# Non-self multivariate contraction mapping principle in Banach spaces

Volume 10, Issue 9, pp 4704--4712
Publication Date: September 09, 2017 Submission Date: April 07, 2017
• 1029 Views

### 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.

### Share and Cite

##### 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.

•  47H05
•  47H10

### References

• [1] R. Agarwal, D. Regan, D. Rahu , Fixed point theory for Lipschitzian-type mappings with applications, Springer, New York (2009)

• [2] A. Amini-Harandi, H. Emami , A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations, Nonlinear Anal., 72 (2010), 2238–2242.

• [3] D. W. Boyd, J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc., 20 (1969), 458–464.

• [4] F. E. Browder, On the convergence of successive approximations for nonlinear functional equations , Nederl. Akad. Wetensch. Proc. Ser. A 71, Indag. Math., 30 (1968), 27–35.

• [5] J. Caristi , Fixed point theorem for mappings satisfying inwardness conditions, Tran. Amer. Math. Soc., 215 (1976), 241– 251.

• [6] M. A. Geraghty, On contractive mappings , Proc. Am. Math. Soc., 40 (1973), 604–608.

• [7] T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal., 65 (2006), 1379–1393.

• [8] J. Harjani, K. Sadarangni , Fixed point theorems for weakly contraction mappings in partially ordered sets , Nonlinear Anal., 71 (2009), 3403–3410.

• [9] J. Harjani, K. Sadarangni , Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Anal., 72 (2010), 1188–1197.

• [10] J. R. Jachymski, Equivalence of some contractivity properties over metrical structures, Proc. Amer. Math. Soc., 125 (1997), 2327–2335.

• [11] J. Jachymski, I. Jóźwik, Nonlinear contractive conditions: a comparison and related problems, in: Fixed Point Theory and its Applications, Banach Center Publisher, 77 (2007), 123–146.

• [12] M. S. Khan, M. Swaleh, S. Sessa, Fixed point theorems by altering distances between the points, Bull. Aust. Math. Soc., 30 (1984), 1–9.

• [13] V. Lakshmikantham, L. Ciric, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal., 70 (2009), 4341–4349.

• [14] J. Nieto, R. Rodriguez-López , Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22 (2005), 223–239.

• [15] J. J. Nieto, R. Rodriguez-López , Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sin., 23 (2007), 2205–2212.

• [16] B. Samet, C. Vetro, P. Vetro , Fixed point theorem for $\alpha-\psi$-contractive type mappings, Nonlinear Anal., 75 (2012), 2154– 2165.

• [17] Y. Su, A. Petruşel, J. Yao , Multivariate fixed point theorems for contractions and nonexpansive mappings with applications, Fixed Point Theory and Appl., 2016 (2016), 19 pages.

• [18] Y. Su, J.-C. Yao, Further generalized contraction mapping principle and best proximity theorem in metric spaces, Fixed Point Theory and Appl., 2015 (2015), 13 pages.

• [19] F. Yan Y. Su, Q. Feng, A new contraction mapping principle in partially ordered metric spaces and applications to ordinary differential equations, Fixed Point Theory Appl., 2012 (2012), 13 pages.