Electronic Journal of Qualitative Theory of Differential Equations 2010, No. 57, 1-12; http://www.math.u-szeged.hu/ejqtde/

Note on multiplicative perturbation of local

C-regularized cosine functions with nondensely

defined generators

Fang Li^{1}^{∗} and James H. Liu^{2} 1School of Mathematics, Yunnan Normal University Kunming 650092, People’s Republic of China fangli860@gmail.com

2Department of Mathematics and Statistics, James Madison University Harrisonburg, VA 22807, USA liujh@jmu.edu

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

Key words : Multiplicative perturbation, local C-regularized cosine functions, second order differential equation

1 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:

\{ \begin{matrix}u′′(t) = Au(t), t ∈ (−∞, ∞) \\ u(0) = u_{0}, u′(0) = u_{1}, \end{matrix}

∗The first author was supported by the NSF of Yunnan Province (2009ZC054M).

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} ∫_{0}^{t} 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.

As a meaningful generalization of the classical cosine operator functions, the C-regularized cosine functions have been investigated extensively (cf., e.g., [2, 4, 5, 9, 10, 13, 15, 16, 20, 21]), where C serves as a regularizing operator which is injective.

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 Ã the part of A in \overline{D(A)}, that is,

Ã ⊂ A, D(Ã) = \{x ∈ D(A)|Ax ∈ \overline{D(A)}\}.

We abbreviate C-regularized cosine function to C-cosine function.

Definition 1.1. Assume τ > 0. A one-parameter family {C(t); |t| ≤ τ} ⊂ L(X) is called a local C-cosine function on X if

(i) C(0) = C and C(t + s)C + C(t − s)C = 2C(t)C(s) (∀|s|, |t|, |s + t| ≤ τ), (ii) C(·)x : [−τ, τ] ⟶ X is continuous for every x ∈ X.

The associated sine operator function S(·) is defined by S(t) := ∫_{0}^{t} C(s)ds (|t| ≤ τ).

The operator A defined by

D(A) = \{x ∈ X; lim_{t→0^{+}} \frac{2}{t^{2}} (C(t)x − Cx) exists and is in R(C)\},
Ax = C^{−1} lim_{t→0^{+}} \frac{2}{t^{2}} (C(t)x − Cx), ∀x ∈ D(A),

is called the generator of {C(t); |t| ≤ τ}. It is also called that A generates {C(t); |t| ≤ τ}.

Lemma 1.2. ([2]) Let A generate a local C-cosine function {C(t); |t| ≤ τ} on X.Then (i)For x ∈ D(A), t ∈ [−τ, τ], C(t)x, S(t)x ∈ D(A), AC(t)x = C(t)Ax, AS(t)x = S(t)Ax; (ii)For x ∈ X, t ∈ [ 0, τ], ∫_{0}^{t} S(s)xds ∈ D(A) and A∫_{0}^{t} S(s)xds = C(t)x − Cx;

(iii)For x ∈ D(A), t ∈ [ 0, τ], ∫_{0}^{t} S(s)Axds = A ∫_{0}^{t} S(s)xds = C(t)x − Cx.

2 Results and proofs

Definition 2.1. Let {C(t); |t| ≤ τ} be a local C-cosine function on X. If a closed linear operator A in X satisfies

(1) C(t)A ⊂ AC(t), |t| ≤ τ,

(2) C(t)x = Cx + A∫_{0}^{t} ∫_{0}^{s} C(σ)xdσds, |t| ≤ τ, x ∈ X,

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.

Remark 2.2. 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. Moreover, if ρ(A) ≠ ∅, then C^{−1}AC = A.

In fact, for x ∈ D(A), we have

lim_{t→0^{+}} \frac{2(C(t)x − Cx)}{t^{2}} = 2 lim_{t→0^{+}} \frac{A ∫_{0}^{t} ∫_{0}^{s} C(σ)xdσds}{t^{2}} = 2 lim_{t→0^{+}} \frac{∫_{0}^{t} ∫_{0}^{s} C(σ)Axdσds}{t^{2}}
= CAx ∈ R(C),

that is x ∈ D(G) and Ax = Gx, i.e., A ⊂ G.

For x ∈ D(C^{−1}AC), then Cx ∈ D(A) and ACx ∈ R(C), since A ⊂ G we have GCx = ACx ∈ R(C), then C^{−1}ACx = C^{−1}GCx = Gx, i.e., C^{−1}AC ⊂ G. On the other hand, for x ∈ D(G), noting

lim_{n→∞} 2n^{2} ∫0^{\frac{1}{n}} ∫_{0}^{s} C(σ)xdσds = lim_{n→∞} \frac{2 ∫0^{\frac{1}{n}} ∫_{0}^{s} C(σ)xdσds}{\frac{1}{n^{2}}} = Cx,

and

lim_{n→∞} A(2n^{2} ∫0^{\frac{1}{n}} ∫_{0}^{s} C(σ)xdσds) = lim_{n→∞} 2n^{2}(C(\frac{1}{n})x − Cx) = CGx,

the closedness of A ensures Cx ∈ D(A) and ACx = CGx, therefore, we have G ⊂ C^{−1}AC.

¿From Proposition 1.4 in [12], we can obtain C^{−1}AC = A if ρ(A) ≠ ∅. Theorem 2.3. Let nondensely defined operator A generate a local C-cosine function \{C(t); |t| ≤ τ\} on X, S(t) = ∫_{0}^{t} C(s)ds, and B ∈ L(\overline{D(A)}). Then

(1) there exists an operator family {E(t); |t| ≤ τ} ⊂ L(X) such that

E(t)x = Cx + A(I + B)∫_{0}^{t} ∫_{0}^{s} E(σ)xdσds, |t| ≤ τ, x ∈ \overline{D(A)},

provided that

(H1)

∥A ∫_{0}^{t} S(t − s)C^{−1}BΦ(s)ds∥ ≤ M ∫_{0}^{t} sup_{0≤s≤σ} ∥Φ(s)∥dσ, t ∈ [ 0, τ],

where Φ ∈ C([ 0, τ], X), and M > 0 is a constant.

(2) (I + B)Ã generates a local C_{1}-cosine function on \overline{D(A)} provided that

(H1’)

∥∫_{0}^{t} Φ(s)C^{−1}BAS(t − s)xds∥ ≤ M∥x∥ ∫_{0}^{t} sup_{0≤s≤σ} ∥Φ(s)∥dσ, t ∈ [ 0, τ],

where x ∈ D(A), Φ ∈ C([ 0, τ], L_{s}(X)), and M > 0 is a constant, (H2) there exists an injective operator C_{1} ∈ L(\overline{D(A)}) such that R(B) ⊂ R(C_{1}) ⊂

C(\overline{D(A)}), C_{1}(I + B)Ã ⊂ (I + B)ÃC_{1}, and C^{−1}C_{1}(D(Ã)) is a dense subspace in D(A), (H3) ρ((I + B)Ã) ≠ ∅.

(3) Ã(I +B) subgenerates a C_{1}-cosine function on \overline{D(A)} provided that C_{1}B = BC_{1}, and (H1’), (H2), and (H3) hold.

Proof. First, we prove the conclusion (2).

Define the operator functions \{\overline{C}_{n}(t)\}_{t∈[ 0, τ]} as follows:

\begin{cases}\overline{C}_{0}(t)x = C(t)x, \\ \overline{C}_{n}(t)x = ∫_{0}^{t} \overline{C}_{n−1}(s)C^{−1}BAS(t − s)xds, x ∈ D(A), t ∈ [0, τ], n = 1, 2, · · · . \end{cases}

By induction, we obtain:

EJQTDE, 2010 No. 57, p. 4 (i) \overline{C}_{n}(t) ∈ C([ 0, τ], L_{s}(\overline{D(A)}));

(ii) ∥\overline{C}_{n}(t)∥ ≤ \frac{M^{n}t^{n}}{n!} sup_{s∈[0, τ]} ∥C(s)∥, t ∈ [0, τ], ∀ n ≥ 0.

It follows that the series ∑_{n=0}^{∞} \frac{M^{n}t^{n}}{n!} converges uniformly on [ 0, τ] and consequently,

\overline{C}(t)x := ∑_{n=0}^{∞} \overline{C}_{n}(t)x ∈ C([ 0, τ], \overline{D(A)}), ∀x ∈ \overline{D(A)},

and satisfies

\overline{C}(t)x = C(t)x + ∫_{0}^{t} \overline{C}(s)C^{−1}BAS(t − s)xds, x ∈ D(A), t ∈ [ 0, τ].  (2.1)

Using (H1’) and Gronwall’s inequality, we can see the uniqueness of solution of (2.1).

Put

Ĉ(t) := \overline{C}(t)C^{−1}C_{1}, t ∈ [0, τ].

It follows from (2.1) and C^{−1}C_{1} ∈ L(\overline{D(A)}) that for x ∈ \overline{D(A)},

Ĉ(t)x ∈ C([ 0, τ], \overline{D(A)}),

and satisfies

Ĉ(t)x = C(t)C^{−1}C_{1}x + ∫_{0}^{t} Ĉ(s)C_{1}^{−1}BAS(t − s)C^{−1}C_{1}xds, x ∈ D(A), t ∈ [ 0, τ]. (2.2)

Note that D(Ã) ⊂ D(C_{1}^{−1}BÃC_{1}) and C^{−1}C_{1} maps D(Ã) into D(Ã). So, for x ∈ D(Ã), by (2.2), we have

∫_{0}^{t} ∫_{0}^{s} Ĉ(σ)C_{1}^{−1}BÃC_{1}xdσds
= ∫_{0}^{t} ∫_{0}^{s} C(σ)C^{−1}BÃC_{1}xdσds
+ ∫_{0}^{t} ∫_{0}^{s} Ĉ(σ)C_{1}^{−1}B[C(s − σ)C^{−1}BÃC_{1}x − BÃC_{1}x]dσds.  (2.3)

Therefore, for x ∈ D(Ã), we have

∫_{0}^{t} ∫_{0}^{s} Ĉ(σ)(I + B)Ãxdσds
= ∫_{0}^{t} ∫_{0}^{s} C(σ)C^{−1}(I + B)ÃC_{1}xdσds
+ ∫_{0}^{t} ∫_{0}^{s} Ĉ(σ)C_{1}^{−1}B[C(s − σ)C^{−1}(I + B)ÃC_{1}x − (I + B)ÃC_{1}x]dσds
= C(t)C^{−1}C_{1}x − C_{1}x + ∫_{0}^{t} ∫_{0}^{s} C(σ)C^{−1}BÃC_{1}xdσds

+ ∫_{0}^{t} ∫_{0}^{s} Ĉ(σ)C_{1}^{−1}BC(s − σ)C^{−1}ÃC_{1}xdσds − ∫_{0}^{t} ∫_{0}^{s} Ĉ(σ)C_{1}^{−1}BÃC_{1}xdσds

+ ∫_{0}^{t} ∫_{0}^{s} Ĉ(σ)C_{1}^{−1}B[C(s − σ)C^{−1}BÃC_{1}x − BÃC_{1}x]dσds
=^{(2.3)} Ĉ(t)x − C_{1}x.  (2.4)

Now we consider the integral equation

v(t)x = C_{1}x + ∫_{0}^{t} ∫_{0}^{s} v(σ)(I + B)Ãxdσds, x ∈ D(Ã), t ∈ [0, τ],  (2.5)

where v(t) ∈ C([0, τ], L_{s}(\overline{D(A)})). Let ṽ(t) satisfy the equation (2.5). Then from (2.5), we obtain, for x ∈ D(Ã),

∫_{0}^{t} ṽ(s)S(t − s)C^{−1}C_{1}xds − C_{1} ∫_{0}^{t} S(s)C^{−1}C_{1}xds
= ∫_{0}^{t} ∫_{0}^{s} ṽ(σ)(I + B)Ã∫_{0}^{s−σ} S(r)C^{−1}C_{1}xdrdσds
= ∫_{0}^{t} ṽ(s)S(t − s)C^{−1}C_{1}xds − ∫_{0}^{t} ∫_{0}^{s} ṽ(σ)C_{1}xdσds
+ ∫_{0}^{t} ∫_{0}^{s} ṽ(σ)BÃ∫_{0}^{s−σ} S(r)C^{−1}C_{1}xdrdσds.

Hence,

∫_{0}^{t} ∫_{0}^{s} ṽ(σ)C_{1}xdσds = C_{1} ∫_{0}^{t} S(s)C^{−1}C_{1}xds + ∫_{0}^{t} ∫_{0}^{s} ṽ(σ)BÃ∫_{0}^{s−σ} S(r)C^{−1}C_{1}xdrdσds,

that is,

(ṽ(t)C)C^{−1}C_{1}x = C_{1}C(t)C^{−1}C_{1}x + ∫_{0}^{t}(ṽ(s)C)C^{−1}BÃS(t − s)C^{−1}C_{1}xds.

Note that C^{−1}C_{1}(D(Ã)) ⊂ D(Ã) is dense in \overline{D(A)}, and the solution w̄(t) of the equation

w(t)y = C_{1}C(t)y + ∫_{0}^{t} w(s)C^{−1}BÃS(t − s)yds, y ∈ C^{−1}C_{1}(D(Ã)), t ∈ [0, τ]

in C([0, τ], L_{s}(\overline{D(A)})) 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

Ĉ(−t)x = Ĉ(t)x, Ĉ(t)C_{1}x = C_{1}Ĉ(t)x, for each x ∈ \overline{D(A)}, t ∈ [0, τ].

Moreover, for t, h, t ± h ∈ [0, τ], we have

Ĉ(t + h)C_{1}x + Ĉ(t − h)C_{1}x

= ∫_{0}^{t+h} ∫_{0}^{s} Ĉ(σ)(I + B)AC_{1}xdσds + ∫_{0}^{t−h} ∫_{0}^{s} Ĉ(σ)(I + B)AC_{1}xdσds + 2C_{1}^{2}x
= ∫_{0}^{h} ∫_{0}^{s} Ĉ(t + σ)(I + B)AC_{1}xdσds + ∫_{0}^{t} ∫_{0}^{t−s} Ĉ(σ)(I + B)AC_{1}xdσds

+ ∫_{0}^{h} ∫_{0}^{t} Ĉ(t − σ)(I + B)AC_{1}xdσds + ∫_{0}^{h} ∫_{0}^{s} Ĉ(t − σ)(I + B)AC_{1}xdσds

+ ∫_{0}^{t} ∫_{0}^{t−s} Ĉ(σ)(I + B)AC_{1}xdσds − ∫_{0}^{h} ∫_{0}^{t} Ĉ(t − σ)(I + B)AC_{1}xdσds +2C_{1}^{2}x

= ∫_{0}^{h} ∫_{0}^{s} [Ĉ(t + σ)C_{1} + Ĉ(t − σ)C_{1}](I + B)Axdσds + 2 ∫_{0}^{t}(t − s)Ĉ(s)(I + B)AC_{1}xds

+2C_{1}^{2}x,

and for all x ∈ D(Ã), t, h ∈ [0, τ], we have

2Ĉ(t)Ĉ(h)x = 2Ĉ(t)[ ∫_{0}^{h} ∫_{0}^{s} Ĉ(σ)(I + B)Ãxdσds + C_{1}x]
= ∫_{0}^{h} ∫_{0}^{s} 2Ĉ(t)Ĉ(σ)(I + B)Axdσds + 2 ∫_{0}^{t}(t − s)Ĉ(s)(I + B)ÃC_{1}xds
+2C_{1}^{2}x.

Therefore, for x ∈ D(Ã), t ∈ [0, τ],

[Ĉ(t + h)C_{1}x + Ĉ(t − h)C_{1}x] − 2Ĉ(t)Ĉ(h)x
= ∫_{0}^{t} ∫_{0}^{s} {[Ĉ(σ + h)C_{1} + Ĉ(σ − h)C_{1}] − 2Ĉ(σ)Ĉ(h)}(I + B)Ãxdσds.

It follows from the uniqueness of solution of (2.5) and the denseness of Ã in \overline{D(A)} that

2Ĉ(t)Ĉ(h) = Ĉ(t + h)C_{1} + Ĉ(t − h)C_{1}

on \overline{D(A)}, for t, h, t ± h ∈ [0, τ]. Therefore, \{Ĉ(t)\}_{t∈[−τ, τ]} is a local C_{1}-cosine operator function on \overline{D(A)}.

Next, we show that the subgenerator of {Ĉ(t)}_{t∈[−τ, τ]} is operator (I + B)Ã.

By the equality (2.4), (H3), the uniqueness of solution of (2.5), we obtain on \overline{D(A)}

(λ − (I + B)Ã)^{−1}Ĉ(t) = Ĉ(t)(λ − (I + B)Ã)^{−1}, t ∈ [0, τ], λ ∈ ρ((I + B)Ã),

therefore,

(I + B)ÃĈ(t)x = Ĉ(t)(I + B)Ãx, x ∈ D(Ã), t ∈ [0, τ],  (2.6)

that is,

Ĉ(t)(I + B)Ã ⊂ (I + B)ÃĈ(t), t ∈ [0, τ].

Moreover, since ρ((I + B)Ã) ≠ ∅, (I + B)Ã is a closed operator. It follows from (2.4) and the closedness of (I + B)Ã that ∫_{0}^{t} ∫_{0}^{s} Ĉ(σ)xdσds ∈ D(Ã) and

Ĉ(t)x = C_{1}x + (I + B)Ã∫_{0}^{t} ∫_{0}^{s} Ĉ(σ)xdσds,  (2.7)

for each x ∈ \overline{D(A)}, t ∈ [0, τ]. Therefore, (I + B)Ã is a subgenerator of \{Ĉ(t)\}_{t∈[−τ, τ]}. By (H3) and remark 2.2, we can see that (I + B)Ã is the generator of {Ĉ(t)}_{t∈[−τ, τ]}. This completes the proof of statement (2).

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).

Next, we prove the conclusion (3).

In view of statement (1) just proved, we can see that (I+B)Ã subgenerates {Ĉ(t)}_{t∈[−τ, τ]} on \overline{D(A)}. Set

Q(t)x = C_{1}x + Ã∫_{0}^{t} ∫_{0}^{s} Ĉ(σ)(I + B)xdσds, |t| ≤ τ, x ∈ \overline{D(A)}.

Obviously, by (2.7) and the fact that the graph norms of Ã and (I + B)Ã are equivalent, we can see that {Q(t)}_{t∈[−τ, τ]} is a strongly continuous operator family of bounded linear operators on \overline{D(A)}. Moreover, by (2.6) and (2.7), for |t| ≤ τ, x ∈ \overline{D(A)}, we obtain

Q(t)Ã(I + B)x = C_{1}Ã(I + B)x + Ã∫_{0}^{t} ∫_{0}^{s} Ĉ(σ)(I + B)Ã(I + B)xdσds,
= Ã(I + B)C_{1}x + Ã(I + B)Ã∫_{0}^{t} ∫_{0}^{s} Ĉ(σ)(I + B)xdσds
= Ã(I + B)Q(t)x

and

(I + B) ∫_{0}^{t} ∫_{0}^{s} Q(σ)xdσds = ∫_{0}^{t} ∫_{0}^{s}(I + B)C_{1}x + ∫_{0}^{t} ∫_{0}^{s}[Ĉ(σ)(I + B)x − C_{1}(I + B)x]dσds
= ∫_{0}^{t} ∫_{0}^{s} Ĉ(σ)(I + B)xdσds.

It follows that for any |t| ≤ τ, x ∈ \overline{D(A)}, (I + B)∫_{0}^{t} ∫_{0}^{s} Q(σ)xdσds ∈ D(Ã) and

Ã(I + B) ∫_{0}^{t} ∫_{0}^{s} Q(σ)xdσds = Ã∫_{0}^{t} ∫_{0}^{s} Ĉ(σ)(I + B)xdσds = Q(x)x − C_{1}x.

According to Definition 2.1, we see that Ã(I + B) subgenerates a local C_{1}-cosine function \{Q(t)\}_{t∈[−τ, τ]} on \overline{D(A)}.

Example 2.4. Let Ω be a domain in R^{n} and write

C_{0}(Ω) :={f ∈ C(Ω) : for each ε > 0 there is a compact Ω_{ε} ⊂ Ω
such that |f(s)| < ε for all s ∈ Ω \ Ω_{ε}}.

Given q ∈ C(Ω) with q(η) ≥ 0, b ∈ C_{0}(Ω) with

be^{τq}, qbe^{τq} ∈ C_{0}(Ω),  (2.8)

and K ∈ L^{1}(Ω), we consider the following Cauchy problem

\{ \begin{matrix}\frac{∂^{2}u(t, η)}{∂t^{2}} & = & q(η) (u(t, η) + b(η) ∫_{Ω} K(σ)u(t, σ)dσ) , \\ u(0, η) & = & f_{1}(η), u′(0, η) = f_{2}(η), η ∈ Ω, 0 ≤ t ≤ τ, \end{matrix}  (2.9)

where f_{1}, f_{2} ∈ C_{0}(Ω).

Set Af =: qf with D(A) =: {f ∈ C_{0}(Ω); qf ∈ C_{0}(Ω)}.

When q is bounded, A generates a classical cosine function C(t) on C_{0}(Ω) (i.e., I-cosine function on [0, ∞)), with

∥C(t)∥ ≤ e^{(sup_{η∈Ω} q(η))t}, t ≥ 0

(for the exponential growth bound of a cosine function (which is closely related to a strongly continuous semigroup in some cases), as well as its relation with the spectral bound of the generator, we refer to, e.g., [1, 16]). Nevertheless, when q is unbounded, A does not generate a global C-cosine function C(t) on C_{0}(Ω) for any C. On the other hand, A generates a local C-cosine function C(t) on C_{0}(Ω):

C(t)f = \{\frac{1}{2} [e^{t\sqrt{q}} + e^{−t\sqrt{q}}] e^{−τ\sqrt{q}}f\}_{t∈[−τ, τ]} ,

with Cf = e^{−τ\sqrt{q}}f. Set

(Bf)(η) = b(η) ∫_{Ω} K(σ)f(σ)dσ, f ∈ C_{0}(Ω).

¿From (2.8), we see the hypothesis (H1) in Theorem 2.3 holds. This means, by Theorem 2.3 (1) and [20, Theorem 2.4], that the Cauchy problem (2.9) has a unique solution in C^{2}([0, τ]; C_{0}(Ω)) for every couple of initial values in a large subset of C_{0}(Ω).

Acknowledgement The authors are grateful to the referees for their valuable suggestions.

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(Received January 26, 2010)