In this paper, we study the well-posedness of the evolution equation of the form u' (t) = Au (t) + Cu (t), t ≥ 0 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 (TA (t) ) ₓ ≥ ₀ satisfying p (TA (t) x) ≤ e^ωtq (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 (T₀+₂ (t) ) ₓ ≥ ₀ satisfying p (T₀+₂ (t) x) ≤ e^ (ω+K) tq (x).
Jawad Ettayb (Tue,) studied this question.