Abstract
For nonautonomous linear equations x′ = A(t)x, we give a complete characterization of general nonuniform contractions in terms of Lyapunov functions. We consider the general case of nonuniform contractions, which corresponds to the existence of what we call nonuniform (D, μ)‐contractions. As an application, we establish the robustness of the nonuniform contraction under sufficiently small linear perturbations. Moreover, we show that the stability of a nonuniform contraction persists under sufficiently small nonlinear perturbations.
Highlights
We consider nonautonomous linear equations x A t x, 1.1 where A : R0 → B X is a continuous function with values in the space of bounded linear operators in a Banach space X
For a large class of nonlinear perturbations f t, x with f t, 0 0 for every t, we show that if 1.1 admits a nonuniform contraction, the zero solution of the equation xAtxft, x
The proof uses the corresponding characterization between the nonuniform contractions and quadratic Lyapunov functions
Summary
We consider nonautonomous linear equations x A t x, 1.1 where A : R0 → B X is a continuous function with values in the space of bounded linear operators in a Banach space X. Our main aim is to characterize the existence of a general nonuniform contraction for 1.1 in terms of Lyapunov functions. Given two growth rates μ, ν, we say that 1.1 admits a nonuniform μ, ν -contraction if there exist constants K, α > 0 and ε ≥ 0 such that. The importance of Lyapunov functions is well established, in the study of the stability of trajectories both under linear and nonlinear perturbations. For a large class of nonlinear perturbations f t, x with f t, 0 0 for every t, we show that if 1.1 admits a nonuniform contraction, the zero solution of the equation xAtxft, x. The proof uses the corresponding characterization between the nonuniform contractions and quadratic Lyapunov functions
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