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Curves of positive solutions of boundary value problems on time-scales. (English) Zbl 1077.34022

The authors prove an existence and uniqueness result for the nonlinear boundary value problem on a time scale \[ -[p(t)u^\Delta(t)]^\Delta + q(t) u^\sigma(t) = \lambda f(t,u^\sigma(t)) \] with Dirichlet conditions \(u(a) = u(b) = 0\) and certain sign and growth conditions on the nonlinearity \(f\). The symbols \(\Delta\) and \(\sigma\) are notions from time scales calculus. A time scale is an arbitrary closed subset of the reals. The result shows that there is a \(C^1\) curve \(\lambda \mapsto u\) of solutions, parameterized by \(\lambda \in [0,\lambda_{\max})\), \(\lambda_{\max}\) being the principal eigenvalue of an associated weighted eigenvalue problem. The main ingredients of the proof are a fixed-point theorem in a cone, maximum principle and generalizations of known techniques to the time scales case.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
39A12 Discrete version of topics in analysis
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[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., Dynamic equations on time scales, (2001), Birkhäuser Boston · Zbl 1021.34005
[3] ()
[4] Brown, K.J.; Ibrahim, M.M.A.; Shivaji, R., S-shaped bifurcation curves, Nonlinear anal., 5, 475-486, (1981) · Zbl 0458.35036
[5] Crandall, M.G.; Rabinowitz, P.H., Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. rational mech. anal., 52, 161-180, (1973) · Zbl 0275.47044
[6] Chyan, C.J.; Davis, J.M.; Henderson, J.; Yin, W.K.C., Eigenvalue comparisons for differential equations on a measure chain, Electron. J. differential equations, 1998, 1-7, (1998)
[7] Dancer, E.N., On the structure of solutions of an equation in catalysis theory when a parameter is large, J. differential equations, 37, 404-437, (1980) · Zbl 0417.34042
[8] Davidson, F.A.; Rynne, B.P., Global bifurcation on time-scales, J. math. anal. appl., 267, 345-360, (2002) · Zbl 0998.34024
[9] Erbe, L.; Hilger, S., Sturmian theory on measure chains, Differential equations dynam. systems, 1, 223-246, (1993) · Zbl 0868.39007
[10] Erbe, L.; Peterson, A., Eigenvalue conditions and positive solutions, J. differ. equations appl., 6, 165-191, (2000) · Zbl 0949.34015
[11] Erbe, L.; Peterson, A.; Mathsen, R., Existence, multiplicity, and nonexistence of positive solutions to a differential equation on a measure chain, J. comput. appl. math., 113, 365-380, (2000) · Zbl 0937.34025
[12] Hilger, S., Analysis on measure chains—a unified approach to continuous and discrete calculus, Results math., 18, 18-56, (1990) · Zbl 0722.39001
[13] Hong, C.-H.; Yeh, C.-C., Positive solutions for eigenvalue problems on a measure chain, Nonlinear anal., 51, 449-507, (2002)
[14] Korman, P., A global solution curve for a class of semilinear equations, Electron. J. differential equations conf., 1, 119-127, (1997) · Zbl 0911.35053
[15] Korman, P.; Shi, J., Instability and exact multiplicity of solutions of semilinear equations, Electron. J. differential equations conf., 5, 311-322, (2000) · Zbl 0970.34015
[16] Laetsch, T., The number of solutions of a nonlinear two point boundary value problem, Indiana univ. math. J., 20, 1-13, (1970) · Zbl 0215.14602
[17] Lakshmikantham, V.; Sivasundaram, S.; Kaymakcalan, B., Dynamical system on measure chains, (1996), Kluwer Academic Dordrecht · Zbl 0869.34039
[18] Rynne, B.P., Solution curves of 2mth order boundary value problems, Electron. J. differential equations, 2004, 1-16, (2004)
[19] Wang, S.-H., On S-shaped bifurcation curves, Nonlinear anal., 22, 1475-1485, (1994) · Zbl 0803.34013
[20] Zeidler, E., Nonlinear functional analysis and its applications, vol. I, (1991), Springer-Verlag New York
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