In this paper, we noticed that the existence of fixed points of F-contractions, in $\mathcal{F}$ -metric space, can be ensured without the third condition (F3) imposed on the Wardowski function $F:(0,\infty )\to \mathbb{R}$ . We obtain fixed points as well as common fixed-point results for Reich-type F-contractions for both single and set-valued mappings in $\mathcal{F}$ -metric spaces. To show the usability of our results, we present two examples. Also, an application to functional equations is presented. The application shows the role of fixed-point theorems in dynamic programming, which is widely used in computer programming and optimization. Our results extend and generalize the previous results in the existing literature.

In this work, we show that the existence of fixed points ofF-contraction mappings in function weighted metric spaces can be ensured without third condition ( F 3 ) (F3) imposed on Wardowski function F :(0, ∞ ) → ℜ F\mathrm{:(0,\hspace{0.33em}}\infty )\to \Re . The present article investigates (common) fixed points of rational typeF-contractions for single-valued mappings. The article employs Jleli and Samet’s perspective of a new generalization of a metric space, known as a function weighted metric space. The article imposes the contractive condition locally on the closed ball, as well as, globally on the whole space. The study provides two examples in support of the results. The presented theorems reveal some important corollaries. Moreover, the findings further show the usefulness of fixed point theorems in dynamic programming, which is widely used in optimization and computer programming. Thus, the present study extends and generalizes related previous results in the literature in an empirical perspective.