Fixed Point Results for Almost Contraction Mappings in Fuzzy Metric Space

Abstract

In certain mathematical, computing, economic, and modeling issues, the presence of a solution to a theoretical or real-world problem is synonymous with the presence of a fixed point (Fp) for an appropriate mapping. Consequently, Fp plays an essential role in a wide variety of mathematical and scientific contexts. In its own right, the theory is a stunning amalgamation of analysis (both pure and applied), geometry, and topology. Recent years have shown the theory of Fps is a highly strong and useful tool in the study of nonlinear events. Fp theorems are concerned with mappings f of a set into itself that, under particular conditions, permit a Fp, that is, a point such that . This work introduces and proves the Fp theorem for various kinds of contraction mappings in a fuzzy metric space ( -space) namely almost -contraction mapping and - almost weakly contraction mapping. At first, the concept of -space and the terms used in the fuzzy setting are recalled. Then the concept of simulation function is given. The concept of simulation function is used to present the notion of almost -contraction mapping. In addition, this notion is used to prove the existence and uniqueness of the Fp for this kind of mapping. After that the notion of -almost weakly contraction mapping is introduced in the framework of -space, as well as the Fp theorem for this kind of mapping. At the end of the paper, some examples are given to support the results.

Keywords

• Almost contraction mappings, Almost Z ̂-contraction mapping, (Ψ ̃ ,Φ ̃)almost weakly contraction mapping, Fixed point, Fuzzy metric space