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Fixed point theory for contractive mappings satisfying $$\Phi$$-maps in G-metric spaces. (English) Zbl 1204.54039
The $$\varphi$$-contraction principle in a complete metric space is well known (see [J. Matkowski, ”Integrable Solutions of Functional Equations,” Dissertationes Math., Warszawa 127, 63 p. (1975; Zbl 0318.39005)]; I. A. Rus, [Math., Rev. Anal. Numér. Théor. Approximation, Math. 24(47), 175–178 (1982; Zbl 0525.54029) and Generalized Contractions and Applications. Cluj-Napoca: Cluj University Press. (2001; Zbl 0968.54029)]; J. Jachymski and M. Styborski, Fixed point theory and its applications. Proceedings of the international conference, Bȩddlewo, Poland, August 1–5, 2005. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 77, 123-146 (2007; Zbl 1149.47044)]).
In this paper a fixed point principle for $$\varphi$$-contractions in a complete G-metric space (see Z. Mustafa and B. Sims, [J. Nonlinear Convex Anal. 7, No. 2, 286–297 (2006; Zbl 1111.54025)]) is presented. Then a theory [I. A. Rus, Fixed Point Theory 9, No. 2, 541–559 (2008; Zbl 1172.54030)] of this metrical fixed point theorem is discussed.

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects) 54E35 Metric spaces, metrizability
##### Keywords:
fixed point; G-metric space; $$\varphi$$-contraction
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##### References:
 [1] Mustafa, Z; Sims, B, A new approach to generalized metric spaces, Journal of Nonlinear and Convex Analysis, 7, 289-297, (2006) · Zbl 1111.54025 [2] Mustafa, Z; Obiedat, H; Awawdeh, F, Some fixed point theorem for mapping on complete [inlineequation not available: see fulltext.]-metric spaces, 12, (2008) [3] Mustafa Z, Sims B: Some remarks concerning -metric spaces. In Proceedings of the International Conference on Fixed Point Theory and Applications, 2004, Yokohama, Japan. Yokohama; 189-198. · Zbl 1079.54017 [4] Mustafa, Z; Sims, B, Fixed point theorems for contractive mappings in complete [inlineequation not available: see fulltext.]-metric spaces, 10, (2009) [5] Mustafa, Z; Shatanawi, W; Bataineh, M, Existence of fixed point results in [inlineequation not available: see fulltext.]-metric spaces, 10, (2009) [6] Matkowski, J, Fixed point theorems for mappings with a contractive iterate at a point, Proceedings of the American Mathematical Society, 62, 344-348, (1977) · Zbl 0349.54032 [7] Mustafa Z: A new structure for generalized metric spaces with applications to fixed point theory, Ph.D. thesis. University of Newcastle, Newcastle, UK; 2005.
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