Global bifurcation on time scales. (English) Zbl 0998.34024

Summary: The authors consider the structure of the solution set of a nonlinear Sturm-Liouville boundary value problem defined on a general time scale. Using global bifurcation theory, they show that unbounded continua of nontrivial solutions bifurcate from the trivial solution at the eigenvalues of the linearization, and that certain nodal properties of the solutions are preserved along these continua. These results extend the well-known results of Rabinowitz for the case of Sturm-Liouville ordinary differential equations.


34B24 Sturm-Liouville theory
34C23 Bifurcation theory for ordinary differential equations
Full Text: DOI Link


[1] Agarwal, R.P.; Bohner, M.; Wong, P.J.Y., Sturm – liouville eigenvalue problems on time scales, Appl. math. comput., 99, 153-166, (1999) · Zbl 0938.34015
[2] Chyan, C.J.; Henderson, J., Eigenvalue problems for nonlinear differential equations on a measure chain, J. math. anal. appl., 245, 547-559, (2000) · Zbl 0953.34068
[3] Coddington, E.A.; Levinson, N., Theory of ordinary differential equations, (1955), McGraw-Hill New York · Zbl 0042.32602
[4] Dancer, E.N., On the structure of solutions of non-linear eigenvalue problems, Indiana univ. math. J., 23, 1069-1076, (1974) · Zbl 0276.47051
[5] Erbe, L.; Hilger, S., Sturmian theory on measure chains, Differential equations dynam. systems, 1, 223-246, (1993) · Zbl 0868.39007
[6] Erbe, L.; Peterson, A., Green’s functions and comparison theorems for differential equations on measure chains, Dynam. continuous, discrete impulsive systems, 6, 121-137, (1999) · Zbl 0938.34027
[7] Erbe, L.; Peterson, A., Positive solutions for a nonlinear differential equation on a measure chain, Math. comput. model., 32, 529-539, (2000) · Zbl 0963.34020
[8] Hilger, S., Analysis on measure chains—A unified approach to continuous and discrete calculus, Results math., 18, 18-56, (1990) · Zbl 0722.39001
[9] Lakshmikantham, V.; Sivasundaram, S.; Kaymakcalan, B., Dynamical system on measure chains, (1996), Kluwer Academic Dordrecht · Zbl 0869.34039
[10] Rabinowitz, P.H., Some global results for nonlinear eigenvalue problems, J. funct. anal., 7, 487-513, (1971) · Zbl 0212.16504
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.