Existence of solutions to nonlinear Hammerstein integral equations and applications

In this paper, we study the existence and multiplicity of solutions of the operator equation Kfu = u in the real Hilbert space L2(G). Under certain conditions on the linear operator K, we establish the conditions on f which are able to guarantee that the operator equation has at least one solution, a unique solution, and infinitely many solutions, respectively. The monotone operator principle and the critical point theory are employed to discuss this problem, respectively. In argument, quadratic root operator K1/2 and its properties play an important role. As an application, we investigate the existence and multiplicity of solutions to fourthorder boundary value problems for ordinary differential equations with two parameters, and give some new existence results of solutions. © 2005 Elsevier Inc. All rights reserved