Journal of Function Spaces
Volume 2018 (2018), Article ID 7469868, 9 pages
https://doi.org/10.1155/2018/7469868

Research Article

Fixed-Point Theorems for Systems of Operator Equations and Their Applications to the Fractional Differential Equations

Xinqiu Zhang,¹ Lishan Liu,1,2 and Yumei Zou³

¹School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China
²Department of Mathematics and Statistics, Curtin University, Perth, WA 6845, Australia
³Department of Statistics and Finance, Shandong University of Science and Technology, Qingdao 266590, China

Correspondence should be addressed to Lishan Liu; mathlls@163.com

Received 20 April 2018; Accepted 26 June 2018; Published 5 August 2018

Academic Editor: Gestur Ólafsson

Copyright © 2018 Xinqiu Zhang et al. 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.

Abstract.

We study the existence and uniqueness of positive solution for a class of nonlinear binary operator equations systems by means of the cone theory and monotone iterative technique, under more general conditions. Also, we give the iterative sequence of the solution and the error estimation of the system. Moreover, we use this new result to study the existence and uniqueness of the solutions for fractional differential equations systems involving integral boundary value conditions in ordered Banach spaces as an application. The results obtained in this paper are more general than many previous results and complement them.

1. Introduction

In this paper, using the cone theory and monotone iterative technique, we consider the following nonlinear nonmonotone binary equations systems in real Banach spaces,

where : , is a subset of real Banach spaces. There have appeared a series of research results concerning the nonlinear operator equation , [1–7], the sum of several classes of mixed-monotone operator equations, and the nonlinear equations systems (1) [8], in recent years. The techniques they used are cone and semiorder [9, 10], the Granas fixed-point index theory [11], the equivalent classes (which are called components) of a real Banach space [12], the Ishikawa iteration process [13, 14], etc.

In [15], Zhang investigated the existence and uniqueness of solutions for a class of nonlinear operator equations in ordered Banach space, by using the cone theory and Banach contraction mapping principle. The assumption they used is

where is a generating normal cone, : is a nonlinear operator, : is a positive linear bounded operator with (where is the spectral radius of ), and is a positive integer.

In [16], Zhang investigated the existence and uniqueness theorems of fixed points to a class of mixed-monotone operators with convexity and concavity, in which suppose that there exist , such that and

But in [15, 16], they did not consider the nonmonotone binary operator equations systems , , where : are two nonlinear operators and is the order interval in . In this article, the existence and uniqueness of positive solution for nonlinear nonmonotone binary operator equations systems are established under more general conditions, even not supposing the generating of cone . Compared with [16], we do not need the assumption of convexity-concavity. Moreover, the operators we consider are -increasing in and the different upper and lower bounds in formula (6) are more general; the results here cannot be obtained by using Banach contraction mapping principle. Also, different from the results in [15], the iterative sequence of the solution and the error estimation of the system are obtained.

As an application, we study the existence and uniqueness of positive solutions and iterative approximation of the unique solution for the following fractional differential equations involving integral boundary value problems:

where , , ,   , , ,     is the standard Riemann-Liouville derivatives and ,   , , , , , , , , , ,   , : is continuous and , is a function of bounded variation, , and denote the Riemann-Stieltjes integral with respect to .

Recently, fractional differential equations, arising in the mathematical modeling of systems and processes, have drawn more and more attention of the research community due to their numerous applications in various fields of science such as engineering, chemistry, physics, and mechanics. Boundary value problems of fractional differential equations have been investigated for many authors [17–22]. Now, there are many papers dealing with the problem for different kinds of boundary value conditions such as multipoint boundary condition [23, 24], integral boundary condition [25–31], and many other boundary conditions [32]. From the application, we can see that the fixed-point theorems in this paper have extensive applied background. The results presented here are more general and complement many previous known results.

2. Preliminaries

Now we present briefly some definitions, lemmas, and basic results that are to be used in the article; for convenience of the reader, we refer the reader to [3, 4, 10, 33, 34] for more details.

Suppose that is a real Banach space, and is the zero element of . Recall that a nonempty closed convex set is a cone if it satisfies (1) ; (2) . The real Banach space can be partially ordered by a cone , i.e., if and only if . If and , then we denote or . Let . Then is a Banach space with the norm , for .

The cone is called normal if there exists a constant such that for all implies , and the smallest is called the normality constant of . If , , the set is called the order interval between and .

Definition 1 ([3, 4]). Let be a subset of a real ordered Banach space ; : is said to be a mixed-monotone operator if is increasing in and decreasing in ; i.e., , , imply . The element is called a fixed point of if .

3. Main Results

Theorem 2. Let be a real Banach space, be a normal cone in , and be the order interval in . Assume that : are two nonlinear operators, : are two positive linear bounded operators and satisfy the following conditions:
  , .
For all , and are decreasing in , i.e., for any , , implies , ; and there exist two positive numbers    such that for all , , ,

   is reversible ( is the identity operator) and .
   and there exist two positive integers , such that (where is the spectral radius of linear bounded operator) and

Then the nonlinear operator equations system (1) has a unique solution in . And for any initial values , , by constructing successively the sequences as follows:

we have , in , as . Moreover, for any , there exists such that

Proof. Since is reversible, let

Then (7) can be written as

By , , it is easy to prove that and satisfy the following conditions:
By , we know : are two mixed-monotone operators.
For all ,

Combining with , it is easy to prove that

where and is the identity operator.
Also, by , we have

Thus, combining with (12), we have

Let , . Thus, by (15), we know

Therefore, by and , using mathematical induction, we can easily prove that

Firstly, we prove that

By , we can easily prove that

i.e., (18) holds for . Suppose that (18) holds for , i.e.,

Then, for , by and we know

By (19)-(21), using mathematical induction, we know (18) holds.
Next we prove that is a Cauchy sequence. From , we have . Then

Consequently, there exists such that

Thus, there exists such that

Then by (17), we have

Consequently, by (18) and (25), we have

Therefore, by (24), (26) and the normality of cone , we have

where is the normality constant of . Consequently, and are Cauchy sequences. Since is complete, thus there exist , such that

And by (17), we know

Thus, . By (18) and (29), we have

Thus, by (26), (30) and the normality of cone , we have

Consequently, . Let ; by and , we have

Let , by (32) and the normality of we have . Therefore, by the definitions of and , we have , ; i.e., is the solution of operator equation (1).
Finally, we prove that is the unique solution of operator equations systems (1) in . In fact, suppose is another solution of equations systems (1) in , then by , using mathematical induction, we can easily see that   . Thus, by (28) and the normality of , we have . Therefore, the operator equations systems (1) have a unique solution in .
Now for any initial points , , we construct successively the sequences

Since , i.e.,

Suppose

Then

Thus, by mathematical induction, we have

By (18) and (37), we have

Thus,

In the same way, we can prove that

Consequently, by (40) and (41) we know that (8) holds.

Taking in Theorem 2, then we get the following corollary.

Corollary 3. Let be a real Banach space, be a normal cone in , and be the order interval in . Assume that : are two nonlinear operators, : are two positive linear bounded operators and satisfy the following conditions:
  , .
For all , and are decreasing in , i.e., for any , , implies , ; and there exist two positive numbers    such that for all , , ,

  , (where is the spectral radius of linear bounded operator) and

Then the nonlinear operator equations system (1) has a unique solution in . And for any initial values , , by constructing successively the sequences as follows:

we have , in , as . Moreover, for any , there exists such that

4. An Application for Fractional Differential Equations Involving Integral Boundary Value Problems

Now we present briefly some definitions, lemmas, and basic results that are to be used in the article for convenience of the reader. We refer the reader to [35–39] for more details.

Definition 4 ([35, 36, 38, 39]). The Riemann-Liouville fractional integral of order of a function : is given by

provided that the right-hand side is pointwise defined on .

Definition 5 ([35, 36, 38, 39]). The Riemann-Liouville fractional derivative of order of a continuous function : is given by

where , denotes the integer part of the number , provided that the right-hand side is pointwise defined on .

Lemma 6 ([35, 36, 38, 39]). (1) If and , then

(2) If and , then has unique solution

where   , .

Lemma 7 ([35, 36, 38, 39]). Let and let be integrable; then

where and is the smallest integer greater than or equal to .

Lemma 8 ([37]). Let , . Assume that the following condition is satisfied:

then

has a unique positive solution

where

in which

Lemma 9 ([37]). If the condition in Lemma 8 is satisfied, the Green function has the following properties:
(1) , for all .
(2) For any , we have

where

In the following, we need the following assumptions:
  , where

   and there exists such that

For all   , , , , ,

There exist four constants () such that for any ,    with , , , ,

Theorem 10. Let be a real Banach space and be a normal cone in . Assume that the conditions and – are satisfied. Then the system of nonlinear differential equations (50) has unique positive symmetry solution , where . Moreover, for any initial functions , there exist monotone iteration sequences , , such that , in , as , where

Proof. It is well known that is a solution of the system (50) if and only if is a solution of the system of nonlinear integral equations

Consider the operators : as follows:

By , for all , , , , we obtain

Consequently, : are mixed-monotone. By , for all , we obtain

By , for any , with ,

where

Set

then Consequently, for any , for any with ,

In the following, we prove . In fact, by , since , there exists a constant such that , for any . Thus, for all , ,

By mathematical induction, we can easily prove that for all natural number ,

therefore,

consequently,

thus, .
Thus, all conditions of Corollary 3 are satisfied; therefore, the conclusions of Theorem 10 hold. Consequently, the proof of Theorem 10 is completed.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgments

The work is supported financially by the National Natural Science Foundation of China (11371221 and 11571296) and the Australian Research Council.

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