Boundary value problems on half‐line for second‐order nonlinear impulsive differential equations

Abstract

We obtain sufficient conditions for existence and uniqueness of solutions of boundary value problems on half‐line for a class of second‐order nonlinear impulsive differential equations. Our technique is different than the traditional ones, as it is based on asymptotic integration method involving principal and nonprincipal solutions. Examples are provided to illustrate the relevance of the results.

Existence of solutions for impulsive boundary value problems on infinite intervals

“…Observe in (1) that impulse effects occur only on the solutions while (2) has continuous solutions as the impulse effects occur only on the derivatives of the solutions. The method of the paper [2] is different from the other studies in the literature as it relies on principal and nonprincipal solutions. A similar approach was applied in [3] and [7], where a particular case of the impulsive BVP (2) was considered in [3], while [7] dealt with a BVP without impulse effects.…”

“…There are many results in the literature regarding the existence of solutions to impulsive BVPs, e.g. [2,3,[10][11][12]. Below, we mention some recent results about impulsive BVPs on unbounded domains.…”

Iterative learning tracking control for second‐order nonlinear hyperbolic impulsive partial differential systems

“…where L j = max{L P 1, j , L P 2, j , L F j }, j = 1, 2, … , m. Adding Equations ( 9) and (10), it can get the following inequality…”

“…The refs. [9] and [10] respectively solved existence and uniqueness theorems of solutions for initial and boundary value problems of second‐order nonlinear impulsive differential equations. Since distributed ILC in the spatial domain leads to high energy consumption and complex structure, ref.…”

“…Hence, controlling the initial state error and accurate tracking are the emphasis and difficulty for partial differential systems. As for the second‐order impulse system, we only get the conclusions of stability [35], controllability [16], existence and uniqueness of solution [36], initial value [9], and boundary value [10] problems, and so on. Meanwhile, the accurate tracking problem of the second‐order impulse system has not achieved relevant results.…”