Abstract
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.
Highlights
The mathematical model of many real-world phenomena can be represented by impulsive equations, and these phenomena have undergone significant changes in the progress
Such equations have a wide range of applications in many areas, including economics, optimal control, dynamic systems, medicine, and many other fields
In 2003, De la Sen and Luo [7] studied the stability of a class of linear time-delay systems
Summary
The mathematical model of many real-world phenomena can be represented by impulsive equations, and these phenomena have undergone significant changes in the progress. Based on the above observation, in this paper, periodic boundary value problems for impulsive difference equations with time delay are considered. We will consider the following first-order impulsive difference equations with time delay:
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