Monotone iterative technique for first order impulsive difference equations with periodic boundary conditions (Q1888524)

The authors investigate a lot of conditions under which the first-order impulsive difference equation with periodic boundary condition \[ \begin{gathered} \Delta x(n)= f(n,x(n)),\quad n\neq n_k,\quad n_k\in \{0,1,\dots, N\}:= J,\\ \Delta x(n_k)= I_k(x(n_k)),\quad k= 1,2,\dots, p,\\ x(0)= x(N),\end{gathered} \] where \(f\in C(J\times\mathbb{R},\mathbb{R})\), \(I_k\in C(\mathbb{R},\mathbb{R})\), \(0< n_1< n_2<\cdots< n_p< N\), and \(N\) is a positive integer. Among others the method of upper and lower solution is used to prove the existence and uniqueness of so-called extremal solutions to the problem under consideration.