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In 1964, A. I. Perov generalized the Banach contraction principle introducing, following the work of ¯D. Kurepa, a new approach to fixed point problems, by defining generalized metric spaces (also known as vector valued metric spaces), and providing some actual results for the first time. Using the recent approach of coordinate representation for a generalized metric of Jachymski and Klima, we verify in this article some natural properties of generalized metric spaces, already owned by standard metric spaces. Among other results, we show that the theorems of Nemytckii (1936) and Edelstein (1962) are valid in generalized metric spaces, as well. A new application to fractional differential equations is also presented. At the end we state a few open questions for young researchers. [ABSTRACT FROM AUTHOR]

A lot of authors have proved various common fixed-point results for pairs of self-mappings under strict contractive conditions in metric spaces. In the case of cone metric spaces, fixed point results are usually proved under assumption that the cone is normal. In the present paper we prove common fixed point results under strict contractive conditions in cone metric spaces using only the assumption that the cone interior is nonempty. We modify the definition of property (E.A), introduced recently in the work by Aamri and Moutawakil (2002), and use it instead of usual assumptions about commutativity or compatibility of the given pair. Examples show that the obtained results are proper extensions of the existing ones.

New fixed point results for a pair of non-self mappings defined on a closed subset of a metrically convex cone metric space (which is not necessarily normal) are obtained. By adapting Assad-Kirk's method the existence of a unique common fixed point for a pair of non-self mappings is proved, using only the assumption that the cone interior is nonempty. Examples show that the obtained results are proper extensions of the existing ones.

In this paper, we discuss the existence of best proximity points of certain mappings via simulation functions in the frame of complete metric-like spaces. Some consequences and examples are given of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we utilize the family F and the notion of ω-distance in an ordered G-metric space and introduce (F, ω)-contractions in order to derive some fixed point results. We also discuss the problems of Ulam-Hyers stability, well-posedness and limit shadowing property. In order to illustrate the use of our results, we apply them to nonlinear integral equations, as well as to some three-point fractional integral boundary value problems, both with numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we define a generalized c-distance in tvs-cone b-metric spaces and introduce some results about its properties. Then we prove some new fixed point and common fixed point results (with the underlying cone which is not normal). Respective results concerning mappings without periodic points are also deduced. Some examples are presented to validate our obtained results. An application to system of Fredholm integral equations is presented. [ABSTRACT FROM AUTHOR]

In this paper, the coincidence point problem of Nadler type is treated for two mappings (one single-valued and one multi-valued) in a metric space. A generalized (α; θ; f)-contractive condition is assumed, together with α-completeness of the space, α-continuity of the multi-valued mapping and α-compatibility of the two mappings. The results are applied to metric spaces endowed with a graph or with a binary relation. [ABSTRACT FROM AUTHOR]

In this paper, several results are presented on common fixed points using (φ1-φ2-φ3)-contraction condition including rational expressions in orbitally 0-complete partial metric spaces. A counterexample illustrates that the results are distinct from some known ones. It is also observed that the work of Kadelburg and Radenovic [Kadelburg, Z., Radenovic, S.: Fixed points under α-α conditions in ordered partial metric spaces, Intern. J. Anal. Appl. 5, 91{101 (2014)] is a particular case of the present investigation. As applications of our results, we prove two theorems for the existence of solutions of certain system of Volterra type integral equations, as well as for a nonlinear fractional differential equation. [ABSTRACT FROM AUTHOR]

In this paper direct proofs of some common fixed point results for two and three mappings under weak contractive conditions are given. Some of these results are improved by using different arguments of control functions. Examples are presented showing that some generalizations can not be obtained and also that our results are distinct from the existing ones. [ABSTRACT FROM AUTHOR]

We prove a common fixed point theorem for a hybrid pair of occasionally coincidentally idempotent mappings via common limit range property. Our result improves some results from the existing literature, especially the ones contained in Sintunavarat and Kumam (2009). Some illustrative and interesting examples to highlight the realized improvements are also furnished.

We discuss topological structure of b-metric-like spaces and demonstrate a fundamental lemma for the convergence of sequences. As an application we prove certain fixed point results in the setup of such spaces for different types of contractive mappings. Finally, some periodic point results in b-metric-like spaces are obtained. Two examples are presented in order to verify the effectiveness and applicability of our main results.