Existence of μ-Pseudo Almost Periodic Solutions to Some Classes of Nonautonomous Partial Evolution Equations

BAAZAOUI Riadh

Department of Mathematics,Faculty of Sciences of Tunis,University Tunis El-Manar,1060 Tunis,Tunisia.

1 Introduction

In this work,we propose to study the existence ofμ-pseudo almost periodic solutions under the measure theory to the class of abstract nonautonomous differential equations

whereA(t)fort∈R is a family of closed linear operators onD(A(t))satisfying the well known Acquist apace-Terreni conditions,B(t),C(t)(t∈R)are families of(possibly unbounded)linear operators,andf：R×X7→,g：R×X7→X areμ-pseudo almost periodic int∈R uniformly in the second variable.Recall that the concept ofμ-pseudo almost periodicity introduced by[5]is a natural generalization of the classical concept of weighted pseudo almost periodicity in the sense of Diagana[12,13].In recent paper[11],results on the existence and uniqueness of weighted pseudo almost periodic solutions for equation(1.1)are developed.Classical definition and properties ofμ-pseudo almost periodic function solutions introduced in[5]are used.

The organization of this works is as follows.In section 2,we introduce the basic notations and recall the definitions and lemmas ofμ-pseudo almost periodic functions introduced in[5],and we introduce the basic notations of evolution family and exponential dichotomy.Some preliminary results on intermediate spaces are also stated there.In Section 3,we study the existence and uniqueness ofμ-pseudo almost periodic mild solution of(1.1).

2 Preliminaries

2.1 μ-pseudo almost periodic functions

Let(X,||.||),(Y,||.||)be two Banach spaces,andBC(R,X)(respectively,BC(R×Y,X))be the space of bounded continuous functionsf：R-→X(respectively,f：R×Y-→X).BC(R,X)equipped with the normis a Banach space.B(X,Y)denotes the Banach spaces of all bounded linear operator from X into Y equipped with natural topology.If Y=X,B(X,Y)is simply denoted byB(X).

Definition 2.1.([6,7])A continuous function f：R7→Xis said to be almost periodic if for every ?＞0there exists a positive number l such that every interval of length l contains a number τ such that

The set of all almost periodic functions fromRtoXwill be denoted by a continuous function f：R×Y7→ Xis said to be almost periodic in t uniformly for y∈ Y,if for every ?＞ 0,and any compact subset K ofY,there exists a positive number l such that every interval of length l contains a number τ such that

We denote the set of such functions APU(R×Y;X).

Notice that(AP(R;X),‖.‖∞),is a Banach space with supremum norm given by

Next,we give the new concept of the ergodic functions developed in[5],and generalizing the ergodicity given before[12,13].

We denote by Bthe Lebesqueσ- field of R and by Mthe set of all positive measuresμon Bsatisfyingμ(R)=+∞andμ([a,b])＜∞,for alla,b∈R(a≤b).

Definition 2.2.([1,5])Letμ∈M.A bounded continuous function f：R→Xis said to beμ-ergodic if

We denote the space of all such functions byE(R;X,μ).

A continuous function f：R→Xis said to beμ-pseudo almost periodic if it is written in the form

where g∈AP(R;X)and h∈ E(R;X,μ).The collection of such functions will be denoted by PAP(R;X,μ).

It is well known[5]that(E(R;X,μ),‖.‖∞)is a Banach space.In the sequel as in[5],we need the following assumptions.

(M1)For alla,bandc∈R,such that 0≤a＜b≤c,there existτ0≥0 andα0≥0 such that

(M2)For allτ∈R,there existβ＞0 and a bounded intervalIsuch that

It is proved in[5]that wheneverμ∈Msatisfies the assumption(M1),the decomposition of aμ-pseudo almost periodic function in the formf=g+h,whereg∈AP(R;X)andh∈E(R;X,μ),is unique.Furthermore,the space(PAP(R;X,μ),‖.‖∞),is a Banach space.Wheneverμ∈Msatisfies the assumption(M2),PAP(R;X,μ)is translation invariant,that isf∈PAP(R;X,μ)impliesfτ=f(.+τ)∈PAP(R;X,μ)for allτ∈R.

Definition 2.3.([5])Letμ∈M.A continuous function f：R×Y→Xis said to beμ-ergodic in t uniformly with respect to y∈Yif the following conditions are true.

(i)For all y∈Y,f(.,y)∈E(R,X,μ).

(ii)f is uniformly continuous on each compact set K inYwith respect to the second variable y.The collection of such functions will be denoted byEU(R×Y;X,μ).

A continuous function f：R×Y→Xis said to be uniformlyμ-pseudo almost periodic if is written in the form

where g∈APU(R×Y;X)and h∈EU(R×Y;X,μ).The collection of such functions denoted by PAPU(R×Y;X,μ).

Theorem 2.1.([5])Letμ∈M,F∈PAPU(R×Y;X,μ)and h∈PAP(R;Y,μ).Assume that,for all bounded subset B ofY,F is bounded onR×B.Then t7-→F(t,h(t))∈PAP(R;X,μ).

2.2 Evolution family and exponential dichotomy

De fi nition 2.4.([8-10])A family of bounded linear operators(U(t,s))t≥s,on a Banach spaceXis called a strongly continuous evolution family if

(1)U(t,r)U(r,s)=U(t,s)and U(s,s)=I for all t≥r≥s and t,r,s∈R,

(2)The map(t,s)→U(t,s)x is continuous for all x∈X,t≥s and t,s∈R.

(3)U(·,s)∈C1((s,∞),B(X)),

for0＜t-s≤1,k=0,1.

(4)

A(t)is as in(1.1).

De fi nition 2.5.An evolution family(U(t,s))t≥son a Banach spaceXis called hyperbolic(or has exponential dichotomy)if there exist projections P(t),t∈R,uniformly bounded and strongly continuous in t,and constants M＞0,δ＞0such that

(1)U(t,s)P(s)=P(t)U(t,s)for t≥s and t,s∈R,

(2)The restriction UQ(t,s)：Q(s)X→Q(t)Xof U(t,s)is invertible for t≥s and t,s∈R(and we set UQ(t,s)=U(s,t)-1).

(3)

for t≥s and t,s∈R.

Here and below we set Q：=I-P.

To introduce the inter and extrapolation spaces forA(t),we need the following assumptions.

(H0)The family of closed linear operatorsA(t)fort∈R on X with domainD(A(t))(possibly not densely defined)satisfy the so-called Acquistapace-Terreni conditions,that is,there exist constantsω∈R,θ∈(,π),K,L≥0 andμ,ν∈(0,1]withμ+ν＞1 such that

and

fort,s∈R,λ∈Σθ：={λ∈C{0}：|argλ|≤θ}.

Note that in the particular case whenA(t)has a constant domainD=D(A(t)),it is wellknown[2]that equation(2.4)can be replaced with the following：There exist constantsLand 0＜γ≤1 such that

2.3 Interpolation Spaces

This setting requires some estimates related toU(t,s).For that,we make extensive use of the real interpolation spaces of order(α,∞)between X andD(A(t)),whereα∈(0,1).We refer the readerto[2-4]for proofs and further information on theses interpolation spaces.

LetAbe a sectorial operator on X(assumption(H0)holds whenA(t)is replaced withA)and letα∈(0,1).De fine the new norm onD(A)(the real interpolation space)by

and which consider the continuous interpolations spacesby the way,is a Banach space when endowed with the norm‖For convenience we further write

and‖x：=‖(ω-A)x‖.Moreover,letX.In particular,we will frequently use the following continuous embedding.

for all 0＜α＜β＜1,where the fractional powers are defined in the usual way.

In general,D(A)is not dense in the spacesand X.However,we have the following continuous injection

for 0＜α＜β＜1.

Given the family of linear operatorsA(t)fort∈R,satisfying(H1),we set

for 0≤α≤1 andt∈R,with the corresponding norms.Then the embedding in(2.7)hold with constants independent oft∈R.These interpolation spaces are of class Jα[[4],Definition 1.1.1]and hence there is a constantc(α)such that

We have the following fundamental estimates for the evolution familyU(t,s).

Proposition 2.1.([11])For x∈X,0≤α≤1and t＞s,the following assertions hold.

(i)There is a constant c(α),such that

(ii)There is a constant m(α),such that

3 Main results

To study the existence and uniqueness ofμ-pseudo almost periodic solutions of equation(1.1)we need the following additional assumptions.

(H1)The evolution family(U(t,s))t≥sgenerated byA(t)has an exponential dichotomy with constantsN＞0,δ＞0,dichotomy projectionsP(t),t∈R.

(H2)R(ω,A(·))∈AP(B(Xα)).Moreover,there exists a functionH：[0,∞)7→ [0,∞)withH∈L1[0,∞)such that for everyε＞ 0 there existsl(ε)such that every interval of lengthl(ε)contains aτwith the property

for allt,s∈R witht＞s.

(H3)There exists 0≤α＜β＜1 such that

for allt∈R,with uniform equivalent norm

If 0≤α＜β＜1,then we letk(α)denote the bound of the embedding Xβ?→Xα,that is

for eachu∈Xβ

(H4)Letμ∈Mand let 0＜α＜β＜1.We supposef：R×X7→Xβbelongs toPAP(X,Xβ,μ)and satisfy

i)For all bounded subsetBof X,fis bounded on R×B.

ii)There existsKf＞0 such that

for allu,v∈X andt∈R.

(H5)Letμ∈Mand let 0＜α＜β＜1.We supposeg：R×X7→X belongs toPAP(X,X,μ)

and satisfy

i)For all bounded subsetBof X,gis bounded on R×B.

ii)There existsKg＞0 such that

for allu,v∈X andt∈R.

(H6)We suppose that the linear operatorsB(t),C(t)：Xα7→X for allt∈R,are bounded and set

Furthermore,t7→B(t)uandt7→C(t)uare almost periodic for eachu∈Xα.

To study the existence and uniqueness of pseudo almost periodic solutions to equation(1.1),we fi rst introduce the notion of mild solution.

Definition 3.1.A function u：R7→Xαis said to be a mild solution to equation(1.1)provided that

the function s→A(s)U(t,s)P(s)f(s,B(s)u(s))is integrable on(s,t),s→A(s)U(t,s)Q(s)f(s,B(s)u(s))is integrable on(t,s)and

for t≥s and for all t,s∈R.

In a fi rst step,we proved the following result.

Theorem 3.1.Assume that assumptions(H0)-(H1)hold and let u be a mild solution of(1.1)onR.Then,for all t∈R

Proof.Letube the mild solution of(1.1)on R.For allt≥sand alls∈R,we have

Multiply both sides of the equality byU(t,s),we get

Henceuis a mild solution of equation(1.1).

Throughout the rest of the paper we denote by Γ1,Γ2,Γ3and Γ4,the nonlinear integral operators defined by

We next ned the following preliminary technical results.

Lemma 3.1.Letμ∈Msatisfying(M1)-(M2)and u∈PAP(R,Xα,μ),if the linear operators C(.)satisfy(H6)then C(.)u(.)∈PAP(R,X,μ).

Proof.Letu∈PAP(R,Xα,μ)thenu=u1+u2whereu1∈AP(R,Xα)andu2∈E(R,Xα,μ).We have,C(t)u(t)=C(t)u1(t)+C(t)u2(t)for allt∈R.Sinceu1∈AP(R,Xα),for every?＞0 there existsl?such that every interval of lengthl?contains aτsuch that

Similarly,sinceC(t)∈AP(B(Xα,X)),we have

Now

and hencet7→C(t)u1(t)belongs toAP(R,X).

To complete the proof,it suffices to prove thatt7→C(t)u2(t)belongs toE(R,X,μ).Indeed,we have

and hence

Lemma 3.2.([11])Assume that assumptions(H0)-(H1)and(H3hold and let0≤θ＜α＜β＜1with2α＞θ+1.Then,there exist two constants m(α,β),n(α,θ)＞0such that

Lemma 3.3.Let assumptions(H0)-(H4)and(H6)hold,then the integral operatorsΓ1andΓ2defined above map PAP(Xα,μ)into itself.

Proof.Letu∈PAP(Xα,μ).From Lemma 3.1 it follows that the functiont7→B(t)u(t)belongs toPAP(X).Using assumption(H4)and Theorem 2.1 it follows thatψ(·)=f(·,Bu(·))is inPAP(Xβ,μ)wheneveru∈PAP(Xα,μ).In particular,

Sinceψ(·)=f(·,Bu(·))is inPAP(Xβ,μ)thenψ=φ1+φ2,whereφ1∈AP(R,Xβ)andφ2∈E(R,Xβ,μ),that is,Γ1ψ=Ξ(φ1)+Ξ(φ2)where

and

Firstly,we show that Ξφ1∈BC(R,Xβ).Indeed,using estimate(3.3),we obtain

Then Ξφ1∈BC(R,Xβ).Next,we prove that Ξ(φ1)∈AP(R,Xα).Sinceφ1∈AP(R,Xβ),then for every?＞0 there existsl(?)＞0 such that every interval of lengthl(?)contains aτwith the property‖φ1(t+τ)-φ1(t)‖β＜?νfor eacht∈R

whereν=Hence,

Using equation(3.3)it follows that

Similarly,using assumption(H2),it follows that

whereTherefore,

for eacht∈R,and hence Ξ(φ1)∈AP(R,Xα).

Now,we show that Ξ(φ2)∈BC(R,Xβ).Using estimate(3.3)and replacing Ξ(φ1)by Ξ(φ2)in the previous case we get the result.To complete the proof,we will prove that Ξ(φ2)∈E(R,Xβ,μ).Now,letr＞0.Again from equation(3.3),we have

Sinceμsatisfy(M2)thent7→φ2(t-s)∈E(R,Xβ,μ)for everys∈R.To complete the proof,we use the well known Lebesgue’s dominated convergence theorem.

The proof for Γ2u(·)is similar to that of Γ1u(·)except that one makes use of equation(3.2)instead of equation(3.3).

Lemma 3.4.Letμ∈Msatisfying(M1)and(M2).Assume further that(H0)-(H3),(H5)and(H6)hold,then the integral operatorsΓ3andΓ4defined above map PAP(R,Xα,μ)into itself.

Proof.Letu∈PAP(R,Xα,μ).From Lemma 3.1 we getC(·)u(·)∈PAP(R,X,μ).Leth(t)=g(t,Cu(t)).Using assumption(H5 and Theorem 2.1 it follows thath∈PAP(R,X,μ).Now writeh=ψ1+ψ2whereψ1∈AP(R,X)andψ2∈E(R,X,μ),that is,Γ3h=Ξ(ψ1)+Ξ(ψ2)where

and

Firstly,we show that Ξψ1∈BC(R,Xβ).Indeed,using estimate(2.9),we obtain

Then Ξψ1∈BC(R,Xβ).Next,we prove that Ξ(ψ1)∈AP(R,Xα).Sinceψ1∈AP(R,Xβ),then for every?＞0 there existsl(?)＞0 such that every interval of lengthl(?)contains aτwith the property

whereUsing equation(2.9)it follows that

Similarly,using assumption(H2).Let?＞0,from[10]we know thatr→Γ(t+r,s+r)∈AP(B(X))fort,s∈R,where we may take the same almost periods fort,swith ‖t-s‖≤h＞0.Hence,there existsl(?)＞0 such that every interval of lengthl(?)contains a numberτ＞0 with the properties that,fort∈R,σ＞0：

and

Therefore,

for eacht∈R,and hence Ξ(ψ1)∈AP(R,Xα).Next,using similar techniquesas previously,we get Ξ(ψ2)∈BC(R,Xβ).In fact,using estimate(2.10)and replacing Ξ(ψ1)by Ξ(ψ2)we get the result.Now,to complete the proof,we will prove that Ξ(ψ2)∈E(R,Xβ,μ).Letr＞0.Again from equation(2.10),we have

Now observe that

Sinceμsatisfy(M2)thent7→ψ2(t-s)∈E(R,Xβ,μ)for everys∈R.Finally the proof is acheived using the as well the Lebesgue’s dominated convergence theorem.

The proof for Γ4u(·)is similar to that of Γ3u(·)except that one makes use of equation(2.9)instead of equation(2.10).

Now,we are able to state our second main result.

Theorem 3.2.Letμ∈Msatisfying(M1)and(M2).Assume further that assumptions(H0)-(H6)hold and that κ＜1.Then,the equation(1.1)has a uniqueμ-pseudo almost periodic mild solution,where

Proof.Consider the nonlinear operator M defined onPAP(Xα,μ)by

for allt∈R.Next,in view of Lemma 3.4 and Lemma 3.3,it follows that M mapsPAP(R,Xα,μ)into itself.To complete the proof one has to show that M is a contractive onPAP(R,Xα,μ).Letu,v∈PAP(R,Xα,μ).Firstly,we have

Next,we have

Now,we have

Finally,we have

Combining previous approximations it follows that

Then M is a contraction map onPAP(R,Xα,μ).Therefore,M has unique fixed point inPAP(R,Xα,μ),that is,there exist uniqueu∈PAP(R,Xα,μ)such that Mu=u.Therefore,(1.1),has a uniqueμ-pseudo almost periodic mild solution.

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