On a higher order evolution equation with a Stepanov bounded solution

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ON A HIGHER-ORDER EVOLUTION EQUATION

WITH A STEPANOV-BOUNDED SOLUTION

ARIBINDI SATYANARAYAN RAO

Received 12 June 2003 and in revised form 5 August 2004

We study strong solutionsu:R→X, a Banach spaceX, of thenth-order evolution equa-tion u(n)−Au(n−1)=f, an infinitesimal generator of a strongly continuous groupA: D(A)⊆X→X, and a given forcing termf:R→X. It is shown that ifX is reflexive,u andu(n−1)_{are Stepanov-bounded, andfis Stepanov almost periodic, thenuand all} deriva-tivesu, . . . , u(n−1)are strongly almost periodic. In the case of a general Banach spaceX, a corresponding result is obtained, proving weak almost periodicity ofu,u, . . . , u(n−1)_.

2000 Mathematics Subject Classification: 34G10, 34C27, 47D03.

1. Introduction. In this paper, we are concerned with annth-order evolution equa-tion of the form

u(n)−Au(n−1)=f . (1.1)

HereA:D(A)⊆X→Xis an infinitesimal generator of a strongly continuous group,

f:R→Xa given forcing term,Xa Banach space with scalar fieldC,na positive integer, andRdenotes the set of reals. We will give suitable assumptions to ensure that almost periodicity of the forcing termf carries over to the solutionuand its derivatives up to order(n−1).

The reason for studying this rather special evolution equation may be classified as a first pilot study of the issue of higher-order evolution equations, which probably has not been studied before.

We first recall the relevant concepts. A continuous functionf:R→X is said to be strongly (or Bochner) almost periodic if, for every givenε >0, there is anr >0 such that any interval inRof lengthrcontains a pointτfor which

sup t∈R

_{f (t+τ)−f (t)≤ε. (1.2)}

Here·denotes the norm inX.

A functionf:R→Xis called weakly almost periodic ifx∗f (·):R→Cis continuous and almost periodic for everyx∗in the dual spaceX∗ofX.

We will call a functionf∈L1loc(R, X)Stepanov-bounded or brieflyS-bounded if

fS:=sup t∈R

t+1

t

We will call a functionf∈L1

loc(R, X)Stepanov almost periodic or brieflyS-almost periodic if, for every givenε >0, there is anr >0 such that any interval inRof length

rcontains a pointτfor which

sup t∈R

t+1

t

f (s+τ)−f (s)ds≤ε. (1.4)

We denote by L(X, X)the set of all bounded linear operators on X into itself. An operator-valued functionT:R→L(X, X)will be called a strongly continuous group if

Tt1+t2

=Tt1

Tt2

∀t1, t2∈R, (1.5)

T (0)=I=the identity operator onX, (1.6)

T (·)x:R→Xis continuous for everyx∈X. (1.7)

We recall (e.g., from Dunford and Schwartz [4]) that the infinitesimal generatorA:

D(A)⊆X→X of a strongly continuous group T :R→L(X, X)is a densely defined, closed linear operator.

An operator-valued functionT :R→L(X, X)is said to be strongly (weakly) almost periodic ifT (·)x:R→Xis strongly (weakly) almost periodic for everyx∈X.

SupposeA:D(A)⊆X→Xis a densely defined, closed linear operator, andf:R→X

is a continuous function. Then a strong solution of the evolution equation

u(n)(t)−Au(n−1)(t)=f (t) a.e. fort∈R (1.8)

is anntimes strongly differentiable functionu:R→Xwithu(n−1)_{(t)∈D(A)for all}

t∈R, and satisfies problem (1.8).

Our first result is as follows (see Zaidman [7,8] for first-order evolution equations).

Theorem1.1. LetXbe reflexive, f:R→X continuous,S-almost periodic,A infini-tesimal generator of a strongly almost periodic groupT:R→L(X, X). In this case, if, for the strong solutionu:R→Xof problem (1.8), bothuandu(n−1)_{areS-bounded onR,} thenu,u, . . . , u(n−1)_{are all strongly almost periodic.}

Our second result refers to a weak variant of our first theorem in the case of a general—not necessarily reflexive—Banach spaceX.

Theorem1.2. Supposef:R→Xis anS-almost periodic (or a weakly almost periodic) continuous function,Aan infinitesimal generator of a strongly continuous groupT:R→

L(X, X)such that the conjugate operator groupT∗:R→L(X∗, X∗)is strongly almost periodic. If, for the strong solutionu:R→X of problem (1.8), bothuandu(n−1) _are

S-bounded onR, thenu,u, . . . , u(n−1)_{are all weakly almost periodic.}

2. Lemmas

Lemma2.1. IfAis the infinitesimal generator of a strongly continuous groupG:R→

L(X, X), then the(n−1)th derivative of any solution of (1.8) has the representation

u(n−1)_(t)=G(t)u(n−1)₍₀₎₊ t

0

G(t−s)f (s)ds fort∈R. (2.1)

Proof. For an arbitrary but fixedt∈R, we have

d ds

G(t−s)u(n−1)_{(s)=G(t−s)u}(n)_(s)−Au(n−1)_(s)

=G(t−s)f (s) a.e. fors∈R, by (1.8).

(2.2)

Now, integrating (2.2) from 0 tot, we obtain t

0

d ds

G(t−s)u(n−1)_(s)ds= t

0

G(t−s)f (s)ds, (2.3)

which gives the desired representation, by (1.6).

Lemma2.2. Ifg:R→Xis a strongly almost periodic function, andG:R→L(X, X)is a strongly (weakly) almost periodic operator-valued function, thenG(·)g(·):R→Xis a strongly (weakly) almost periodic function.

For the proof ofLemma 2.2, see [6, Theorem 1] for weak almost periodicity.

Lemma2.3. Ifg:R→X is anS-almost periodic continuous function, and G:R→

L(X, X)is a weakly almost periodic operator-valued function, thenx∗G(·)g(·):R→C is anS-almost periodic continuous function for everyx∗∈X∗.

Proof. By our assumption, for an arbitrary but fixedx∗∈X∗, the functionx∗G(·)x:

R→C is almost periodic, and so is bounded on R, for every x∈X. Hence, by the uniform-boundedness principle,

sup t∈R

_x∗_{G(t)=K <∞. (2.4)}

We note that the functionx∗G(·)g(·)is continuous onR(see [6, proof of Theorem 1]). Consider the functionsgηgiven by

gη(t)= 1

η η

0

g(t+s)ds forη >0, t∈R. (2.5)

SincegisS-almost periodic fromRtoX,gη is strongly almost periodic fromRtoX for every fixedη >0. Further, as shown forC-valued functions in [2, pages 80-81], we can prove thatgη→gasη→0+in theS-sense, that is,

sup t∈R

t+1

t

g(s)−gη(s)ds →0 asη →0+. (2.6)

Now we have

x∗G(s)g(s)=x∗G(s)g(s)−gη(s)

and, by (2.4) and (2.6),

sup t∈R

t+1

t

x∗G(s)g(s)−gη(s)ds

≤Ksup t∈R

t+1

t

g(s)−gη(s)ds →0 asη →0+.

(2.8)

ByLemma 2.2, the functionsx∗G(·)gη(·)are almost periodic fromRtoC. Therefore, it follows from (2.7)-(2.8) thatx∗G(·)g(·)isS-almost periodic fromRtoC.

Lemma2.4. Ifg:R→X is anS-almost periodic continuous function, and G:R→

L(X, X)is a strongly almost periodic operator-valued function, thenG(·)g(·):R→Xis anS-almost periodic continuous function.

The proof of this lemma parallels that ofLemma 2.3and may therefore be safely omitted.

Lemma2.5. In a reflexive spaceX, assumeh:R→Xis anS-almost periodic continu-ous function, and

H(t)= t

0

h(s)ds fort∈R. (2.9)

IfHisS-bounded, then it is strongly almost periodic fromRtoX.

For the proof ofLemma 2.5, see [5, Notes (ii)].

Lemma2.6. For an operator-valued functionG:R→L(X, X), supposeG∗(t)is the conjugate (adjoint) of the operatorG(t)for t∈R. If G∗ :R→L(X∗, X∗) is strongly almost periodic, andg:R→Xis weakly almost periodic, thenG(·)g(·):R→Xis weakly almost periodic.

For the proof ofLemma 2.6, see [6, Remarks (iii)].

3. Proof ofTheorem 1.1. By (2.1), we have

T (−t)u(n−1)(t)=u(n−1)(0)+ t

0T (−s)f (s)ds fort∈R. (3.1)

Evidently,T (−·):R→L(X, X)is a strongly almost periodic group. Therefore,T (−·)x:

R→Xis strongly almost periodic, and so is bounded onR, for everyx∈X. Hence, by the uniform-boundedness principle,

sup t∈R

_{T (−t)<∞. (3.2)}

Consequently, T (−·)u(n−1)_{(·) is S-bounded on R (by our assumption, u}(n−1) _{is S} -bounded onR).

Now consider a sequence(αk)k=1,2,...of infinitely differentiable nonnegative functions onRsuch that

αk(t)=0 for|t| ≥ 1

k,

1/k

−1/kαk(t)dt=1. (3.3)

The convolution ofuandαkis defined by

u∗αk(t)=

Ru(t−s)αk(s)ds=

Ru(s)αk(t−s)ds fort∈R. (3.4)

We set

Cα_k= max

|t|≤1/kαk(t). (3.5)

Then we have

u∗αk

(t)= 1

−1

u(t−s)αk(s)ds ≤Cαk

t+1

t−1

u(ρ)dρ

≤2CαkuS fort∈R,by (1.3).

(3.6)

That is,u∗αkis bounded onR.

We note that, form=1,2, . . . , n−1 andk=1,2, . . .,

u∗αk (m)

(t)=u(m)∗αk

(t) fort∈R. (3.7)

Further, sinceu(n−1)_{is strongly almost periodic fromRtoX,(u}∗_α

k)(n−1)=(u(n−1)∗αk) is strongly almost periodic fromRtoX. Consequently, by [3, corollary to Lemma 5],

u∗αk, u∗αk, . . . , u(n−2)∗αkare all strongly almost periodic fromRtoX.

Withu(n−1)_{being bounded onR,u}(n−2)_{is uniformly continuous onR. Therefore, the} sequence of convolutions(u(n−2)∗_αk)(t)→u(n−2)_{(t)ask→ ∞, uniformly fort∈R.} Henceu(n−2) _{is strongly almost periodic fromRtoX. We thus conclude successively} thatu(n−2)_{, . . . , u, uare all strongly almost periodic fromRtoX, completing the proof}

of the theorem.

4. Proof ofTheorem 1.2. By our assumption, for an arbitrary but fixedx∗∈X∗,

x∗T (·)=T∗(·)x∗:R→X∗ is strongly almost periodic, and sox∗T (·)x:R→C is almost periodic for everyx∈X. Therefore, it follows thatT:R→L(X, X)is a weakly almost periodic group.

By (3.1), we have

x∗T (−t)u(n−1)_(t)=x∗_u(n−1)₍₀₎₊ t

0

x∗T (−s)f (s)ds fort∈R. (4.1)

For the sequence(αk)k=1,2,...defined by (3.3),(x∗u∗αk)=x∗(u∗αk)is bounded on

R(by (3.6)). Further, form=1,2, . . . , n−1 andk=1,2, . . . ,we have

x∗u∗αk (m)

(t)=x∗u(m)∗αk

(t) fort∈R. (4.2)

Now the rest of the proof is obvious.

If f :R→X is weakly almost periodic, then byLemma 2.6,T (−·)f (·):R→X is weakly almost periodic.

Remark4.1. IfT (t)≡Ifort∈R, and soA=0, then problem (1.8) reduces to

u(n)_{(t)=f (t) a.e. fort∈R. (4.3)}

(i) In a reflexive spaceX, supposefis defined as inTheorem 1.1. Ifu:R→Xis anS -bounded strong solution of problem (4.3), thenu, u, . . . , u(n−1)_{are all strongly almost} periodic fromRtoX.

(ii) Assumef:R→Xis a weakly almost periodic continuous function. Ifu:R→X

is anS-bounded strong solution of problem (4.3), thenu, u, . . . , u(n−1) _{are all weakly} almost periodic fromRtoX.

These statements are clearly special cases of Theorems1.1and1.2if we take into account that the assumptionu(n−1) _{S-bounded can be omitted, since, by (4.3),u}(n) _is

S-almost periodic, and sou(n−1) _{is strongly (weakly) uniformly continuous onR(by} Amerio and Prouse [1, Theorem 8, page 79]).

References

[1] L. Amerio and G. Prouse,Almost-Periodic Functions and Functional Equations, Van Nostrand Reinhold, New York, 1971.

[2] A. S. Besicovitch,Almost Periodic Functions, Dover Publications, New York, 1955.

[3] R. Cooke,Almost periodicity of bounded and compact solutions of differential equations, Duke Math. J.36(1969), 273–276.

[4] N. Dunford and J. T. Schwartz,Linear Operators. I. General Theory, Pure and Applied Math-ematics, vol. 7, Interscience Publishers, New York, 1958.

[5] A. S. Rao,On differential operators with Bohr-Neugebauer type property, J. Differential Equa-tions13(1973), 490–494.

[6] ,On weakly almost-periodic families of linear operators, Canad. Math. Bull.18(1975), no. 1, 81–85.

[7] S. Zaidman,Some asymptotic theorems for abstract differential equations, Proc. Amer. Math. Soc.25(1970), 521–525.

[8] ,Some remarks on almost-periodicity, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 106(1972), 63–67.

[9] ,Abstract differential equations with almost-periodic solutions, J. Math. Anal. Appl. 107(1985), no. 1, 291–299.

Aribindi Satyanarayan Rao: Department of Computer Science, Vanier College, 821 Avenue Ste Croix, St. Laurent, Quebec, Canada H4L 3X9