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Abstract  

We define an alternate convexically nonexpansive map T on a bounded, closed, convex subset C of a Banach space X and prove that if X is a strictly convex Banach space and C is a nonempty weakly compact convex subset of X, then every alternate convexically nonexpansive map T : CC has a fixed point. As its application, we give an existence result for the solution of an integral equation.

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Abstract  

We extend the notion of R-weak commutativity and its variants to probabilistic metric spaces and prove common fixed point theorems concerning them. Examples are included to reflect upon the distinctiveness of the types of mappings defined in the paper.

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Dhage, B. C. , Generalized Metric Space and Mappings With Fixed Point, Bull. Cal. Math. Soc. , 84 (1992), 329–336. MR 93k :54075 Dhage B. C

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Abstract  

The main result of this paper is a fixed point theorem of self-mappings in Menger spaces which satisfy certain inequality. This inequality involves a class of real functions which we call Φ-functions. As a corollary we obtain a result in the corresponding metric spaces. The result is supported by an example. The class of real functions we have used is the conceptual extension of altering distance functions used in metric fixed point theory.

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Abstract  

We prove new common fixed point theorems for weakly compatible mappings on uniform spaces. Also, an application to locally convex spaces is presented.

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Goebel, K. and Kirk, W. A. , A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972), 171–174. MR 45 #7552 Kirk W. A. A fixed

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Abstract  

The aim of this paper is to prove some fixed point theorems which generalize well known basic fixed point principles of nonlinear functional analysis. Moreover, we investigate the class of mappings f: X→ X, where X is a Banach space, for which one of the main conditions in the metric fixed point theory, namely the condition (1), is satisfied. We obtain essential applications of this fact. All our results are illustrated by suitable examples.

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Summary  

Amini-Harandi proved that alternate convexically nonexpansive mappings on non-empty weakly compact convex subsets of strictly convex Banach spaces have fixed points. We prove that Amini-Harandi's result holds also in Banach spaces with the Kadec--Klee property and the result is true for a larger class of mappings. Moreover, we show that the Alspach mapping in L 1[0,1] is not a 2-alternate convexically nonexpansive mapping.

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Acta Mathematica Hungarica
Authors: Lj. B. Ćirić, Lj. B. Ćirić, Lj. B. Ćirić, N. T. Nikolić, N. T. Nikolić, N. T. Nikolić, Ume J. S., Ume J. S., and Ume J. S.

Summary  

Recently, Pathak [13] has made an extension of the notion of compatibility to weak compatibility, and extended the coincidence theorem for compatible mappings in Kaneko and Sessa [11] to weakly compatible mappings [13]. In the present paper, we define a new class of weakly compatible mappings (Definition 4) and prove some common fixed point theorems for these mappings, which satisfy Condition (2) below. Although our main theorem is formulated for weakly compatible mappings, its corresponding formulation for commutative mappings is also a new result, thus presenting a generalization of some theorems of Fisher, Das and Naik, Khan and Kubiaczyk, Reich, Ćirić and Rhoades and Watson.

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