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Nodal solutions to nonlinear eigenvalue problems on time scales. (English) Zbl 1103.34006
Summary: Using global bifurcation theory, we obtain the existence of solutions with specified numbers of simple generalized zeros for the nonlinear eigenvalue problem on time scales $\bbfT$ $$-u^{\Delta\Delta}(t)+q(t)u^\sigma(t)= rf\bigl( u^\sigma(t)\bigr),\quad t\in\bbfT,\quad u(0)=u(1)=0,$$ where $r>0$ is a given constant. In addition, we argue that our existence theorem is a generalization of a previous result which shows the existence of at least one positive solution for the above problem.

MSC:
34B15Nonlinear boundary value problems for ODE
39A10Additive difference equations
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References:
[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] Bohner, M.; Peterson, A. C.: Dynamic equations on time scales. (2001) · Zbl 0978.39001
[3] Bohner, M.; Peterson, A. C.: Advances in dynamic equations on time scales. (2003) · Zbl 1025.34001
[4] 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
[5] Davidson, F. A.; Rynne, B. P.: Curves of positive solutions of boundary value problems on time scales. J. math. Anal. appl. 300, 491-504 (2004) · Zbl 1077.34022
[6] Davidson, F. A.; Rynne, B. P.: Global bifurcation on time scales. J. math. Anal. appl. 267, 345-360 (2002) · Zbl 0998.34024
[7] Hilger, S.: Analysis on measure chains--a unified approach to continuous and discrete calculus. Results math. 18, 18-56 (1990) · Zbl 0722.39001
[8] Rabinowitz, P. H.: Nonlinear Sturm--Liouville problems for second order ordinary differential equations. Commun. pure. Appl. math. 23, 939-961 (1970) · Zbl 0206.09706
[9] Rabinowitz, P. H.: Some global results for nonlinear eigenvalue problems. J. funct. Anal. 7, 487-513 (1971) · Zbl 0212.16504
[10] Ma, R.: Bifurcation from infinity and multiple solutions for periodic boundary value problems. Nonlinear anal. 42, 27-39 (2000) · Zbl 0966.34015
[11] Ma, R.; Thompson, B.: Multiplicity results for second-order two-point boundary value problems with superlinear or sublinear nonlinearities. J. math. Anal. appl. 303, 726-735 (2005) · Zbl 1075.34017
[12] Ma, R.; Thompson, B.: Nodal solutions for nonlinear eigenvalue problems. Nonlinear anal. 59, 707-718 (2004) · Zbl 1059.34013
[13] Ma, R.; Thompson, B.: Global behavior of positive solutions of nonlinear three-point boundary value problems. Nonlinear anal. 60, 685-701 (2005) · Zbl 1069.34016