On a nonlinear impulsive system of integro-differential equations with degenerate kernel and maxima

Abstract

A nonlocal boundary value problem for a system of ordinary integro-differential equations with impulsive effects, degenerate kernel and maxima is investigated. The boundary value problem is given by the integral condition. The method of successive approximations in combination with the method of compressing mapping is used. The existence and uniqueness of the solution of the boundary value problem are proved. The continuous dependence of the solution on the right-hand side of the boundary value condition is shown. KEYWORDS impulsive integro-differential equations, nonlocal condition, successive approximations, existence and uniqueness, continuous dependence of solution FOR CITATION Yuldashev T.K., Fayziev A.K. On a nonlinear impulsive system of integro-differential equations with degenerate kernel and maxima.

Periodic solutions for an impulsive system of integro-differential equations with maxima

“…where 𝑥(𝑡, 𝑥 0 ) = lim 𝑘→∞ 𝑥 𝑘 (𝑡, 𝑥 0 ) = 𝑥 ∞ (𝑡, 𝑥 0 ) is the solution of the non-linear system (13). Therefore, 𝑥 ∞ (𝑡, 𝑥 0 ) is the solution of the system of impulsive integro-differential equations (1) for Δ(𝑥 0 ) = 0 through 𝑥 0 at 𝑡 = 0.…”

“…. , 𝑝, then this solution can be proven to be based on the system of nonlinear functional-integral equations (13). The questions of the existence of a solution of the system of impulsive differential equations (1) we reduce to the questions of the existence of zeros of function Δ(𝑥 0 ) in ( 21) and we solve this problem by finding zeros of function Δ 𝑘 (𝑥 0 ) in ( 22).…”

“…Such differential and integro-differential equations are called equations with impulsive effects. Various publications are appearing on the study of differential and integro-differential equations with impulsive effects, describing many natural and technical processes (see for instance [1][2][3][4][5][6][7][8][9][10][11][12][13]).…”

Mixed problem for a linear differential equation of parabolic type with nonlinear impulsive conditions

“…[29][30][31][32][33][34][35][36][37][38]. A lot of publications are devoted to study differential equations with impulsive effects, describing many natural and technical processes (see, for example, [39][40][41][42][43][44][45][46][47][48][49][50]). The questions of existence and uniqueness of periodic solutions of differential and integro-differential equations were studied in [51][52][53][54][55].…”

Inverse problem for a second order impulsive system of integro-differential equations with two redefinition vectors and mixed maxima

“…Such differential equations with "discontinuities" at fixed or non-fixed time moments are called differential equations with impulsive effects. There are a lot of publications of devoted to differential equations with impulsive effects, which describe many natural and technical processes [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25].…”