Multivariate fixed point theorems for contractions and nonexpansive mappings with applications
 Yongfu Su^{1},
 Adrian Petruşel^{2} and
 JenChih Yao^{3}Email author
https://doi.org/10.1186/s1366301504930
© Su et al. 2016
Received: 14 October 2015
Accepted: 21 December 2015
Published: 13 January 2016
Abstract
The first purpose of this paper is to prove an existence and uniqueness result for the multivariate fixed point of a contraction type mapping in complete metric spaces. The proof is based on the new idea of introducing a convenient metric space and an appropriate mapping. This method leads to the changing of the nonselfmapping setting to the selfmapping one. Then the main result of the paper will be applied to an initialvalue problem related to a class of differential equations of first order. The second aim of this paper is to prove strong and weak convergence theorems for the multivariate fixed point of a Nvariables nonexpansive mapping. The results of this paper improve several important works published recently in the literature.
Keywords
contraction mapping principle complete metric spaces multivariate fixed point multiply metric function multivariate mapping differential equation strong and weak convergence1 Introduction
On the other hand, in 2015, Su and Yao [7] proved the following generalized contraction mapping principle.
Theorem SY
In particular, the study of the fixed points for weak contractions and generalized contractions was extended to partially ordered metric spaces in [8–18]. Among them, some results involve altering distance functions. Such functions were introduced by Khan et al. in [19], where some fixed point theorems are presented.
The first purpose of this paper is to prove an existence and uniqueness result of the multivariate fixed point for contraction type mappings in complete metric spaces. The proof is based on the new idea of introducing a convenient metric space and an appropriate mapping. This ingenious method leads to the changing of the nonselfmapping setting to the selfmapping one. Then the main result of the paper will be applied to an initialvalue problem for a class of differential equations of first order. The second aim of this paper is to prove strong and weak convergence theorems for the multivariate fixed point of Nvariables nonexpansive mappings. The results of this paper improve several important results recently published in the literature.
2 Contraction principle for multivariate mappings
We will start with some concepts and results which are useful in our approach.
Definition 2.1
 (1)
\(\triangle(a_{1},a_{2},\ldots,a_{N})\) is nondecreasing for each variable \(a_{i}\), \(i\in\{1,2,3, \ldots,N \}\);
 (2)
\(\triangle(a_{1}+b_{1},a_{2}+b_{2},\ldots,a_{N}+b_{N})\leq \triangle(a_{1},a_{2},\ldots,a_{N})+\triangle(b_{1},b_{2},\ldots,b_{N})\);
 (3)
\(\triangle(a,a,\ldots,a)=a\);
 (4)
\(\triangle(a_{1},a_{2},\ldots,a_{N})\rightarrow0 \Leftrightarrow a_{i}\rightarrow0\), \(i\in\{1,2,3,\ldots, N \}\), for all \(a_{i},b_{i}, a \in\mathbb{R}\), \(i\in\{1,2,3, \ldots,N \}\), where \(\mathbb{R}\) denotes the set of all real numbers.
The following are some basic examples of multiply metric functions.
Example 2.2
(1) \(\triangle_{1}(a_{1},a_{2},\ldots,a_{N})=\frac{1}{N} \sum_{i=1}^{N}a_{i}\). (2) \(\triangle_{2}(a_{1},a_{2},\ldots,a_{N})=\frac{1}{h} \sum_{i=1}^{N}q_{i} a_{i}\), where \(q_{i}\in[0,1)\), \(i\in\{1,\ldots, N \}\), and \(0< h:= \sum_{i=1}^{N}q_{i}<1\).
Example 2.3
\(\triangle_{3}(a_{1},a_{2},\ldots,a_{N})=\sqrt{\frac{1}{N} \sum_{i=1}^{N}a_{i}^{2}}\).
Example 2.4
\(\triangle_{4}(a_{1},a_{2},\ldots,a_{N})=\max \{a_{1},a_{2},\ldots,a_{N}\}\).
An important concept is now presented.
Definition 2.5
In the following, we prove the following theorem, which generalizes the Banach contraction principle.
Theorem 2.6
Proof
 (i)
\(D((x_{1},x_{2},\ldots,x_{N}), (y_{1},y_{2},\ldots,y_{N}))=0 \Leftrightarrow (x_{1},x_{2},\ldots,x_{N})= (y_{1},y_{2},\ldots,y_{N})\);
 (ii)
\(D( (y_{1},y_{2},\ldots,y_{N})), (x_{1},x_{2},\ldots ,x_{N})=D((x_{1},x_{2},\ldots,x_{N}), (y_{1},y_{2},\ldots,y_{N}))\), for all \((x_{1},x_{2},\ldots,x_{N}), (y_{1},y_{2},\ldots,y_{N})\in X^{N}\).
Notice that taking \(N=1\), \(\triangle(a)=a\) in Theorem 2.6, we obtain Banach’s contraction principle.
Some other consequences of the above general result are the following corollaries.
Corollary 2.7
Notice that the above corollary is related to the wellknown Prešić’s fixed point theorem (see [21]).
Prešić’s theorem
Choosing \(\Delta:=\Delta_{2}\), \(h:= \sum_{i=1}^{N}q_{i}\), and \(x=(x_{1},x_{2},\ldots,x_{N}), y=(x_{2},x_{3},\ldots ,x_{N+1})\in X^{N}\), the contraction condition given in Theorem 2.6 leads to Prešić’s contraction type condition.
Corollary 2.8
Corollary 2.9
Notice also here that the above corollary is related to a multivariate fixed point theorem of Ćirić and Prešić (see [22]), which reads as follows.
ĆirićPrešić’s theorem
Choosing \(\Delta:=\Delta_{4}\), \(h\in(0,1)\), and \(x=(x_{1},x_{2},\ldots,x_{N}), y=(x_{2},x_{3},\ldots ,x_{N+1})\in X^{N}\), the contraction condition given in Theorem 2.6 leads to the above ĆirićPrešić’s contraction type condition.
It is worth to mention that the above results are in connection with a very interesting multivariate fixed point principle proved by Tasković in [23]. More precisely, Tasković’s result is as follows.
Tasković’s theorem
Notice here that \(\triangle(a_{1},\ldots, a_{n}):=f(a_{1},\ldots, a_{n})\) satisfies part of the axioms of the multiply metric. More connections with the above mentioned results will be given in a forthcoming paper.
The following result is another multivariate fixed point theorem for a class of generalized contraction mappings related to the SY theorem. The proof of it can be obtained by Theorem SY, in the same way as was used in the proof of Theorem 2.6.
Theorem 2.10
In [7], Su and Yao also gave some examples of functions \(\psi(t)\), \(\phi(t)\). Here we recall some of them.
Example 2.11
([7])
For example, if we choose \(\psi_{5}(t)\), \(\phi_{5}(t)\) in Theorem 2.10, then we can get the following result.
Theorem 2.12
Using the following notions it is easy to prove another consequence of our main results.
Remark 2.13
 (i)
\(\psi(0)=\phi(0)\);
 (ii)
\(\psi(t)>\phi(t)\), \(\forall t>0\);
 (iii)
ψ is lower semicontinuous and ϕ is upper semicontinuous.
Then \(\psi(t)\), \(\phi(t)\) satisfy the above mentioned conditions (1) and (2).
Corollary 2.14
3 An application to an initialvalue problem related to a first order differential equation
4 N variable nonexpansive mappings in normed spaces
We will introduce first the concept of N variable nonexpansive mapping.
Definition 4.1
Some useful results are the following.
Lemma 4.2
Proof
 (1)
\(\langle x,x\rangle^{*}=\frac{1}{N}\sum_{i=1}^{N}\langle x_{i},x_{i}\rangle\geq0\) and \(\langle x,x\rangle^{*}=0 \Leftrightarrow x=0\), \(\forall x=(x_{1},x_{2},\ldots,x_{N}) \in X^{N}\);
 (2)
\(\langle x,y\rangle^{*}=\langle y,x\rangle^{*}\), \(\forall x,y \in X^{N}\);
 (3)
\(\langle\lambda x,y\rangle^{*}=\frac{1}{N}\sum_{i=1}^{N}\langle \lambda x_{i},y_{i}\rangle= \lambda\frac{1}{N} \sum_{i=1}^{N}\langle x_{i},y_{i}\rangle=\lambda\langle x,y\rangle^{*} \), \(\forall x,y \in X^{N}\);
 (4)
\(\langle x+y,z\rangle^{*}=\langle x,z\rangle^{*}+\langle y,z\rangle^{*}\), \(\forall x,y,z \in X^{N}\).
Hence \((X^{N}, \langle\cdot,\cdot\rangle^{*})\) is an inner product space.
Lemma 4.3
 (1)
\((X^{N})^{*}=X^{*}\times X^{*}\times\cdots\times X^{*}\);
 (2)\(f \in(X^{N})^{*}\) if and only if there exist \(f_{i} \in X^{*}\), \(i\in\{1,2,3, \ldots,N \}\) such that$$f(x)=\frac{1}{N}\sum_{i=1}^{N}f_{i}(x_{i}), \quad \forall x=(x_{1},x_{2},\ldots,x_{N}) \in X^{N}. $$
Proof
Theorem 4.4
 (C_{1}):

\(\lim_{n\rightarrow\infty}\alpha_{n}=0\);
 (C_{2}):

\(\sum_{n=1}^{\infty}\alpha_{n}=+\infty\);
 (C_{3}):

\(\sum_{n=1}^{\infty}\alpha_{n+1}\alpha _{n}<+\infty\).
Proof
If the condition (C_{3}) can be replaced by the condition (C_{4}) [25] or the condition (C_{5}) [26], then Theorem 4.4 still holds.
If T is a nonexpansive mapping with at least one fixed point and if the control sequence \(\{\alpha_{n}\}\) is chosen so that \(\sum_{n=0}^{\infty}\alpha_{n}(1\alpha_{n})=+\infty\), then the sequence \(\{x_{n}\}\) generated by Mann’s algorithm (4.3) converges weakly, in a uniformly convex Banach space with a Fréchet differentiable norm (see [27]), to a fixed point of T.
Next we prove a weak convergence theorem for a Nvariables nonexpansive mapping in Hilbert spaces.
Theorem 4.5
Then the sequence \(\{x_{i}^{n}\}\) converges weakly to a multivariate fixed point p of T.
Proof
Remark
The above presented method can successfully be applied for several other iterative schemes in order to prove weak and strong convergence theorems for the multivariate fixed points of Nvariables nonexpansive type mappings.
Declarations
Acknowledgements
For this work, the second author benefits from the financial support of a grant of the Romanian National Authority for Scientific Research, CNCSUEFISCDI, project number PNIIIDPCE201130094. The third author was partially supported by the Grant MOST 1032923E039001MY3.
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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