The phrase "a generator of a solution" is correct and usable in written English.
It can be used in contexts where you are describing something that produces or creates a solution to a problem or challenge.
Example: "The new software acts as a generator of a solution for optimizing our workflow and increasing productivity."
Alternatives: "a source of a solution" or "a creator of a solution".
then is a generator of a solution operator, which is given by (2.6).
Hence is a generator of a solution operator satisfying the estimate (2.7) on.
where, is a generator of a solution operator,, and stand for the right and left limits of at, respectively, and, are appropriate functions to be specified later.
New existence and uniqueness results in the case when is a generator of a solution operator, under various criterions, are proved.
We focus on a Cauchy problem for impulsive integrodifferential equations involving nonlocal initial conditions, where the linear part is a generator of a solution operator on a complex Banach space.
We call A the generator of a solution operator if there exist ω ˜ ∈ R and a strong continuous function S α : R + → L ( X ) such that { λ α : R e λ > ω ˜ } ⊂ ρ ( A ) and λ α − 1 ( λ α − A ) − 1 x = ∫ 0 ∞ e − λ t S α ( t ) x d t, R e λ > ω ˜, x ∈ X.
We call A the generator of a solution operator if there exist (muin {mathbb{R}}) and a strong continuous function (S_{alpha}: {mathbb{R}}^rightarrow L X)) such that ({lambda^{alpha}:operatorname{Re}lambda>mu }subsetrho(A)) and lambda^{alpha-1}bigl(lambda^{alpha}-Abigr)^{-1}x=int _{0}^{infty }e^{-lambda t}S_{alpha}(t)x,dt,quad operatorname{Re}lambda>mu, xin X.
We say that A is the generator of a solution operator if there exist (omegainmathbb{R}) and a strongly continuous function (S_{alpha}:R_tomathcal{L}(mathbb {X})) such that ({lambda^{alpha}:operatorname{Re}lambda>omega} subsetrho(A)) and lambda^{alpha-1} bigl(lambda^{alpha}I-A bigr)^{-1}x= int^{+infty }_{0}e^{-lambda t}S_{alpha}(t)x,dt, quadoperatorname{Re}lambda >omega, xinmathbb{X}.
If A is sectorial of type μ with (0
In this case, S α ( t ) is called the solution operator generated by A. Note that if A is sectorial of type ω ˜ with 0 < θ < π ( 1 − α / 2 ), then A is the generator of a solution operator given by S α ( t ) : = 1 2 π i ∫ γ e λ t λ α − 1 ( λ α − A ) − 1 d λ, where γ is a suitable path lying outside the sector ω ˜ + S θ [32].
We call A is the generator of a solution operator if there are (omegain mathbb{R}) and a strongly continuous function S_{alpha}: mathbb{R}^rightarrow L X) such that bigl{ lambda^{alpha}: operatorname{Re}lambda>omega bigr} subseteq rho(A) and lambda^{alpha-1}bigl(lambda^{alpha}-Abigr)^{-1}x=int _{0}^{infty }e^{-lambda t}S_{alpha}(t)x, mathrm{d}t, quadoperatorname{Re}lambda>omega, xin X.
