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Some fixed point results on a metric space with a graph. (English) Zbl 1237.54042
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.
MSC:
54H25Fixed-point and coincidence theorems in topological spaces
54E40Special maps on metric spaces
05C63Infinite graphs
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
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.
[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 · doi:10.1016/j.camwa.2010.06.003
[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 · doi:10.1007/s11228-009-0104-5
[4]Diestel, R.: Graph theory, (2000)
[5]Echenique, F.: A short and constructive proof of Tarski’s fixed point theorem, Internat. J. Game theory 33, No. 2, 215-218 (2005) · Zbl 1071.91002 · doi:10.1007/s001820400192
[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 · doi:10.1016/j.topol.2005.03.001
[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 · doi:10.1016/j.jmaa.2009.03.023
[8]Jachymski, J.: The contraction principle for mappings on a metric space with a graph, Proc. amer. Math. soc. 136, No. 4, 1359-1373 (2008)
[9]Kelisky, R. P.; Rivlin, T. J.: Iterates of Bernstein polynomials, Pacific J. Math. 21, 511-520 (1967)
[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, No. 5, 1435-1443 (2003) · Zbl 1060.47056 · doi:10.1090/S0002-9939-03-07220-4
[11]Reich, S.; Zaslavski, A. J.: Convergence of inexact iterative schemes for nonexpansive set-valued mappings, Fixed point theory appl. (2010) · Zbl 1214.47074 · doi:10.1155/2010/518243
[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) · Zbl 1189.54037 · doi:10.1155/2010/351531
[14]Rus, I. A.: Iterates of Bernstein operators, via contraction principle, J. math. Anal. appl. 292, 259-261 (2004) · Zbl 1056.41004 · doi:10.1016/j.jmaa.2003.11.056
[15]Suzuki, T.: A generalized Banach contraction principle that characterizes metric completeness, Proc. amer. Math. soc. 136, 1861-1869 (2008) · Zbl 1145.54026 · doi:10.1090/S0002-9939-07-09055-7
[16]Suzuki, T.: A new type of fixed point theorem in metric space, Nonlinear anal. 71, 5313-5317 (2009) · Zbl 1179.54071 · doi:10.1016/j.na.2009.04.017