Özet:

In this paper a general fixed point theorem for mappings satisfying an phi - implicit relation is proved, which generalize the results from [3] and [13].In [4], [5] Dhage introduced a new class of generalized metric space, called D - metric space. Mustafa and Sims [12], [13] proved that most of the claims concerning the fundamental topological structures of D - metric spaces are incorrect and introduced an appropriate notion of generalized metric space, named G - metric space. In fact, Mustafa and other authors [3], [7] – [15], [18] studied many fixed point results for self mappings in a G - metric space under certain conditions. In [6] and [18] some fixed point theorems for mappings satisfying φ - maps are proved. In [16], [17], Popa initiated the study of fixed points for mappings satisfying implicit relations. In [2] Altun and Turkoglu introduced a new type of implicit relations satisfying a φ - map. The purpose of this paper is to prove a general fixed point theorem in G - metric spaces for mappings satisfying an φ - implicit relation which generalize the results from [3] and [14].   

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