In this writing, first, we disclose the first and second category of a ΓτF-fuzzy proximal contraction for a mapping O:U→V which is nonself and also declare a fuzzy q-property to confirm the existence of the best proximity point for nonself function O. Then, we discover a few results using the ΓτF-fuzzy proximal contraction of the first category for a continuous and discontinuous nonself function O in a non-Archimedean fuzzy metric space. Later, we discuss another result for the ΓτF-fuzzy proximal contraction of the second category as well. In between the fuzzy proximal theorems, many examples are presented in support of the definitions and theorems proved in this writing.

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Best Proximity Point for ΓτF-Fuzzy Proximal Contraction

Thispaper recognizes a general approach related to recent fixed point results about the classes of interpolative and hybrid contractions in metric space and general metric spaces. Considering auxiliary functions, so called Wardowski functions, and a rich set of implicit relations, we introduce types of (αvq,ϕ,F)−contractions and r−order hybrid (αvq,ϕ,F)−contractions in the setting of b−metric-like spaces. They generate and simplify many forms of contractions widely used in the literature. The resulting theorems significantly extend, generalize, and unify an excellent work on fixed point theory.

General New Results on (ϕ,F)−Contractions in b−Metric-like-Spaces

Since 1964, when I.A. Perov introduced the so-called generalized metric space where \(d(x,y)\) is an element of the vector space \(\mathbb{R}^m\). Since then, many researchers have considered various contractive conditions on this type of spaces. In this paper, we generalize, extend and unify some of those established results. It is primarily about examining the existence of a fixed point of some mapping from \(X\) to itself, but if \((x,y)\) belong to some relation \(R\) on the set \(X\). Then the binary relation \(R\) and some \(F\) contraction defined on the space cone \(\mathbb{R}^m\) are combined. We start our consideration on the paper [1] and give strict critical remarks on the results published in there. Also, we improve their result by weakening one condition. 

Remarks on “Perov Fixed-Point Results on F-contraction Mappings Equipped with Binary Relation”

Since 1964, when I.A. Perov introduced the so-called generalized metric space where d(x,y) is an element of the vector space Rm, many researchers have considered various contractive conditions in this type of space. In this paper, we generalize, extend and unify some of those established results. We are primarily concerned with examining the existence of a fixed point of some mapping from X to itself, but if (x,y) belongs to some relation R on the set X, then the binary relation R and some F contraction defined on the space cone Rm are combined. We start our consideration with the recently announced results and give them strict, critical remarks. In addition, we improve several announced results by weakening some of the given conditions.

/pdf/remarks-on-perov-fixed-point-results-on-f-contraction-29cib40g.pdf

Remarks on “Perov Fixed-Point Results on F-Contraction Mappings Equipped with Binary Relation”

D. Wardowski proved in 2012 a generalization of Banach Contraction Principle by introducing F-contractions in metric spaces. In the next ten years, a great number of researchers used Wardowski's approach, or some of its modifications, to obtain new fixed point results for single- and multivalued mappings in various kinds of spaces. In this review article, we present a survey of these investigations, including some improvements, in particular concerning conditions imposed on function F entering the contractive condition.

Vesna Šešum Čavić

Papers

On F-Contractions: A Survey