Representation of solutions and its periodicity of quaternion difference equations with variable coefficients

In this paper, we investigate general solutions and periodic solutions of quaternion difference equations (QDCEs) with variable coefficients. To begin with, we provide general solutions for linear homogeneous QDCEs (LHQDCEs), an algorithm for computing the fundamental matrix and its properties, and derive general solutions for linear nonhomogeneous QDCEs (LNHQDCEs) using the variation of constants formula as well as for semilinear QDCEs (SLQDCEs) using the fixed-point theorem. Secondly, the conditions that ensure the existence of periodic solutions for LHQDCEs are presented, thereafter, periodic solutions of LNHQDCEs under different conditions are derived using the Green function and adjoint system, respectively. Moreover, we establish the existence and uniqueness of periodic solutions of SLQDCEs. Finally, several examples are presented to demonstrate the correctness of the theoretical results. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2025.