Existence of solutions to nonlinear Hammerstein integral equations and applications

• DOI: 10.1016/j.jmaa.2005.10.014
• Publication Date: 2005-12-08T12:14:25Z
• Authors: Fuyi Li; Yuhua Li; Zhanping Liang
• Keywords: Strongly monotone operator principle; Fourth-order boundary value problem; PS condition; Applied Mathematics; Linking theorem; 0101 mathematics; Mountain pass lemma; 01 natural sciences; Analysis

AbstractIn 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 fourth-order boundary value problems for ordinary differential equations with two parameters, and give some new existence results of solutions.