end{aligned} Since sA is convergent to zero, by Theorem 1.2, we see that ((I-sA)) is nonsingular and (I-sA)^{-1}=I+sA +cdots+ sA)^{p}+cdots.
If (( omega_{n} )_{ngeq1}) is convergent to zero and lim_{nrightarrowinfty}n^{k} omega_{n}- omega_{n+1})=lin mathbb{R}, with (k>1), then there exists the limit lim_{nrightarrowinfty}n^{k-1}omega_{n}= frac{l}{k-1}.
Note that | y ( θ t ω ) | is tempered, and by (4.23 - 4.24), the integrand of the second term on the right-hand side of (4.22) is convergent to zero exponentially as s → − ∞.
Lemma 1 If the sequence ( x n ) n ∈ N is convergent to zero and there exists the limit lim n → + ∞ n s ( x n − x n + 1 ) = l ∈ (2.1).
Let be a multivalued operator with closed (with respect to ) graph, such that the following conditions are satisfied: (a), for each and each ; (b there exist such that the matrix is convergent to zero such that for each, for each and all, there exists with.
If the sequence ((x_{n})_{ninmathbb{N}}) is convergent to zero and there exists the limit lim_{nrightarrow+infty}n^{s}(x_{n}-x_{n+1})=l in[-infty,+infty], (2.1) with (s>1), then lim_{nrightarrow+infty}n^{s-1}x_{n}= frac{l}{s-1}.
If the sequence ((x_{n})_{nin mathbb{N}}) is convergent to zero and there exists the limit begin{aligned} lim_{nrightarrow +infty }n^{s}(x_{n}-x_{n+1})=l in [-infty,+ infty ] end{aligned} (1.13) with (s>1), then begin{aligned} lim_{nrightarrow +infty }n^{s-1}x_{n}=frac{l}{s-1}.
If ( λ n ) n ≥ 1 is convergent to zero and there exists the limit lim n → ∞ n k ( λ n − λ n + 1 ) = l ∈ R, with k > 1, then there exists the limit lim n → ∞ n k − 1 λ n = l k − 1. Lemma 1 gives a method for measuring the speed of convergence.
If the sequence ((x_{n})_{ninmathbb{N}}) is convergent to zero and there exists the limit lim_{nrightarrow+infty}n^{s}(x_{n}-x_{n+1})=l in[-infty,+infty] (2.1) with (s>1), then there exists the limit lim_{nrightarrow+infty}n^{s-1}x_{n}= frac{l}{s-1}.
