Periodic boundary value problems for first-order impulsive difference equations with time delay

This paper focuses on a certain type of periodic boundary value problems for first-order impulsive difference equations with time delay. Notions of lower and upper solutions are introduced, with which two new comparison theorems are established. Using Schaefer's fixed point theorem, sufficient conditions for the existence and uniqueness of solutions to the corresponding linear problem of the boundary value problem are derived. By utilizing monotone iterative methods combined with the methods of lower and upper solutions, an existence theorem of extremal solutions to first-order impulsive difference equations with delay is obtained. These results extend some existing results in the literature. An interesting example is also given to verify the results obtained.