A survey: F -contractions with related fixed point results
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Journal of Fixed Point Theory and Applications
A survey: F -contractions with related fixed point results Erdal Karapınar, Andreea Fulga and Ravi P. Agarwal Abstract. In this note, we aim to review the recent results on F contractions, introduced by Wardowski. After examining the fixed point results for such operators, we collect the sequent results in this direction in a different setting. One of the aims of this survey is to provide a complete collection of several fixed generalizations and extensions of F -contractions. Mathematics Subject Classification. 47H10, 54H25. Keywords. F-contraction, metric space, contraction.
1. Introduction Most scientific resources acknowledge that the metric fixed point theory was initiated by Banach [26]. Although, particularities of “the fixed point theory” can be found in some earlier results, such as the papers of Liouville [57] and Picard [69], the fixed point theorem was provided by Banach. The main success of the Banach is the abstraction of the “successive approximation” method in these earlier results and provide an elegant self-contained fixed point results in the setting of a complete normed space. Based on this observation, in some sources, the well-known Banach contraction mapping principle was called the “Picard–Banach Theorem.” It is an indispensable fact that the approach of Banach strongly affected the (nonlinear) functional analysis. Both the statement and the proof of Banach’s fixed point theorem were expressed in a perfect way. The theorem not only guarantees the existence and uniqueness of a fixed point for “the contraction mapping” but also indicates how to find the desired point. The certainty of the method brings a wide application potential not only in various branches of mathematics, but also in distinct disciplines, including economics, computer science, and so on. Roughly speaking, the Banach fixed point result is settled on the relation between the distances of two pairwise points and the image of the corresponding points; namely, d(x, y) and d(T x, T y) for all x, y ∈ X, where T is a self-mapping on a metric space (X, d). Further, in the construction of 0123456789().: V,-vol
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Banach fixed point theorem, the considered self-mapping T is necessarily continuous. Based on these inspections, research has been done to get some new relations by regarding other possible distances, d(x, T x), d(x, T y), d(y, T y) and d(y, T x), together with d(x, y) and d(T x, T y). Finding a fixed point of discontinuous mapping has been an interesting research topic. Indeed, the discussion above can be considered the basic and initial attempt to improve the result of Banach. In the literature, one can find several different questions and approaches aimed at extending and generalizing this pioneer work that belongs to Banach. Since then, Banach’s result has been generalized and extended in several directions and among these results, we shall focus on one of the most interesting extensions of Banach’s fixed point theorem that was provided by Wardowski [88
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