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Existence of solutions to nonlinear Hammerstein integral equations and applications. (English) Zbl 1104.45003
Authors’ abstract: The authors study the existence and multiplicity of solutions of the operator equation \(Kfu=u\) in the real Hilbert space \(L^{2}(G)\). Under certain conditions on the linear operator \(K\), they 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. The quadratic root operator \(K^{1/2}\) and its properties play an important role. As an application, the authors 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.

MSC:
45G10 Other nonlinear integral equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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