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approaches zero attracts compact sets Ax(t Banach space bounded set Bx(t Chapter characteristic equation characteristic multiplier compact set consider the equation continuous function continuously differentiable Corollary decomposition deﬁned deﬁnition delays difference equations differential difference equations discuss eigenspace eigenvalues Equation 1.1 equivalent example existence exponential fact ﬁnite ﬁrst derivative ﬁxed point following result formal adjoint formal adjoint equation functional differential equations Furthermore given Hale hypotheses implicit function theorem implies Inequality inﬁnite initial data initial value integral Lemma Liapunov functionals linear equation linear operator linear systems manifold matrix neighborhood neutral equations NFDE NFDE(D obtain ordinary differential equations periodic orbit periodic solution precompact proof of Theorem properties real number RFDE roots satisﬁes Equation satisfy scalar Section semigroup solution of Equation solution operator speciﬁcally subset subspace Suppose Theorem 3.1 theory uniformly asymptotically stable unique solution variation-of-constants formula w-periodic process