A Sum Operator Method for the Existence and Uniqueness of Positive Solutions to a System of Nonlinear Fractional Integral Equations

and Applied Analysis 3 (S2) for all i, g i (t, x 1 , x 2 , . . . , τx i , . . . , x n ) ≥ τg i (t, x 1 , x 2 , . . . , x i , . . . , x n ) for τ ∈ (0, 1), t ∈ [0, 1], x i ∈ [0, +∞) and there exist constants γ i ∈ (0, 1) such that f i (t, x 1 , x 2 , . . . , τx i , . . . , x n ) ≥ τ γ f i (t, x 1 , x 2 , . . . , x i , . . . , x n ) (12) for τ ∈ (0, 1), t ∈ [0, 1], x i ∈ [0, +∞), i = 1, 2, . . . , n; (S3) there exists δ i > 0 such that f i (t, x 1 , x 2 , . . . , x i , . . . , x n ) ≥ δ i g i (t, x 1 , x 2 , . . . , x i , . . . , x n ) , t ∈ [0, 1] , xi ≥ 0, i = 1, 2, . . . , n. (13) Then problem (4) has a unique positive solution x in P h . Moreover, for any initial value x = (x 1 , x (0) 2 , . . . , x (0) n ) ∈ P h , constructing successively the sequence


Introduction
Fractional calculus has been used for the study of problems in various fields of sciences, such as Abel integral equation and viscoelasticity, analysis of feedback amplifiers, capacitor theory, fractances, generalized voltage dividers, and engineering and biological sciences. In [1], Kilbas et al. give a survey of research in fractional calculus and its applications in mathematical analysis such as ODEs, PDEs, convolution integral equations, and theory of generating equations. Particularly, fractional differential equations have successful applications in nonlinear oscillation analysis of earthquakes, seepage flow in porous media [2], and fluid dynamic models for traffic flow [3], as the fractional derivatives can eliminate the deficiency of continuum traffic flow.
Open problems in this field are finding easy and effective methods for solving the equations. In recent years, many techniques of functional analysis, such as the fixed point theory, the Banach contraction principle, and the Leray-Schauder theory, are applied for solving the nonlinear fractional differential equations [4][5][6][7][8][9][10][11]. Iterative techniques [12][13][14] and the upper and lower solution method [15,16] are also introduced to investigate the existence and uniqueness of the solutions to nonlinear fractional order differential equations with various boundary conditions.
Recently, prompted by the applications in physics, the following nonlinear quadratic system of integral equations and its generalizations have provoked some interest: Salem [17] applied Krasnoselskii's fixed point theorem to obtain the existence of solutions for the system: has been studied in [18,19]. The aim of this paper is to study the existence and uniqueness of positive solutions for the following Volterra nonlinear fractional system of integral equations: Our main interest is to give some alternative answers to the main results of papers [17][18][19]. By using a fixed point theorem of a sum operator, we not only obtain the existence and uniqueness of positive solutions for the system (4), but also construct some sequences for approximating the unique solution.

Basic Definitions and Preliminaries
For the convenience of the reader, we present here some definitions, lemmas, and basic results that will be used in the proofs of our main results.
Suppose that is a real Banach space which is partially ordered by a cone ⊂ ; that is, ≤ if and only if − ∈ . If ≤ and ̸ = , then we denote < or > . By we denote the zero element of . Recall that a nonempty closed convex set ⊂ is a cone if it satisfies (i) ∈ , ≥ 0 ⇒ ∈ ; (ii) ∈ , − ∈ ⇒ = . Let ∘ = { ∈ | is an interior point of }, and then a cone is said to be solid if ∘ is nonempty. Moreover, is called normal if there exists a constant > 0 such that, for all , ∈ , ≤ ≤ implies ‖ ‖ ≤ ‖ ‖; in this case is called the normality constant of . If 1 , 2 ∈ , the set [ 1 , 2 ] = { ∈ | 1 ≤ ≤ 2 } is called the order interval between 1 and 2 . We say that an operator : . For all , ∈ , the notation ∼ means that there exist > 0 and > 0 such that ≤ ≤ . Clearly, ∼ is an equivalence relation. Given ℎ > (i.e., ℎ ≥ and ℎ ̸ = ), we denote by ℎ the set ℎ = { ∈ | ∼ ℎ}. It is easy to see that ℎ ⊂ . Definition 2. Let = or = ∘ and be a real number with 0 ≤ < 1. An operator : → is said to beconcave if it satisfies Definition 3. An operator : → is said to be homogeneous if it satisfies An operator : → is said to be subhomogeneous if it satisfies In the recent paper [20], Zhai and Anderson considered the following sum operator equation: where is an increasing -concave operator, is an increasing subhomogeneous operator, and is a homogeneous operator. They established the existence and uniqueness of positive solutions for the above equation, and when is a null operator, they present the following interesting result.
Lemma 4 (see [20]). Let be a normal cone in a real Banach space , let : → be an increasing -concave operator, and let : → be an increasing subhomogeneous operator. Assume that (2) there exists a constant > 0 such that ≥ , for all ∈ .
In what follows, we establish the existence and uniqueness of positive solutions for the following system of quadratic integral equations of the fractional type: