We introduce a new type of interpolative proximal contractive condition that ensures the existence of the best proximity points of fuzzy mappings in the complete non-archimedean fuzzy metric spaces. We establish certain best proximity point theorems for such proximal contractions. We improve and generalize the fuzzy proximal contractions by introducing fuzzy proximal interpolative contractions. The obtained results improve and generalize the best proximity point theorems published in Fuzzy Information and Engineering, 5 (2013), 417–429. Moreover, we provide many nontrivial examples to validate our best proximity point theorem.

We introduce a Ćirić type contraction principle in a vector-valued $b$-metric space that generalizes Perov's contraction principle. We investigate the possible conditions on the mappings $W,E:G\to G$ ( $G$ is a non-empty set), for which these mappings admit a unique common fixed point in $G$ subject to a nonlinear operator $\mathbf{F}:{\mathbb{P}}^m\to {\mathbb{R}}^m$. We illustrate the hypothesis of our findings with examples. We consider an infectious disease model represented by the system of delay integro-differential equations and apply the obtained fixed point theorem to show the existence of a solution to this model.

In this paper, we prove some results on the Hausdorff partial $b$-metrics. We prove some new Lemmas regarding convergence of the sequences in the Hausdorff partial b-metric spaces. The obtained results generalize and improve many existing fixed-point results. The examples are given for the explanation of theory. The existence of the solution to the boundary value problem is proved via fixed-point approach.