Note on Multiplicative Perturbation of Local C-regularized Cosine Functions with Nondensely Defined Generators

In this note, we obtain a new multiplicative perturbation theorem for local C-regularized cosine function with a nondensely defined generator A. An application to an integrodifferential equation is given.


Introduction and preliminaries
Let X be a Banach space, A an operator in X.It is well known that the cosine operator function is the main propagator of the following Cauchy problem for a second order differential equation in X: The first author was supported by the NSF of Yunnan Province (2009ZC054M).
EJQTDE, 2010 No. 57, p. 1 which controls the behaviors of the solutions of the differential equations in many cases (cf., e.g., [2, 4-10, 13, 15, 16, 19-21]); if A is the generator of a C-regularized cosine function {C(t)} t∈R , then u(t) = C −1 C(t)u 0 + C −1 t 0 C(s)u 1 ds is the unique solution of the above Cauchy problem for every pair (u 0 , u 1 ) of initial values in C(D(A))(see [5,16,20]).So it is valuable to study deeply the properties of the cosine operator functions.
Stimulated by these works as well as the works on integrated semigroups and Cregularized semigroups ([3, 11, 14, 17, 18]), we study further the multiplicative perturbation of local C-regularized cosine functions with nondensely defined generators, in the case where (1) the range of the regularizing operator C is not dense in a Banach space X; (2) the operator C may not commute with the perturbation operator.
Throughout this paper, all operators are linear; L(X, Y ) denotes the space of all continuous linear operators from X to a space Y , and L(X, X) will be abbreviated to L(X); L s (X) is the space of all continuous linear operators from X to X with the strong operator topology; C([0, t], L s (X)) denotes all continuous L(X)-valued functions, equipped with the norm F ∞ = sup r∈[ 0, t] F (r) .Moreover, we write D(A), R(A), ρ(A), respectively, for the domain, the range and the resolvent set of an operator A. We denote by A the part of A in D(A), that is, We abbreviate C-regularized cosine function to C-cosine function.The operator A defined by 2 Results and proofs In fact, for x ∈ D(A), we have lim ¿From Proposition 1.4 in [12], we can obtain provided that (H1) where Φ ∈ C([ 0, τ ], X), and M > 0 is a constant. ( where x ∈ D(A), Φ ∈ C([ 0, τ ], L s (X)), and M > 0 is a constant, (H2) there exists an injective operator  Proof.First, we prove the conclusion (2).
Define the operator functions {C n (t)} t∈[ 0, τ ] as follows: By induction, we obtain: EJQTDE, 2010 No. 57, p. 4 It follows that the series M n t n n! converges uniformly on [ 0, τ ] and consequently, and satisfies Using (H1') and Gronwall's inequality, we can see the uniqueness of solution of (2.1). Put It follows from (2.1) and and satisfies Now we consider the integral equation where v(t) ∈ C([0, τ ], L s (D(A))).Let v(t) satisfy the equation (2.5).Then from (2.5), we obtain, for x ∈ D( A), Hence, , and the solution w(t) of the equation ) is unique, we can see the solution of (2.5) is also unique.
By the uniqueness of solution of (2.5), we can obtain that Moreover, for t, h, t ± h ∈ [0, τ ], we have and for all x ∈ D( A), t, h ∈ [0, τ ], we have It follows from the uniqueness of solution of (2.5) and the denseness of EJQTDE, 2010 No. 57, p. 7 Next, we show that the subgenerator of { C(t)} t∈[−τ, τ ] is operator (I + B) A.
By the equality (2.4), (H3), the uniqueness of solution of (2.5), we obtain on D(A) that is, By a combination of similar arguments as above and those given in the proof of [11,Theorem 2.1], we can obtain the conclusion (1).
In view of statement (1) just proved, we can see that (I+B) Obviously, by (2. According to Definition 2.1, we see that A(I + B) subgenerates a local C 1 -cosine function Let Ω be a domain in R n and write Given q ∈ C(Ω) with q(η) ≥ 0, b ∈ C 0 (Ω) with be τ q , qbe τ q ∈ C 0 (Ω), and K ∈ L 1 (Ω), we consider the following Cauchy problem where f 1 , f 2 ∈ C 0 (Ω).

Remark 2 . 2 .
then we say that A subgenerates a local C-cosine function on X, or A is a subgenerator of a local C-cosine function on X.The generator G of a local C-cosine function {C(t); |t| ≤ τ } is a subgenerator of {C(t); |t| ≤ τ }.But for each subgenerator A, one has A ⊂ G and G = C −1 AC.
On the other hand, for x ∈ D(G), noting lim n→∞ Cx) = CGx, the closedness of A ensures Cx ∈ D(A) and ACx = CGx, therefore, we have G ⊂ C −1 AC.