The notion of Banach operator pair is used as a new class of noncommuting maps. The common fixed point theorems for Banach operator pair have been proved. The concept of g-nonexpansive, g-asymptotically nonexpansive, asymptotically regular, p-starshaped sets have been used. An attempt has been made to obtain the best approximation to the common fixed point of Banach operator pair.The concept of orbital continuity has been used to obtain the existence of common fixed points of Banach operator pair. Generalized contractive type condition for three mappings is defined and obtained some common fixed point results for these mappings. Some applications of fixed point theory to functional difference equation are given. The existence of solutions to boundary value problems of functional difference equation between its lower and upper solutions is obtained. The Schauder’s fixed point theorem has been used in proving the existence of solution. Using maximum principles the existence of extremal solutions of fourth order BVP of functional difference equations between its lower and upper solutions are obtained.