Non-self multivariate contraction mapping principle in Banach spaces

Volume 10, Issue 9, pp 4704--4712 http://dx.doi.org/10.22436/jnsa.010.09.13

Publication Date: September 09, 2017 Submission Date: April 07, 2017

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Authors

Yanxia Tang - Department of Mathematics, College of Science, Hebei North University, Zhangjiakou 075000, China. Jinyu Guan - Department of Mathematics, College of Science, Hebei North University, Zhangjiakou 075000, China. Yongchun Xu - Department of Mathematics, College of Science, Hebei North University, Zhangjiakou 075000, China. Yongfu Su - Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China.

Abstract

The purpose of this article is to prove the non-self multivariate contraction mapping principle in a Banach space. The main result is the following: let \(C\) be a nonempty closed convex subset of a Banach space \((X,\|\cdot\|)\). Let \(T: C \rightarrow X\) be a weakly inward \(N\)-variables non-self contraction mapping. Then \(T\) has a unique multivariate fixed point \(p\in C\). That is, there exists a unique element \(p \in C\) such that \(T(p,p,\cdots ,p)=p\). In order to get the non-self multivariate contraction mapping principle, the inward and weakly inward \(N\)-variables non-self mappings are defined. In addition, the meaning of \(N\)-variables non-self contraction mapping \(T: C \rightarrow X\) is the following: \[ \|Tx-Ty\|\leq h \nabla (\|x_1-y_1\|, \|x_2-y_2\|,\cdots ,\|x_N-y_N\|) \] for all \(x=(x_1,x_2, \cdots, x_N), \ y=(y_1,y_2, \cdots, y_N)\in C^N\), where \(h \in (0,1)\) is a constant, and \(\nabla\) is an \(N\)-variables real function satisfying some suitable conditions. The results of this article improve and extend the previous results given in the literature.

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ISRP Style

Yanxia Tang, Jinyu Guan, Yongchun Xu, Yongfu Su, Non-self multivariate contraction mapping principle in Banach spaces, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 9, 4704--4712

AMA Style

Tang Yanxia, Guan Jinyu, Xu Yongchun, Su Yongfu, Non-self multivariate contraction mapping principle in Banach spaces. J. Nonlinear Sci. Appl. (2017); 10(9):4704--4712

Chicago/Turabian Style

Tang, Yanxia, Guan, Jinyu, Xu, Yongchun, Su, Yongfu. "Non-self multivariate contraction mapping principle in Banach spaces." Journal of Nonlinear Sciences and Applications, 10, no. 9 (2017): 4704--4712

Keywords

Non-self mapping
Caristi’s fixed point theorem
contraction mapping principle
multivariate fixed point
inward condition
weakly inward condition
iterative sequence.

MSC

47H05
47H10

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