Relaxing the Φ-Family Auxiliary Functions and Related Results

This article establishes the existence of fixed points and common fixed points for set-valued mappings satisfying an implicit-type contraction inequality involving a new auxiliary function in a complete metric space equipped with a binary relation. Through a novel family of functions referred to as the Δ-family, which simplifies the axioms in comparison to the previously defined Φ-family, the study unifies a few classical fixed-point theorems. The practical relevance of the theoretical findings is demonstrated by applying the results to investigate the existence of solutions for a system of integral equations.