The current study provides significant findings on coincidence the best proximity points for proximal-contractions in the context of extended $b$-metric spaces. To substantiate our assertions, we present illustrative examples across several circumstances. The conclusions presented in this study provide an expanded and more nuanced viewpoint, extending and generalizing multiple previous findings in optimal proximity theory. These findings present a new method for comprehending proximal coincidence locations in multi-valued mappings, representing a substantial advancement in the existing research landscape. This work gives practical benchmarks for implementing optimal proximity results, provided that existence and uniqueness constraints are met.

Helmholtz’s problem helps us to completely understand how sound behaves in a cylinder that is closed from one of its ends and opened at another. This paper aims to employ some novel convergence results to the Helmholtz problem with mixed boundary conditions and demonstrate the existence and uniqueness of the solution by applying graph-controlled contractions. For this purpose, we enunciate graphically Reich type and graphically Ćirić type contractions in the realm of graphical-controlled metric type spaces. In our study, we showcase the existence and uniqueness of fixed point results by employing these graphical contractions. This is demonstrated through extensive examples that a graphicalcontrolled metric-type space is distinct from traditional controlled metric-type spaces. We also exhibit an example of a graphically Reich contraction that is not a classical Reich contraction. Similarly, a decent example of graphical Ćirić contraction is presented, which is distinct from the classical Ćirić contraction. Concrete illustrative examples solidify our theoretical framework.

In this paper, we introduce the Euler sequence space e r(p) of nonabsolute type and prove that the spaces e r(p) and l(p) are linearly isomorphic. Besides this, we compute the α-, β- and γ-duals of the space e r(p). The results proved herein are analogous to those in [ALTAY, B.—BASŠAR, F.: On the paranormed Riesz sequence spaces of non-absolute type, Southeast Asian Bull. Math. 26 (2002), 701–715] for the Riesz sequence space r q(p). Finally, we define a modular on the Euler sequence space e r(p) and consider it equipped with the Luxemburg norm. We give some relationships between the modular and Luxemburg norm on this space and show that the space e r(p) has property (H) but it is not rotund (R).