Existence Theorem for a Nonlinear Functional Integral Equation and an Initial Value Problem of Fractional Order in L 1 ( R + )

The aim of this paper is to study the existence of integrable solutions of a nonlinear functional integral equation in the space of Lebesgue integrable functions on unbounded interval, L1(R+). As an application we deduce the existence of solution of an initial value problem of fractional order that be studied only on a bounded interval. The main tools used are Schauder fixed point theorem, measure of weak noncompactness, superposition operator and fractional calculus.


Introduction
The class of functional integral equations of various types plays very important role in numerous mathematical research areas.An interesting feature of functional integral equations is its role in the study of many problems of functional differential Equations [1][2][3][4].
In this work we study the solvability of the following initial value problem where D y  denotes the fractional derivative of order  of with y   0,1   .Such initial value problem of arbitrary order (1) was investigated in [5][6][7].To achieve this goal, let us consider the integral equation which is different from that studied in [2].Section 2 contains some basic results.Our main result will be given in Section 3. Solvability of the considered initial value problem will be discussed in Section 4.

Basic Concepts
This section is devoted to recall some notations and known results that will be needed in the sequel.
A If is a Lebesgue measurable subset of the set of real numbers then we use the symbol R   meas.A to denote the Lebesgue measure of A   1 L A .Let be the space of all real functions defined and Lebesgue measurable on the set A   then the norm of x is defined as:

The Superposition Operator
An important operator called the superposition operator can be investigated in the theories of differential integral and functional equations [4,[8][9][10].It can be defined as follows: Definition 1. Assume that : f I R R   satisfies Carathéodory conditions, that is it is measurable in for any t x and continuous in x for almost all where t , t I x R   .Then for every measurable function x on the interval I we assign the function: The operator F defined in this way is called the superposition operator generated by the function f .Carathéodory [11] gave the first contribution to the theory of the superposition operator and proved its measurability according to the measurability of f .We state the following result giving the necessary and sufficient condition so that the superposition operator F generated by f will map continuously into itself [12]. 1 L Theorem 2. Let f satisfy the conditions in Definition 1.The superposition operator F generated by the function f maps continuously the space into itself if and only if: , where is a function that belongs to and is a nonnegative constant.
It is known that a real valued continuous function is measurable and that the converse is not necessarily true.However, for the converse we have the following results due to Dragoni [13].
L Theorem 3. Let I be a bounded interval and : be a function satisfying Caratheodory conditions.Then for each 0 is continuous.

Volterra Integral Operator
We proceed by recalling some basic facts concerning the linear Volterra integral operator in the Lebesgue space 1 Suppose is a given function which is measurable with respect to both variables where For an arbitrary function 1 x L  define Volterra integral operator as follows: then it is continuous [4,9].
In general, it is rather difficult to find necessary and sufficient conditions for the function   , k t s guarantee- ing that the integral operator K transforms the space 1 into itself.Some special cases of this problem were discussed in [4,14].In this direction we state the next result [15] empty and is nonempty and ker The family ker µ is said to be the kernel of the measure of weak noncompactness µ .Let us observe that the intersection set X  from 5) belongs to ker µ .Indeed, We can construct a useful measure of weak noncompactness in the space 1 that based on the following criterion for weak noncompactness due to Dieudonné We state here some results concerning the above mentioned operators: [17,18].Theorem 6.A bounded set X is relatively weakly compact in 1 if and only if the following two conditions are satisfied: L a) for any 0 b) for any 0 Now, for a nonempty and bounded subset X of the space let us define: where It can be shown [17] that the function  is a measure of weak noncompactness in the space 1 such that

Fractional Calculus
The definitions of both differential operator and the integral operator of fractional order are stated as follows [19,20].

Existence Theorem
Consider the integral Equation ( 2) and let H denotes the operator determined by the right hand side of this equation, i.e., where t R   In fact the operator H can be written as the product and the superposition operator Therefore Equation ( 4) can be written as: To establish our main result concerning existence of an integrable solution of Equation ( 2) we impose suitable conditions on the functions involved in that equation.Namely we assume 1) The functions : f R R R    satisfy the Caratheodory conditions and there exist functions 2) The functions satisfy the Caratheodory conditions and the linear Volterra operators  is increasing, absolutely continuous and there exists a constant such that Now we can state our main result in the next theorem.Theorem 10.Under the above assumptions the Equation (2) has at least one solution 1 .

x L 
Proof.Since H is a nonlinear operator defined by Equation ( 5), then based on assumptions i) and ii) if

Hx L
 Moreover, from Equation ( 5), and noting that 1 2 , K K according to our assumptions are indeed bounded, we have .
The above estimate shows that the operator H maps into itself, where Moreover, according to Theorem 2, we deduce that the operator H is continuous on the space .L₁ Next, to prove that H is a contraction, let X be a nonempty subset of B Fix .
where the symbol D  denotes the operator norm acting from the space   1 L D into itself.Also in the above calculation we used the fact that for t From the absolute continuity of the function  and the obvious equality and using Theorem 6 we obtain Furthermore, fixing we can deduce that 0 where the symbol T denotes the operator norm acting from the space   1 , L T  into itself.Now according to the fact that the set consisting of one element is weakly compact, by using Theorem 6 and the formula According to Equation (3), combining ( 6) and ( 7), we get   . Clearly, according to assumption iv) 1 q  .Consider the sequence of sets In view of Theorem 3 we can find a closed subset ) and such that the functions   are continuous.Hence we infer that In what follows we show that   n y is an equicontinuous on D  , for that let us take arbitrarily 1 2 , t t D   .
Without loss of generality we can assume that 1 2 t t  .
Then, keeping in mind our assumptions, for an arbitrary fixed we obtain: where denotes the modulus of continuity of the function on the set By rearranging the order of double integrations, we get From the above estimate and the consideration of the fact that we obtain .
n y Now, utilizing the fact that the sequence is weakly compact and taking into account Theorem 6 we can show that the number is arbitrarily small provided the number   taken to be sufficiently small (it is a consequence of the fact that a one element set is weakly compact in ). 1 Finally, from ( 10) and ( 11  L L R   Hence we conclude that the set HY is relativelycompact in this space.
In the last step of the proof let us consider the set

Nonlinear Equation of Convolution Type
Assume that is an integrable function.For an arbitrary function This operator K is a linear integral operator of convolution type and maps into itself continuously.
1 Now, consider the following condition , Then we have the following Corollary Corollary 11.Let the hypotheses i)-v) are satisfied.Then a nonlinear equation of convolution type Copyright © 2013 SciRes.AM has at least one integrable solution 1 x L  .
In the next subsection, we prove an existence theorem for integral equation of fractional order as a special form of Equation (12).

Initial Value Problems of Fractional Order
As a special case of Equation ( 14), we consider where and       .Equation ( 13) is an integral equation of fractional order that can be written in the form Obviously, Equation ( 14) has at least one integrable solution

Conclusion
The existence theorem of functional integrable equation in the space of Lebesgue integrable functions on unbounded interval is presented and proved.As an application of this theorem, we investigated the existence of solution of the suggested initial value problems of fractional order.
g t be an absolutely continuous function on   , a b .Then the fractional derivative of or- der quence of uniformly bounded and equicontinuous functions on D  .Hence, in view of Ascoli-Arzela theorem we deduce that the sequence   n Hy is relatively compact subset in the space   C D  .Further observe that the above reasoning does not depend on the choice of  .Thus we can construct a se- quence of closed subsets of the interval In what follows, utilizing the fact that the set HY 0 is weakly compact, let us choose a number   such that for each closed subset D  of the interval  

Definition 12 .
By a solution of the initial value problem (1) we mean an absolutely continuous function x satisfies the initial value problem (1).i)-iii) and v) are satisfied, then the initial value problem (1) has at least one solution .we deduce that is an absolutely continuous function satisfies the initial value problem(1).Hence the proof is complete. y

.3. Measures of Weak Noncompactness
: To apply the classical Schauder fixed point theorem, we need to prove that the set HY is relatively compact in 1 .For this aim let us consider the functions L