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We associate a discrete-time equation to an exponentially bounded semigroup and we characterize the exponential instability of the semigroup in terms of the complete admissibility of the pair (l ∞ (N, X), l ∞ (N, X)). As a consequence, we obtain that in certain conditions a C 0-semigroup is exponentially unstable if and only if the pair (C b (R + , X), C b… (More)
In this paper, we use the Leray–Schauder degree theory to establish new results on the existence and uniqueness of anti-periodic solutions for a class of nonlinear nth-order differential equations with two deviating arguments of the form x (n) (t) + f (t, x (n−1) (t)) + g 1 (t, x(t − τ 1 (t))) + g 2 (t, x(t − τ 2 (t))) = e(t). As an application, we also… (More)
We prove that a general system of variational difference equations is uniformly exponentially stable if and only if certain associated sets are of the second category. We also deduce necessary and sufficient conditions for uniform exponential stability of systems with uniformly bounded coefficients. We apply our results for the study of exponential… (More)
In this paper, we investigate some properties of the operator exponential solutions of the initial value problem X ∆ (t) = A(t)X(t), t ∈ T X(s) = I, where I is the identity operator on a Banach space X, s is an initial point in a time scale T and A(t) is a bounded linear operator on X, t ∈ T. Also, we give an example of differentiable 2 × 2 matrices A(t)… (More)
This paper is concerned with systems with control whose state evolution is described by linear skew-product semiflows. The connection between uniform exponential stability of a linear skew-product semiflow and the stabilizability of the associated system is presented. The relationship between the concepts of exact con-trollability and complete… (More)
In this paper we investigate results on finite-time blow-up of solutions to a nonlocal quasi-linear parabolic problem. We extend some known results to higher dimensions .