The exact solution for solving a class nonlinear operator equations in the reproducing kernel space

Abstract

In this paper, the author will introduce the Sobolev space W 1 2(Ω n) firstly and then give the reproducing properties of the space W 1 2(Ω n) . As these results of those, a class of the nonlinear operator equation ∑ n i=1 ∏ m i j=1 ( A ij u)= f is transformed to the n-dimensional linear operator equation Au= f. Finally, we also give the exact solution of this nonlinear operator equation.