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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 731065, 6 pages
A Sum Operator Method for the Existence and Uniqueness of Positive Solutions to a System of Nonlinear Fractional Integral Equations
1School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, Sichuan 610074, China
2School of Information and Engineering, Wenzhou Medical College, 325035, Wenzhou, Zhejiang, China
Received 29 April 2013; Revised 7 June 2013; Accepted 27 June 2013
Academic Editor: Yonghong Wu
Copyright © 2013 Jing Wu and Tunhua Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper is concerned with the existence and uniqueness of positive solutions for a Volterra nonlinear fractional system of integral equations. Our analysis relies on a fixed point theorem of a sum operator. The conditions for the existence and uniqueness of a positive solution to the system are established. Moreover, an iterative scheme is constructed for approximating the solution. The case of quadratic system of fractional integral equations is also considered.
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 , 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 , and fluid dynamic models for traffic flow , 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 [4–11]. Iterative techniques [12–14] 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  applied Krasnoselskii’s fixed point theorem to obtain the existence of solutions for the system: under the assumptions that is continuous nondecreasing for all variables, and is continuous nonincreasing for all variables, where denotes the -products and . For the physical point of view, only positive solutions are interesting. A simple form of the system (2): 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 [17–19]. 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.
2. 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.
Definition 1 (see ). The fractional integral of order of a function is given by provided that the right-hand side is defined pointwisely on , and denotes the gamma function.
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) , ; (ii) , .
Let , and then a cone is said to be solid if is nonempty. Moreover, is called normal if there exists a constant such that, for all , implies ; in this case is called the normality constant of . If , the set is called the order interval between and . We say that an operator is increasing (decreasing) if implies . For all , the notation means that there exist and such that . Clearly, is an equivalence relation. Given (i.e., and ), we denote by the set . It is easy to see that .
Definition 2. Let or and be a real number with . An operator is said to be -concave 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 , 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 ). 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(1)there is such that ;(2)there exists a constant such that , for all .
Then the operator equation , has a unique solution in . Moreover, constructing successively the sequence . for any initial value , we have , as .
3. Main Results
Now by , we mean the Banach space of continuous functions on with the usual max-norm . Also, recall the Banach space of the cartesian product equipped by the norm . Notice that this space can be equipped with a partial order: Set , the standard cone. It is clear that is a normal cone in and the normality constant is 1. Take and ,
Theorem 5. Assume that(S1) for all are continuous and increasing with respect to the arguments , and for any ;(S2) for all for , , and there exist constants such that for , , ;(S3) there exists such that Then problem (4) has a unique positive solution in . Moreover, for any initial value , constructing successively the sequence then as .
Proof. To begin with, we define the following operators by
Thus is the positive solution of problem (4) if and only if . From and , we know that , . In the sequel we check that satisfy all assumptions of Lemma 4.
Firstly, we prove that are two increasing operators. In fact, by and , for with , we know that , and obtain that is, . Similarly, .
Next we show that is a -concave operator and is a subhomogeneous operator. In fact, for any and , by , we obtain Consequently, , where . Also, for any and , by and , we obtain that is, for , . So the operator is a subhomogeneous operator.
Now we show that , . In fact, by , we have and thus take then . Let It follows from that So , and then , hence . Similarly, from and , we easily prove . Hence the condition (1) of Lemma 4 is satisfied.
In the following we show that the condition (2) of Lemma 4 is satisfied. For , from , we have Take and then we have , . By Lemma 4, the operator equation has a unique solution ; of course, is also a unique solution of problem (4). In addition, by we know that the unique solution is also positive.
Now for any initial value , let us construct successively the sequence and we have as , and then problem (4) has a unique positive solution in ; that is, for any initial value , constructing successively the sequence: then as .
Corollary 6. Assume that(A1) for all is continuous and increasing with respect to the arguments , and for any ;(A2) for all , there exists constant such that for , , . Then the problem has a unique positive solution in . Moreover, for any initial value , constructing successively the sequencethen as .
In what follows, we establish the existence and uniqueness of positive solutions for the following system of quadratic integral equations of the fractional type:
Corollary 7. The system (31) has a unique positive solution.
This work is supported by the Natural Science Foundation of China (no. 21207103), Natural Science Foundation of Zhejiang Province (no. LY13H180012), Public Benefit Project of Zhejiang Province (no. 2012C31025), Scientific Research Project of Zhejiang Education Department (no. Y201222932), and Scientific Research Project of Wenzhou (no. G20110004).
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