Multivariate contraction mapping principle with the error estimate formulas in locally convex topological vector spaces and application
-
1773
Downloads
-
2708
Views
Authors
Yongchun Xu
- 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.
Yanxia Tang
- 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 paper is to present the concept of multivariate contraction mapping in a locally convex topological
vector spaces and to prove the multivariate contraction mapping principle in such spaces. The neighborhood-type error estimate
formulas are also established. The results of this paper improve and extend Banach contraction mapping principle in the new
idea.
Share and Cite
ISRP Style
Yongchun Xu, Jinyu Guan, Yanxia Tang, Yongfu Su, Multivariate contraction mapping principle with the error estimate formulas in locally convex topological vector spaces and application, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 3, 1064--1074
AMA Style
Xu Yongchun, Guan Jinyu, Tang Yanxia, Su Yongfu, Multivariate contraction mapping principle with the error estimate formulas in locally convex topological vector spaces and application. J. Nonlinear Sci. Appl. (2017); 10(3):1064--1074
Chicago/Turabian Style
Xu, Yongchun, Guan, Jinyu, Tang, Yanxia, Su, Yongfu. "Multivariate contraction mapping principle with the error estimate formulas in locally convex topological vector spaces and application." Journal of Nonlinear Sciences and Applications, 10, no. 3 (2017): 1064--1074
Keywords
- Contraction mapping principle
- locally convex
- topological vector spaces
- fixed point
- error estimate formula.
MSC
References
-
[1]
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.
-
[2]
D. W. Boyd, J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc., 20 (1969), 458–464.
-
[3]
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.
-
[4]
M. A. Geraghty, On contractive mappings, Proc. Am. Math. Soc., 40 (1973), 604–608.
-
[5]
T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. , 65 (2006), 1379–1393.
-
[6]
J. Harjani, K. Sadarangani, Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear Anal., 71 (2009), 3403–3410.
-
[7]
J. Harjani, K. Sadarangani, Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Anal., 72 (2010), 1188–1197.
-
[8]
J. R. Jachymski, Equivalence of some contractivity properties over metrical structures, Proc. Amer. Math. Soc., 125 (1997), 2327–2335.
-
[9]
J. R. Jachymski, I. Jóźwik, Nonlinear contractive conditions: a comparison and related problems, Fixed point theory and its applications, Banach Center Publ., Polish Acad. Sci. Inst. Math., Warsaw, 77 (2007), 123–146.
-
[10]
M. S. Khan, M. Swaleh, S. Sessa, Fixed point theorems by altering distances between the points, Bull. Austral. Math. Soc., 30 (1984), 1–9.
-
[11]
V. Lakshmikantham, L. Ćirić, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal., 70 (2009), 4341–4349.
-
[12]
P. Liu, Basis of topological vector spaces, (Chinese) Wuhan University Press, (2002)
-
[13]
J. J. Nieto, R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22 (2005), 223–239.
-
[14]
J. J. Nieto, R. Rodríguez-López, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sin. (Engl. Ser.), 23 (2007), 2205–2212.
-
[15]
D. O’Regan, A. Petruşel, Fixed point theorems for generalized contractions in ordered metric spaces, J. Math. Anal. Appl., 341 (2008), 1241–1252.
-
[16]
B. Samet, C. Vetro, P. Vetro, Fixed point theorems for \(\alpha -\psi\)-contractive type mappings, Nonlinear Anal., 75 (2012), 2154–2165.
-
[17]
Y.-F. Su, A. Petruşel, J.-C. Yao, Multivariate fixed point theorems for contractions and nonexpansive mappings with applications, Fixed Point Theory Appl., 2016 (2016), 19 pages.
-
[18]
Y.-F. Su, J.-C. Yao, Further generalized contraction mapping principle and best proximity theorem in metric spaces, Fixed Point Theory Appl., 2015 (2015), 13 pages.
-
[19]
Y.-X. Tang, J.-Y. Guan, P.-C. Ma, Y.-C. Xu, Y.-F. Su, Generalized contraction mapping principle in locally convex topological vector spaces, J. Nonlinear Sci. Appl., 9 (2016), 4659–4665.
-
[20]
F.-F. Yan, Y.-F. Su, Q.-S. 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.