Periodic Solutions to a Third-Order Conditional Difference Equation over the Integers

As mentioned in 1 , difference equations appear naturally as a discrete analogue and as a numerical solution of differential and delay differential equations having applications in various scientific branches, such as biology, ecology, physics, economy, technics, and sociology. The stability, asymptotic behavior, and periodic property of solutions to difference equations had been widely investigated, such as 2–14 . Recently, the study of max-type difference equation attracted a considerable attention, for example, 7, 11, 15–25 . Although max-type difference equations are relatively simple in form, it is unfortunately extremely difficult to understand thoroughly the behavior of their solutions. The max operator arises naturally in certain models in automatic control theory. On the other hand, there exists another kind of difference equations called conditional difference equations, which also have simple forms, but it is difficult to understand clearly the behavior of their solutions. From 2, 5 , we know that the following conditional difference equation