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About differential inequalities for nonlocal boundary value problems with impulsive delay equations. (English) Zbl 1349.34256
The paper deals with the non-local boundary value problem for the first-order impulsive delay differential equation with an essentially bounded right-hand side $\begin{gathered} x'(t)=\sum_{i=1}^mp_i(t)x(t-\tau_i(t))+f(t),\quad t\in[a,b],\\ x(\zeta)=0,\quad\zeta\not\in[a,b],\\ x(t_j)=\beta_jx(t_j-0),\quad j=1,\ldots,k,\\ \int_a^b\varphi(s)x'(s)ds+\theta x(a)=c.\end{gathered}$ The main result (namely, Theorem 2.1) provides general assertions, which are equivalent to the property $G(t,s)\leq0\quad\text{for}\;s,t\in[a,b],\quad G(t,s)<0\quad\text{for}\;a\leq s<t\leq b$ of Green’s function of the problem considered under the assumptions that $$p_i(t)\leq0$$ ($$i=1,\ldots,m$$), $$\theta\neq0$$, and $G_0(t,s)\leq0\quad\text{for}\;s,t\in[a,b],\quad G_0(t,s)<0\quad\text{for}\;a\leq s<t\leq b,$ where $$G_0$$ is Green’s function of the auxiliary problem $\begin{gathered} x'(t)=f(t),\quad t\in[a,b],\\ x(t_j)=\beta_jx(t_j-0),\quad j=1,\ldots,k,\\ \int_a^b\varphi(s)x'(s)ds+\theta x(a)=c.\end{gathered}$

##### MSC:
 34K10 Boundary value problems for functional-differential equations 34K06 Linear functional-differential equations 34K45 Functional-differential equations with impulses 34K38 Functional-differential inequalities 34K12 Growth, boundedness, comparison of solutions to functional-differential equations
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