Perturbations of local C-cosine functions - KUO, Chung-Cheng (Studia Universitatis Babeș-Bolyai Mathematica, 2020)
We show that A+B is a closed subgenerator of a local C-cosine function T (·) on a complex Banach space X defined by ∞ t T (t)x = '\" Bn n=0 0 jn−1 (s)jn(t − s)C(|t − 2s|)xds for all x ∈ X and 0 ≤ t < T0, if A is a closed subgenerator of a local C-cosine function C(·) on X and one of the following cases holds: (i) C(·) is exponentially bounded, and B is a bounded linear operator on D(A) so that BC = CB on D(A) and BA ⊂ AB; (ii) B is a bounded linear operator on D(A) which commutes with C(·) on D(A) and BA ⊂ AB; (iii) B is a bounded linear operator on X which commutes with C(·) on X. Here jn(t) = t { t j−1(s)j0(t − s)C(|t − 2s|)xds = C(t)x 0 for all x ∈ X and 0 ≤ t < T0. Mathematics Subject Classification (2010): 47D60, 47D62.