Existence results for time scale boundary value problem. (English) Zbl 1107.34018

This paper deals with the existence of a positive solution for dynamic equations on a time scale \[ x^{\triangle \triangle}(t) + m(t) f(t, x(\sigma(t))) = 0, \quad t \in [a,b], \] and \[ x^{\triangle \triangle}(t) + m(t) f(t, x(\sigma(t)), x^{\triangle}(\sigma(t))) = 0, \quad t \in [a,b], \] satisfying the boundary conditions \[ \alpha x(a) - \beta x^{\triangle}(b) = 0, \quad \gamma x(\sigma(b)) + \delta x^{\triangle}(\sigma(b)) = 0, \] where \(f\) is continuous, \(m: (a,\sigma(b)) \to [0,\infty)\) is right-dense continuous and may be singular at both \(t=a\) and \(t=\sigma(b)\), \(\gamma \beta + \alpha \delta + \alpha \gamma (\sigma(b) - a) > 0\) and \(\delta = \gamma (\sigma^2(b)-a) \geq 0\). The existence of at least one positive solution is based on an application of the Krasnosel’skiĭ fixed-point theorem. Certain norm estimates needed to apply the cone-theoretic theorem of Krasnosel’skiĭ are obtained from the following assumption on \(f\), \[ f(t,x) \leq p(t) + q(t) x, \quad (t,x) \in [a,\sigma^2(b)] \times [0,\infty), \] where \(q: [a,\sigma(b)] \to [0,\infty)\) is continuous, in the case of the first dynamic equation. Similar condition on \(f\) are imposed for the second dynamic equation. The results in this case are based on in-depth analysis of the Green function.


34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
39A12 Discrete version of topics in analysis
Full Text: DOI


[1] Anderson, D. R., Eigenvalue intervals for a two-point boundary value problem on a measure chain, J. Comput. Appl. Math., 141, 57-64 (2002) · Zbl 1134.34310
[3] Bohner, M.; Guseinov, G. Sh., Improper integrals on time scales, Dynamic Systems Appl., 12, 45-65 (2003) · Zbl 1058.39011
[6] Chyan, C. J.; Henderson, J., Eigenvalues problems for nonlinear differential equations on a measure chain, J. Math. Anal. Appl., 245, 547-559 (2000) · Zbl 0953.34068
[7] Erbe, L. H.; Peterson, A., Green’s functions and comparison theorems for differential equation on measure chains, Dynamics Continuous, Discrete Impulsive Systems, 6, 121-137 (1999) · Zbl 0938.34027
[8] Erbe, L. H.; Peterson, A., Positive solutions for nonlinear differential equation on a measure chain, Math. Comput. Modelling, 32, 571-585 (2000) · Zbl 0963.34020
[9] Erbe, L. H.; Peterson, A., Eigenvalue conditions and positive solutions, J. Differ. Equations Appl., 6, 165-191 (2000) · Zbl 0949.34015
[10] Erbe, L. H.; Wang, H. Y., On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc., 120, 743-748 (1994) · Zbl 0802.34018
[11] Henderson, J.; Wang, H. Y., Positive solutions for nonlinear eigenvalue problems, J. Math. Anal. Appl., 208, 252-259 (1997) · Zbl 0876.34023
[12] Hilger, S., Analysis on measure chains—a unified approach to continuous and discrete calculus, Result Math., 18, 18-56 (1990) · Zbl 0722.39001
[13] Hilger, S., Differential and difference calculus-unified, Nonlinear Anal., 30, 2683-2694 (1997) · Zbl 0927.39002
[14] Hong, C. H.; Yeh, C. C., Positive solutions for eigenvalue problems on a measure chain, Nonlinear Anal., 51, 499-507 (2002) · Zbl 1017.34018
[15] Kaymakcalan, B.; Lakshmikantham, V.; Sivasundaram, S., Dynamical Systems on Measure Chains (1996), Kluwer Academic: Kluwer Academic Boston · Zbl 0869.34039
[16] Krasnosel’skii, M. A., Positive Solutions of Operator Equations (1964), Noordhoff: Noordhoff Groningen · Zbl 0121.10604
[17] Lian, W. C.; Wong, W. F.; Yeh, C. C., On the existence of positive solutions of nonlinear differential equations, Proc. Amer. Math. Soc., 124, 1117-1126 (1996) · Zbl 0857.34036
[18] Liang, J.; Xiao, T. J.; Hao, Z. C., Positive solutions of singular differential equations on measure chain, Comput. Math. Appl., 49, 651-663 (2005) · Zbl 1085.34019
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.