1) Invent new sufficient and necessary conditions for some nonlinear mapping with an approximate algorithms finding the existence of minimal elements of equilibrium problems to establish enhanced EVP in generalized distance function.
2) To apply our results to investigate optimization problems non-convex minimization problems and Caristis fixed point theorem and its equivalent theorems.
3) To obtain and study EVP for single valued mapping and multi-valued operator which is generalized Banach contraction mapping with generalized fitting function in generalized distance function and also provide some example support our result.
