Jleli and Samet introduced the notion of $\mathcal{F}$-metric space as a generalization of traditional metric space and proved Banach contraction principle in the setting of these generalized metric spaces. The aim of this article is to utilize $\mathcal{F}$-metric space and establish some common $\alpha$-fuzzy fixed point theorems for rational ( $\beta$- $\varphi )$-contractive conditions. Our results extend, generalize and unify some well-known results in the literature. As application of our main result, we discuss the solution of fuzzy integrodifferential equations in the setting of a generalized Hukuhara derivative.

In this research article, we give the notion of graphic $F$-contraction in the setting of $\mathcal{F}$-metric space and establish some fixed point results. We also supply some examples to demonstrate the brilliance of the established results. We also establish some fixed point results for orbitally continuous and orbitally $G$ -continuous graphic $F$-contractions as applications of our main results. Also, we discuss the existence and the uniqueness of solutions of functional equations involving in dynamic programming.

This paper introduces the concept of set-valued almost $\mathrm{{\rm Y}}$-contractions in $\mathcal{F}$-metric spaces, aiming to obtain the best proximity point results for set-valued mappings. The newly proposed idea of set-valued almost $\mathrm{{\rm Y}}$-contractions includes various contractive conditions like set-valued almost contractions, set-valued $\mathrm{{\rm Y}}$-contractions, and traditional $\mathrm{{\rm Y}}$-contractions. Consequently, the results presented here extend and unify numerous established works in this domain. To illustrate the practical significance of the theoretical findings, a specific example is provided.