In this paper, we introduce the notion of $\omega$-interpolative Ćirić-Reich-Rus-type proximal contraction. We obtain some best proximity point results for these mappings using the concept of $\omega$-admissibility in complete metric spaces. Some best proximity results are extended to partial ordered metric spaces and graphical metric spaces. Several new definitions are presented by considering the special cases of aforementioned results. The application of these results in fixed point theory is also discussed. The acquired results extend $\omega$-interpolative Ćirić-Reich-Rus-type contraction for obtaining fixed points.

In this article, we define a new space named the modular-like space with its related concepts to prove the existence of a fixed point and a point of coincidence for mappings on this space. Also, we defined $\acute{C}$iri $\acute{c}$-Reich-Rus type weakly contractive mappings on modular-like spaces and discussed some conditions that guarantee the existence of the fixed points for these kind of mappings. Some examples are also provided to elaborate the usability of our main results. It is worth mentioning that a modular-like space is a generalization of a modular space, thus our theorems are more general and applicable than the fixed point theorems on modular spaces.