A problem with impulse actions for nonlinear ODEs

Abstract

In the paper, we consider problem with impulse actions for system of nonlinear ordinary differential equations (ODEs) in the interval. For solving this problem we use a modification of parameterization method by Dulat Dzhumabaev. In the modification of the method we introduce the parameters as the values of the unknown function at the middle of the subintervals of the partition of the considered interval. The problem with impulse actions transfers to an equivalent problem system of nonlinear ODEs with parameters. Conditions for an existence of solution to the equivalent problem are obtained. Existence theorem for solutions this problem is established by one generalizes a theorem of Hadamard. We also constructed an algorithm for finding of solution to this problem. Finally, we found conditions for solvability to the problem with impulse actions for system of nonlinear ODEs in terms of special matrix composed by the initial data. This method can be applied to various types of nonlinear problems with impulse actions for ODEs.