Very recently, by considering a self-mapping T on a complete metric space satisfying a general contractivity condition of the form $\psi \left(d\right(Tx,Ty\left)\right)\le \phi \left(d\right(x,y\left)\right)$, Proinov proved some fixed-point theorems, which extended and unified many existing results in the literature. Accordingly, inspired by Proinov-type contraction conditions, Roldán López de Hierro et al. introduced a novel family of contractions in fuzzy metric spaces (in the sense of George and Veeramani), whose main advantage is the very weak constraints imposed on the auxiliary functions that appear in the contractivity condition. They also proved the existence and uniqueness of fixed points for the discussed family of fuzzy contractions in the setting of non-Archimedean fuzzy metric spaces. In this paper, we introduce a new family of fuzzy contractions based on Proinov-type contractions for which the involved auxiliary functions are not supposed to satisfy any monotonicity assumptions; further, we establish some new results about the existence and uniqueness of fixed points. Furthermore, we show how the main results in the above-mentioned paper can be deduced from our main statements. In this way, our conclusions provide a positive partial solution to one of the open problems posed by such authors for deleting or weakening the hypothesis of the nondecreasingness character of the auxiliary functions.

In this paper, we introduce two new concepts, generalized $\alpha$-$\eta$-$\mathcal{FZ}$-contraction and modified $\alpha$-$\eta$-$\mathcal{FZ}$-contraction, which unify several types of contractions in the context of fuzzy metric spaces. We discuss the existence and uniqueness results of such mappings in the setting of a complete fuzzy metric space in the sense of George and Veeramani and present several significant consequences of our obtained results by using variant examples for $\mathcal{FZ}$-simulation functions and admissible mappings. Some examples are provided to illustrate the usability of our main results.

In this paper, firstly, we introduce some new generalizations of $ F- $contraction, $ F- $Suzuki contraction, and $ F- $expanding mappings. Secondly, we prove the existence and uniqueness of the fixed points for these mappings. Finally, as an application of our main result, we investigate the existence of a unique solution of an integral boundary value problem for scalar nonlinear Caputo fractional differential equations with a fractional order (1, 2).

In this paper, we study the existence and uniqueness of common fixed point of six self-mappings in Menger spaces by using the common limit range property (denoted by (CLR_ST)) of two pairs. Our results improve, extend, complement and generalize several existing results in the literature. Also, some examples are provided to illustrate the usability of our results.