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Journal ArticleDOI

Partial b-Metric Spaces and Fixed Point Theorems

01 May 2014-Mediterranean Journal of Mathematics (Springer Basel)-Vol. 11, Iss: 2, pp 703-711
TL;DR: In this paper, the concept of partial b-metric spaces is introduced as a generalization of partial metric and b-measure spaces, and an analog to Banach contraction principle, as well as a Kannan type fixed point result is proved in such spaces.
Abstract: The purpose of this paper is to introduce the concept of partial b-metric spaces as a generalization of partial metric and b-metric spaces. An analog to Banach contraction principle, as well as a Kannan type fixed point result is proved in such spaces. Some examples are given which illustrate the results.
Citations
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Journal ArticleDOI
TL;DR: In this article, the authors generalize a series of fixed point results in the framework of b-metric spaces and exemplify it by extending Nadler's contraction principle for set-valued functions.
Abstract: In this paper, we indicate a way to generalize a series of fixed point results in the framework of b-metric spaces and we exemplify it by extending Nadler’s contraction principle for set-valued functions (see Nadler, Pac J Math 30:475–488, 1969) and a fixed point theorem for set-valued quasi-contraction functions due to Aydi et al. (see Fixed Point Theory Appl 2012:88, 2012).

85 citations

Journal ArticleDOI
TL;DR: In this article, the authors give a survey of recent results on reducing fixed point theorems on generalized metric spaces to fixed-point theorem on metric spaces and then investigate this fact in other generalized metric spaces.
Abstract: There have been many attempts to generalize the definition of a metric space in order to obtain possibilities for more general fixed point results. In this paper, we give a survey of recent results on reducing fixed point theorems on generalized metric spaces to fixed point theorems on metric spaces and then investigate this fact in other generalized metric spaces. We show that many generalized metric spaces are topologically equivalent to certain metric spaces or to previously generalized metric spaces. Also, the fixed point theory in these generalized metric spaces may be a consequence of the fixed point theory in certain metric spaces or in previously generalized metric spaces.

84 citations


Cites background from "Partial b-Metric Spaces and Fixed P..."

  • ...complex valued b-metric spaces [102], quasi-metric-like spaces [143], M-metric spaces [22], quaternion-valued metric spaces [12], partial b-metric spaces [135] and others....

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  • ...There are other generalizations of metric spaces, with the same idea as partial metric spaces and cone metric spaces, such as partial cone metric spaces [99], partial rectangular metric spaces [134], quasi-partial metric spaces [87], partial b-metric spaces [107,135]....

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Journal ArticleDOI
TL;DR: In this paper, it was shown that every b -metric space with the topology induced by its convergence is a semi-metrizable space and thus many properties of b-metric spaces used in the literature are obvious.

79 citations

Journal ArticleDOI
TL;DR: In this article, a modified version of ordered partial b-metric spaces is introduced, and a fundamental lemma for the convergence of sequences in such spaces is demonstrated, and some fixed point and common fixed point results for (ψ, ϕ)-weakly contractive mappings are shown.
Abstract: In this paper, we introduce a modified version of ordered partial b-metric spaces. We demonstrate a fundamental lemma for the convergence of sequences in such spaces. Using this lemma, we prove some fixed point and common fixed point results for (ψ , ϕ)-weakly contractive mappings in the setup of ordered partial b-metric spaces. Finally, examples are presented to verify the effectiveness and applicability of our main results. MSC: 47H10; 54H25

71 citations

Posted Content
TL;DR: In this paper, the authors generalize a series of fixed point results in the framework of b-metric spaces and exemplify it by extending Nadler's contraction principle for set-valued functions.
Abstract: In this paper we indicate a way to generalize a series of fixed point results in the framework of b-metric spaces and we exemplify it by extending Nadler's contraction principle for set-valued functions (see Multi-valued contraction mappings, Pac. J. Math., 30 (1969), 475-488) and a fixed point theorem for set-valued quasi-contractions functions due to H. Aydi, M.F. Bota, E. Karapinar and S. Mitrovic (see A fixed point theorem for set-valued quasi-contractions in b-metric spaces, Fixed Point Theory Appl. 2012, 2012:88).

71 citations


Cites background from "Partial b-Metric Spaces and Fixed P..."

  • ...In the last period many mathematicians obtained fixed point results for single-valued or set-valued functions, in the setting of b-metric spaces (see, for example, [1], [8], [9], [10], [17], [23], [24], [29], [30] and the references therein)....

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References
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Journal ArticleDOI
TL;DR: This paper presents a symmetric generalised metric for such topologies, an approach which sheds new light on how metric tools such as Banach's Theorem can be extended to non‐Hausdorff topologies.
Abstract: Metric spaces are inevitably Hausdorff and so cannot, for example, be used to study non-Hausdorff topologies such as those required in the Tarskian approach to programming language semantics. This paper presents a symmetric generalised metric for such topologies, an approach which sheds new light on how metric tools such as Banach's Theorem can be extended to non-Hausdorff topologies.

1,090 citations

01 Jan 1993
TL;DR: In this article, a generalization of Banach's fixed point theorem in so-called b-metric spaces is presented, where the convergence of measurable functions with respect to measure leads to a generalisation of the notion of metric.
Abstract: Some generalizations of well known Banach's fixed point theorem in so-called b-metric spaces are presented. 1991 Mathematics Subject Classification: 47H10 1. Some problems, particurarly the problem of the convergence of measurable functions with respects to measure lead to a generalization of notion of metric. Using this idea we shall present generalization of some fixed point theorems of Banach type. Lex X be a spece and let R+ denotes the set of all nonnegative numbers. A function d : X x X —> R+ is said to be an b-metric iff for all x, y, z 6 X and all r > 0 the following conditions are satisfacted: d{x, y) = 0iffx = y (1) d{x, y) = d{y, x) (2) d{x, y) 0. Let us consider the following condition: d{y, z) R+ an b-metric iff the conditions (1) (2) and (5) are satisfied. For T : X -> X we denote by T then n-th iterate of T. 2. Now we present following Theorem 1. Let {X, d) be e a complete b-metric space and let T : X —> X satisfy d[T(x), T(y)} R+ is increasing function such that lirnn-+co P{t) = 0 for each fi.xed > 0. Then T has exactly one fixed point u and limn-+00d[T {x), u] = 0

952 citations

01 Jan 1968

777 citations

Journal ArticleDOI
TL;DR: Some results on fixed points were discussed in this article, where the authors proposed a method for computing fixed points in a fixed point set, using fixed points as the fixed point function.
Abstract: (1969). Some Results on Fixed Points—II. The American Mathematical Monthly: Vol. 76, No. 4, pp. 405-408.

519 citations