Basic existence, uniqueness and approximation results for positive solutions to nonlinear dynamic equations on time scales
This paper studies the existence and uniqueness of solutions of the second order nonlinear dynamic equations \[ \big(r(t)\,x^\Delta\big)^\Delta+F(t,x^\sigma)=0, \quad t\in[a,\infty)\cap {\mathbb T}, \] \[ \big(r(t)\,x^\Delta\big)^\Delta+F(t,x^\sigma,x^{\Delta\sigma})=0, \quad t\in[a,\infty)\cap {\mathbb T}, \] where \({\mathbb T}\) is a time scale (i.e., a nonempty closed subset of \({\mathbb R}\)). These equations generalize and unify the corresponding second order differential and difference equations. The authors use the Banach fixed point theorem to prove existence and uniqueness of solutions with an asymptote, i.e., \(\lim_{t\to\infty}x(t)=M>0\) (the so-called terminal value problem). The function \(F\) is assumed to be nonnegative and \(r\) is positive. In the time scale environment one also needs to assume that \(r(t)\) is bounded away from zero, i.e., \(\inf\{r(t),\;t\in[a,b]\cap{\mathbb T}\}>0\) for all \(b\in(a,\infty)\cap{\mathbb T}\) (so that the function \(1/r(\cdot)\) is rd-continuous) as e.g. in [\textit{R.~Hilscher} and \textit{C.~C.~Tisdell}, ``Terminal value problems for first and second order nonlinear equations on time scales'', Electron. J. Differ. Equ. 2008, Paper No.~68, 21 pp. (2008)]. Note also that the set \(\chi_M\) is a closed subset of the Banach space \(C_{\mathbb T}^1\) (and not a Banach space itself).
scientific article; zbMATH DE number 5324201
scientific article