Executive Summary : | The study of linear and nonlinear operators on abstract spaces is one of the most exciting and vital topics in applied and pure mathematics. In particular, the study of existence, uniqueness, and approximation of fixed point and common fixed point of an operator and pair of operators on normed spaces plays a crucial role in finding the solution of equations arising from various branches of mathematics. For example, the problem of finding a steady-state of the Markov process, the problem of existence and uniqueness solution of the differential equation, the integral equation arising from various mathematical models can be seen as a fixed point problem in suitable abstract spaces. Physicists and engineers frequently encounter boundary value problems and integral equations in the study of systems associated with model problems, so the solutions of such problems are of keen interest and are at the centre of several research activities. The proposed work aims to establish the existence and approximate the solutions of initial, boundary value problems and integral equations with which an operator or pair of operators on a suitable normed space are associated. The fixed point techniques are applicable on those problems which can be expressed in the form of equation Tx=x, consisting of an operator T and some space X on which the operator T acts. Fortunately, many of the real-life problems can be modelled in the said form, and so, to obtain the solution of such problems the fixed point techniques can be applied. For instance, scientific models like; the governing boundary value problem of a heavy hanging cable, convergence of Markov process, in economics the stability of Markets etc. can be expressed as a fixed point problem. After the completion of this research work it is expected that some new generalized fixed point theorems for operators and approximation techniques of fixed point of operators will be established; and the solution of differential and integral equations occurs in scientific models will be constructed with it. |