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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]

We give a proof for the common fixed point theorem of Jungck in rectangular b-metric spaces. As a corollary, we obtain well known common fixed point theorems in b-metric spaces. [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. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notions of α – ψ ϕ -contractive and cyclic α – ψ ϕ -contractive mappings and establish the existence and uniqueness of fixed points for such mappings in complete metric-like spaces. The results presented here substantially generalize and extend several comparable results in the existing literature, in particular those of Karapinar and Salimi (2013). As an application, we prove new fixed point results for ψ ϕ -graphic and cyclic ψ ϕ -graphic contractive mappings. Moreover, some examples and an application to integral equations are given here to illustrate the usability of the obtained results. [ABSTRACT FROM AUTHOR]