Functional integral and differential equations of different types play an im- portant and a fascinating role in nonlinear analysis and finding various ap- plications in describing of several real world problems         .
Nonlinear functional integral equations have been discussed in the literature extensively, for a long time. See for example, Subramanyam and Sundersanam  , Ntouyas and Tsamatos  , Dhage and O’Regan  and the references therein.
Dhage  and  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  studied the existence of the nonlinear functional integral equation (in short FIE)
by using fixed point theorems concerning the nonlinear alternative of Leray- Schauder type which are proved in  .
Banaś and Sadarangani  discussed the existence of solutions for a general NLFIE
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  . The existence of solutions of the equation
are proved in (see   -  , and the references therein). These studies were mainly based on the convexity and the closure of the bounded domain, the Schauder fixed point theorem   .
In this paper, instead of using the technique of measure of noncompactness in Banach algebra, we shall use Dhage fixed point theorem  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
which appears in in radiative transfer, neutron transport and the kinetic theory of gases    .
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.
In this section, we collect some definitions and theorems which will be needed in our further considerations.
Let and denotes the space of all continuous real- valued functions on J equipped with the norm . Clearly, 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 we have
A complete normed algebra is called a Banach algebra.
Let be the class of Lebesgue integrable functions on J with the standard norm.
Definition 1.  A mapping is called totally bounded if is a totally bounded subset of X for any bounded subset S of X. Again a map 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.  A mapping is called -Lipschitzian if there exists a continuous and nondecreasing function satisfying
for all where .
Sometimes, we call for the function to be a D-function of the mapping A on X. In the special case when , A is called a Lipschitz constant . Obsviously, every Lipschitzian mapping is D-Lipschitzian. In particular if , A is called a contraction with a contraction constant . Further, if then A is called nonlinear contraction on X  .
Theorem 1.  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 and respectively,
2) B is completely continuous, and,
3) , for all .
Then the operator equation has a solution whenever , where .
3. 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 that satisfies Equation (1), where stands for the space of continuous real-valued functions on J.
Consider the following assumptions:
1) satisfies Carathéodory condition (i.e. measurable in t for all and continuous in x for almost all ). There exist a positive constant k and a function such that:
2) are continuous and bounded with respectively.
3) There exist two positive constants and satisfying
for all and .
4) is continuous for all and . Moreover,
5) There exists a constant satisfying
for all and .
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:
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 . We shall prove that A, B and C satisfy all the conditions of Theorem 1.
Let us define a subset S of by
Obviously, S is nonempty, bounded, convex and closed subset of .
For every since then we have
Then, and hence .
First. we start by showing that C is Lipschitzian on S. To see that, let So
for all Taking supremum over t
for all This shows that C is a Lipschitzian mapping on S with the Lipschitz constant .
By a similar way we can deduce that
for all This shows that A is a Lipschitzizan mapping on S with the Lipschitz constant .
Secondly, we show that B is continuous and compact operator on S. First we show that B is continuous on S. To do this, let us fix arbitrary and let be a sequence of point in S converging to point Then we get
Furthermore, let us assume that Then, by assumption 4) and Lebesgue dominated convergence theorem, we obtain the estimate:
for all Thus, as uniformly on J and hence B is a continuous operator on S into S. Now by 1) and 2)
for all Then for all This shows that is a uniformly bounded sequence in .
Now, we proceed to show that it is also equi-continuous. Let (with- out loss of generality assume that ), then we have
Then, we obtain
As a consequence, as . This shows that is an equicontinuous sequence in S. Now an application of Arzela-Ascoli theorem yields that has a uniformly convergent subsequence on the the compact subset J. without loss of generality, call the subsequence it self. We can easily show that is Cauchy in S.
Hence 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 has a fixed point in S. +
4. 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
As a particular case of Theorem 2 (when and , is positive constant) we can obtain theorem on the existence of solutions belonging to the space for the quadratic integral equation
The usually existence of solutions of (4) is proved under the additional as- sumption that that the so-called characteristic function is an even poly- nomial in s  .
If is a function in and , then the quadratic in- tegral equation (4) has at least one solution in .
In case of Then and . Therefore, the quadratic in-
has at least one solution in
In our work, we prove the existence of solutions of Equation (4) under much weaker assumptions ( need not to be continuous).
Equation (1) includes the well known functional equation 
Example 4.3: For Then Equation (1) has reduced to the form
Example 4.4: For and Then Equation (1) has the form
Example 4.5: Consider the quadratic integral equation
We can easily verify that and satisfy all the assumptions of Theorem 2.
5. 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  .
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 and satisfy
1) For each , is D- Lipschitzian with a D-function ;
2) For each , is continuous and is precompact;
3) For each , implies that .
Then, the operator equation has a solution provided that
Equation (6) is investigated under the assumptions:
1) satisfy Carathéodory condition (i.e. measurable in t for all and continuous in x for almost all ) such that:
and for all such that
2) are continuous and bounded
3) There exist constants satisfying
for all and
Theorem 4. Let the assumptions 1)-3) be satisfied. Furthermore, if
then the general quadratic integral equation
(6) has at least one solution in the space .
Consider the mapping and on defined by:
Then the integral Equation (6) can be written in the form:
we shall show that and satisfy all the conditions of Theorem 3.
Let us define a subset of by
Obviously, is nonempty, bounded, convex and closed subset of .
As done before in the proof of Theorem 2 we can get, For every we have
Then, and hence .
Easily, we can deduce that
for all This shows that are a Lipschitz mapping on with the Lipschitz constants . Also, we can prove that the operators are con- tinuous and compact operator on for all and for all .
Since all conditions of Theorem 3 are satisfied, then the operator
has a fixed point in . +
As particular cases of Theorem 4 we can obtain theorems on the existence of solutions belonging to the space for the following integral equations:
1) Let , then we have
2) Let with then we have
3) Let , then we have
where are functions in and are positive constants.
5) Let then we have
are two functions in and are positive con- stants.
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.
The authors gratefully acknowledge Qassim University, represented by the Deanship of Scientific Research, on the material support for this research under the number (2915) during the academic year 1436 AH/2015 AD.
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