existence and uniqueness of pseudo almost periodic solutions of semilinear cauchy problems with non...

Nonlinear Analysis 45 (2001) 73–83

Existence and uniqueness of pseudo almostperiodic solutions of semilinear Cauchy problems

with non dense domain

Claudio Cuevasa ;1, Manuel Pintob;∗;2

aDepartamento de Matem�aticas, Universidad de la Frontera, Casilla 54-D Temuco, ChilebDepartamento de Matem�aticas, Facultad de Ciencias, Universidad de Chile, Casilla 653,

Santiago, Chile

Received 14 September 1998; accepted 13 May 1999

Keywords: Almost periodic functions; Pseudo almost periodic solutions; Semilinear Cauchyproblem; Hille–Yosida operator; nondense domain

1. Introduction

As for the almost periodic functions, pseudo-almost periodic functions (p.a.p.) havemany applications in several problems for example in theory of functional di1erentialequations, integral equations and partial di1erential equations. Most of these problemsneed to be studied in abstract spaces and the operators are de3ned over nondensedomain. In this context very scarce literature exists and we refer the reader to AitDads [1], Ait Dads and Ezzimbi [2,3], Hale–Lunel [10] and Pazy [13] for the detail.In 1992, Zhang [15,16] has introduced an extension of the concept of almost periodic

functions, the so-called pseudo-almost periodic functions. In paper [16] Zhang investi-gated the existence of pseudo-almost periodic solutions of p.a.p. nonlinear perturbationof a linear autonomous ordinary di1erential equation.

∗ Corresponding author.E-mail address: [email protected] (M. Pinto).1 Supported by Direcci>on de Investigaci>on, y Desarrollo. Project UFRO No. IN-16=98.2 Partially Supported by Fondecyt 1980835 and 1970723.

0362-546X/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved.PII: S0362 -546X(99)00330 -2

74 C. Cuevas, M. Pinto / Nonlinear Analysis 45 (2001) 73–83

In this paper, we study the existence and uniqueness of pseudo-almost periodicsolutions of the following equation

x′(t) = Ax(t) + f(t; x(t)); t ∈ R; (1.1)

where A is an unbounded linear operator, assumed to be a Hille–Yosida (see De3nition2.4) with the negative type having the domain D(A) not necessarily dense on someBanach space X , and f : R× X0 → X is a continuous function, where X0 = D(A).Recently, Amir and Maniar [4,5] proved that the following Cauchy problem:

x′(t) = Ax(t) + f(t); t ∈ R; (1.2)

has a unique p.a.p. solution if the inhomogeneous term has the same property. In thispaper, we prove that Eq. (1.1) has a unique p.a.p. solution if the function f(t; x) isp.a.p. and L(t)-Lipschitzian with respect to the second variable. L(t) satis3es severalsuitable conditions. We extend the results obtained in the recent article of Amir andManiar [4], where L is a suJciently small constant. We obtain existence and uniquenessresults to the more general case of evolutionary systems. Our results can be appliedto the theory of delay and partial di1erential equations [6,10]. In Banach spaces thereexists not much work concerning the study of the existence of p.a.p. solutions ofEq. (1.1), where A is a generator of C0-semigroup.The article is organized as follows: In Section 2 we develop the necessary prelim-

inaries and in Section 3 we study the existence and uniqueness of p.a.p. solutions. InSection 4 we present some examples to illustrate the usefulness of the results.

2. Preliminaries and basic results

Let us recall the notion of almost periodicity and pseudo-almost periodicity whichshall come into play later on.

De�nition 2.1. (Zaidman [14]). Let (X; ‖ · ‖) be a Banach space. Then f : R→ X iscalled almost periodic if f is continuous, and for each �¿ 0 there exists l(�)¿ 0 suchthat every interval I of length l(�) contains a number � with the property that

‖f(t + �)− f(t)‖¡� for all t ∈ R:

An almost periodic function is bounded and uniformly continuous on R.

De�nition 2.2. Let X and Y be two Banach space. Then f : R × Y → X is calledalmost periodic in t uniformly for x ∈ Y if f is continuous, and for each �¿ 0 andany compact K of Y there exists l(�)¿ 0 such that every interval I of length l(�)contains a number � with the property that

‖f(t + �; x)− f(t; x)‖¡� for all t ∈ R; x ∈ K:

The sets of such functions will be denoted, respectively, by AP(X ) and AP(R ×Y; X ): We have the following standard result in the theory of almost periodic functions(see [9]).

C. Cuevas, M. Pinto / Nonlinear Analysis 45 (2001) 73–83 75

Proposition 2.1. Let f ∈ AP(R×Y; X ) and h ∈ AP(Y ). Then the function f(· ; h(·)) ∈AP(X ):

Let PAP0(X ) denote the space of all continuous functions ’ : R→ X such that

limr→∞

12r

∫ r

−r‖’(t)‖ dt = 0

and PAP0(R× Y; X ) denote the space of all continuous functions ’ : R× Y → X suchthat ’(·; x) is bounded for all x ∈ Y and

limr→∞

12r

∫ r

−r‖’(t; x)‖ dt = 0;

uniformly in x ∈ Y .

De�nition 2.3. (Zhang [15,16]). A function f∈Cb(R; X ) (respectively, f∈C(R ×Y; X )) is called pseudo-almost periodic (pseudo-almost periodic in t ∈ R; uniformly inx ∈Y ) if

f = g+ ’;

where g ∈ AP(X )(AP(R× Y; X )) and ’ ∈ PAP0(X ) (PAP0(R× Y; X )):

The functions g and ’ are called the almost periodic component and, respectively,the ergodic perturbation of the function f. The set of all functions will be denoted byPAP(X ) (resp. PAP(R× Y; X )):For the theory of extrapolation spaces we refer to [7,12]. We only recall here some

notions and properties that will be essential for us.

De�nition 2.4. Let X be a Banach space and A be a linear operator with domain D(A).We say that (A;D(A)) is a Hille–Yosida operator on X if there exist ! ∈ R and apositive constant M ≥ 1 such that (!;+∞)⊆ �(A) (�(A) is the resolvent set of A) and

‖(�I − !)−1‖ ≤ M(� − !)n

�¿!; n ∈ N:

If the constant ! can be chosen smaller than zero, A is called a Hille–Yosida operatorof the negative-type.It follows from the Hille–Yosida Theorem that any Hille–Yosida operator gener-

ates a C0-semigroup on the closure of its domain. More precisely (see [11]), thepart (A0; D(A0)) of A in X0 de3ned by A0x = Ax; x∈D(A0) with D(A0) = {x∈D(A):Ax ∈ X0} generates a C0-semigroup (T0(t))t≥0.We assume that (A;D(A)) is a Hille–Yosida operator of the negative type on X .

This implies that A−1 ∈ L(X ).On X0 we introduce a new norm by the expression ‖x‖−1 = ‖A−1

0 x‖, for allx ∈ X0. Then the completion of (X0; ‖ · ‖−1) will be called the extrapolation spaceof X0 associated to A0 and will be denoted by X−1.


One can show that, for each t ≥ 0, the operator T0(t) can be extended to a uniquebounded operator on X−1 denoted by T−1(t). The family (T−1(t))t≥0 is a C0-semigroupon X−1 which will be called the extrapolated semigroup of (T0(t))t≥0.Next, we state the fundamental lemma, which will be used in the sequel (see [4]).

Lemma 2.1. For f ∈ Cb(R; X ); the following properties hold:(i)∫ t−∞ T−1(t − s)f(s) ds ∈ X0; for all t ∈ R;

(ii) ‖ ∫ t−∞ T−1(t − s)f(s) ds‖ ≤ Cewt

∫ t−∞ e

−ws‖f(s)‖ ds,where −w and C are positive constants independent of t and f; and T−1(t) is theextrapolated semigroup.

The next three propositions are extensions of Amir and Maniar’s Theorem 3.1[4, p. 10].

Proposition 2.2. Let f ∈ PAP(R× Y; X ) satisfy the Lipschitz condition:

‖f(t; x)− f(t; y)‖ ≤ L(t)‖x − y‖ for all x; y ∈ Y and t ∈ R; (2.1)

where L ∈ L1(R): If h ∈ PAP(Y ); then the function f(·; h(·)) ∈ PAP(X ):

Proof. Since f∈PAP(R × Y; X ), then f = g + ’, where g∈AP(R × Y; X ) and ’ ∈PAP0(R×Y; X ). Moreover, let h= h1 + h2, with h1 ∈ AP(Y ) and h2 ∈ PAP0(Y ). Sincef is bounded, f(·; h(·)) ∈ Cb(R; X ). On the other hand,

f(·; h(·)) = g(·; h1(·)) + f(·; h(·))− g(·; h1(·))= g(·; h1(·)) + f(·; h(·))− f(·; h1(·)) + ’(·; h1(·)): (2.2)

By Proposition 2.1, g(·; h1(·)) ∈ AP(X ). Let

F(t) = f(t; h(t))− f(t; h1(t)); (2.3)

from (2.1) and h2 bounded, we obtain

12r

∫ r

−r‖F(t)‖ dt ≤

(C̃2r

)‖L‖1: (2.4)

Thus, we conclude that F(·) ∈ PAP0(X ). f(·; h(·)) ∈ PAP(X ), follows from

limr→∞

12r

∫ r

−r‖’(t; h1(t))‖ dt = 0: (2.5)

Since h1(R) is relatively compact in Y , for �¿ 0, one can 3nd a 3nite number of openballs Ok with center xk ∈ h1(R) and radius less than �=3 such that h1(R)⊆

⋃mk=1 Ok :

Any set Bk = {x ∈ R=h1(x) ∈ Ok} is open and R =⋃m

k=1 Bk . Let Ek = Bk −⋃k−1

i=1 Bi

and E1 =B1, then Ei ∩Ej = ∅ for i �= j. Using the fact that ’ ∈ PAP0(R× Y; X ), thereis a number r0¿ 0 such that

12r

∫ r

−r‖’(t; xk)‖ dt ¡ �

3mfor all r ≥ r0 and k ∈ {1; : : : ; m}: (2.6)


Furthermore, since g ∈ AP(R × Y; X ) is uniformly continuous in x ∈ h1(R), one canobtain

‖g(t; xk)− g(t; x)‖¡�3

for x ∈ Ok and k = 1; : : : ; m (2.7)

uniformly in t ∈ R. Let Ak :=Ek ∩ [− r; r], where choosing r suJciently large, we have

12r

m∑k=1

∫Ak

‖’(t; h1(t))‖ dt ≤ 12r

m∑k=1

∫Ak

(‖’(t; h1(t))− ’(t; xk)‖+ ‖’(t; xk)‖) dt

≤ 12r

m∑k=1

∫Ak

(L(t)‖h1(t)− xk‖

+ ‖g(t; h1(t))− g(t; xk)‖) dt

+m∑

k=1

12r

∫ r

−r‖’(t; xk)‖ dt: (2.8)

Since, for any t ∈ Ak; h1(t) ∈ Ok we have

12r

m∑k=1

∫Ak

L(t)‖h1(t)− xk‖ dt ≤ �3

(‖L‖12r

)≤ �3: (2.9)

So, from (2.6)–(2.9), (2.5) is true and this completes the proof of the proposition.

Proposition 2.3. Let f ∈PAP(R× Y; X ) satisfy (2:1) and

limt→∞

12t

∫ t

−tL(s) ds= 0: (2.10)

If h ∈ PAP(Y ); then the function f(·; h(·)) ∈ PAP(X ):

Proof. We proceed in a similar way to Proposition 2.2. In fact, let F(t) be as in (2.3),from (2.4) and(2.10), we conclude that F(·) ∈ PAP0(X ): On the other hand, inequality(2.9) is transformed in

12r

m∑k=1

∫Ak

L(t)‖h1(t)− xk‖ dt ≤ �3

(12r

∫ r

−rL(t) dt

)≤ �3:

The remainder of the proof is similar to that of Proposition 2.2.

Proposition 2.4. Let f ∈ PAP(R× Y; X ) be as in (2:1) with L such that

,(t) = e!t∫ t

−∞e−!sL(s) ds (2.11)

is bounded. If h ∈ PAP(Y ); then the function f(·; h(·)) ∈ PAP(X ):


Proof. The proof is based on Proposition 2.2. We note that L(t) = ,′(t)− !,(t). LetF(t) be as in (2.3), taking into account that h2 and , are bounded functions, we obtain

12r

∫ r

−r‖F(t)‖ dt ≤ 1

2r

∫ r

−r,′(t)‖h2(t)‖ dt +

(−!2r

)∫ r

−r,(t)‖h2(t)‖ dt

≤(

C̃12r

)∫ r

−r,′(t) dt +

(C̃2(−!)2r

)∫ r

−r‖h2(t)‖ dt

≤ C̃1C̃2r

+(

C̃2(−!)2r

)∫ r

−r‖h2(t)‖ dt:

Since h2 ∈ PAP0(Y ), we conclude that F(·) ∈ PAP0(X ): For �¿ 0, one can 3nd a 3nitenumber of open balls Ok with center xk ∈ h1(R) and radius less than �=3(1+(−!)C̃2)such that h1(R)⊆

⋃mk=1 Ok , where C̃2 is a positive number such that ,(t) ≤ C̃2 for all

t ∈ R. On the other hand, formula (2.9) is transformed in

12r

m∑k=1

∫Ak

L(t)‖h1(t)− xk‖ dt

≤ �

3(1 + (−!)C̃2)

(12r

∫ r

−r,′(t) dt +

(−!)2r

∫ r

−r,(t) dt

)

≤ �

3(1 + (−!)C̃2))

(C̃2r+ (−!)C̃2

)≤ �3:

This completes the proof of the proposition.

3. Existence and uniqueness of p.a.p. solutions

Let us consider Banach spaces X and Y , and a bounded continuous functionf :R× Y → X . We have the following result.

Theorem 3.1. Let f∈PAP(R × X0; X ) be a bounded function as in (2:1); with L∈L1(R). Then Eq. (1:1) admits one and only one pseudo-almost periodic solutionon R.

Proof. Let y be a function in PAP(X0). Then using Proposition 2.2, the functiong(·) :=f(·; y(·)) is in PAP(X ). From [4], the Cauchy problem

x′(t) = Ax(t) + g(t); t ∈ R; (3.1)

has a unique solution x in PAP(X0), which is given by

x(t) = (Fy)(t) =∫ t

−∞T−1(t − s)f(s; y(s)) ds; t ∈ R: (3.2)


It suJces now to show that this operator F has a unique 3xed point in PAP(X0). Forthis, let x and y be in PAP(X0). Using Lemma 2.1, we have

‖(Fx)(t)− (Fy)(t)‖ ≤Ce!t∫ t

−∞e−!s‖f(s; x(s))− f(s; y(s))‖ ds

≤C(∫ t

−∞L(s) ds

)‖x − y‖∞

≤C‖L‖1‖x − y‖∞: (3.3)

In general, we get

‖(Fnx)(t)− (Fny)(t)‖ ≤ Cn

(n− 1)!

(∫ t

−∞L(s)

(∫ s

−∞L(�) d�

)n−1ds

)‖x − y‖∞

≤ Cn

n!

(∫ t

−∞L(�) d�

)n

‖x − y‖∞

≤ (C‖L‖1)nn!

‖x − y‖∞: (3.4)

Hence, since (C‖L‖1)n=n!¡ 1 for n suJciently large, by the contraction principle Fhas a unique 3xed point y ∈ PAP(X0).

Now, we wish to extend the existence and uniqueness results of pseudo-almostperiodic solutions to the more general case of equations. For our purposes we in-troduce a new class of functions L which do not necessarily belong to L1. We havethe following result.

Theorem 3.2. Let f be as in Theorem 3:1; where L satis9es (2:10); and one of thefollowing two conditions is true:(i) ‖L‖M¡(1−e!)=C; whereC is given by Lemma 2:1 and ‖L‖M =supt∈R

∫ t+1t L(s) dsC;

(ii) The integral∫ t−∞ L(s) ds exists for all t ∈ R.

Then the conclusion of Theorem 3:1 holds.

Proof. Under condition (i): Let F be the operator de3ned by (3.2). For x; y ∈ PAP(X0),using Lemma 2.1, we have

‖(Fx)(t)− (Fy)(t)‖ ≤C∫ t

−∞e!(t−s)L(s)‖x(s)− y(s)‖ ds

≤C

( ∞∑m=0

∫ t−m

t−(m+1)e!(t−s)L(s) ds

)‖x − y‖∞

≤ (C‖L‖M=(1− e!))‖x − y‖∞:

By the contraction principle F has a unique 3xed point y ∈ PAP(X0).


Under condition (ii): De3ne a new norm by ‖|x‖|=Supt∈R {v(t)‖x(t)‖}, where v(t)=

e−k∫ t

−∞ L(s) dsand k is a 3xed positive constant greater than C. Let x and y be in

PAP(X0), then we have

v(t)‖(Fx)(t)− (Fy)(t)‖ ≤C(∫ t

−∞e!(t−s)v(t)v(s)−1L(s) ds

)‖|x − y‖|

≤ Ck

(∫ t

−∞ek∫ s

tL(�)d� k L(s) ds

)‖|x − y‖|

≤ Ck‖|x − y‖|:

Hence, since C=k ¡ 1; F has a unique 3xed point y ∈ PAP(X0).

Corollary 3.1. If L belongs to Lp with p-norm su:ciently small; then Eq. (1:1) hasa unique p:a:p: solution.

Proof. Using the HPolder inequality, it is easy to see that ‖L‖M ≤ ‖L‖p and ‖L‖t ≤(1=(2t)1=p)‖L‖p.

Corollary 3.2. If L satis9es condition (ii) of Theorem 3:2 and belongs to Lp; thenEq. (1:1) has a unique p:a:p: solution.

Remark 1. Observe that Theorem 3.2 does not include the cases where L is a constantor a bounded function. However, Theorems 3.3 and 3.4 apply in the case where L isa constant suJciently small studied in [4], or a bounded integrable function.

Theorem 3.3. Let f ∈ PAP(R × X0; X ) be a bounded function as in (2:1); with Lsuch that the following conditions holds:(i) ,(t) = e!t

∫ t−∞ e

−!sL(s) ds is bounded;

(ii) The integral∫ t−∞ ,(�) d� exists for all t ∈ R.

Then Eq. (1:1) has a unique p:a:p: solution x such that

‖x(t)‖ ≤(

C−!

)‖f(·; 0)‖∞ek

∫ t

−∞ L(s) ds;

where the constant C is given by Lemma 2:1 and k is any constant such that k ¿C

Proof. De3ne a new norm in PAP(X0) by

|x|= Supt∈R

{v(t)‖x(t)‖}; (3.5)


where v(t)=Exp[− k(,(t)−!∫ t−∞ ,(�) d�)] and k is a 3xed positive constant greater

than C. Let x and y in PAP(X0), then

v(t)‖(Fx)(t)− (Fy)(t)‖ ≤C(∫ t

−∞e!(t−s)v(t)v(s)−1L(s) ds

)‖|x − y‖|

=Cv(t)

k

(∫ t

−∞e!(t−s) d

ds[v(s)−1] ds

)‖|x − y‖|

= C(1k+ !

∫ t

−∞e!(t−s)v(t)v(s)−1 ds

)‖|x − y‖|:

Therefore,

‖|Fx − Fy ‖| ≤ Ck‖|x − y‖|:

By the contraction principle, there is a unique p.a.p. solution of

x(t) =∫ t

−∞T−1(t − s)f(s; x(s)) ds; t ∈ R;

which is by de3nition the unique p.a.p. solution of (1.1). On the other hand, we get

e−!t‖x(t)‖ ≤ C∫ t

−∞e−!s‖f(s; 0)‖ ds+ C

∫ t

−∞L(s)e−!s‖x(s)‖ ds:

Using Gronwall–Bellman’s inequality, we obtain

e−!t‖x(t)‖ ≤ Cek∫ t

−∞ L(s) ds(∫ t

−∞e−!s‖f(s; 0)‖ ds

):

The proof is complete.

Corollary 3.3. If L is either bounded or Lp or ‖L‖M ¡∞; and the integral∫ t−∞ L(s) ds

exists for all t ∈ R; then Eq. (1:1) has a unique p:a:p: solution.

Remark 2. Note that Theorem 3.3 does not include the cases where L is a constantor with p-norm suJciently small.

Theorem 3.4. Let f be as in Theorem 3:1. Suppose also that there exists a constant/¿ 0 such that C,(t) ≤ /¡ 1; for all t ∈ R; where the constant C is given byLemma 2:1 and ,(t) =

∫ t−∞ e

!(t−s)L(s) ds. Then equation (1:1) has a unique p:a:p:solution.

We obtain Amir–Maniar’s Theorem 3.2 [4, p. 13] as a corollary of Theorem 3.4.

Corollary 3.4. If L is a su:ciently small constant; then Eq. (1:1) has a unique p:a:p:solution.

4. Examples

We complete this work with an application to the partial di1erential equations.


Example 1. We consider the following partial di1erential equation

@tu(t; x) = @xu(t; x)− ,u(t; x) + 2(sin(t) +

11 + t2

)sin(u(t; x)); (4.1)

where ,; 2 ∈ R and , ≥ 0.Let X = L∞(R) with the supremum norm ‖ · ‖∞ and the operator A de3ned on

X by

Ax = x′ − ,x;

with domain

D(A) = {x ∈ X : x is absolutely continuous and x′ ∈ X }:We can easily show that A is a Hille–Yosida operator of type ! = −,¡ 0 with a

nondense domain (see [8]).Eq. (4.1) can be formulated by the following inhomogeneous Cauchy problem:

u′(t) = Au(t) + F(t; u(t)); t ∈R; (4.2)

where u(t) = u(t; ·); F(t; ’)(x) = 2b(t)sin(’(x)) for all ’ ∈ X and x; t ∈ R withb(t) = b1(t) + b2(t). We observe that b1 is almost periodic function and b2 is anergodic function because

limr→∞

12r

∫ r

−r

11 + t2

dt = limr→∞

1rarctan(r) = 0:

On the other hand, F is a pseudo-almost periodic function such that (2.1) holdsfor L(t) = |2‖b(t)|. If we assume that |2|¡ − !=2C, then for Theorem 3.4 the semi-linear Cauchy problem (4.2) has one and only one pseudo-almost periodic solution.Consequently, the partial di1erential equation (4.1) admits a unique p.a.p. solution.

Example 2. We consider a general situation in Example 1, taking the functionb= b1 + b2 satisfying one of the following three conditions:(a) b1 is almost periodic and b2 is a bounded integrable function and 2 is a real number

such that C|2|(‖b1‖∞=(−!)+‖b2‖1)¡ 1 or C|2| ‖b‖M=(1−exp!)¡ 1, where theconstant C is given by Lemma 2.1.

(b) b1 is almost periodic and b2 is a bounded Lp-integrable function and 2 is areal number such that C|2|(‖b1‖∞=(−!) + (1=q(−!))q

−1‖b2‖p)¡ 1, wherep−1 + q−1 = 1.

(c) b1 is almost periodic and b2 is a bounded ergodic function and 2 is a real numbersuch that C|2|¡ (−!)=C‖b‖∞.

Then from Theorem 3.4 the partial diferential equation

@tu(t; x) = @xu(t; x)− ,u(t; x) + 2b(t) sin(u(t; x)); (4.3)

admits a unique p.a.p. solution.

Acknowledgements

The authors wish to thank the referee for useful comments and suggestions.


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existence and uniqueness of pseudo almost periodic solutions of semilinear cauchy problems with non...

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