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 the present paper, we introduce a new type of quartic functional equation and examine the Hyers–Ulam stability in fuzzy normed spaces by employing the direct method and fixed point techniques. We provide some applications in which the stability of this quartic functional equation can be controlled by sums and products of powers of norms. In particular, we show that if the control function is the fuzzy norm of the product of powers of norms, the quartic functional equation is hyperstable.

In this work, we investigate the generalized Hyers-Ulam stability of the Apollonius type additive functional equation in modular spaces with or without ${\Delta }_2$ -conditions. We study the same problem in fuzzy Banach spaces and $\beta$ -homogeneous Banach spaces. We show the hyperstability of the functional equation associated with the Jordan triple product in fuzzy Banach algebras. The obtained results can be applied to differential and integral equations with kernels of non-power types.

In this research article, by introducing a mapping $\phi$ defined on ${\left[0,\infty \right)}^4$ , with some axioms, we define two generalized contractions called $F_{H^{+}}^{\phi }$ -contractions and $\phi H_{\mathcal{P}}^{+}$ -contractions. We investigate their mutual relation and establish an existence theorem addressing $F_{H^{+}}^{\phi }$ -contractions with some applications.