A shadowing lemma with applications to semilinear parabolic equations

The property of hyperbolic sets that is embodied in the Shadowing Lemma is of great importance in the theory of dynamical systems. In this paper a new proof of the lemma is presented, which applies not only to the usual case of a diffeomorphism in finite-dimensional space but also to a sequence of possibly noninvertible maps in a Banach space. The approach is via Newton’s method, the main step being the verification that a certain linear operator is invertible. At the end of the paper an application to parabolic evolution equations is given.