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

In this paper, the author will introduce the Sobolev space <f>W12(Ωn)</f> firstly and then give the reproducing properties of the space <f>W12(Ωn)</f>. As these results of those, a class of the nonlinear operator equation <f>∑ni=1∏mij=1(Aiju)=f</f> is transformed to the <f>n</f>-dimensional linear operator equation <f>Au=f</f>. Finally, we also give the exact solution of this nonlinear operator equation. [Copyright &y& Elsevier]