About Positivity of Green's Functions for Nonlocal Boundary Value Problems with Impulsive Delay Equations

Abstract

The impulsive delay differential equation is considered (Lx)(t) = x′(t) + ∑i=1 m p i(t)x(t − τ i(t)) = f(t), t ∈ [a, b], x(t j) = β j x(t j − 0), j = 1,…, k, a = t 0 < t 1 < t 2 < ⋯<t k < t k+1 = b, x(ζ) = 0, ζ ∉ [a, b], with nonlocal boundary condition lx = ∫a b φ(s)x′(s)ds + θx(a) = c, φ ∈ L ∞[a, b]; θ, c ∈ R. Various results on existence and uniqueness of solutions and on positivity/negativity of the Green's functions for this equation are obtained.

About differential inequalities for nonlocal boundary value problems with impulsive delay equations

“…Existence of Green's function G 0 (t, s) for problem (2.9), (2.4) was discussed in [12]. On the basis of estimates of Green's function G 0 (t, s) of problem (2.9), (2.4), sufficient conditions of positivity of Green's function G(t, s) of nonlocal boundary value problems of the type (2.1)-(2.4) were obtained in [11]. In this paper we propose theorems about differential inequalities that allow us to obtain results about positivity/negativity of Green's function of the nonlocal boundary value problem (2.1)-(2.4) based only on sign-constancy of G 0 (t, s) and without knowledge of the explicit formula for G 0 (t, s).…”

Sign-constancy of Green’s functions for impulsive nonlocal boundary value problems

“…was considered. Positivity of Green's functions for the first order impulsive functional differential equations with nonlocal boundary conditions was studied in [15][16][17]. Nonlocal boundary value problems for systems of impulsive functional differential equations were considered in [6].…”

Nonnegative solutions for a system of impulsive BVPs with nonlinear nonlocal BCs

“…. 12 The existence of multiple solutions of the system (4.1) follows from Theorem 4.1. Then, for ρ 1 = 1/8, ρ 2 = 1 and ρ 3 = 11, we have (the constants that follow have been rounded to 2 decimal places unless exact) (1, 1, 1…”

“…studied for example [12,20,28,31,32,34,41,44,49,50,53,54]. In the case of impulsive equations, nonlocal BCs have been studied by many authors, see for example [5,6,8,13,14,18,31,32,39,56] and references therein.…”