondecreasing solution to a nonlinear quadratic integral equation of Urysohn type by applying the technique of weak noncompactness. Also, the asymptotic stability of solutions for that quadratic integral equation is studied.

#### 1. Introduction

Integral equations play an important role in many branches of linear and nonlinear functional analysis and their applications in the theory of elasticity, engineering, mathematical physics, and contact mixed problems, and the theory of integral equations is rapidly developing with the help of several tools of functional analysis, topology, and fixed point theory. For details, we refer to [1–23].

Quadratic integral equations often appear in many applications of real world problems, for example, in the theory of radiative transfer, kinetic theory of gases, in the theory of neutron transport, and in the traffic theory (see [12]). The quadratic integral equation can be very often encountered in many applications (see [1, 2, 6–10, 13–26]). However, in most of the previous literature, the main results are realized with the help of the technique associated with the measure of noncompactness. Instead of using the technique of measure of noncompactness, the Tychonoff fixed point theorem is used for some quadratic integral equations [20, 26]. Picard and Adomian decomposition methods are used to compare approximate and exact solutions for quadratic integral equations [13, 19, 22]. Also, nondecreasing solution of a quadratic integral of Urysohn-Stieltjes type is studied in [10].

Let be the class of Lebesgue integrable functions on with the standard norm.

Here, we are concerned with the nonlinear quadratic functional integral equation and we prove the existence of monotonic solutions in by using the technique of measure of noncompactness. The results of this work generalize those obtained in [18]. Finally, the asymptotic stability of solutions for the quadratic integral equation (1) is studied.

#### 2. Preliminaries

In this section, we collect some definitions and results needed in our further investigations. Assume that the function satisfies Carathèodory condition that is measurable in for any and continuous in for almost all . Then, to every function being measurable on the interval , we may assign the function

The operator defined in such a way is called the superposition operator. This operator is one of the simplest and most important operators investigated in the nonlinear functional analysis. For this operator, we have the following theorem due to Krasnosel’skii [3].

Theorem 1. *The superposition operator maps into itself if and only if
**
and , where is a function from and is a nonnegative constant.*

Now, let be a Banach space with zero element and a nonempty bounded subset of . Moreover denote by the closed ball in centered at and with radius . In the sequel, we will need some criteria for compactness in measure; the complete description of compactness in measure was given by Banaś [3], but the following sufficient condition will be more convenient for our purposes (see [3]).

Theorem 2. *Let be a bounded subset of . Assume that there is a family of subsets of the interval (a,b) such that meas for every , and for every , , ; then, the set is compact in measure.*

The measure of weak noncompactness defined by De Blasi [11, 27] is given by

The function possesses several useful properties which may be found in [11].

The convenient formula for the function in was given by Appell and De Pascale (see [27]) as follows: where the symbol stands for Lebesgue measure of the set .

Next, we shall also use the notion of the Hausdorff measure of noncompactness (see [3]) defined by

In the case when the set is compact in measure, the Hausdorff and De Blasi measures of noncompactness will be identical. Namely, we have the following (see [11, 27]).

Theorem 3. *Let be an arbitrary nonempty bounded subset of . If is compact in measure, then .*

Finally, we will recall the fixed point theorem due to Banaś [5].

Theorem 4. *Let be a nonempty, bounded, closed, and convex subset of , and let be a continuous transformation which is a contraction with respect to the Hausdorff measure of noncompactness ; that is, there exists a constant such that for any nonempty subset of . Then, has at least one fixed point in the set .*

#### 3. Existence Theorem

Let the integral operator be defined as

Then, (1) may be written in operator form as where .

Consider the following assumptions.(i) are functions such that . Moreover, the functions satisfy Carathèodory condition (i.e., are measurable in for all and continuous in for all ), and there exist two functions and constants such that Apart from this, the functions and are nondecreasing in both variables.(ii) is such that for , and satisfies Carathéodory condition (i.e., it is measurable in for all and continuous in for almost all ). (iii)There exist a positive constant , a function , and a measurable (in both variables) function such that and the integral operator , generated by the function and defined by maps continuously into on .(iv) is a.e. nondecreasing on for almost all fixed and for each .(v) is continuous. (vi), , are increasing, absolutely continuous functions on , and there exist positive constants , , such that a.e. on .(vii)Let , where .

Now, let be a positive root of the equation and define the set

For the existence of at least one -positive solution of the quadratic functional integral equation (1), we have the following theorem.

Theorem 5. *Let the assumptions (i)–(vii) be satisfied.**If , then the quadratic integral equation (1) has at least one solution which is positive and a.e. nondecreasing on .*

* Proof. *Take an arbitrary ; then, we get
which implies that

From this estimate, we show that the operator maps the ball into itself with

From assumption (vii) we have
which implies that

Then, is positive which implies that is a positive constant.

Now, let denote the subset of consisting of all functions which are positive and a.e. nondecreasing on .

The set is nonempty, bounded, convex, and closed (see [3, page 780]). Moreover, this set is compact in measure (see Lemma 2 in [4, page 63]).

From the assumptions, we deduce that the operator maps into itself. Since the operator is continuous (Theorem 1 in Section 2), then the operator is continuous, and, hence, the product is continuous. Also, is continuous. Thus, the operator is continuous on .

Let be a nonempty subset of . Fix , and take a measurable subset such that meas . Then, for any , using the same reasoning as in [3, 4], we get

Since

we obtain

This implies that
where is the De Blasi measure of weak noncompactness.

Keeping in mind Theorem 3, we can write (22) in the form
where is the Hausdorff measure of noncompactness.

Since , from Theorem 4 follows that is contraction with respect to the measure of noncompactness . Thus, has at least one fixed point in which is a solution of the quadratic functional integral equation.

#### 4. Asymptotic Stability of the Quadratic Integral Equation

We shall show that the solution of the quadratic integral equation (1) is asymptotically stable on .

*Definition 6. *The function is said to be asymptotically stable solution of (1) if for any there exists such that for every and for every other solution of (1),

*Proof. *Let be defined by (16), and consider the following assumptions.) There exist constants and satisfying that
() .

For solutions and of (1) in , by the assumptions () and (), we deduce that
for any , using (), we have
Then,

That is, the solution of (1) is asymptotically stable on . This completes the proof.

#### 5. Applications

As particular cases of Theorem 5, we can obtain theorems on the existence of positive and a.e. nondecreasing solutions belonging to the space of the following quadratic integral equations.(1)If , then we obtain the quadratic integral equation (2)If and , then we obtain the quadratic integral equation (3)If , , , and , then we obtain the quadratic integral equation which was proved by Banaś in [4]. (4)If , then we obtain the functional equation which is the same results proved by Banaś in [3]. (5)If , and , then we obtain the quadratic integral equation which is the same result proved in [16]. (6)If , , and , then we obtain the quadratic integral equation which is the same result proved in [18].

*Example 7. *Let us consider the quadratic integral equation of Urysohn type having the form

This equation represents the Hammerstein counterpart of the famous Chandrasekhar quadratic integral equation which has numerous application (cf. [1, 2, 6, 24]). It arose originally in connection with scattering through a homogeneous semi-infinite plane atmosphere [24].

In case and is a positive constant. Then, (35) has the form

In order to apply our results, we have to impose an additional condition that the so-called “characteristic” function is continuous on .

In this case, , and the assumption (vii) may be reduced to where .

*Example 8. *Consider the following quadratic functional integral equation:

Taking
then we can easily deduce that (i) and , , , and which implies that , and , ; (ii), , and , and then , , and , , , and .

Now, we will calculate .

Then, and .

Thus, all the assumptions of Theorem 5 are satisfied; so, the quadratic functional integral equation (37) possesses at least one solution being positive, a.e. nondecreasing, and integrable in .

#### Acknowledgments

This work is supported by Deanship for Scientific Research, Qassim University. The authors express their gratitude to Deanship for Scientific Research, Qassim University, for their hospitality and their support. The authors are thankful to Professor A. M. A. El-Sayed for his help and encouragement.