Periodic and Pseudo Periodic Solutions for Matrix Sylvester Dynamic System on Measure Chains

In this paper, we investigate the existence conditions for matrix Sylvester dynamic system solutions on measure chains, both periodic and pseudo-periodic. We convert the matrix Sylvester dynamic system on measure chains to an equivalent Kronecker product dynamic system on measure chains using vectorization operator. There are two steps in this process. For this system, an exponential dichotomy is admitted by one of the existing homogeneous and non-homogeneous equations. To expand on the findings, we develop another pseudo-periodic solution. Finally, we apply the Banach contraction mapping principle to obtain the results.