zbMATH — the first resource for mathematics

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 \[ \begin{gathered} x'(t)=f(t),\quad t\in[a,b],\\ x(t_j)=\beta_jx(t_j-0),\quad j=1,\ldots,k,\\ \int_a^b\varphi(s)x'(s)ds+\theta x(a)=c.\end{gathered} \]

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
Full Text: Link