In this paper to obtain exponential stability conditions we use the substitution x t)= Wz) (t):= int_{t_{0}}^{t} Y t,s) z(s),ds, where (Y t,s)) is the fundamental function of exponentially stable autonomous ordinary differential equation of the second order and then analyze the operator equation z=Tz+f in some functional Banach spaces on semi-axes.
Assume that U is an admissible function, ν is a non-negative non-degenerate Borel measure on [ 0, ∞ ), and φ is the fundamental function of ν with respect to U q and σ ∈ Ω U p. If { x k } is a discretizing sequence for φ with respect to U q, then ∫ [ 0, ∞ ) φ ( t ) r q − 1 σ ( t ) r p d ν ( t ) ≈ ∑ k ∈ Z φ ( x k ) r q σ ( x k ) r p. Lemma 2.2 ([[7], Corollary 2.13]).
Assume that U is an admissible function, f ∈ Ω U, ν is a non-negative non-degenerate Borel measure on [ 0, ∞ ) and φ is the fundamental function of ν with respect to U q. If { x k } is a discretizing sequence for φ with respect to U q, then ( ∫ [ 0, ∞ ) ( f ( t ) U ( t ) ) q d ν ( t ) ) 1 q ≈ ( ∑ k ∈ Z ( f ( x k ) U ( x k ) ) q φ ( x k ) ) 1 q. Lemma 2.3 ([[7], Lemma 3.5]).
While a lot of news coverage has used a partisan framework to address the new voter restrictions ("Stricter Voting Laws Carve Latest Partisan Divide," front page, Sept. 26), what's really at stake is the fundamental functioning of democracy.
If is a Banach sequence space with then is called the fundamental function of.
Although the function of DnaT in the recruitment of DnaB helicase has been proposed, the fundamental function of DnaT for the replication restart primosome assembly is not widely known.
The mapping is called the fundamental function of the space.
Denote by the Cauchy function of the equation which is also the fundamental function of (3.1).
Comparing this representation with the variation-of-constant formula of Hale [[3], Chapter 6, Section 6.1], we see that they almost coincide; the kernel in the integral term is called the fundamental function by his numerous followers.
It is obvious that the fundamental function X can be written as a linear combination of linearly independent vectors.
