Optimal pair of fixed points for a new class of noncyclic mappings under a $(\phi ,{\mathcal{R}}^t)$-enriched contraction condition

In the present study, we commenced by presenting a new class of maps, termed noncyclic $(\phi ,{\mathcal{R}}^t)$-enriched quasi-contractions within metric spaces equipped with a transitive relation ${\mathcal{R}}^t$. Subsequently, we identified the conditions for the existence of an optimal pair of fixed points pertaining to these mappings, thereby extending and refining a selection of contemporary findings documented in some articles. Specifically, our analysis will encompass the outcomes pertinent to reflexive and strictly convex Banach spaces.