Functional Differential Equations of Retarded Type |
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adjoint equation autonomous ax(t b₁ Banach space bilinear form Bx(t By(t characteristic equation characteristic multiplier choose closed bounded sets compact set completely continuous completes the proof Consider the equation constants formula continuous function continuous linear contradiction converges decomposed decomposition defined eigenspace eigenvalues equation x(t exists fact finite dimensional fixed point follows functional differential equations Furthermore implies initial value Lemma linear systems Lyapunov function mapping multiplier of 32.1 nondecreasing open set ordinary differential equations periodic solution positive eigenvectors proves the theorem real number satisfies semigroup sequence sets into bounded solution of 3.1 spectral radius stability subspace Suppose t₁ takes closed bounded uniformly asymptotically stable uniformly stable unique solution V(x+ variation of constants vector w-periodic x²(t zero ΑΦ Фє