ON NONAUTONOMOUS SECOND-ORDER DIFFERENTIAL EQUATIONS ON BANACH SPACE

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ON NONAUTONOMOUS SECOND-ORDER DIFFERENTIAL EQUATIONS ON BANACH SPACE

NGUYEN THANH LAN (Received 7 June 1999)

Abstract.We show the existence and uniqueness of classical solutions of the nonau- tonomous second-order equation:u(t)=A(t)u(t)+B(t)u(t)+f (t), 0≤t≤T;u(0)=x0, u(0)=x1on a Banach space by means of operator matrix method and apply to Volterra integrodifferential equations.

2000 Mathematics Subject Classification. 34G10, 47D06.

1. Introduction. In this paper, we study nonautonomous second-order Cauchy problems

u(t)=A(t)u(t)+B(t)u(t)+f (t), 0≤t≤T , u⁽ⁱ⁾(0)=xi, i=0,1 (1.1) on a Banach spaceE, whereA(t)andB(t), 0≤t≤T, are linear operators onEandf is a continuous function from[0,T ]toE.

Definition1.1. A functionu(·):[0,T ]→Eis said to be a solution of (1.1) if it is twice continuously differentiable on[0,T ], A(t)u(t), andB(t)u(t)are defined and continuous int, and (1.1) is satisfied.

Our idea is to reduce (1.1) to a differential equation of first-order. It is motivated by the work of N. Tanaka [12], who studied the first-order abstract Cauchy problem

u(t)=A(t)u(t), 0≤t≤T , u(0)=x0. (1.2) In his paper, Tanaka showed the existence and uniqueness of classical solutions of (1.2), when family{A(t)}0≤t≤T of linear operators inEsatisfies the conditions which are usually referred as the “hyperbolic” condition, except for the density of the com- mon domainDofA(t).

The purpose of this paper is to show the existence and uniqueness of classical solu- tions of (1.1) on the basis of Tanaka’s result in [12] and the operator matrix method. We will consider two cases: the damped case, whenA(t)is more unbounded thanB(t)and the undamped one, whenB(t)is more unbounded thanA(t). For both cases, we use an operator matrix method to reduce (1.1) into a first-order differential equation of the form of (1.2) and then apply Tanaka’s result. The two cases reduce in different ways, but the technique in each reduction is quite straightforward. In the undamped case, our result obtained improves Kozak’s one [4] by requiring much weaker assumptions.

In the damped case, we generalize Neubrander’s result [6] to the nonautonomous ver- sion. Our proof is simpler and more natural than Oka’s one in [8]. This proof creates a

new framework to deal with the abstract higher-order differential equations on Banach spaces, which we will discuss in a subsequent paper.

In the following, for a linear operatorAon a Banach spaceE, we denote the resolvent set of an operatorAbyρ(A)and the resolvent(λ−A)⁻¹ byR(λ,A). By L(E,F)we denote the set of all linear, bounded operators fromE toF. Finally, for short, we write the family ofA(t), 0≤t≤T by{A(t)}. First we recall the fundamental results obtained by Tanaka [12].

Theorem1.2(see [12, Theorem 1.8]). A family of operatorsA(t),(0≤t≤T )satis- fies the hyperbolic condition if

(H1)The common domainD:=D(A(t))is a Banach space for the norm·D. More- over, there existsc0>0such that

c⁻¹0 xD≤ x+A(t)x≤c0xD (1.3) for allt∈[0,T ]andx∈D.

(H2)The family(A(t))t∈[0,T ]is stable, that is, there exist constantsM≥1andω∈R such that

(ω,∞)⊂ρ A(t)

∀t∈[0,T ], (1.4)

k j=1

R λA

tj

≤M(λ−ω)^−k ∀λ > ω, (1.5) and any finite sequence0≤t1≤t2≤ ··· ≤tk≤T.

(H3)The mappingtA(t)yis continuously differentiable inEfor everyy∈D.

If the family{A(t)}satisfies the hyperbolic condition, then there is an evolution family {U(t,s)}0≤s≤t≤T onD¯with the following properties.

(1)U(t,s)D(s)⊂D(t)for all0≤s≤t≤T, where the setD(r )is defined by D(r ):=

x∈D:A(r )x∈D¯

. (1.6)

(2) The mapping t → U(t,s)x is continuously differentiable in E on [s,T ] and (∂/∂t)U(t,s)x=A(t)U(t,s)xforx∈D(s)andt∈[s,T ].

If there is such an evolution family {U(t,s)}0≤s≤t≤T, then, for every initial value u0∈D(0),u(t):=U(t,0)u0is the unique solution of (1.2).

Generally, it is not trivial to show the stability of a family of operators. Thus, the following two lemmas, which will be used frequently, are very useful tools to verify this condition.

Lemma 1.3(see [11, Theorem 5.2.3]). Let{A(t)}0≤t≤T be stable and {B(t)}0≤t≤T

be a family of uniformly bounded operators. Then the family{A(t)+B(t)}0≤t≤T with D(A(t)+B(t)):=D(A(t))is stable.

Lemma 1.4 (see [2, Proposition 4.4]). Let {A(t)}0≤t≤T be a stable family and {S(t)}0≤t≤T be a family of isomorphismsS(t)∈L(E), which is strongly continuously differentiable. Then the family{A(t)}˜ 0≤t≤T:= {S(t)A(t)S⁻¹(t)}0≤t≤T withD(A(t))˜ = {x:S⁻¹(t)x∈D(A(t))}is stable.

For further information on evolution equations, evolution family and the theory of operator matrices, we refer to, for example, [5,11,12].

2. The damped second-order equations. We now consider the damped second- order differential equations. First, we start with the homogeneous version

u(t)=A(t)u(t)+B(t)u(t), 0≤t≤T , u(0)=x0∈E, u(0)=x1∈E, (2.1) where operatorsA(t)have common domainDandD(B(t))⊃D. For our purpose, we assume that there exists an invertible operatorAfromDontoEwithA⁻¹∈L(E). We introduce new variables by defining

v0:=Au, v1:=u. (2.2)

Then we havev0 =Au=v1 and v1 =u =A(t)u+B(t)u=A(t)v1+B(t)A⁻¹v0. Moreover,v0(0)=Ax0andv1(0)=x1. Thus, we can write a differential equation for ᐂ:=(v0,v1)^T in the Banach spaceE²as follows:

ᐂ(t)=Ꮽ(t)ᐂ(t), 0≤t≤T , ᐂ(0)=ᐂ0∈E², (2.3) whereᏭ(t):= _{0 A}

B(t)A⁻¹A(t)

withD(Ꮽ(t)):=E×Dandᐂ0:=(Ax0,x1)^T. We easily see that if{A(t)}satisfies the hyperbolic condition, we can chooseA:=(A(0)−λ)for a λ > ω. We have the following lemma.

Lemma2.1. Forxi∈D (i=0,1), the following statements hold true.

(1)Ifu(t)is a solution of (2.1), thenu⁽ⁱ⁾(t)∈D,(i=0,1)and(Au(t),u(t))^T is a solution of (2.3).

(2)Conversely, if(v0(t),v1(t))^T is a solution of (2.3), thenu(t):= ₀^tv1(s)ds+x0is a solution of (2.1).

Proof. (1)⇒(2). Letu(t)be a solution of (2.1) withu0∈Dandu1∈D. In view of the closedness ofAwe have

_t

0Au(τ)dτ=A _t

0u(τ)dτ=A

u(t)−x0

. (2.4)

Sincex0∈D, it followsu(t)∈D, hence the functiontAu(t)is continuously dif- ferentiable and(d/dt)Au(t)=Au(t). Therefore,(v0(t),v1(t))^T :=(Au(t),u(t))^T is continuously differentiable and satisfies (2.3), and thus, is a solution of (2.3).

(2)⇒(1). Conversely, suppose that(v0(t),v1(t))^Tis a solution of (2.3). We define the functionuby

u(t):=

_t

0v1(r )dr+x0. (2.5) Thenu(t)is twice continuously differentiable. Furthermore, from (2.3) we have

v0(t)= _t

0Av1(r )dr+x0=A _t

0v1(r )dr+A⁻¹v0(0)

=Au(t). (2.6) Thus,

u(t)=v₁(t)=B(t)A⁻¹v0(t)+A(t)v1(t)=A(t)u(t)+B(t)u(t). (2.7)

Finally, it is easy to see thatu(0)=x0andu(0)=x1. Therefore,u(t)is a solution of (2.1), and the lemma is proved.

Now we are in a position to express the main result of this section.

Theorem 2.2. For the second-order differential equation (2.1) we assume that {A(t)}satisfies the hyperbolic condition and{B(t)}is a family of linear operators with D(B(t))⊇Dsuch thatB(t)∈L(D,E)andtB(t)xis continuously differentiable for eachx∈D. Then it has a unique solution for every initialx0∈D,x1∈D, such that (B(0)x0+A(0)x1)∈D.¯

Proof. ByLemma 2.1and Tanaka’s theorem (Theorem 1.2), to show the existence of solutions of (2.1), we only have to prove that the family{Ꮽ(t)}satisfies the hyper- boliccondition. It is easy to see that{Ꮽ(t)}satisfies items (H1) and (H3) of this condi- tion. It remains to show its stability. To do that, we assume, without loss of generality, thatω <0. ThenA(t)is invertible for everyt∈[0,T ]. SinceB(t)∈L(D,E),B(t)A⁻¹ is bounded inE. Because of the strongly continuous differentiability of{B(t)A⁻¹}, by the principle of uniform boundedness, this family is uniformly bounded. In addition, ifA(t)is strongly continuously differentiable, then so isA⁻¹(t). Thus, the strongly continuous differentiability ofAA⁻¹(t)follows from that of its inverseA(t)A⁻¹. Using elementary matrix rules we have

0 A B(t)A⁻¹ A(t)

=

I AA⁻¹(t)

0 I

0 0 0 A(t)

I −AA⁻¹(t)

0 I

+

0 0 B(t)A⁻¹ 0

. (2.8)

By the stability of family{A(t)}, the family ˜Ꮽ(t):=_{0 0}

0A(t)

withD(Ꮽ˜(t)):=E×Dis stable inE². From the above observation, the family of isomorphisms

I AA⁻¹(t)

0 I

and of their inverses,

I−AA⁻¹(t)

0 I

, are strongly continuously differentiable. Using Lemmas 1.3and1.4we conclude that the family{Ꮽ(t)}is stable.

The uniqueness of the solutions of (2.1) follows, byLemma 2.1, from that of the solutions of (2.3), completing the proof of the theorem.

Remark2.3. In the above proof, for convenience, we assumedω <0, that is,A(t)is invertible. Actually, this assumption can be removed. Indeed, if family{A(t)}satisfies the hyperboliccondition and ifA∈L(D,E), then by the identity

0 A 0 A(t)

=

I −AR λ,A(t)

0 I

0 0 0 A(t)−λ

I AR λ,A(t)

0 I

(2.9)

for aλ > ω, we see that_{0 A}

0A(t)

is stable by the same argument.

We now consider the inhomogeneous equation (1.1). To this end, we recall that in [7] we considered the first-order inhomogeneous equation

u(t)=A(t)u(t)+f (t), 0≤t≤T , u(0)=x0, (2.10)

for a hyperbolicfamily {A(t)} and f ∈ W^1,1(R+,E). We showed that the family {Ꮽ(t)}0≤t≤T :=

A(t) δ0 0 d/dx

0≤t≤T with D(Ꮽ(t)):=D×W^1,1(R+,E) satisfies the hy- perbolic condition. Moreover, the first component of the solution of the problem

ᐁ(t)=Ꮽ(t)ᐁ(t), 0≤t≤T , ᐁ(0)= x0

f

, (2.11)

is the solution of (2.10). By combining this result andLemma 2.1we have the following.

Theorem 2.4. Suppose that the assumptions of Theorem 2.2 are satisfied. Then (1.1) has a unique solution for everyf∈W^1,1(R+,E)andxi∈D,(i=0,1), satisfying (B(0)x0+A(0)x1+f (0))∈D.¯

Proof. OnE×E×L¹(R+,E)we consider the equation





 v w φ







(t)=Ꮿ(t)





 v w φ





(t), 0≤t≤T ,





 v w φ





(0)=





 Au0

u1

f





, (2.12)

where

Ꮿ(t):=







0 A 0

B(t)A⁻¹ A(t) δ0

0 0 d

dx





, (2.13)

andD(Ꮿ(t)):=E×D×W^1,1(R+,E).

We writeᏯ(t)= _{0 Ꮽ}

Ꮾ(t)Ꮽ(t)0

withᏭ0:=(A,0),Ꮾ(t):=

B(t)A⁻¹ 0

, andᏭ(t):=

A(t) δ0 0 d/dx

. From the above consideration, the family {Ꮽ(t)}satisfies the hyperboliccondi- tion. As inTheorem 2.2, and in view ofRemark 2.3, we conclude that{Ꮿ(t)}is stable and thus satisfies the hyperboliccondition. Therefore, by Tanaka’s theorem, problem (2.12) has a unique solution for every initial value ᐂ(0):=(Ax0,x1,f )^T(0)∈Ᏸ:=

E×D×W^1,1(R+,E), such thatᏯ(0)ᐂ(0)∈Ᏸ¯, or in other words,(B(0)x0+A(0)x1+ f (0))∈D.¯

Let ᐂ(t):=(v(t),w(t),φ(t))^T be a solution of (2.12). Obviouslyφ(t) =Tr(t)f, whereTr(t)f (θ):=f (t+θ). We now define a functionuby

u(t):=

_t

0w(r )dr+x0. (2.14)

Then, with the same procedure as inTheorem 2.2, we have thatu(·)is twice contin- uously differentiable,v(t)=Au(t),u(t)=w(t), and

u(t)=w(t)=B(t)A⁻¹(0)v(t)+A(t)w(t)+φ(t)(0)

=A(t)u(t)+B(t)u(t)+f (t). (2.15) Moreover,u(0)=x0 andu(0)=w(0)=x1. Therefore,u(t) is a solution of (1.1).

The uniqueness of this solution follows from the uniqueness of the solution of the homogeneous equation and the theorem is proved.

3. The undamped second-order equations. This section is devoted to second- order differential equations in whichB(t) is more unbounded thanA(t). We start with the following problem:

u(t)=B(t)u(t)+f (t) 0≤t≤T , u(0)=x0, u(0)=x1, (3.1) and carry out the substitution:

v0:=u, v1:=u. (3.2)

Then we can rewrite (3.1) in matrix form as v0(t)

v1(t)

=

0 I B(t) 0

v0(t) v1(t)

+ 0

f (t)

, 0≤t≤T ,

v0(0) v1(0)

= x0

x1

, (3.3) onE². To investigate the Cauchy problem (3.3), we make the following assumptions toB(t).

Assumption3.1. (A1)For eacht∈[0,T ], there exists a linear operatorC(t):E→E such thatB(t)=C²(t)withD(Cⁱ(t))≡Di,i=1,2, independent oft.

(A2){C(t)}0≤t≤Tand{−C(t)}0≤t≤TwithD(C(t))=D1are stable families.

(A3)The maptCⁱ(t)xis continuously differentiable for everyx∈Diandi=1,2.

Without loss of generality, we assumeω <0, where(M,ω)are the stability con- stants of the family {C(t)}0≤t≤T. On the subsetsD1 and D2of E, we establish the following norms

x[D1]:=C(0)x forx∈D1, x[D2]:=C²(0)x forx∈D2. (3.4) Then it is easy to see that([D1],·[D1])and([D2],·[D2])are Banach spaces. More- over,C(t)andC²(t) are bounded operators from[D1]and [D2]toE, respectively, for everyt∈[0,T ]. From the above assumptions, we obtain some information.

Lemma3.2. (i)EachᏮ(t), where Ꮾ(t):=

0 I B(t) 0

(3.5) withD(Ꮿ(t)):=D2×D1on the Banach space[D1]×Eis similar to the operator

Ꮿ(t):=

0 C(t) C(t) 0

(3.6)

withD(Ꮿ(t)):=D1×D1onE². (ii)LetQ:=(1/√

2)_I−I

I I

andQ⁻¹:=(1/√ 2)_{I I}

−I I

onE². Then 0 C(t)

C(t) 0

=Q

C(t) 0 0 −C(t)

Q⁻¹ (3.7)

with the same domain.

Proof. (ii) is trivial and (i) follows from the identity 0 C(t)

C(t) 0

=

C(t) 0

0 I

0 I C²(t) 0

C(t)⁻¹ 0

0 I

, (3.8)

where_C(t)0

0 I

are isomorphisms from[D1]×EtoE². Now we prove the main results of this section.

Theorem3.3. Let the operatorsB(t)satisfyAssumption 3.1. Then the second-order Cauchy problem (3.1) has a unique solution withu(t)∈D2,u(t)∈D¯2in[D1]-norm andu(t)∈D¯1for every initial value(u(0),u(0))^T =(x0,x1)^T ∈D2×D1andf∈ W^1,1(R+,E)such thatB(0)x0+f (0)∈D¯1andx1∈D¯2in[D1]-norm.

Proof. Consider the family{Ꮾ(t)}0≤t≤T in problem (3.3). As {B(t)} is strongly continuously differentiable, so is{Ꮾ(t)}. We now show that{Ꮾ(t)}0≤t≤Tis stable.

By assumption, {C(t)}, and {−C(t)} are stable families, and so is the family _{C(t) 0}

0 −C(t)

. ByLemma 3.2(ii), the family _{0 C(t)}

C(t) 0

is stable since it is similar to a stable family. Now, using Lemmas1.4and3.2(i), for which we notice that the families C(t)0

0 I

and

C⁻¹(t)0 0 I

are strongly continuously differentiable by assumption, we see that the family{Ꮾ(t)}is stable and therefore satisfies the hyperboliccondition.

By Tanaka’s theorem, equation (3.3) has a unique solution for each(x0,x1)^T∈D2×D1

andf∈W^1,1(R+,E)such thatB(0)x0+f (0)∈D¯1andx1∈D¯2in[D1]-norm.

Letᐂ(·)=(v0,v1)^T be a solution of (3.3). Then we havev₀(t)=v1(t)in[D1]and v₁(t)=B(t)v0+f (t)in E. Since the norm in [D1]is finer than the norm ofE, the above equations also hold inE. This impliesv0(t)∈D2,v1(t)∈D1,tv0(t)is twice continuously differentiable andv₀(t)=B(t)v0(t)+f (t)fort∈[0,T ]. That means thatv0is a solution of the second-order Cauchy problem (3.1).

To prove the uniqueness of the solution of (3.1), we again apply Tanaka’s theorem.

We first assume thatf≡0. OnD2×D1, the closure ofD2×D1on Banach space[D1]×E, we consider the evolution family{ᐂ(t,s)}0≤s≤t≤Tgenerated by{Ꮿ(t)}, where

ᐂ(t,s)=

V1,1(t,s) V1,2(t,s) V2,1(t,s) V2,2(t,s)

. (3.9)

Then we have V1,1(t,t)≡I on[D1] and V1,2(t,t)≡0. Moreover, by [3, Lemma 4], V1,1(t,s)is bounded inE. From the identity

∂

∂τᐂ(t,τ)ᐁ=ᐂ(t,τ)Ꮾ(τ)ᐁ (3.10) forᐁ∈D(τ):= {(u1,u2)^T∈D2×D1,Ꮾ(τ)(u1,u2)^T∈D2×D1}, we obtain

∂

∂τV1,1(t,τ)u1= −V1,2(t,τ)B(τ)u1 (3.11) foru1∈D2andB(τ)x1∈D¯1, and

∂

∂τV1,2(t,τ)u2= −V1,1(t,τ)u1 (3.12) foru2∈D¯2in[D1]-norm.

Now let u be a solution of (3.1), then u(τ)∈D2, B(τ)u(τ)=u(τ)∈D¯1, and u(τ)∈D¯2in[D1]-norm forτ∈[0,T ]. From the above equations it follows that

∂

∂τ

V1,1(t,τ)u(τ)

= −V1,2(t,τ)B(τ)u(τ)+V1,1(t,τ)u(τ),

∂

∂τ

V1,2(t,τ)u(τ)

= −V1,1(t,τ)u(τ)+V1,2(t,τ)u(τ)

= −V1,1(t,τ)u(τ)+V1,2(t,τ)B(τ)u(τ).

(3.13)

Adding these equations, we obtain

∂

∂τ

V1,1(t,τ)u(τ) + ∂

∂τ

V1,2(t,τ)u(τ)

=0. (3.14)

Integrating both sides from 0 tot, we have

u(t)=V1,1(t,0)x0+V1,2(t,0)x1. (3.15) Thus,uis uniquely determined byx0andx1. The uniqueness of the solutions for in- homogenous Cauchy problem follows from that of the solutions for the homogenous one, and the theorem is proved.

Corollary3.4. For the complete second-order Cauchy problem

u(t)=A(t)u(t)+B(t)u(t)+f (t), 0≤t≤T , u⁽⁰⁾(s)=xi, i=0,1, (3.16) we assume that the operatorsB(t)satisfyAssumption 3.1and that{A(t)}0≤t≤T is a family of bounded operators such that tA(t)x is continuously differentiable for allx∈D1. Then the second-order Cauchy problem (3.16) has a unique solution with u(t)∈D2,u(t)∈D¯2in[D1], andu(t)∈D¯1for every initial value(u(0),u(0))^T= (x0,x1)^T∈D2×D1andf∈W^1,1(R+,E)satisfyingB(0)x0+A(0)x1+f (0)∈D¯1and x1∈D¯2in[D1]-norm.

Proof. On the Banach space[D1]×E, we consider the initial value problem ᐂ(t)=Ꮾ1(t)ᐂ(t)+F(t), 0≤t≤T , ᐂ(s)=

x0,x1_T

, (3.17)

whereᐂ(t):=(v0(t),v1(t))^T,F(t):=(0,f (t))^T, and Ꮾ1(t):=

0 I B(t) A(t)

(3.18) withD(Ꮾ(t)):=D2×D1. We can writeᏮ1(t)as

Ꮾ1(t)=

0 I B(t) 0

+

0 0 0 A(t)

=Ꮾ(t)+Ꮽ(t), (3.19) that is, the sum of Ꮾ(t) and a bounded operator Ꮽ(t). Applying Lemma 1.3 and Theorem 3.3, we see that {Ꮾ1(t)}is stable and thus satisfies the hyperboliccondi- tion. As in the proof ofTheorem 3.3, the first component of the solution of (3.17) is a solution of the second-order Cauchy problem (3.16).

Remark3.5. As in the proof ofTheorem 3.3we show that in the inhomogenous case, the solution of (3.1) and (3.16) has the form

u(t)=V1,1(t,s)x0+V1,2(t,s)x1+ _t

sV1,2(t,τ)f (τ)dτ. (3.20) Remark3.6. IfD1is dense inE, thenD2is dense in[D1]. Therefore, inTheorem 3.3 andCorollary 3.4, we can drop all compatibility conditions.

Applications. (1) We first consider the autonomous second-order Cauchy prob- lem

u(t)=Bu(t)+f (t), t≥0, u(0)=x0, u(0)=x1. (3.21) ByTheorem 3.3, ifB=C²,such thatCis the generator of aC0-group, or, equivalently, ifBis the generator of a cosinus family, then (3.21) has a unique solution for every initial value(x0,x1)^T∈D(C²)×D(C). This is a classical result on the “wellposedness”

of second-order Cauchy problems (see [1]).

(2) We are now concerned with the second-order Volterra integrodifferential equa- tion

u(t)=B(t)u(t)+

_t

0C(t,s)u(s)ds+f (t), 0≤t≤T , u⁽ⁱ⁾(0)=xi, i=0,1. (3.22) The autonomous version of (3.22) was studied by Oka [9] forB(t)≡BandC(t,s)= C(t−s). For the first-order Volterra integrodifferential equations, Oka and Tanaka [10] showed that under the conditions

(A) the family{B(t)}0≤t≤Tsatisfies the hyperbolic condition with constant domain D, which is not necessarily dense inE,

(B) {C(t,s)}0≤s≤t≤T is a family of bounded linear operators fromDtoEsuch that for everyy∈D,C(t,s)y is continuous on the set∆:= {(t,s): 0≤s≤t≤T} and continuously differentiable with respect tot, then the Volterra integrodif- ferential equation

u(t)=B(t)u(t)+ _t

0C(t,s)u(s)ds+f (t), 0≤t≤T , u(0)=x0 (3.23) has a unique solution for every initial valuex0∈Dandf∈W^1,1(R+,E), such thatB(0)x0+f (0)∈D.¯

Combining this and our result, we obtain the existence and uniqueness of the solu- tions of (3.22). More precisely, we have the following theorem.

Theorem 3.7. Consider the nonautonomous second-order Volterra integrodiffer- ential equation (3.22), where the families{B(t)}0≤t≤T and{C(t,s)}0≤s≤t≤T have the properties

(i) the family{B(t)}satisfiesAssumption 3.1,

(ii) {C(t,s)}is a family of bounded linear operators from [D2]toE such that for everyy∈D2, C(t,s)y is continuous on the set∆:= {(t,s): 0≤s≤t≤T}and continuously differentiable with respect tot.

Then (3.22) has a unique solution for every initial value (x0,x1)^T ∈D2×D1 and every inhomogenous termf∈W^1,1(R+,E)satisfyingB(0)x0+f (0)∈D¯1andx1∈D¯2

in[D1].

Proof. On the basis of our substitution, we convert our second-order problem into a first-order system on[D1]×Eas follows:

ᐁ(t)=Ꮾ(t)ᐁ(t)+ _t

0Ꮿ(t,s)ᐁ(s)ds+F(t), 0≤t≤T , ᐁ(0)= x0,x1_T

(3.24) on[D1]×E, whereᐁ:=(u,u)^T,F:=(0,f )^T,Ꮾ(t)as defined in (3.5) and

Ꮿ(t,s):=

0 0 C(t,s) 0

. (3.25)

We can now check that the families{Ꮾ(t)}and{Ꮿ(t,s)}0≤s≤t≤T satisfy the conditions (A) and (B). Using the result of Oka and Tanaka, we obtain the existence and uniqueness of the solutions of (3.24) and then those of (3.22).

Acknowledgement. This paper was partially written while the author was visit- ing Tübingen, Germany. The author thanks K.-J. Engel and R. Nagel for helpful com- ments.

References

[1] H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, North- Holland Mathematics Studies, vol. 108, North-Holland Publishing, Amsterdam, 1985.MR 87b:34001. Zbl 564.34063.

[2] T. Kato,Linear evolution equations of “hyperbolic” type, J. Fac. Sci. Univ. Tokyo Sect. I17 (1970), 241–258.MR 43#5347. Zbl 222.47011.

[3] M. Kozak,An abstract linear second-order temporally inhomogeneous differential equa- tion. I, Univ. Iagel. Acta Math. 1137 (1994), no. 31, 21–30. MR 95j:34085.

Zbl 829.34047.

[4] , An abstract second-order temporally inhomogeneous linear differential equa- tion. II, Univ. Iagel. Acta Math. 1169 (1995), no. 32, 263–274. MR 96g:34091.

Zbl 829.34048.

[5] R. Nagel,Towards a “matrix theory” for unbounded operator matrices, Math. Z. 201 (1989), no. 1, 57–68.MR 90c:47004. Zbl 672.47001.

[6] F. Neubrander,Well-posedness of higher order abstract Cauchy problems, Trans. Amer.

Math. Soc.295(1986), no. 1, 257–290.MR 88a:34087. Zbl 589.34004.

[7] T. L. Nguyen and G. Nickel, Time-dependent operator matrices and inhomogeneous Cauchy problems, Rend. Circ. Mat. Palermo (2)47(1998), no. 1, 5–24.MR 99f:34087.

Zbl 916.47032.

[8] H. Oka,A class of complete second order linear differential equations, Proc. Amer. Math.

Soc.124(1996), no. 10, 3143–3150.MR 96m:34114. Zbl 880.34064.

[9] ,Second order linear Volterra integrodifferential equations, Semigroup Forum53 (1996), no. 1, 25–43.MR 97d:45004. Zbl 856.45016.

[10] H. Oka and N. Tanaka, Nonautonomous integro-differential equations of hyperbolic type, Differential Integral Equations 8(1995), no. 7, 1823–1831.MR 97a:45017.

Zbl 826.45006.

[11] A. Pazy,Semigroups of Linear Operators and Applications to Partial Differential Equa- tions, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.

MR 85g:47061. Zbl 516.47023.

[12] N. Tanaka,Quasilinear evolution equations with non-densely defined operators, Differen- tial Integral Equations9(1996), no. 5, 1067–1106.MR 97b:34069. Zbl 942.34053.

Nguyen Thanh Lan: Department of Mathematics, Ohio University, Athens OH45701, USA

E-mail address:[email protected]

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