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 Open Access
Multivariate systems of nonexpansive operator equations and iterative algorithms for solving them in uniformly convex and uniformly smooth Banach spaces with applications
 Yongchun Xu^{1},
 Jinyu Guan^{1},
 Yanxia Tang^{1} and
 Yongfu Su^{2}Email author
https://doi.org/10.1186/s1366001816297
© The Author(s) 2018
 Received: 10 October 2017
 Accepted: 7 February 2018
 Published: 13 February 2018
Abstract
We prove some existence theorems for solutions of a certain system of multivariate nonexpansive operator equations and calculate the solutions by using the generalized Mann and Halpern iterative algorithms in uniformly convex and uniformly smooth Banach spaces. The results of this paper improve and extend the previously known ones in the literature.
Keywords
 Uniformly convex
 Uniformly smooth
 Banach space
 Systems of nonexpansive operator equations
 Solution
 Iterative algorithms
1 Introduction and preliminaries
Multivariate mathematical analysis is an important branch in mathematical fields and applied science fields. A system of nonlinear operator equations is an essential tool in the broader fields of science and technology. It is also an important method in pure and applied mathematics. Many structures of mathematics can be expressed in the form of fixed point equations. For example, equilibrium problems, variational inequalities, convex optimization, split feasibility problems, and inclusion problems are equivalent to relatively fixed point problems. Furthermore, generalized equilibrium problems, generalized variational inequalities, generalized convex optimization, generalized split feasibility problems, and generalized inclusion problems are equivalent to relatively fixed point equation or systems of nonlinear operator equations (see [1–9]).
Recently, multivariate fixed point theorems of Nvariable nonlinear mappings have been studied by some authors. Many interesting results and their applications have been also given. In 2014, Lee and Kim [5] proved multivariate coupled fixed point theorems on ordered partial metric spaces. In 2016, Su, Petruşel, and Yao [7] presented the concept of a multivariate fixed point and proved a multivariate fixed point theorem for Nvariable contraction mappings, which further generalizes the Banach contraction mapping principle. In 2016, Luo, Su, and Gao [6] presented the concept of a multivariate best proximity point and proved multivariate best proximity point theorems in metric spaces for Nvariable contraction mappings. In 2017, Xu et al. [8] presented the concept of a multivariate contraction mapping in a locally convex topological vector space and proved the multivariate contraction mapping principle in such spaces. In 2017, Guan et al. [4] studied a certain system of Nfixed point operator equations with Npseudocontractive mapping in reflexive Banach spaces and proved existence theorems of solutions. In 2017, Tang et al. [9] studied a certain system of Nvariable variational inequalities and proved existence theorems of solutions.
Our purpose in this paper is to prove some existence theorems for solutions of a certain system of the multivariate nonexpansive operator equations and to calculate the solutions by using the generalized Mann and Halpern iterative algorithms in uniformly convex and uniformly smooth Banach spaces. The results of this paper improve and extend the previously known ones in the literature.
The following classical theorems are useful for the results of this paper.
Theorem 1.1
(Browder and Göhde fixed point theorem [10])
Let X be a real uniformly convex Banach space, and let C be a nonempty closed convex bounded subset of X. Then every nonexpansive mapping \(T: C\rightarrow C\) has a fixed point.
Mann’s iterative process was initially introduced in 1953 by Mann [11]. Mann’s iterative scheme is an important iterative scheme to study the class of nonexpansive mappings. The following is a representative result in recent years.
Theorem 1.2
 (C_{1}):

\(\sum_{n=1}^{\infty}\alpha_{n}=\infty\);
 (C_{2}):

\(\lim_{n\rightarrow\infty}\alpha_{n}=0\);
 (C_{3}):

\(\lim_{n\rightarrow\infty}\frac{\alpha_{n+1}\alpha _{n}}{\alpha^{2}_{n+1}}=0\).
 (C_{1}):

\(\sum_{n=1}^{\infty}\alpha_{n}=\infty\);
 (C_{2}):

\(\lim_{n\rightarrow\infty}\alpha_{n}=0\);
 (C_{4}):

\(\lim_{n\rightarrow\infty}\sum_{n=1}^{\infty}\alpha _{n+1}\alpha_{n}<+ \infty\).
 (C_{5}):

\(\lim_{n\rightarrow\infty}\frac{\alpha_{n+1}\alpha_{n}}{\alpha_{n}}=0\).
Theorem 1.3
([19])
 (C_{1}):

\(\sum_{n=1}^{\infty}\alpha_{n}=\infty\);
 (C_{2}):

\(\lim_{n\rightarrow\infty}\alpha_{n}=0\);
 (C_{5}):

\(\lim_{n\rightarrow\infty}\frac{\alpha_{n+1}\alpha_{n}}{\alpha_{n}}=0\).
2 Cartesian product of uniformly convex Banach spaces
Definition 2.1
Theorem 2.2
Proof
From Theorem 2.2 we get the following corollary.
Corollary 2.3
Theorem 2.4
Proof
Theorem 2.5
Proof
Open question 2.6
3 Cartesian product of uniformly smooth Banach spaces
Definition 3.1
Theorem 3.2
Proof
From Theorem 3.2 we get the following corollary.
Corollary 3.3
Theorem 3.4
Proof
Theorem 3.5
Proof
Open question 3.6
4 Results and discussion
Definition 4.1
Theorem 4.2
Proof
Lemma 4.4
Proof
Theorem 4.5
 (1)there exists an element \(p=(p_{1},p_{2}, \ldots, p_{N})\in C^{N}\) such that$$\textstyle\begin{cases} T_{1}(p_{1},p_{2},\ldots,p_{N})=p_{1}, \\ T_{2}(p_{1},p_{2},\ldots,p_{N})=p_{2}, \\ \ldots, \\ T_{i}(p_{1},p_{2},\ldots,p_{N})=p_{i}, \\ \ldots, \\ T_{N}(p_{1},p_{2},\ldots,p_{N})=p_{N}; \end{cases} $$
 (2)for any \(x_{0}=(x_{1,0},x_{2,0},x_{3,0},\ldots,x_{N,0}) \in C^{N}\), the iterative sequence \(\{x_{i,n}\}\subset X\) defined byconverges weakly to \(p_{i}\) for all \(i=1,2,\ldots,N\), where \(0< a\leq\alpha _{n}\leq b<1\) for two constants a, b.$$x_{i, n+1}=\alpha_{n} x_{i,n}+(1\alpha_{n})T_{i}(x_{1,n},x_{2,n}, \ldots ,x_{N,n}),\quad n=0,1,2, \ldots, $$
Proof
Theorem 4.6
 (C_{1}):

\(\sum_{n=1}^{\infty}\alpha_{n}=\infty\);
 (C_{2}):

\(\lim_{n\rightarrow\infty}\alpha_{n}=0\);
 (C_{3}):

\(\lim_{n\rightarrow\infty}\frac{\alpha_{n+1}\alpha_{n}}{\alpha_{n}}=0\).
Proof
The concept of a coupled fixed point was introduced by Chang and Ma [20] in 1991. Since then, the concept has been of interest to many researchers in metrical fixed point theory [21–24]. In 2006, Bhaskar and Lakshmikantham [24] introduced the concept of a coupled fixed point in the setting of singlevalued mappings and established some coupled fixed point results and found its application to the existence and uniqueness of solutions for periodic boundary value problems. In 2011, Berinde and Borcut [25] introduced the concept of a tripled fixed point for nonlinear mappings in complete metric spaces.
Definition 4.7
([20])
Definition 4.8
([25])
Definition 4.9
([7])
Form the above results we get the following corollaries.
Corollary 4.10
Proof
By using the same way as in Corollary 4.10, we can get Corollary 4.11.
Corollary 4.11
Corollary 4.12
Proof
By Theorem 4.5 we get the following three corollaries.
Corollary 4.13
 (1)
T has a coupled fixed point;
 (2)for any given \(x_{0}=(x_{1,0},x_{2,0}) \in C^{2}\), the iterative sequences \(\{x_{1,n}\}, \{x_{2,n}\}\subset X\) defined byand$$x_{1,n+1}=\alpha_{n} x_{1,n}+(1\alpha_{n})T(x_{1,n},x_{2,n}) $$converge weakly to two elements \(p_{1}\) and \(p_{2}\), respectively, and \((p_{1},p_{2})\) is a coupled fixed point of T, where \(0< a\leq\alpha _{n}\leq b<1\) for two constants a, b.$$x_{2,n+1}=\alpha_{n} x_{2,n}+(1\alpha_{n})T(x_{2,n},x_{1,n}) $$
Corollary 4.14
 (1)
T has a tripled fixed point;
 (2)for any given \(x_{0}=(x_{1,0},x_{2,0},x_{3,0}) \in C^{3}\), the iterative sequences \(\{x_{1,n}\}, \{x_{2,n}\}, \{x_{3,n}\}\subset X\) defined byconverge weakly to three elements \(p_{1}\), \(p_{2}\), \(p_{3}\), respectively, and \((p_{1},p_{2},p_{3})\) is a tripled fixed point of T, where \(0< a\leq\alpha _{n}\leq b<1\) for two constants a, b.$$\begin{aligned}& x_{1,n+1}=\alpha_{n} x_{1,n}+(1\alpha_{n})T(x_{1,n},x_{2,n}, x_{3,n}), \\& x_{2,n+1}=\alpha_{n} x_{2,n}+(1\alpha_{n})T(x_{2,n}, x_{3,n}, x_{1,n}), \\& x_{3,n+1}=\alpha_{n} x_{3,n}+(1\alpha_{n})T(x_{3,n}, x_{2,n},x_{1,n}) \end{aligned}$$
Corollary 4.15
 (1)
T has a multivariate fixed point;
 (2)for any given \(x_{0}=(x_{1,0},x_{2,0},x_{3,0},\ldots,x_{N,0}) \in C^{N}\), the iterative sequence \(\{x_{n}\}\subset X\) defined byconverges weakly to a multivariate fixed point of T, where \(0< a\leq \alpha_{n}\leq b<1\) for two constants a, b.$$x_{n+1}=\alpha_{n} x_{n}+(1\alpha_{n})T(x_{1,n},x_{2,n}, \ldots ,x_{N,n}),\quad n=0,1,2, \ldots, $$
By Theorem 4.6 we get the following three corollaries.
Corollary 4.16
 (C_{1}):

\(\sum_{n=1}^{\infty}\alpha_{n}=\infty\);
 (C_{2}):

\(\lim_{n\rightarrow\infty}\alpha_{n}=0\);
 (C_{3}):

\(\lim_{n\rightarrow\infty}\frac{\alpha_{n+1}\alpha_{n}}{\alpha_{n}}=0\).
Corollary 4.17
 (C_{1}):

\(\sum_{n=1}^{\infty}\alpha_{n}=\infty\);
 (C_{2}):

\(\lim_{n\rightarrow\infty}\alpha_{n}=0\);
 (C_{3}):

\(\lim_{n\rightarrow\infty}\frac{\alpha_{n+1}\alpha_{n}}{\alpha_{n}}=0\).
Corollary 4.18
 (C_{1}):

\(\sum_{n=1}^{\infty}\alpha_{n}=\infty\);
 (C_{2}):

\(\lim_{n\rightarrow\infty}\alpha_{n}=0\);
 (C_{3}):

\(\lim_{n\rightarrow\infty}\frac{\alpha_{n+1}\alpha_{n}}{\alpha_{n}}=0\).
Applied example 4.19
Discussion 4.20
An important contribution of this paper is to prove some existence theorems for solutions of certain systems of the multivariate nonexpansive operator equations and to calculate the solutions by using the generalized Mann and Halpern iterative algorithms in uniformly convex and uniformly smooth Banach spaces. To get the desired results, we first need to study the convexity and smoothness of Cartesian products of uniformly convex Banach spaces and uniformly smooth Banach spaces, respectively. On the other hand, we used a clever way to prove the main results. This method converts a nonselfmapping into a selfmapping such that the classical results can be used. Of course, in the main theorems, the assumptions that “\((X^{N}, \\cdot\_{*})\) is a uniformly convex Banach space” and “\((X^{N}, \\cdot\_{*})\) is a uniformly smooth Banach space” are still a limitation.
5 Conclusions
Conclusion 5.1
Conclusion 5.2
Conclusion 5.3
 (C_{1}):

\(\sum_{n=1}^{\infty}\alpha_{n}=\infty\);
 (C_{2}):

\(\lim_{n\rightarrow\infty}\alpha_{n}=0\);
 (C_{3}):

\(\lim_{n\rightarrow\infty}\frac{\alpha_{n+1}\alpha_{n}}{\alpha_{n}}=0\).
Declarations
Acknowledgements
The authors wish to thank the referees for their valuable comments on an earlier version of this paper. This project is supported by the major project of Hebei North University under grant No. ZD201304.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript. YS found the main reference of this paper in the literature study and read it with YT. Then JG put forward the main problem and some ideas and methods to deal with the problem. Finally, YX and YS carried out concretely the above ideas and methods, and YT accomplished this paper.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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