APPROXIMATIONS TO THE SOLUTION OF QUADRATIC FRACTIONAL INTEGRAL EQUATION (QFIE) WITH GENERALIZED MITTAG-LEFFLER Q FUNCTION

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, we will find the solution to the quadratic fractional integral equation involving the Q function which is the generalization of Mittag-Leffler function with the help of forming the sequence of solutions converging to the solution of the fractional integral equation involving the Q function. We will study in this paper the existence and convergence of a nonlinear quadratic fractional integral equation with the new Q function which is the generalization of Mittag-Leffler function, on a closed and bounded interval of the real line with the help of some conditions.


INTRODUCTION
Linear and nonlinear integral equations form an essential class of problems in mathematics.
The theory of integral operators and integral equations is an imperative part of nonlinear analysis. It is initiated by the fact that this theory is often applicable in other branches of mathematics and some equations define mathematical models in physics, engineering or biology as well in describing problems linked with real world. Many authors have demonstrated applications of fractional calculus in the nonlinear oscillation of earthquakes [13], fluid-dynamic traffic model [14], to model frequency dependent damping behavior of many viscoelastic materials [15,16], continuum and statistical mechanics [17], colored noise [18], solid mechanics [19], economics [20], bioengineering [21,22,23], anomalous transport [24], and dynamics of interfaces between nanoparticles and substrates [25]. There are also such equations whose interest lies in other branch of pure mathematics. Integral equations of fractional order create an interesting and important branch of the theory of integral equations. The theory of such integral equations is developed intensively in recent years together with the theory of differential equations of fractional order ( [1,2,3,4,5,6,7]). On the other hand the theory of quadratic integral equations is also intensively studied and finds numerous applications in describing real world problems ([8, 9, 10, 11]). Let us mention that this theory was initiated by considering a quadratic integral equation of Chandrasekhar type ( [2,11,12]).In this paper we prove the existence as well as approximations of the solutions of a certain generalized quadratic integral equation via an algorithm based on successive approximations under weak partial Lipschitz and compactness type conditions. Given a closed and bounded interval J = [0, T ] of the real line R for some T > 0, we consider the quadratic fractional integral equation (in short QFIE) where f : J × R → R and q : J → R are continuous functions, 1 ≤ q < 2 and Γ is the Euler gamma function, and Q γ,q,r α,β ,δ (x) is generalized mittag leffler function. By a solution of the QFIE (1.1) we mean a function x ∈ C(J, R) that satisfies the equation

AUXILIARY RESULTS
Unless otherwise mentioned, throughout this paper that follows, let E denote a partially ordered real normed linear space with an order relation and the norm · . It is known that E is regular if {x n } n∈N is a nondecreasing (resp. nonincreasing) sequence in E such that x n → x * as n → ∞, then x n x * (resp. x n x * ) for all n ∈ N. Clearly, the partially ordered Banach space C(J, R) is regular and the conditions guaranteeing the regularity of any partially ordered normed linear space E may be found in Heikkilä and Lakshmikantham [39] and the references therein.
In this section,we present some basic definitions and preliminaries which are useful in further discussion.
Let (E, , · ) be a partially ordered normed linear algebra. Denote

MAIN RESULT
The QFIE (1.1) is considered in the function space C(J, R) of continuous real-valued functions defined on J. We define a norm · and the order relation ≤ in C(J, R) by for all t ∈ J respectively. Clearly, C(J, R) is a Banach algebra with respect to above supremum norm and is also partially ordered w.r.t. the above partially order relation ≤. The following lemma in this connection follows by an application of Arzelá-Ascoli theorem.
for all t ∈ J and x, y ∈ R,x ≤ y.
for all t ∈ J, where x 0 = v, converges monotonically to x * .
Proof. Set E = C(J, R). Then, from Lemma 3.1 it follows that every compact chain in E possesses the compatibility property with respect to the norm · and the order relation ≤ in E.
Define two operators A and B on E by is equivalent to the operator equation We shall show that the operators A and B satisfy all the conditions of Theorem 2.12. This is achieved in the series of following steps.
Step I: A and B are nondecreasing on E.
Let x, y ∈ E be such that x ≤ y. Then by hypothesis (A 3 )and (A 4 ), we obtain and for all t ∈ J. This shows that A and B are nondecreasing operators on E into E. Thus, A and B are nondecreasing positive operators on E into itself.
Step II: A is partially bounded and partially D-Lipschitz on E.
Let x ∈ E be arbitrary. Then by (A 2 ), for all t ∈ J. Taking supremum over t, we obtain A x ≤ M and so, A is bounded. This further implies that A is partially bounded on E.
Now, let x, y ∈ E be such that x ≤ y. Then, by hypothesis, for all t ∈ J. Taking supremum over t, we obtain for all x, y ∈ E with x ≤ y. Hence A is partially nonlinear D-Lipschitz operators on E which further implies that it is also a partially continuous on E into itself.
Step III: B is a partially continuous operator on E.
Let {x n } n∈N be a sequence in a chain C of E such that x n → x for all n ∈ N. Then, by dominated convergence theorem, we have for all t ∈ J. This shows that Bx n converges monotonically to Bx pointwise on J.
Next, we will show that {Bx n } n∈N is an equicontinuous sequence of functions in E. Let Since the functions Q γ,q,r α,β ,δ , q are continuous on compact interval J and interval is continuous on compact set J × J, they are uniformly continuous there. Therefore, from the above inequality (3.7) it follows that |Bx n (t 2 ) − Bx n (t 1 )| → 0 as n → ∞ uniformly for all n ∈ N. This shows that the convergence Bx n → Bx is uniform and hence B is partially continuous on E.
Step IV: B is uniformly partially compact operator on E.
Let C be an arbitrary chain in E. We show that B(C) is a uniformly bounded and equicontinuous set in E. First we show that B(C) is uniformly bounded. Let y ∈ B(C) be any element.
Then there is an element x ∈ C be such that y = Bx. Now, by hypothesis (A 1 ), for all t ∈ J. Taking the supremum over t, we obtain y ≤ Bx ≤ r for all y ∈ B(C). Hence, B(C) is a uniformly bounded subset of E. Moreover, B(C) ≤ r for all chains C in E. Hence, B is a uniformly partially bounded operator on E.
Next, we will show that B(C) is an equicontinuous set in E. Let t 1 ,t 2 ∈ J with t 1 < t 2 . Then, for any y ∈ B(C), one has  B is a uniformly partially compact operator on E into itself.
Step V: v satisfies the operator inequality v ≤ A v + Bv.
By hypothesis (A 5 ), the QFIE (1.1) has a lower solution v defined on J. Then, we have