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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].

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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. ...

In this paper, we investigate the existence of ϕ− fixed point for Banach orbital contraction over U−metric space. Also a fixed point result has been established via asymptotic regularity property over such generalized metric space. Our fixed point theorems have also been applied to the fixed circle problem. Moreover, we give some new solutions to the open problem raised by Özgür and Taş on the geometric properties of ϕ-fixed points of self-mappings and the existence and uniqueness of ϕ-fixed circles and ϕ-fixed discs for various classes of self-mappings.

... In 2005, Lj. B.Ćirić [3] established the following result. Theorem 1. ([3]) Let (X, d) be a complete metric space and T be a self-mapping on X satisfying the following condition: for all x, y in X, where ai = ai(x, y) (i = 1, 2, 3, 4, 5) are nonnegative functions for which there exist three constants K > 0 and λ1, λ2 ∈ (0, 1) such that the following inequalities, a1(x, y), a2(x, y) ≤ K, (1.2) a4(x, y) + a5(x, y) ≤ λ1, (1.3) a3(x, y) + 2a5(x, y) ≤ λ2, (1.4) are satisfied for all x, y in X. ...

In 2005, Lj. B. Ciric studied fixed points of asymptotically regular mappings [Math. Commun. 10(2005), 111-114]. The results of that study were extended by several authors. The aim of this article is to extend the main results to the context of g-metric spaces with order n, which was introduced by H. Choi, S. Kim, and S. Y. Yang in 2018
asymptotically regular mapping

... 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. ...

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.

... Results similar to the above corollaries are available in usual metric spaces (see, e.g., [8]). In the following we illustrate the existence of self map which satisfies contractive condition of Corollary 10, in partial metric space but not in usual metric space. ...

The notion of asymptotically regular mapping in partial metric spaces is introduced, and a fixed point result for the mappings of this class is proved. Examples show that there are cases when
new results can be applied, while old ones (in metric space) cannot. Some common fixed point theorems for sequence of mappings in partial metric spaces are also proved which generalize and improve some known results
in partial metric spaces.

... 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. ...

Existence of common fixed points is established for two self-mappings satisfying a generalized contractive
condition. The presented results generalize several well known comparable results in the literature. We also study well-posedness of a common fixed point problem related to these mappings.

In this chapter, we generalize the recently introduced notions of properties S and S0 by introducing the properties Sg and S0g respectively and utilize these notions to prove a common fixed point theorem in fuzzy metric spaces. As the application of our main result, we prove the existence of unique common solution to the system of Fredholm integral equations. We also provide some non-trivial examples to support the claim and usability of the present results. In this way, our main result generalizes the fixed point theorem of Shukla et al. [1] for a pair of
mappings using Z-contraction.

Professor Ljubomir B. Ćirić, one of the pioneers in the study of fixed point theory and nonlinear analysis in Serbia, died on Saturday, 23th July 2016. Here, we present his brief biography, some comments on main streams of his research work and complete bibliography.

In this paper, we prove common coupled fixed point theorems for mappings T : X × X → X and g : X → X satisfying a generalized contractive condition on a metric space. We provide examples of new concepts introduced herein. We also study the well posedness of a common coupled fixed point problem. Our results generalize several well known comparable results in the literature.

Let T be a self-mapping on a complete metric space(X, d). Then T has a fixed point if there exist self-mappings a1( a2, a3, a4, a5 on [0, co) such that (a) a1(i)-r-a2(i)+a3(i)-r-a4(i)+a5(i)for 0, (b) each o^ is upper semicontinuous from the right, (c)d(T(x), T(y)) axd(x, T(x)) + a, d(y, T(y)) + a, d(x, T(y))+ atd(y, T(x)) + a5d(x, y)for all pairs of distinct x, y in X, where ai=txi(d(x, y))/d(x, y).Related results are obtained for two mappings and mappings on abounded convex subset of a uniformly convex Banach space.

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].