Existence of solutions to nonlinear Hammerstein integral equations and applications.

*(English)*Zbl 1104.45003Authors’ 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.

Reviewer: Yves Cherruault (Paris)

##### MSC:

45G10 | Other nonlinear integral equations |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

##### Keywords:

strongly monotone operator principle; mountain pass Lemma; linking theorem; fourth-order boundary value problem; nonlinear integral equations; existence of solution; Hilbert space; critical point theory
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\textit{F. Li} et al., J. Math. Anal. Appl. 323, No. 1, 209--227 (2006; Zbl 1104.45003)

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