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
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