A fixed point method to solve differential equation and Fredholm integral equation

Abstract

The purpose of this research is to explore a fixed point method to solve a class of functional equations, Tu = f, where T is a differential or an integral operator on a Sobolev space H2(Ω), where Ω is an open set in Rn. First, T is converted into a sum of I+ λA with λ > 0, where A is a continuous linear operator and I is identity mapping. Then it is shown that T is a contraction on the prescribed Sobolev space and norm of A is estimated on the prescribed Sobolev space. By means of the theory of inverse operator of I+ λA and by choosing the appropriate value of λ, the solution u of differential or integral operator is obtained. Some practical problems concerning the linear differential equation and Fredholm integral equation are solved by virtue of the fixed point method.