On perturbation of closed operators in a Banach space

A perturbation result of m-accretive linear operators in Hilbert spaces

“…Note that this result cover the case of relatively bounded perturbation, see [13,Remark 4.4]. There are many papers on the question of such perturbation, see [15,16,17,19,21] for more results. The aim of this paper is to establish a new perturbation results on the m-accretivity of the operator T + A.…”

“…Many papers have been devoted to this problem and most results treat pairs T , A of relatively bounded or resolvent commuting operators. We refer the reader to [2,3,5,6,15,17,18,20,21,22]. Since T is closed it follows that there are two nonnegative constants a, b such that Ax 2 ≤ a x 2 + b T x 2 , for all x ∈ D(T ) ⊂ D(A).(.1) In this case, A is called relatively bounded with respect to T or simply T -bounded, and refer to b as a relative bound.…”

On the solvability of fourth-order boundary value problems with accretive operators

“…Such results are interesting, firstly because they provide an existence, uniqueness and L p -maximal regularity of the solution for the fourth order abstract boundary value problem (1.1) under some necessary and sufficient conditions on the data. Moreover, they are of interest regarded as an application of the perturbation theory of m-accretive operators, [10], [11], [13], [20], [21], [22], [32]. This allows us to find various sufficient conditions on the accretive operators B and C, under which our results remains valid.…”

“…(4) Assume (A.4). Due to Proposition 2.12, p. 55 in [32], the lower bound β in (3.10) is equal to sup t>0 C(B 2 + tI) −1 . Hence (A.4) implies (A.2), see [21], Theorem 1, p. 851.…”

Perturbations of linear 𝑚-accretive operations

“…Remark 4.3. Theorem 4.1 is essentially proved by Yoshikawa though the closedness of B is assumed in his statement (see [11,Theorem 3.10 and Remark 3.12]). Thus, Corollary 1 to Theorem 1 and Theorem 2 of [9] are covered by Theorem 4.1.…”