The intent of this manuscript is to establish some common fixed point theorems in a complete metric space under weak contraction condition for two pairs of discontinuous weak compatible maps. The results proved herein are the generalization of some recent results in literature. We give an example to support our results. MSC: 47H10; 54H25

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.

In this paper, we first introduce a new family of functions like an implicit function called Γ-functions. Furthermore, we introduce a new concept of α-ΓF-fuzzy contractive mappings, which is weaker than the class of fuzzy F-contractive mappings. Then, the existence and uniqueness of the fixed point are established for a new type of fuzzy contractive mappings in the setting of fuzzy metric spaces. Moreover, some examples and an application to nonlinear fractional differential equation are given, and these show the importance of the introduced theorems in fuzzy settings.

In the piece of this note, we mention various Suzuki-type fuzzy contractive inequalities in 1-Z-complete fuzzy metric-like spaces for uniqueness and existence of a fixed point and prove a few fuzzy fixed point theorems, which are appropriate generalizations of some of the latest famed results in the literature. Mainly, we generalize fuzzy Θ-contraction in terms of Suzuki-type fuzzy Θ-contraction and also fuzzy ϓ-contractive mapping in view of Suzuki-type. For this new group of Suzuki-type functions, acceptable conditions are formulated to ensure the existence of a unique fixed point. The attractive beauty of this fuzzy distance space lies in the symmetry of its variables, which play a crucial role in the construction of our contractive conditions to ensure the solution. Furthermore, a lot of considerable examples are presented to illustrate the significance of our results. In the end, we have discussed an application in an extensive way for the solution of a nonlinear fractional differential equation via Suzuki-type fuzzy contractive mapping.

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