Analytical Study of a φ − Fractional Order Quadratic Functional Integral Equation

: Quadratic integral equations of fractional order have been studied from different views. Here we shall study the existence of continuous solutions of a φ − fractional-orders quadratic functional integral equation, establish some properties of these solutions and prove the existence of maximal and minimal solutions of that quadratic integral equation. Moreover, we introduce some particular cases to illustrate our results.

Quadratic integral equations have been appeared in many useful application and problems of the real world. For example, in the theory of radiative transfer, the kinetic theory of gases, the theory of neutron transport, the queuing theory and the traffic theory [2,5,6,12].
In [13], we generalized the Carathéodory theorem for the nonlinear quadratic integral equation x(t) = a(t) + t 0 f (s, x(s)) ds t 0 g(s, x(s)) ds, (1) and proved the existence of at least one positive nondecreasing continuous solution to the Equation (1) under the assumption that the functions f and g satisfy the conditions of the Carathèodory Theorem [14]. Furthermore, we proved the existence of the maximal and minimal solutions of the quadratic integral Equation (1). Let J = [0, T], φ 1 , φ 2 : J → R be increasing and absolutely continuous and ψ i : J → J, i = 1, 2 be continuous. Let α, β ∈ (0, 1] and t ∈ J.
Consider the φ− fractional-orders quadratic functional integral equation f 2 (s, x(ψ 2 (s))) φ 2 (s) ds, t ∈ J, α, β ∈ (0, 1]. Now, we shall generalize these results and obtain similar ones for the fractional quadratic φ− integral Equation (2), which in turn gives the existence as well as the existence of many key integral and functional equations that arise in nonlinear analysis and its applications. Finally, we discuss the existence of maximal and minimal solutions of (2). Now, we shall denote by L 1 φ = L 1 φ [0, T] the space of all real functions defined on J. such that φ (t) f (t) ∈ L 1 (J) and T 0 | φ (t) f (t) | dt ≤ ∞. Where φ is an increasing function and absolutely continuous on J and we introduce the norm [9] || f (t) || L 1 Definition 1 ([9]). The φ− fractional integral of order α ≥ 0 of the function f (t) ∈ L 1 φ is defined as I α φ may be known as the fractional integral of the function f (t) with respect to φ(t), which is defined for any monotonic increasing function φ(t) ≥ 0, with a continuous derivative.

Main Results
Consider the functional quadratic φ− integral equation of fractional order (2) under the following assumptions: (i) a : J → R + is continuous and sup t∈J |a(t)| = k; (ii) f 1 , f 2 : J × R → R + satisfy the Carathéodory condition (i.e., measurable in t for all x ∈ R and continuous in x for all t ∈ J). (iii) There exist two functions m 1 , m 2 ∈ L 1 and nonnegative constants (vii) r is a positive solution of the inequality: With the aim of proving the existence of at least one solution for the Equation (2), firstly we construct an iterative scheme (as done in the original Carathéodory theorem) and secondly we apply the Schauder fixed point theorem.

Existence Results of QFIE (2) via Iterative Scheme
Theorem 1. Let assumptions (i)-(vii) be satisfied, then the functional quadratic integral equation of fractional order (2) has at least one positive solution x ∈ C(J).
Proof. Consider the ball S r in the space C(J) defined as Also, the sequence is equi-continuous.
Then we obtain . .
This implies and this proves the equi-continuity of the sequence {x n (t)}. Hence, {x n (t)} is a sequence of equi-continuous and uniformly bounded functions. By Arzela-Ascoli Theorem [14], then there exists a subsequence {x n k (t)} of continuous functions which converges uniformly to a continuous function x as k → ∞. Now we show that this limit function is the required solution.
Similarly we have which proves the existence of a positive solution x ∈ C(J) of the quadratic integral Equation (2).

Existence Results of QFIE (2) via the Fixed Point Theorem
In this subsection, we shall prove another existence result for the functional quadratic φ− integral of fractional order (2) by applying the Schauder fixed point.

Theorem 2. Let assumptions (i)-(vii) hold. Then the φ− fractional-orders quadratic functional integral Equation (2) has at least one solution x ∈ C(J).
Proof. Fix a number r > 0 and the ball S r in the space C(J) as defined above.
Let T be the operator defined on S r by the formula Then, in view of our assumptions, for x ∈ S r and t ∈ J we obtain Hence, in view of the assumption (vii), we have that T transforms the ball S r into itself. Now, for t 1 and t 2 ∈ I (without loss of generality assume that t 1 < t 2 ), we have Then we obtain . . .
This means that the functions from TS r are equi-continuous on J. Then, by the Arzela-Ascoli Theorem [14], the closure of TS r is compact.
It is clear that the set S r is nonempty, bounded, closed and convex. Assumptions (ii) and (iv) imply that T : S r → C(J) is a continuous operator in x.
Since all conditions of the Schauder fixed-point theorem hold, then T has a fixed point in S r .

Special Cases and Remarks
In Section 2, we prove an existence result for the functional quadratic φ− integral equation of fractional order (2) which in turn gives the existence as well as the existence of many key integral and functional equations that arise in nonlinear analysis and its applications. Corollary 1. Let the assumptions (i)-(vii) be satisfied with ψ 1 = ψ 2 = ψ then there exists at least one solution for the functional quadratic φ− integral equation of fractional order Corollary 2. Let the assumptions (i)-(vii) be satisfied with ψ 1 = ψ 2 = ψ, φ 1 = φ 2 = φ and α = β then there exists at least one solution for the functional quadratic φ− integral equation of fractional order Corollary 3. Let the assumptions (i)-(vii) be satisfied with φ 1 (t) = φ 2 (t) = φ(t), ψ 1 (t) = ψ 2 (t) = t then there exists at least one solution for the functional quadratic φ− integral equation of fractional order Corollary 5. Let the assumptions (i)-(vii) be satisfied with φ 1 (t) = φ 2 (t) = t m , m > 0 and ψ 1 (t) = ψ 2 (t) = t then there exists at least one solution for the Erdélyi-Kober functional quadratic equation of fractional order Corollary 6. Let the assumptions (i)-(vii) be satisfied with φ 1 (t) = φ 2 (t) = t, ψ 1 = ψ 2 = ψ then there exists at least one solution for the functional quadratic integral equation of fractional order The same result is obtained in [15].

Properties of Solutions
In this section, we give the sufficient conditions for the uniqueness of the solution of the quadratic integral Equation (2) and study some of its properties.

Uniqueness of Solutions of QFIE (2)
Let us assume the following assumptions (i*) a : J → R + is continuous and sup t∈J |a(t)| = k; (ii*) f 1 , f 2 : J × R → R + satisfy the Carathéodory condition (i.e., measurable in t for all x ∈ R and continuous in x for all t ∈ J). (iii*) There exist two nonnegative constants λ 1 , λ 2 ∈ L 1 such that Theorem 3. Let the assumptions (i)-(vii) be satisfied. If then the quadratic integral Equation (2) has a unique positive solution x ∈ C(J). Proof.
Therefore, the quadratic integral Equation (5) has a continuous solution x (t) according to Theorem 2.
From the uniqueness of the maximal solution (see [16,17]), it is clear that x (t) tends to q(t) uniformly in t ∈ J as → 0.
In a similar way we can prove that there exists a minimal solution of (2).
In this work, we discussed a φ− fractional order quadratic integral equation. Some exiting results were established by constructing an iterative scheme in aim of proving the analogous result for the Carathéodory theorem [14], and by applying Banach contraction mapping to demonstrate the existence of the unique solution of that equation. Furthermore, the existence of maximal and minimal solutions of the φ− fractional order quadratic integral equation is proved.