How to solve nonlinear operator equation A(v2)+Cv=f

In this paper, we discuss the problems of how to solve linear operator equation in Hilbert spaces and a kind of nonlinear operator equation A (v(2)) + Cv = f in reproducing kernel space W-2(1)[a, b]. For linear operator equation, we obtain a criterion about the existence of solutions. If it has solutions, we get the analytic representation of its minimum norm solution and the structure of its solution space; For nonlinear operator equation A (v(2)) + Cv = f, by using the method of solving linear operator equation, one exact solution is given. Besides, we obtain the representation of its solution set. Final examples show our methods are effective. (C) 2003 Elsevier Inc. All rights reserved.