Asymptotic Stability of Solutions to a Nonlinear Urysohn Quadratic Integral Equation

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 L 1 = L 1 [0, T] be the class of Lebesgue integrable functions on I = [0, T] with the standard norm. Here, we are concerned with the nonlinear quadratic functional integral equation


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 .
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][7][8][9][10][13][14][15][16][17][18][19][20][21][22][23][24][25][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].

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].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]).
The measure of weak noncompactness defined by De Blasi [11,27] is given by  () = inf ( > 0; there exists a weakly compact subset The function () possesses several useful properties which may be found in [11].
The convenient formula for the function () in  1 was given by Appell and De Pascale (see [27]) as follows: where the symbol meas  stands for Lebesgue measure of the set .
Next, we shall also use the notion of the Hausdorff measure of noncompactness  (see [3]) defined by  () = inf ( > 0; there exists a finite subset  of  such that  ⊂  +   ) .(6) 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  1 .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  ∈ [0,1) such that () ≤ () for any nonempty subset  of .Then,  has at least one fixed point in the set .
(iii) There exist a positive constant  3 , a function  3 ∈  1 , and a measurable (in both variables) function (, ) =  :  ×  →  + such that and the integral operator , generated by the function  and defined by maps continuously  1 into  ∞ on .
Now, let  be a positive root of the equation and define the set For the existence of at least one  1 -positive solution of the quadratic functional integral equation ( 1), we have the following theorem.
, then the quadratic integral equation (1) has at least one solution  ∈  1 which is positive and a.e.nondecreasing on .
Proof.Take an arbitrary  ∈  1 ; then, we get which implies that 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  > 0, and take a measurable subset  ⊂  such that meas  ≤ .Then, for any  ∈ , using the same reasoning as in [3,4], we get 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 ( 1 / 1 ) + ( 2 || 3 ||  1 / 2 ) + ( 2  3 / 2  3 ) < 1, 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.

Asymptotic Stability of the Quadratic Integral Equation
We shall show that the solution of the quadratic integral equation ( 1) is asymptotically stable on R + .Definition 6.The function  is said to be asymptotically stable solution of (1) if for any  > 0 there exists   =   () > 0 such that for every  ≥   and for every other solution  of (1), Proof.Let  be defined by ( 16), and consider the following assumptions.

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  1 () of the following quadratic integral equations. ( which is the same result proved in [18]. 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 semiinfinite plane atmosphere [24].
In case () = 1 and (, , ()) = ()(),  is a positive constant.Then, (35) has the form  () = 1 +  () ∫ In order to apply our results, we have to impose an additional condition that the so-called "characteristic" function  is continuous on .
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 [0, 1].

Theorem 1 .
The superposition operator  maps  1 into itself if and  ∈ , where () is a function from  1 and  is a nonnegative constant.
is positive which implies that  is a positive constant.Now, let   denote the subset of   ∈  1 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]).