Some fixed point results on a metric space with a graph. (English) Zbl 1237.54042

Aleomraninejad, S. M. A.; Rezapour, Sh.; Shahzad, N.

Topology Appl. 159, No. 3, 659-663 (2012).

The main results of this paper (Theorems 2.1–2.4) deal with the stability (not explicitly called so) for the Picard iteration associated to \(G\)-contractions or \(G\)-nonexpansive mappings defined on a metric space endowed with a graph \(G\).

Reviewer: Vasile Berinde (Baia Mare)

MSC:

54H25	Fixed-point and coincidence theorems (topological aspects)
54E40	Special maps on metric spaces
05C63	Infinite graphs
47J25	Iterative procedures involving nonlinear operators
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Keywords:

metric space; fixed point; graph; \(G\)-contractive mapping; \(G\)-nonexpansive mapping; iterative scheme

References:

[1]	S.M.A. Aleomraninejad, Sh. Rezapour, N. Shahzad, Convergence of an iterative scheme for multifunctions, J. Fixed Point Theory Appl. (2011), doi:10.1007/s11748-011-0046-z, in press. · Zbl 1282.47072
[2]	Beg, I.; Butt, A.R.; Radojević, S., The contraction principle for set valued mappings on a metric space with a graph, Comput. math. appl., 60, 1214-1219, (2010) · Zbl 1201.54029
[3]	De Blasi, F.S.; Myjak, J.; Reich, S.; Zaslavski, A.J., Generic existence and approximation of fixed points for nonexpansive set-valued maps, Set-valued var. anal., 17, 97-112, (2009) · Zbl 1183.47055
[4]	Diestel, R., Graph theory, (2000), Springer-Verlag New York · Zbl 0945.05002
[5]	Echenique, F., A short and constructive proof of tarskiʼs fixed point theorem, Internat. J. game theory, 33, 2, 215-218, (2005) · Zbl 1071.91002
[6]	Espinola, R.; Kirk, W.A., Fixed point theorems in R-trees with applications to graph theory, Topology appl., 153, 1046-1055, (2006) · Zbl 1095.54012
[7]	Gwozdz-Lukawska, G.; Jachymski, J., IFS on a metric space with a graph structure and extensions of the kelisky-Rivlin theorem, J. math. anal. appl., 356, 453-463, (2009) · Zbl 1171.28002
[8]	Jachymski, J., The contraction principle for mappings on a metric space with a graph, Proc. amer. math. soc., 136, 4, 1359-1373, (2008) · Zbl 1139.47040
[9]	Kelisky, R.P.; Rivlin, T.J., Iterates of Bernstein polynomials, Pacific J. math., 21, 511-520, (1967) · Zbl 0177.31302
[10]	Ran, A.C.M.; Reurings, M.C.B., A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. amer. math. soc., 132, 5, 1435-1443, (2003) · Zbl 1060.47056
[11]	Reich, S.; Zaslavski, A.J., Convergence of inexact iterative schemes for nonexpansive set-valued mappings, Fixed point theory appl., (2010), Article ID 518243, 10 pp · Zbl 1214.47074
[12]	Reich, S.; Zaslavski, A.J., Approximating fixed points of contractive set-valued mappings, Commun. math. anal., 8, 70-78, (2010) · Zbl 1171.47056
[13]	Reich, S.; Zaslavski, A.J., Existence and approximation of fixed points for set-valued mappings, Fixed point theory appl., (2010), Article ID 351531, 10 pp · Zbl 1189.54037
[14]	Rus, I.A., Iterates of Bernstein operators, via contraction principle, J. math. anal. appl., 292, 259-261, (2004) · Zbl 1056.41004
[15]	Suzuki, T., A generalized Banach contraction principle that characterizes metric completeness, Proc. amer. math. soc., 136, 1861-1869, (2008) · Zbl 1145.54026
[16]	Suzuki, T., A new type of fixed point theorem in metric space, Nonlinear anal., 71, 5313-5317, (2009) · Zbl 1179.54071