Fixed points of a new type of contractive mappings in complete metric spaces

Abstract

In the article, we introduce a new concept of contraction and prove a fixed point theorem which generalizes Banach contraction principle in a different way than in the known results from the literature. The article includes an example which shows the validity of our results, additionally there is delivered numerical data which illustrates the provided example. MSC: 47H10; 54E50

Fixed Points of Ordered F-Contractions

“…Then, we give a fixed point theorem for such mapping. To support our result, we give an example showing that our main theorem is applicable, but both results of Ran and Reurings [12] and Wardowski [16] are not. …”

An observation on $$\alpha$$ α -type F-contractions and some ordered-theoretic fixed point results

“…Further, on varying the elements of F suitably, a variety of known contractions in the literature can be deduced. Example 1.1 [1] Consider F 2 F given by FðsÞ ¼ ln s. Then each self-mapping f on X satisfying inequality (1.1) is an F-contraction such that dðfx; fyÞ e Às dðx; yÞ; where x; y 2 X and x 6 ¼ y. Observe that this inequality holds trivially if x ¼ y.…”

“…Some well-known members of F are FðsÞ ¼ ln s, FðsÞ ¼ s þ ln s, FðsÞ ¼ À1 ffi ffi s p and FðsÞ ¼ lnðs 2 þ sÞ. Moreover, Wardowski [1] proved that every F-contraction mapping on a complete metric space possesses a unique fixed point. Further, on varying the elements of F suitably, a variety of known contractions in the literature can be deduced.…”

“…Using Ć irić-type generalized contraction in Definition 1.1, Wardowski and Van Dung [2] (also independently Mnak et al [3]) introduced the notion of F-weak contraction and utilize the same to generalize the main result of [1] as well as several other results of the existence literature. Definition 1.2 [2,3] Let ðX; dÞ; s and F be as in Definition 1.1.…”

“…In 2012, Wardowski [1] generalized Banach contraction principle in a novel way by introducing a new type of contraction called F-contraction: for all x; y 2 X; where F : R þ ! R is a mapping satisfying the following conditions:…”