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We are interested in the question whether the-semigroup converges to 0 with respect to the operator norm as, i.e. whetheras. In this case,for someand all, so the semigroup is said to beuniformly exponentially stable. A necessary condition for the uniform exponential stability of a-semigrou...
Let U be an invertible operator on a Banach space Y. U is said to be trigonometrically well-bounded provided the sequence {U n } n ∞ =∞ is the Fourier-Stieltjes transform of a suitable projection-valued function E(·): [0, 2π]→(Y). This class of operators is known to apply ...
An n × n-matrix-valued function A(t) on T is called regressive if I + µ(t)A(t) is invertible for all t ∈ T. The set of functions being both regressive and rd-continuous is denoted by R = R(T) = R(T, R)(R(T, R n×n )). The set of all regressive functions ...
Sets, relations, and functions: Domain, co-domain, and range-functions: Into, on to, one-one in to, one-one on to functions; constant function; identity function; composition of functions; invertible functions; binary operations Complex numbers: Complex numbers in the form a +ib; real and ...
P function g will The ? be called a g-function if g is a function from to (0; 1) and k=01 g(ix) = 1. We i will only consider those g which are continuous here. The cylinder of those points which agree with x for the rst n + 1 places (that is the set fy: yi = xi ...
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Then, a reciprocally convex combination of these functions over D is a function of form (2.3) where the real numbers αi satisfy αi > 0 and ∑iαi = 1. Lemma 2.4 (see [1].)For any symmetric positive define matrix R > 0, scalars γ2 > γ1 > 0 and vector function x : [...
There are also other criteria for exponential dichotomy in terms of the existence of solutions of inhomogeneous equations in certain spaces (Massera and Schäffer [5], Coppel [2]) or in terms of the existence of a Lyapunov function (Coppel [2]) but these ...
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