A generation theorem for the perturbation of exponentially equicontinuous C₀-semigroups on locally convex spaces

In this paper, we study the well-posedness of the evolution equation of the form u'(t) = Au(t) + Cu(t), t ≥ ۰ where A is the infinitesimal generator of an exponentially equicontinuous C₀-semigroup and C is a (possibly unbounded) linear operator in a sequentially complete locally convex Hausdorff space X. In particular, we demonstrate that if A generates an exponentially equicontinuous C₀-semigroup (T_A(t))_{t ≥ ۰} satisfying p(T_A(t)x) ≤ e^{ωt}q(x) and C is a linear operator on X such that D(A) ⊂ D(C) and {K⁻¹(μ-ω)ⁿ(CR(μ, A))ⁿ; μ > ω, n ∈ ℕ} is equicontinuous, then the above-mentioned evolution equation is well-posed, that is, A + C generates an exponentially equicontinuous C₀-semigroup (T_{A+C}(t))_{t ≥ ۰} satisfying p(T_{A+C}(t)x) ≤ e^{(ω+K)t}q(x).

کلیدواژه ها:

C₀-semigroups ، continuous linear operators ، locally convex spaces