Viable periodic solutions in state-dependent impulsive problems
This nice paper presents a topological tool, the impulsive index, suitable for detecting periodic trajectories for state-dependent impulsive ODEs, i.e. impulses occur at variable times, under the additional requirement that these trajectories are viable in a prescribed closed subset \(K\). For the construction of the impulsive index the fixed point index on ANRs is applied to an induced discrete semidynamical system on a barrier, where jumps occur. The author shows in Section 2, that the impulsive index satisfies the main properties which one could expect: existence, additivity, homotopy. Section 2 is dedicated to present sufficient conditions for the homotopy property. Then, in Section 4, the Poincaré map technique is adapted to compute the impulsive index and an example is presented to illustrate it. Finally, in the last section of the paper, the author discusses advantages and disadvantages of an alternative construction of the impulsive index.
scientific article; zbMATH DE number 6482868