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A general initial value problem
k Existence lU 5 Continuation of solutions
31 other sections not shown
adjoint equation autonomous ax(t Banach space belongs bilinear form Bx(t characteristic equation characteristic multiplier choose closed bounded sets compact set compact subset completely continuous completes the proof Consequently Consider the equation continuous first derivative continuous function continuous linear contradiction converges cp(o cr(A decomposed decomposition defined eigenspace eigenvalues equation x(t exists fact finite dimensional fixed point Floquet theory follows functional differential equations Furthermore inequalities initial value integral equation L(xt Lebesgue integrable Lemma linear systems Lyapunov function nondecreasing NONLINEAR nonzero obtain open set ordinary differential equations periodic solution positive definite positive eigenvectors proves the theorem real number satisfies semigroup sequence sets into bounded solution of 26.2 solution of 3.1 spectral radius subspace takes closed bounded uniformly asymptotically stable uniformly stable unique solution unstable V(xt variation of constants vector x(cp Xl)k zero