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Fixed points of asymptotically regular mappings

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Abstract

Two general fixed point theorems for asymptotically regular self-mappings on a metric space X which satisfy the contractive condition (1) below are proved. Our results extend and generalize results of Sharma and Yuel [4] and Guay and Singh [3].

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... Sharma and Yuel [16] and Guay and Singh [12] were among the first who used concept of asymptotic regularity to prove fixed point theorems for a wider class of mappings than a class of mappings introduced and studied bý Cirić in [5]. In (2005),Ćirić (see [6]) has generalized the results of [16] and [12] by the following result. Theorem 1.1. ...
... The following theorem unifies and generalizes two theorems obtained by Sharma and Yuel in [16] and by Lj.Ćirić in [6]. ...
... Theorem 3.1 generalizes Theorem 1.1 obtained by Lj. B. Cirić in[6]. ...
... The concept of asymptotic regularity of a mapping is an useful tool for researchers working in the area of fixed point theory. Several researchers have used this concept to find fixed point of different types of mappings in metric spaces (see [6], [19], [22]). Definition 1.11. ...
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... As an exotic result we mention the following one obtained by Górnicki [72]. We end this section with a result concerning operators which are not necessarily continuous, obtained by Guay and Singh [82], see also [44], [125], [142]. for all x, y ∈ M, where 0 ≤ a, c; a + 2c < 1 and b + c < 1. ...
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Let (M,d) be a metric space. In this paper we survey some of the most relevant results which relate the three concepts involved in the title: a) the asymptotic regularity; b) the existence (and uniqueness) of fixed points and c) the convergence of the sequence of successive approximations to the fixed point(s), for a given operator f : M ? M or for two operators f,g : M ? M connected to each other in some sense.
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... Since then, many interesting coincidence and common fixed point theorems of compatible and weakly compatible maps under various contractive conditions and assuming the continuity of at least one of the mappings, have been obtained by a number of authors. ´ Ciri´c [5] studied necessary conditions to obtain a fixed point result of asymptotically regular mappings on complete metric spaces. The purpose of this paper is to present a common fixed point theorem for two mappings satisfying a generalized contractive condition. ...
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Two general fixed point theorems for asymptotically regular self-mappings on a metric space X which satisfy the contractive condition (1) below are proved. Our results extend and generalize results of Sharma and Yuel [4] and Guay and Singh [3].