Stability of Solutions in the Broad Sense for Evolutionary Systems With Periodic Conditions

ABSTRACT This paper investigates periodic solutions in the broad sense of evolutionary systems subject to periodic conditions. For the corresponding homogeneous system, normalized fundamental matrices are constructed. For nonhomogeneous linear systems, conditions ensuring the existence and uniqueness of periodic solutions in the broad sense are obtained, and the asymptotic stability of these solutions with respect to the time variable is investigated. The analysis is then extended to nonlinear evolutionary systems of triangular‐block type. Using the contraction mapping principle, sufficient conditions for the existence and uniqueness of periodic solutions in the broad sense are established, and their asymptotic stability is proved. An example is presented to verify the obtained sufficient conditions and illustrate the asymptotic stability of the periodic solution. The results are applicable to a wide class of evolutionary systems subject to periodic conditions.