In this paper the persistence of bounded solutions to degenerate evolution equations of SobolevGalpern type is discussed. In order to define the evolution operator well, we study the existence and uniqueness of solutions to its linear form. On this basis we discuss exponential dichotomies of the evolution operator and give the Fredholm alternative result for bounded solutions of nonhomogeneous linear degenerate equations. This result enables us to give a condition for the persistence of bounded solutions of a general degenerate nonlinear autonomous equation under a nonautonomous perturbation.

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