Studies on anti-periodic boundary value problems for two classes of special second order impulsive differential equations (Q2874191)

In this paper, the authors consider two anti-periodic boundary value problems NEWLINE\[NEWLINE\begin{cases} x''(t)=f(t,x(t)),\;t\in[0,T],\;t\neq t_1,\dots,t_p, \\ \Delta x(t_k)=l_k(x(t_k)), \;\Delta x'(t_k)=l_k^*(x'(t_k)),\;k=1,2,\dots,p, \\ x(0)=-x(T),\;x'(0)=-x'(T),\end{cases} \tag{1}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\begin{cases} x''(t)=f(t,x'(t)),\;t\in[0,T],\;t\neq t_1,\dots,t_p,\\ \Delta x(t_k)=l_k(x(t_k)),\;\Delta x'(t_k)=l_k^*(x'(t_k)),\;k=1,2,\dots,p,\\ x(0)=-x(T),\;x'(0)=-x'(T),\end{cases} \tag{2}NEWLINE\]NEWLINE where \(l_{k},l_{k}^*\in C(\mathbb R,\mathbb R)\), \(0=t_0<t_1<\dots<t_p<t_{p+1}=1\), \(\Delta x(t_k)=x(t_k^+)-x(t_k)\), \(\Delta x'(t_{k})=x'(t_k^+)-x'(t_k)\) and \(f:[0,T]\times\mathbb R\to\mathbb R\) is a Carathéodory function. Using impulsive differential inequalities and monotone iterative technique coupled with lower and upper solutions, the authors obtain the existence of solutions of (1) and (2). Moreover, they give two examples.