Fixed Point Theorems For Non-Self Mappings With Nonlinear Contractive Condition In Strictly Convex Menger $$PM\mbox{-}$$
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FIXED POINT THEOREMS FOR NON-SELF MAPPINGS WITH NONLINEAR CONTRACTIVE CONDITION IN STRICTLY CONVEX MENGER PM-SPACES ´2 ´ 1,∗ and V. T. RISTIC R. M. NIKOLIC 1 2
Belgrade Metropolitan University, Tadeuˇsa Koˇs´ cuˇska 63, 11000 Beograd, Serbia e-mail: [email protected]
Faculty of Education, University in Kragujevac, Milana Mijalkovi´ ca 14, 35000 Jagodina, Serbia e-mail: [email protected] (Received February 28, 2020; revised July 4, 2020; accepted July 7, 2020)
Abstract. We prove existence and uniqueness of a fixed point for non-self mappings with nonlinear contractive condition defined in strictly convex Menger PM-spaces.
1. Introduction The Banach Contraction Principle [2] is one of the most important theorems in functional analysis. There are many generalizations of the Banach Contraction Principle for classical metric spaces. One of the most important of them is the introduction of a nonlinear contractive principle by Boyd and Wong [3]. The notion of statistical metric spaces, as a generalization of metric spaces, with non-deterministic distance, was introduced by Menger [15] in 1942. Schweizer and Sklar [18,19] studied the properties of spaces introduced by Menger and gave some basic results on these spaces. They studied topology, convergence of sequences, continuity of mappings, defined the completeness of these spaces, etc. The first result from the fixed point theory in probabilistic metric spaces was obtained by Sehgal and Bharucha–Reid [20] as a generalization of the classical Banach Contraction Principle. ∗ Corresponding
author. This research is supported by the Ministry of Education, Science and Technological Development of Republic of Serbia, institutionally funded through the Faculty of Mathematics, University of Belgrade. Key words and phrases: Menger PM-space, strictly convex structure, fixed point, non-self mapping. Mathematics Subject Classification: 47H10.
0236-5294/$20.00 © 2020 Akade ´miai Kiado ´, Budapest, Hungary
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´ and V. T. RISTIC ´ R. M. NIKOLIC NIKOLIC RISTIC
Takahashi [22] defined convex and normal structures for sets in metric spaces and generalized some important fixed point theorems previously proved for Banach spaces. Hadˇzi´c [10] introduced the notion of a convex structure for sets in Menger probabilistic metric spaces (briefly, Menger PM-spaces) and proved the fixed point theorem for mappings in this spaces with a convex structure. Recently, Jeˇsi´c et al. [12] have defined a strictly convex structure in Menger PM-space. Furthermore, cases occur in convex spaces where the involved function is not necessarily a self-mapping of a closed subset. Assad and Kirk [1] first considered non-self mappings in metric spaces (X, d). They proved that for a non-self (single-valued) mapping f : C → X, which satisfies the well-known Banach Contraction Mapping Principle d(f x, f y) ≤ λd(x, y) for some λ ∈ (0, 1) and all x, y ∈ C, where X is a complete metrically convex space in the sense of Menger, the condition f (∂C) ⊆ C is sufficient to guarantee the existence of a fixed point for the mapping f
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