Kaplun Y. Asymptotic solutions to the singular perturbed differential equations with impulsive effects

Object of research: singularly perturbed equations under the conditions of impulsive effects at fixed moments of time are studied. The main new results: the algorithm of constructing asymptotical solution to equation with a small parameter at the highest derivative is developed for the case when the condition of the impulsive effects contains a small parameter as well as when the condition of impulsive effects does not depend on a small parameter. There is proved analogious of A.B.Vasilyeva's theorem on the order with that asymptotical solution satisfies the initial problem. There is considered unperturbed problem for the singularly perturbed one under the condition of impulsive effects. This problem contains implicit function equation and condition of impulsive effects. The necessary and sufficient conditions on existence of discontinuous solutions to the implicit function equation (under the condition of impulsive effects) are found. Periodicity of the solutions is studied. For singularly perturbed equation with impulsive effects when the condition of impulsive effects contains a small parameter there is considered nonperturbed problem being implicit func-tion equation. For such equation a problem on existence of global solution is considered. Algorithm of determination of maximal interval is given and global implicit function theorem is proved. The piecewise smooth (continuous) solutions to the problem mentioned above are studied and sufficient conditions on existence of periodical solutions to the problem are found. Application of global implicit function theorem for investigating of problem on prolongation of solution to the first order ordinary differential equation to and via the singular set, that is the set where the right part of the equation is not defined, is demonstrated.