FIXED POINT THEOREMS AND STABILITY RESULTS FOR NONLINEAR OPERATORS WITH APPLICATIONS TO DIFFERENTIAL AND INTEGRAL EQUATIONS

Abstract

The theory of fixed point theorems is a core of the study of nonlinear operators, and offers the key to existence and uniqueness proofs of solutions to differential and integral equations. This paper discusses three numerical experiments that exercise the theoretical and practical applications of these theorems through matlab. The first experiment describes how Banach iteration converts to a unique fixed point. The second looks at the asymptotic stability of a linear ordinary differential equation, and it compares the numerical solutions with the exact analytical solution. The third is a picture of Picard successive approximation to an integral equation of volterra, which shows the consistency between numerical and analytical solutions. The experiments all point to the interaction of theory and computation, and show the usefulness of fixed point methods in nonlinear problems and in studying stability. Future applications of the work can include more complex partial and integro-differential equations, engineering, physics, and economics applications.