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Existence of solutions to a class of nonlinear second order two-point boundary value problems. (English) Zbl 1088.34012
Existence and multiplicity results are obtained for the following boundary value problem $$-u''(t)=f(t,u(t)), \quad t\in [0,1],\quad u(0)=u'(1)=0,$$ where $f: [0,1]\times {\Bbb R}\to {\Bbb R}$ is continuous. By using the strongly monotone operator principle and the critical point theory, the authors establish some conditions for $f$ which guarantee that the boundary value problem has a unique solution, at least one nonzero solution, and infinitely many solutions.

34B15Nonlinear boundary value problems for ODE
47H07Monotone and positive operators on ordered topological linear spaces
47J30Variational methods (nonlinear operator equations)
Full Text: DOI
[1] Davis, J. M.; Henderson, J.; Wong, P. J. Y.: General lidstone problems: multiplicity and symmetry of solutions. J. math. Anal. appl. 251, 527-548 (2000) · Zbl 0966.34023
[2] Deimling, K.: Nonlinear functional analysis. (1985) · Zbl 0559.47040
[3] Erbe, L. H.; Hu, S.; Wang, H.: Multiple positive solutions of some boundary value problems. J. math. Anal. appl. 184, 640-648 (1994) · Zbl 0805.34021
[4] Erbe, L. H.; Wang, H.: On the existence of positive solutions of ordinary differential equations. Proc. amer. Math. soc. 120, 743-748 (1994) · Zbl 0802.34018
[5] Guo, D.: Nonlinear functional analysis. (2001)
[6] Henderson, J.; Wang, H.: Positive solutions for nonlinear eigenvalue problems. J. math. Anal. appl. 208, 252-259 (1997) · Zbl 0876.34023
[7] Lan, K.; Webb, J. R. L.: Positive solutions of semilinear differential equations with sigularities. J. differential equations 148, 407-421 (1998) · Zbl 0909.34013
[8] Li, F.; Han, G.: Generalization for amann’s and Leggett -- Williams three-solution theorems and applications. J. math. Anal. appl. 298, 638-654 (2004) · Zbl 1073.47055
[9] Li, F.; Zhang, Y.: Multiple symmetric nonnegative solutions of second-order ordinary differential equations. Appl. math. Lett. 17, 261-267 (2004) · Zbl 1060.34009
[10] Liu, Z.; Li, F.: Multiple positive solutions of nonlinear two-point boundary value problems. J. math. Anal. appl. 203, 610-625 (1996) · Zbl 0878.34016
[11] Rabinowitz, P. H.: Minimax methods in critical point theory with applications to differential equations. CBMS regional conf. Ser. in math. 65 (1986)
[12] Struwe, M.: Variational methods: applications to nonlinear partial differential equations and Hamiltonian systems. (1996) · Zbl 0864.49001
[13] Taylor, A. E.; Lay, D. C.: Introduction to functional analysis. (1980) · Zbl 0501.46003
[14] Zeidler, E.: Nonlinear functional analysis and its applications, III: Variational methods and optimization. (1985) · Zbl 0583.47051
[15] Zeidler, E.: Nonlinear functional analysis and its applications, I: Fixed-point theorems. (1986) · Zbl 0583.47050
[16] Zeidler, E.: Nonlinear functional analysis and its applications, II/B: nonlinear monotone operators. (1990) · Zbl 0684.47029