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Eigenvalue problems for nonlinear differential equations on a measure chain. (English) Zbl 0953.34068

Summary: Values of \(\lambda\) are determined for which there exist positive solutions to the second-order differential equation on a measure chain, \[ u^{\Delta\Delta}(t)+\lambda a(t) f(u(\sigma(t)))= 0,\quad t\in [0,1], \] satisfying either the conjugate boundary conditions \(u(0)= u(\sigma(1))= 0\) or the right focal boundary conditions \(u(0)= u^\Delta(\sigma(1))= 0\), where \(a\) and \(f\) are positive valued, and both \(\lim_{x\to 0^+} {f(x)\over x}\) and \(\lim_{x\to \infty}{f(x)\over x}\) exist.

MSC:

34L05 General spectral theory of ordinary differential operators
34B24 Sturm-Liouville theory
34B15 Nonlinear boundary value problems for ordinary differential equations
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[1] R. P. Agarwal, and, M. Bohner, Basic calculus on time scales and some of its applications, preprint. · Zbl 0927.39003
[2] Aulback, B.; Hilger, S., Linear dynamic processes with inhomogeneous time scale, Nonlinear dynamics and quantum dynamical systems, Math. res., 59, (1990), Akademie Verlag Berlin
[3] Deimling, K., Nonlinear functional analysis, (1985), Springer-Verlag New York · Zbl 0559.47040
[4] Eloe, P.W.; Henderson, J., Positive solutions and nonlinear (k,n−k) conjugate eigenvalue problems, Differential equations dynam. systems, 6, 309-317, (1998) · Zbl 1003.34018
[5] Erbe, L.H.; Hilger, S., Sturmian theory on measure chains, Differential equations dynam. systems, 1, 223-246, (1993) · Zbl 0868.39007
[6] Erbe, L.H.; Peterson, A., Green’s functions and comparison theorems for differential equations on measure chains, Dynam. contin., discrete impuls. systems, 6, 121-137, (1999) · Zbl 0938.34027
[7] L. H. Erbe, and, A. Peterson, Positive solutions for a nonlinear differential equation on a measure chain, preprint. · 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] Henderson, J.; Wang, H., Positive solutions for nonlinear eigenvalue problems, J. math. anal. appl., 208, 252-259, (1997) · Zbl 0876.34023
[10] Kaymakcalan, B.; Lakshmikantham, V.; Sivasundaram, S., Dynamical systems on measure chains, (1996), Kluwer Academic Boston · Zbl 0869.34039
[11] Krasnosel’skii, M.A., Positive solutions of operator equations, (1964), Noordhoff Groningen · Zbl 0121.10604
[12] S. Lauer, Positive Solutions of Boundary Value Problems for Nonlinear Difference Equations, Ph.D. dissertation, Auburn University, 1997. · Zbl 0883.39003
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