Search of minimal metric structure in the context of fixed point theorem and corresponding operator equation problems

The paper contains a brief summary of the generalization of metrical structure regarding the fixed point theorem and corresponding operator equation problems. We observed that many researcher either tried to weaken the metrical structure, the contraction condition, or both. The idea behind this paper is to look for a minimal metrical structure to establish fixed point theorems. In this connection, we present new variants of the known fixed point theorem under non -triangular metric space (namely F -contraction, (A, S) -contraction, (psi, phi)-contraction). We also apply the obtain result in solving various types of operator equation problems. e.g., high -order fractional differential equation with non -local boundary conditions and non-linear integral equation problems.