Set-valued mappings and best proximity points: A study in $\mathcal{F}$-metric spaces

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.