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On Green’s functions and positive solutions for boundary value problems on time scales. (English) Zbl 1007.34025
The authors investigate the second-order dynamic equation on a time scale \[ -[p(t)y^\Delta]^\nabla+q(t)y=h(t). \tag{*} \] Recall that a time scale \(\mathbf T\) is any closed subset of \(\mathbb{R}\), where the jump operators \(\sigma(t)=\inf\{s>t, s\in \mathbf T\}\), \(\rho(t)=\sup\{s<t, s\in \mathbf T\}\) are defined. Using these operators, the so-called \(\Delta\)-derivative and \(\nabla\)-derivative of a function \(f:\mathbf T\to \mathbb{R}\) are defined by \[ f^\Delta(t)=\lim_{s\to t}\frac{f(s)-f(\sigma(t))}{s-\sigma(t)},\quad f^\nabla(t)=\lim_{s\to t}\frac{f(s)-f(\rho(s))}{s-\rho(t)}. \] First, the basic properties of solutions to (*) are established (Wronskian-type identity for homogeneous equation, variation of parameters formula,…). Then, the attention is focused on the Sturm-Liouville boundary value problem associated with (*), in particular, properties of Green’s function to this boundary value problem are investigated. In the last part of the paper, these properties of Green’s function are used to study the existence of positive solutions to (*).

34B24 Sturm-Liouville theory
34B27 Green’s functions for ordinary differential equations
39A10 Additive difference equations
Full Text: DOI
[1] Agarwal, R.P.; Bohner, M., Basic calculus on time scales and some of its applications, Results math., 35, 3-22, (1999) · Zbl 0927.39003
[2] Agarwal, R.P.; Bohner, M.; Wong, P.J., Sturm – liouville eigenvalue problems on time scales, Appl. math. comput., 99, 153-166, (1999) · Zbl 0938.34015
[3] Ahlbrandt, C.D.; Bohner, M.; Ridenhour, J., Hamiltonian systems on time scales, J. math. anal. appl., 250, 561-578, (2000) · Zbl 0966.39010
[4] Aslim, G.; Guseinov, G.Sh., Weak semirings, ω-semirings, and measures, Bull. allahabad math. soc. (India), 14, 1-20, (1999) · Zbl 1102.28301
[5] Atici, F.M., Existence of positive solutions of nonlinear discrete sturm – liouville problems, Math. comput. modelling, 32, 599-607, (2000) · Zbl 0965.39009
[6] Atici, F.M.; Guseinov, G.Sh., Positive periodic solutions for nonlinear difference equations with periodic coefficients, J. math. anal. appl., 232, 166-182, (1999) · Zbl 0923.39010
[7] Atici, F.M.; Guseinov, G.Sh., On the existence of positive solutions for nonlinear differential equations with periodic boundary conditions, J. comput. appl. math., 132, 341-356, (2001) · Zbl 0993.34022
[8] Atici, F.M.; Guseinov, G.Sh.; Kaymakçalan, B., On Lyapunov inequality in stability theory for Hill’s equation on time scales, J. inequal. appl., 5, 603-620, (2000) · Zbl 0971.39005
[9] B. Aulbach, S. Hilger, Linear dynamic processes with inhomogeneous time scale, in: Nonlinear Dynamics and Quantum Dynamical Systems, Akademie Verlag, Berlin, 1990, pp. 9-20. · Zbl 0719.34088
[10] Erbe, L.H.; Hilger, S., Sturmian theory on measure chains, Differential equations dynamics systems, 1, 223-246, (1993) · Zbl 0868.39007
[11] Erbe, L.H.; Hu, S.; Wang, H., Multiple positive solutions of some boundary value problems, J. math. anal. appl., 184, 640-648, (1994) · Zbl 0805.34021
[12] Erbe, L.H.; Peterson, A., Green’s functions and comparison theorems for differential equations on measure chains, Dynamics contin. discrete impuls. systems, 6, 121-137, (1999) · Zbl 0938.34027
[13] Erbe, L.H.; Peterson, A., Positive solutions for a nonlinear differential equation on a measure chain, Math. comput. modelling, 32, 571-585, (2000) · Zbl 0963.34020
[14] Erbe, L.H.; Peterson, A., Eigenvalue condition and positive solutions, J. differ. equations appl., 6, 165-191, (2000) · Zbl 0949.34015
[15] Erbe, L.H.; Wang, H., On the existence of positive solutions of ordinary differential equations, Proc. amer. math. soc., 120, 743-748, (1994) · Zbl 0802.34018
[16] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press San Diego · Zbl 0661.47045
[17] G.Sh. Guseinov, B. Kaymakçalan, On a disconjugacy criterion for second order dynamic equations on time scales, J. Comput. Appl. Math., this issue.
[18] Hilger, S., Analysis on measure chains—a unified approach to continuous and discrete calculus, Results math., 18, 18-56, (1990) · Zbl 0722.39001
[19] Krasnosel’skii, M.A., Positive solutions of operator equations, (1964), Noordhoff Groningen
[20] Lakshmikantham, V.; Sivasundaram, S.; Kaymakçalan, B., Dynamic systems on measure chains, (1996.), Kluwer Academic Publishers Boston
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