Building infinitely many solutions for some model of sublinear multipoint boundary value problems. (English) Zbl 1343.34054

Summary: We show that the sublinearity hypothesis of some well-known existence results on multipoint Boundary Value Problems (in short BVPs) may allow the existence of infinitely many solutions by using Tietze extension theorem. This is a qualitative result which is of concern in Applied Analysis and can motivate more research on the conditions that ascertain the existence of multiple solutions to sublinear BVPs. The idea of the proof is of independent interest since it shows a constructive way to have ordinary differential equations with multiple solutions.


34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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[1] Agarwal, R. P., Boundary Value Problems from Higher Order Differential Equations (1986), Singapore: World Scientific, Singapore
[2] Gregus, M., Third Order Linear Differential Equations. Third Order Linear Differential Equations, Mathematics and Its Applications (1987), D. Reidel · Zbl 0878.34025
[3] Degla, G. A., A unifying maximum principle for conjugate boundary value problems, Advanced Nonlinear Studies, 1, 1, 121-131 (2001) · Zbl 1006.34013
[4] Elias, U., Oscillation Theory of Two-Term Differential Equations. Oscillation Theory of Two-Term Differential Equations, Mathematics and Its Applications, 396 (1997), Dordrecht, The Netherlands: Kluwer Academic Publishers, Dordrecht, The Netherlands · Zbl 0878.34022
[5] Coppel, W. A., Disconjugacy. Disconjugacy, Lecture Notes in Mathematics, 220 (1971), New York, NY, USA: Springer, New York, NY, USA
[6] Erbe, L. H.; Wang, H., On the existence of positive solutions of ordinary differential equations, Proceedings of the American Mathematical Society, 120, 3, 743-748 (1994) · Zbl 0802.34018
[7] Eloe, P. W.; Henderson, J., Positive solutions for \((n\) − 1, 1) conjugate boundary value problems, Nonlinear Analysis, Theory, Methods & Applications, 28, 10, 1669-1680 (1997) · Zbl 0871.34015
[8] Brezis, H.; Oswald, L., Remarks on sublinear elliptic equations, Nonlinear Analysis. Theory, Methods & Applications, 10, 1, 55-64 (1986) · Zbl 0593.35045
[9] Erbe, L. H.; Hu, S. C.; Wang, H., Multiple positive solutions of some boundary value problems, Journal of Mathematical Analysis and Applications, 184, 3, 640-648 (1994) · Zbl 0805.34021
[10] Ma, R., Multiple positive solutions for nonlinear \(m\)-point boundary value problems, Applied Mathematics and Computation, 148, 1, 249-262 (2004) · Zbl 1046.34030
[11] Davis, J. M.; Henderson, J.; Wong, P. J. Y., General lidstone problems: multiplicity and symmetry of solutions, Journal of Mathematical Analysis and Applications, 251, 2, 527-548 (2000) · Zbl 0966.34023
[12] Avery, R. I.; Henderson, J., Three symmetric positive solutions for a second-order boundary value problem, Applied Mathematics Letters, 13, 3, 1-7 (2000) · Zbl 0961.34014
[13] Liu, Z.; Li, F., Multiple positive solutions of nonlinear two-point boundary value problems, Journal of Mathematical Analysis and Applications, 203, 3, 610-625 (1996) · Zbl 0878.34016
[14] Degla, G., Positive nonlinear eigenvalue problems for conjugate BVPs, Nonlinear Analysis: Theory, Methods & Applications, 55, 5, 617-627 (2003) · Zbl 1042.34044
[15] Coyle, J.; Eloe, P. W.; Henderson, J., Bifurcation from infinity and higher order ordinary differential equations, Journal of Mathematical Analysis and Applications, 195, 1, 32-43 (1995) · Zbl 0845.34045
[16] Yao, Q., Existence, multiplicity and infinite solvability of positive solutions to a nonlinear fourth-order periodic boundary value problem, Nonlinear Analysis. Theory, Methods & Applications, 63, 2, 237-246 (2005) · Zbl 1082.34025
[17] Dugundji, J., Topology (1989), Dubuque, Iowa, USA: Wm. C. Brown Company Publishers, Dubuque, Iowa, USA
[18] Eschrig, H., Topology and Geometry for Physics. Topology and Geometry for Physics, Lecture Notes in Physics, 822 (2011), Berlin, Germany: Springer, Berlin, Germany · Zbl 1222.53001
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