Browder and Göhde fixed point theorem for \(G\)-nonexpansive mappings
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Authors
Monther Rashed Alfuraidan
- Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia.
Sami Atif Shukri
- Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia.
Abstract
In this paper, we prove the analog to Browder and Göhde fixed point theorem for \(G\)-nonexpansive
mappings in complete hyperbolic metric spaces uniformly convex. In the linear case, this result is refined.
Indeed, we prove that if X is a Banach space uniformly convex in every direction endowed with a graph \(G\),
then every \(G\)-nonexpansive mapping \(T : A \rightarrow A\), where \(A\) is a nonempty weakly compact convex subset of
\(X\), has a fixed point provided that there exists \(u_0 \in A\) such that \(T(u_0)\) and \(u_0\) are \(G\)-connected.
Share and Cite
ISRP Style
Monther Rashed Alfuraidan, Sami Atif Shukri, Browder and Göhde fixed point theorem for \(G\)-nonexpansive mappings, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 6, 4078--4083
AMA Style
Alfuraidan Monther Rashed, Shukri Sami Atif, Browder and Göhde fixed point theorem for \(G\)-nonexpansive mappings. J. Nonlinear Sci. Appl. (2016); 9(6):4078--4083
Chicago/Turabian Style
Alfuraidan, Monther Rashed, Shukri, Sami Atif. "Browder and Göhde fixed point theorem for \(G\)-nonexpansive mappings." Journal of Nonlinear Sciences and Applications, 9, no. 6 (2016): 4078--4083
Keywords
- Directed graph
- fixed point
- G-nonexpansive mapping
- hyperbolic metric space
- Mann iteration
- uniformly convex space.
MSC
References
-
[1]
M. R. Alfuraidan, The contraction principle for multivalued mappings on a modular metric space with a graph, Canad. Math. Bull., 59 (2015), 3-12.
-
[2]
M. R. Alfuraidan, Fixed points of monotone nonexpansive mappings with a graph, Fixed Point Theory Appl., 2015 (2015), 6 pages.
-
[3]
M. R. Alfuraidan, On monotone Ćirić quasi-contraction mappings with a graph, Fixed Point Theory Appl., 2015 (2015), 11 pages.
-
[4]
B. Beauzamy, Introduction to Banach spaces and their geometry, North-Holland Mathematics Studies, Amsterdam (1985)
-
[5]
B. A. Bin Dehaish, M. A. Khamsi, Browder and Göhde fixed point theorem for monotone nonexpansive mappings, Fixed Point Theory Appl., 2016 (2016), 9 pages.
-
[6]
F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A., 54 (1965), 1041-1044.
-
[7]
H. Busemann, Spaces with non-positive curvature, Acta. Math., 80 (1948), 259-310.
-
[8]
A. L. Garkavi, The best possible net and the best possible cross-section of a set in a normed space, Amer. Math. Soc., Transl. II, 39 (1964), 111-132.
-
[9]
D. Göhde, Zum Prinzip der kontraktiven Abbildung, (German) Math. Nachr., 30 (1965), 251-258.
-
[10]
K. Goebel, W. A. Kirk, Iteration processes for nonexpansive mappings, Topological methods in nonlinear functional analysis, Amer. Math. Soc. Providence, RI, (1983), 115-123.
-
[11]
K. Goebel, W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (1990)
-
[12]
K. Goebel, S. Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York (1984)
-
[13]
J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc., 136 (2008), 1359-1373.
-
[14]
M. A. Khamsi , On metric spaces with uniform normal structure, Proc. Amer. Math. Soc., 106 (1989), 723-726.
-
[15]
W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly, 72 (1965), 1004-1006.
-
[16]
W. A. Kirk, Fixed point theory for nonexpansive mappings, Springer, Berlin, 886 (1981), 485-505
-
[17]
W. A. Kirk, Fixed point theorems in CAT(0) spaces and R-trees, Fixed Point Theory Appl., 2004 (2004), 309-316.
-
[18]
L. Leustean, A quadratic rate of asymptotic regularity for CAT(0) spaces, J. Math. Anal. Appl., 325 (2007), 386-399.
-
[19]
W. R. Mann, Mean value method in iteration, Proc. Amer. Math. Soc., 4 (1953), 506-510.
-
[20]
K. Menger, Untersuchungen über allgemeine Metrik, (German) Math. Ann., 100 (1928), 75-163.
-
[21]
J. J. Nieto, R. L. Pouso, R. Rodríguez-López, Fixed point theorems in ordered abstract spaces, Proc. Amer. Math. Soc., 135 (2007), 2505-2517.
-
[22]
A. C. M. Ran, M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc., 132 (2004), 1435-1443.
-
[23]
S. Reich, I. Shafrir , Nonexpansive iterations in hyperbolic spaces , Nonlinear Anal., 15 (1990), 537-558.
-
[24]
J. Tiammee, A. Kaewkhao, S. Suantai, On Browder's convergence theorem and Halpern iteration process for G-nonexpansive mappings in Hilbert spaces endowed with graphs, Fixed Point Theory Appl., 2015 (2015), 12 pages.