Solutions of the Difference Equation 𝑥𝑛+1=𝑥𝑛𝑥𝑛−1−1

Abstract

Our goal in this paper is to investigate the long-term behavior of solutions of the following difference equation: 𝑥𝑛+1=𝑥𝑛𝑥𝑛−1−1, 𝑛=0,1,2,…, where the initial conditions 𝑥−1 and 𝑥0 are real numbers. We examine the boundedness of solutions, periodicity of solutions, and existence of unbounded solutions and how these behaviors depend on initial conditions.