In this study, the $ (h $-$ arphi)_R $ Wood Print and $ (h $-$ arphi)_M $-contractions with two metrics endowed with a directed graph are examined using auxiliary functions.We propose a set of criteria that guarantees the existence of common fixed points for our contractions.This leads to a generalization of previous results in the Protection Boot Bag literature.
Towards our accomplishments, we establish affirmative results that demonstrate solutions to a class of nonlinear two-term fractional differential equations with nonlocal boundary conditions.To further corroborate our major findings, we also provide instances.