Solvability of Chandrasekhar ’ s Quadratic Integral Equations in Banach Algebra

In this paper, we prove some results concerning the existence of solutions for some nonlinear functional-integral equations which contain various integral and functional equations that are considered in nonlinear analysis. Our considerations will be discussed in Banach algebra using a fixed point theorem instead of using the technique of measure of noncompactness. An important special case of that functional equation is Chandrasekhar’s integral equation which appears in radiative transfer, neutron transport and the kinetic theory of gases [1].


Introduction
Nonlinear functional integral equations have been discussed in the literature extensively, for a long time.See for example, Subramanyam and Sundersanam [10], Ntouyas and Tsamatos [11], Dhage and O'Regan [12] and the references therein.
Dhage [12] and [13] initiated the study of nonlinear integral equations in a Banach algebra via fixed point techniques instead of using the technique of measure of noncompactness.
Dhage [14] studied the existence of the nonlinear functional integral equation ∫ by using fixed point theorems concerning the nonlinear alternative of Leray-Schauder type which are proved in [14].
Banaś and Sadarangani [15] discussed the existence of solutions for a general NLFIE ( ) x t f t v t s x s s x t g t u t s x s s x t α β = ⋅ ∫ ∫ using the technique of measure of noncompactness in Banach algebra.Also, an existence results for Chandrasekhar's integral equation was deduced.
A fixed point theorem involving three operators in a Banach algebra by blending the Banach fixed point theorem with that Schauder's fixed point principle was proved by B. C. Dhage in [16].The existence of solutions of the equation x AxBx Cx = + are proved in (see [14] [17]- [22], and the references therein).These studies were mainly based on the convexity and the closure of the bounded domain, the Schauder fixed point theorem [13] [14].
In this paper, instead of using the technique of measure of noncompactness in Banach algebra, we shall use Dhage fixed point theorem [20] to prove an existence theorem for a nonlinear functional integral equation An important special case of the functional Equation ( 1) is Chandrasekhar's integral equation Our paper is organized as: In Section 2, we introduce some preliminaries and use them to obtain our main results in Section 3. In Section 4, we provide some examples and special cases that verify our results.In the last section, further existence results has been proved.

Preliminaries
In this section, we collect some definitions and theorems which will be needed in our further considerations.

Let
[ ] ( )  denotes the space of all continuous realvalued functions on J equipped with the norm sup  is a complete normed algebra with respect to this supremum norm.
A normed algebra is an algebra endowed with a norm satisfying the following property, for all , x y X ∈ we have .

x y x y ⋅ ≤ ⋅
A complete normed algebra is called a Banach algebra.
be the class of Lebesgue integrable functions on J with the standard norm.
Definition 1. [20] A mapping : is a totally bounded subset of X for any bounded subset S of X. Again a map : T X X → is completely continuous if it is continuous and totally bounded on X.It is clearly that every compact operator is totally bounded, but the converse may not be true, however the two notions are equivalent on bounded subsets of a Banach space X.
Definition 2. [20] A mapping : A X X → is called  -Lipschitzian if there exists a continuous and nondecreasing function : for all , x y X ∈ where ( ) Sometimes, we call for the function φ to be a D-function of the mapping A on X.In the special case when ( ) [20].Theorem 1. [20] Let S be a closed convex and bounded subset of a Banach algebra X and let , : be three operators such that: 1) A and C are Lipschitzian with constants α and β respectively, 2) B is completely continuous, and, 3) x AxBy Cx x S = + ⇒ ∈ , for all y S ∈ .
Then the operator equation AxBx Cx x + = has a solution whenever

Main Results
The main object of this section is to apply Theorem 1 to discuss the existence of solutions to the functional quadratic integral Equation (4).
Definition 3. By a solution of the quadratic functional integral Equation (1) We mean a function ( ) 1), where ( ) stands for the space of continuous real-valued functions on J.
Consider the following assumptions: 1) : u J × →   satisfies Carathéodory condition (i.e.measurable in t for all x∈ and continuous in x for almost all t J ∈ ).There exist a positive constant k and a function 3) There exist two positive constants 1 L and 2 L satisfying for all t J ∈ and x ∈  .
Theorem 2. Let the assumptions 1)-5) be satisfied.Furthermore, if ( ) + < then the quadratic functional integral equation ( 1) has at least one solution in the space ( ) Consider the mapping A, B and C on ( )  , defined by:

Cx t f t x t =
Then functional integral Equation (1) can be written in the form: Hence the existence of solutions of the FIE (1) is equivalent to finding a fixed point to the operator Equation ( 7) in ( ) , J   .We shall prove that A, B and C satisfy all the conditions of Theorem 1.
Let us define a subset S of ( ) Obviously, S is nonempty, bounded, convex and closed subset of ( ) Then, Tx S ∈ and hence TS S ⊂ .
First.we start by showing that C is Lipschitzian on S. To see that, let , .
( ) Furthermore, let us assume that .

t J ∈
Then, by assumption 4) and Lebesgue dominated convergence theorem, we obtain the estimate: as n → ∞ uniformly on J and hence B is a continuous operator on S into S. Now by 1) and 2) ( ) is a uniformly bounded sequence in ( ) Now, we proceed to show that it is also equi-continuous.Let 1 2 , t t J ∈ (with- out loss of generality assume that 1 2 t t < ), then we have ( ) ( ) , ( )   Hence ( ) B S is relatively compact and consequently B is a continuous and compact operator on S.
Since all conditions of Theorem 1 are satisfied, then the operator T C AB = + has a fixed point in S. 

Examples and Remarks
In this section, we present some examples and particular cases in nonlinear analysis.
As a particular case of Theorem 2, an existence theorem of solutions to the following quadratic integral equation of Chandrasekhar type ( , λ is positive constant) we can obtain theorem on the existence of solutions belonging to the space ( ) The usually existence of solutions of ( 4) is proved under the additional assumption that that the so-called characteristic function φ is an even poly- nomial in s [1].
, then the quadratic in- tegral equation ( 4) has at least one solution in ( ) r ≤ .Therefore, the quadratic integral In our work, we prove the existence of solutions of Equation (4) under much weaker assumptions ( φ need not to be continuous).

Example 4.2:
Equation (1) includes the well known functional equation [24] ( ) We can easily verify that , , f g ψ and u satisfy all the assumptions of Theorem 2.

Further Existence Results
Consider now the quadratic integral equation Also, the existence of solutions for the Equation ( 6) can be proved by a direct application of the following fixed point theorem [25].Theorem 3. Let n be a positive integer, and  be a nonempty, closed, convex and bounded subset of a Banach algebra X. Assume that the operators 3) For each y ∈  , Then, the operator equation

∫
Then the integral Equation ( 6) can be written in the form: ( ) ( ) ( ) we shall show that i A and i B satisfy all the conditions of Theorem 3.
Let us define a subset  of ( ) Obviously,  is nonempty, bounded, convex and closed subset of ( ) , J   .As done before in the proof of Theorem 2 we can get, For every x ∈ we have ( )( ) x y∈  This that i A are a Lipschitz mapping on  with the Lipschitz constants i L .Also, we can prove that the operators i B are con- tinuous and compact operator on  for all t J ∈ and ( ) Since all conditions of Theorem 3 are satisfied, then the operator

Conclusion
In this paper, we proved an existence theorem for some functional-integral equations which includes many key integral and functional equations that arise in nonlinear analysis and its applications.In particular, we extend the class of characteristic functions appearing in Chandrasekhar's classical integral equation from astrophysics and retain existence of its solutions.Finally, some examples and remarks were illustrated.