Investigation of solutions of state-dependent multi-impulsive boundary value problems

Abstract

We describe a reduction technique allowing one to combine an analysis of the existence of solutions with an efficient construction of approximate solutions for a state-dependent multi-impulsive boundary value problem which consists of non-linear system of differential equationsu^{\prime}(t)=f(t,u(t))\quad\text{for a.e. }t\in[a,b],subject to the state-dependent impulse conditionu(t+)-u(t-)=\gamma_{t}(u(t-))\quad\text{for }t\in(a,b)\text{ such that }g(t,u(% t-))=0,and the non-linear two-point boundary conditionV(u(a),u(b))=0.

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

“…By Lemma 1, the first term approaches zero as z → y. The convergence to zero of the second term is established similarly to (14).…”

“…Another approach to the study of this kind of boundary value problems was proposed in [11,13,14]. It is based on the ideas of the Samoilenko numerical-analytical method.…”

𝒮‐asymptotically 𝓌‐periodic mild solutions for noninstantaneous impulsive integro‐differential equations with state‐dependent delay

“…Detailed information and applications; e.g. earlier studies [1][2][3][4][5][6][7][8][9][10] and the cited references. However, some phenomena in real life cannot be described by the action of instantaneous impulses, for instance, earthquakes and tsunamis.…”

Existence of solutions for fractional instantaneous and non-instantaneous impulsive differential equations with perturbation and Dirichlet boundary value

“…Detailed information and applications, see e.g. [3,6,14,5,12,15,7,11,16,13,24,17] and the cited references. However, not all the phenomena in real life could be described by instantaneous impulses, for example, earthquakes and tsunamis.…”

“…We will mainly use the Weierstrass theorem to get the existence of solutions for problem (1). Compared with the existing works, the paper has the following new sights: Firstly, we focus on the fractional instantaneous and non-instantaneous impulsive differential equations with perturbation, where g(t)|z(t)| p−2 z(t) is the disturbing term and is sublinear; Secondly, when p = 2, the problem (1) is reduced to the problem in [16] and when α = 1, problem (1) is simplified to an integer differential equation, which can be viewed as a supplement and extension of the problem in [16]. Finally, we give a definition of classical solution and then give the proof that the weak solution is also the classical solution for problem (1).…”