In this paper, we introduce the notion of fuzzy $\mathcal{R}-\psi -$contractive mappings and prove some relevant results on the existence and uniqueness of fixed points for this type of mappings in the setting of non-Archimedean fuzzy metric spaces. Several illustrative examples are also given to support our newly proven results. Furthermore, we apply our main results to prove the existence and uniqueness of a solution for Caputo fractional differential equations.

In this paper, we define almost ${\mathcal{R}}_g$-Geraghty type contractions and utilize the same to establish some coincidence and common fixed point results in the setting of $b_2$-metric spaces endowed with binary relations. As consequences of our newly proved results, we deduce some coincidence and common fixed point results for almost g-$\alpha$-$\eta$ Geraghty type contraction mappings in $b_2$-metric spaces. In addition, we derive some coincidence and common fixed point results in partially ordered $b_2$-metric spaces. Moreover, to show the utility of our main results, we provide an example and an application to non-linear integral equations.

The aim of this paper is to prove some common fixed point theorems for four mappings satisfying $(psi,varphi)$-weak contractions involving rational expressions in ordered partial metric spaces. Our results extend, generalize and improve some well-known results in the literature. Also, we give two examples to illustrate our results.

In this paper, we establish quadruple fixed point theorems for compatible type mappings in partially ordered \(b\) -metric spaces without the mixed \(g\) -monotone property under some conditions. Also, an example is given to show our results are real generalization of known results in quadruple fixed point theory.