Existence results for second order boundary value problem of impulsive dynamic equations on time scales
The authors establish existence results for an impulsive second-order dynamic equation on time scales \(-y^{\Delta\Delta}(t)=f(t,y(t))\), \(t\in[0,1]\subseteq{\mathbb T}\), \(t\neq t_k\), \(k=1,\dots,m\), where \({\mathbb T}\) is a time scale (nonempty closed subset of \({\mathbb R}\)) and where the solution \(y(t)\) satisfies \(y(0)=0=y(1)\), and \(y(t_k^+)-y(t_k^-)\) and \(y^\Delta(t_k^+)-y^\Delta(t_k^-)\) are functions of \(y(t_k^-)\) and \(y^\Delta(t_k^-)\), respectively. The solution \(y(t)\) does not need to be continuous at \(t=t_k\). The technique used is that of the fixed-point theory. Hypothesis (H1) of the paper contains the \textit{continuity} of the forward jump function \(\sigma:{\mathbb T}\to{\mathbb T}\) (which excludes time scales with left-dense/right-scattered points). It seems that this condition is not needed.
scientific article; zbMATH DE number 2093817
scientific article