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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.

45G10 Other nonlinear integral equations
34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text: DOI
[1] Bai, Z.; Wang, H., On positive solutions of some nonlinear fourth-order beam equations, J. math. anal. appl., 270, 357-368, (2002) · Zbl 1006.34023
[2] Deimling, K., Nonlinear functional analysis, (1985), Springer-Verlag Berlin · Zbl 0559.47040
[3] Guo, D., Nonlinear functional analysis, (2001), Shandong Sci. & Tec. Press, (in Chinese)
[4] Li, F.; Liang, Z.; Zhang, Q., Existence of solutions to a class of nonlinear second order two-point boundary value problems, J. math. anal. appl., 312, 357-373, (2005) · Zbl 1088.34012
[5] Li, F.; Zhang, Q.; Liang, Z., Existence and multiplicity of solutions of a kind of fourth-order boundary value problem, Nonlinear anal., 62, 803-816, (2005) · Zbl 1076.34015
[6] Li, Y., Positive solutions of fourth-order boundary value problems with two parameters, J. math. anal. appl., 281, 477-484, (2003) · Zbl 1030.34016
[7] Rabinowitz, P.H., Minimax methods in critical point theory with applications to differential equations, () · Zbl 0152.10003
[8] Rabinowitz, P.H., Some critical point theorems and applications to semilinear elliptic partial differential equations, Ann. scuola normale sup. Pisa, class scienza, 4, 215-223, (1978) · Zbl 0375.35026
[9] Rabinowitz, P.H., Some minimax theorems and applications to nonlinear partial differential equations, (), 161-177
[10] Struwe, M., Variational methods: applications to nonlinear partial differential equations and Hamiltonian systems, (1996), Springer-Verlag Berlin · Zbl 0864.49001
[11] Taylor, A.E.; Lay, D.C., Introduction to functional analysis, (1980), Wiley
[12] Willem, M., Minimax theorems, (1996), Birkhäuser · Zbl 0856.49001
[13] Zeidler, E., Nonlinear functional analysis and its applications, III: variational methods and optimization, (1985), Springer-Verlag New York · Zbl 0583.47051
[14] Zeidler, E., Nonlinear functional analysis and its applications, I: fixed-point theorems, (1986), Springer-Verlag New York
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