Existence of Periodic Solutions to Quaternion-Valued Impulsive Differential Equations

In this paper, periodic solutions of quaternion-valued impulsive differential equations (QIDEs) are considered. First, the sufficient and necessary conditions to guarantee periodic solutions are given for linear homogeneous QIDEs. Second, the representations of periodic solutions are derived by constructing Green functions in one case, and the sufficient and necessary conditions to guarantee periodic solutions are presented by means of adjoint systems in the other case for linear nonhomogeneous QIDEs. In addition, the existence and uniqueness of periodic solutions are studied by virtue of fixed point theorems for semilinear QIDEs. All results in the sense of complex-valued and quaternion-valued are equivalent to each other due to the adjoint matrix of quaternion matrix and the isomorphism between quaternion vector space and complex variables space. Finally, examples and simulations are provided to demonstrate the validity of our main results.