# zbMATH — the first resource for mathematics

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 (*).

##### MSC:
 34B24 Sturm-Liouville theory 34B27 Green’s functions for ordinary differential equations 39A10 Additive difference equations
Full Text:
##### References:
  Agarwal, R.P.; Bohner, M., Basic calculus on time scales and some of its applications, Results math., 35, 3-22, (1999) · Zbl 0927.39003  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  Ahlbrandt, C.D.; Bohner, M.; Ridenhour, J., Hamiltonian systems on time scales, J. math. anal. appl., 250, 561-578, (2000) · Zbl 0966.39010  Aslim, G.; Guseinov, G.Sh., Weak semirings, ω-semirings, and measures, Bull. allahabad math. soc. (India), 14, 1-20, (1999) · Zbl 1102.28301  Atici, F.M., Existence of positive solutions of nonlinear discrete sturm – liouville problems, Math. comput. modelling, 32, 599-607, (2000) · Zbl 0965.39009  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  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  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  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  Erbe, L.H.; Hilger, S., Sturmian theory on measure chains, Differential equations dynamics systems, 1, 223-246, (1993) · Zbl 0868.39007  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  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  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  Erbe, L.H.; Peterson, A., Eigenvalue condition and positive solutions, J. differ. equations appl., 6, 165-191, (2000) · Zbl 0949.34015  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  Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press San Diego · Zbl 0661.47045  G.Sh. Guseinov, B. Kaymakçalan, On a disconjugacy criterion for second order dynamic equations on time scales, J. Comput. Appl. Math., this issue.  Hilger, S., Analysis on measure chains—a unified approach to continuous and discrete calculus, Results math., 18, 18-56, (1990) · Zbl 0722.39001  Krasnosel’skii, M.A., Positive solutions of operator equations, (1964), Noordhoff Groningen  Lakshmikantham, V.; Sivasundaram, S.; Kaymakçalan, B., Dynamic systems on measure chains, (1996.), Kluwer Academic Publishers Boston
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.