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Impulsive dynamic equations on a time scale. (English) Zbl 1184.34091
After some preparations concerning the time scales calculus the authors study the impulsive initial value problem $$ y^\Delta = -a(t) y^\sigma + f(t,y), \quad y(0) = 0, \quad y(t_i^+) = y(t_i^-) + I (t_i,y(t_i)), \tag $*$ $$ where $t \in [0,T]$ and $i\in\{ 1,\dots,n \}$. By using a contraction type fixed point theorem of Krasnoselskii it is shown that the $(*)$ has a solution. Finally the authors modify conditions and techniques in their proofs in order to assure uniqueness and stability of the solution.

34N05Dynamic equations on time scales or measure chains
34A37Differential equations with impulses
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
47N20Applications of operator theory to differential and integral equations
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