Application of the method of averaging to boundary value problems for differential equations with non-fixed moments of impulse

Abstract

The method of averaging is applied to study the existence of solutions of boundary value problems for systems of differential equations with non-fixed moments of impulse action. It is shown that if an averaged boundary value problem has a solution, then the original problem is solvable as well. Here the averaged problem for the impulsive system is a simpler problem of ordinary differential equations.

A problem with impulse actions for nonlinear ODEs

“…Then, we make a change of functions: 𝑥𝑥𝑥𝑥 𝑟𝑟𝑟𝑟 (𝑡𝑡𝑡𝑡) = 𝑦𝑦𝑦𝑦 𝑟𝑟𝑟𝑟 (𝑡𝑡𝑡𝑡) + 𝜉𝜉𝜉𝜉 𝑟𝑟𝑟𝑟 on each 𝑟𝑟𝑟𝑟-th subinterval. We transfer problem ( 4)-( 6) to the equivalent problem with parameters 𝜉𝜉𝜉𝜉 𝑟𝑟𝑟𝑟 : ���������� , are continuously differentiable on [𝑡𝑡𝑡𝑡 𝑟𝑟𝑟𝑟−1 , 𝑡𝑡𝑡𝑡 𝑟𝑟𝑟𝑟 ) and satisfy system of nonlinear ODEs (7), conditions (8), and relations ( 9), (10) with 𝜉𝜉𝜉𝜉 𝑟𝑟𝑟𝑟 = 𝜉𝜉𝜉𝜉 𝑟𝑟𝑟𝑟 * , 𝑟𝑟𝑟𝑟 = 1, 𝑘𝑘𝑘𝑘 + 1 ���������� . The equivalence of problems (1)-( 3) and ( 7)- (10) is understood in the following sense.…”

“…Vice versa, if a pair �𝑦𝑦𝑦𝑦 �[𝑡𝑡𝑡𝑡], 𝜉𝜉𝜉𝜉 ̃� with elements In contrast to problem ( 4)-( 6), problem with parameters ( 7)- (10) have conditions (8) for the values of the desired functions in the middle of the subintervals…”

“…This is the modification of the parameterization method. We get the problem with parameters ( 7)- (10), where instead of the initial conditions for the desired function [12], appear the conditions (8) with the values of the desired function in the middle of the subintervals…”

“…Then the sequence of pairs (𝑦𝑦𝑦𝑦 (𝑘𝑘𝑘𝑘) [𝑡𝑡𝑡𝑡], 𝜉𝜉𝜉𝜉 (𝑘𝑘𝑘𝑘) ), 𝑘𝑘𝑘𝑘 ∈ 𝑁𝑁𝑁𝑁, determined by the algorithm belongs to 𝑆𝑆𝑆𝑆�𝑦𝑦𝑦𝑦 (0) [𝑡𝑡𝑡𝑡], 𝜌𝜌𝜌𝜌 𝑦𝑦𝑦𝑦 � × 𝑆𝑆𝑆𝑆(𝜉𝜉𝜉𝜉 (0) , 𝜌𝜌𝜌𝜌 𝜉𝜉𝜉𝜉 ) , converges to (𝑦𝑦𝑦𝑦 * [𝑡𝑡𝑡𝑡], 𝜉𝜉𝜉𝜉 * ) is an isolated solution of problem ( 7)- (10) in 𝑆𝑆𝑆𝑆�𝑦𝑦𝑦𝑦 (0) [𝑡𝑡𝑡𝑡], 𝜌𝜌𝜌𝜌 𝑦𝑦𝑦𝑦 � × 𝑆𝑆𝑆𝑆(𝜉𝜉𝜉𝜉 (0) , 𝜌𝜌𝜌𝜌 𝜉𝜉𝜉𝜉 ) and the following estimates hold:…”

“…The problem with impulse actions holds a very significant place in the theory of discontinuous differential equations (see [1][2][3][4] and references cited therein). Many authors investigate the existence and uniqueness of problem impulse actions for differential equations by using various methods [5][6][7][8][9][10][11]. To the best of our knowledge only monography [4] investigate impulsive system with variable time of the impulse action via numericalanalytic method, Lyapunov direct method and Green's function method.…”