LEVITAN/BOHR ALMOST PERIODIC AND ALMOST
AUTOMORPHIC SOLUTIONS OF SECOND-ORDER MONOTONE
DIFFERENTIAL EQUATIONS
TOM´
AS CARABALLO AND DAVID CHEBAN
Abs ac . The aim o his pape is o p o e he exis ence o Le i an/Boh
almos pe iodic, almos au omo phic, ecu en and Poisson s able solu ions
o he second o de di e en ial equa ion
(1) x′′ = (σ( , y), x, x′),(y∈Y)
whe e Yis a comple e me ic space and (Y, R, σ) is a dynamical sys em (also
called a d i ing sys em). When he unc ion in (1) is inc easing wi h espec
o i s second a iable, he exis ence o a leas one quasi pe iodic ( espec i ely,
Boh almos pe iodic, almos au omo phic, ecu en , pseudo ecu en , Le i-
an almos pe iodic, almos ecu en , Poisson s able) solu ion o (1) is p o ed
unde he condi ion ha (1) admi s a leas one solu ion ϕsuch ha ϕand
ϕ′a e bounded on he eal axis.
1. In oduc ion
The aim o his pape is o analyze he exis ence o Le i an/Boh almos pe iodic,
almos au omo phic, ecu en and Poisson s able solu ions o he second o de
di e en ial equa ion
(2) x′′ = (σ( , y), x, x′),(y∈Y)
whe e Yis a comple e me ic space, and (Y, R, σ) is a (d i ing) dynamical sys em.
The exis ence o Boh almos pe iodic solu ions o equa ion
(3) x′′ = ( , x, x′)
wi h Boh almos pe iodic igh hand-side wi h espec o ime, uni o mly wi h
espec o he a iables x, x′on e e y compac subse in R2, was s udied by C.
Co duneanu in [15] (see also [1]), whe e i was es ablished ha , i ∂ ( ,x,u)
∂x ≥k > 0
o all ( , x, u)∈R3,equa ion (3) admi s a unique Boh almos pe iodic solu ion.
When he unc ion ( , x, u) is only inc easing (in he la ge sense), he same p oblem
was s udied by Z. Opial in [21], whe e he ollowing esul was es ablished.
Theo em 1.1. (Z. Opial [21]) Suppose ha he ollowing condi ions a e ul illed:
Da e: Oc obe 16, 2010.
1991 Ma hema ics Subjec Classi ica ion. p ima y:34C11, 34C30, 34C35, 34D45, 37C55,
37C60, 37C65, 37C70, 37C75.
Key wo ds and ph ases. Non-au onomous dynamical sys ems; skew-p oduc sys ems; cocycles;
quasi-pe iodic, Boh /Le i an almos pe iodic, almos au omo phic, pseudo- ecu en solu ions,
mono one second o de equa ion.
1
2 TOM´
AS CARABALLO AND DAVID CHEBAN
(i) ∈C(R3,R)and is inc easing in he la ge sense wi h espec o he a i-
able x, i.e., he inequali y x1≤x2implies ( , x1, u)≤ ( , x2, u) o all
u, ∈R;
(ii) o all > 0, he e exis s a numbe L( )>0such ha | ( , x1, u1)−
( , x2, u2)| ≤ L( )(|x1−x2|+|u1−u2|) o all |xi|,|ui| ≤ (i= 1,2) and
∈R.
Then, he ollowing s a emen s hold:
(i) I equa ion (3) admi s a solu ion usuch ha uand i s i s de i a i e u′a e
bounded on R, hen his equa ion admi s a leas one Boh almos pe iodic
solu ion.
(ii) I u( )and ( )a e wo Boh almos pe iodic solu ions o equa ion (3),
hen he e exis s a cons an c∈Rsuch ha u( )− ( ) = c o all ∈R.
(iii) I he unc ion is s ic ly inc easing wi h espec o he second a iable
x∈R, hen equa ion (3) admi s a mos one Boh almos pe iodic solu ion.
Some gene aliza ion o Theo em 1.1 when (3) is a ec o ial equa ion (i.e., ∈
C(Y×Rn×Rn,Rn) (n≥2) a e es ablished in [16] and [13].
In [12], P. Cieu a s udied he exis ence o bounded and Boh almos pe iodic
solu ions o he ollowing Li´ena d equa ion
(4) x′′ + (x)x′+g(x) = p( ),
whe e p:R→Ris a Boh almos pe iodic unc ion, (x)≥0 and gis a s ic ly
dec easing unc ion. Namely, i was p o ed in [12] ha e e y solu ion, which is
bounded on R+, is asymp o ically Boh almos pe iodic, and he e exis s a unique
Boh almos pe iodic solu ion o equa ion (4). A ypical model o such equa ion
(4) is
x′′ +cx′+ 1/xα=p( ),(x∈(0,+∞)),
whe e c≥0, α > 0 and pis Boh almos pe iodic.
Recen ly, he exis ence o almos au omo phic solu ions o equa ion (4) wi h almos
au omo phic o cing e m pwas s udied by Cieu a e al. in [14], whe e hey p o ed
he asymp o ically almos au omo phy o e e y solu ion which is bounded on R+,
and he exis ence o a unique almos au omo phic solu ion o equa ion (4).
In he pe iodic case (i.e. when pis pe iodic), he dynamics o equa ion (5) was
in ensi ely s udied by P. Ma ´ınez-Amo es and P. J. To es [20] and J. Campos
and P. J. To es [4].
Desheng Li and Jinqiao Duan [19] analyzed he s uc u e o he se o bounded
solu ions o equa ion (2). In pa icula , hey p o ed he exis ence o a unique
pe iodic ( espec i ely, quasi-pe iodic, Boh almos pe iodic) solu ion o equa ion
(2) i he poin y∈Yis also pe iodic ( espec i ely, quasi-pe iodic, Boh almos
pe iodic), and he unc ion is s ic ly inc easing wi h espec o i s second a iable.
Namely, he ollowing heo em was p o ed in [19].
Theo em 1.2. [19] Suppose ha he ollowing condi ions a e ul illed:
(i) (H, ρ)is a compac comple e me ic space and (H, R, θ)is a minimal dy-
namical sys em on H;
LEVITAN/BOHR ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS ... 3
(ii) :H×I×R→Ris a con inuous map, whe e I:= (a, b)⊆R;
(iii) Fo any compac subse V⊂(a, b)×R, he e exis s an L > 0such ha
| (h, x, p)− (h, y, q)| ≤ L(|x−y|+|p−q|) o all (x, p),(y, q)∈Vand
h∈H;
(i ) (h, x, p)is s ic ly inc easing in x;
( ) Fo any compac in e al I⊂I he e exis s c0>0such ha | (h, x, p)| ≤
c0(1 + |p|2) o all (h, x, p)∈H×I×R.
Then:
(i) The e exis s a con inuous map Γ : H7→ Rsuch ha o each h∈H,
γh( ) := Γ(θ( , h)) is he unique solu ion o equa ion
(5) x′′ = (θ( , h), x, x′)
which is bounded on R;
(ii) Fo each h∈H, he e exis s a con inuous dec easing unc ion Φhde-
ined on a maximal nonemp y open in e al D(Φh)⊂I, such ha o any
x∈D(Φh),x( ) := ψh( , x, Φh(x)) is he unique solu ion o (5), which
is bounded on R+, ha sa is ies x(0) = x, whe e ψ( , x, x′)deno es he
unique solu ion o equa ion (5) passing h ough he poin (x, x′)∈I×Ra
he ini ial momen = 0;
(iii) Fo any compac in e al D⊂D(Φh)
lim
→+∞(|ψh( , x, Φh(x)) −γh( )|+|ψ′
h( , x, Φh(x)) −γ′
h( )|) = 0
uni o mly wi h espec o x∈D.
We no e ha , in all o he p e iously ci ed wo ks (wi h he excep ion o [21]),
an assump ion o s ic mono ony is imposed. In he p esen pape , we conside
equa ion (2) when he unc ion is inc easing wi h espec o i s second a iable
in he la ge sense. All o ou esul s will be o mula ed and p o ed o his case
which includes, o cou se, he s ic ly inc easing one.
The pape is o ganized as ollows.
In Sec ion 2, we collec some no ions (quasi pe iodici y, Le i an/Boh almos pe i-
odici y, almos au omo phy, ecu ence, pseudo ecu ence, Poisson s abili y) ac s
and cons uc ions (Bebu o dynamical sys ems, skew-p oduc dynamical sys ems,
cocycles e c) om he heo y o dynamical sys ems which will be necessa y in his
pape .
Sec ion 3 is dedica ed o he s udy o a special class o non-au onomous dynamical
sys ems (NDS): he so-called NDS wi h con e gence. The main esul in his sec ion
is Theo em 3.9 which p o ides su icien condi ions o he con e gence o a NDS.
An applica ion o Theo em 3.9 o s udy he dynamics o he scala one-dimensional
equa ion x′= (σ( , y), x) (y∈Y) wi h pseudo ecu en base (Y, R, σ) (d i ing
sys em) is ca ied ou in Sec ion 4. The main esul o his sec ion is Theo em 4.2.
Le i an almos pe iodic and almos au omo phic solu ions o a second o de equa-
ion o he o m x′′ = (σ( , y), x, x′) and wi h inc easing (in he la ge sense)
4 TOM´
AS CARABALLO AND DAVID CHEBAN
a e analyzed in Sec ion 5. The main esul s o his sec ion a e Theo em 5.4 and
Co olla y 5.5.
Sec ion 6 is de o ed o he exis ence o quasi-pe iodic, Boh almos pe iodic and
ecu en solu ions (in he sense o Bi kho ) o he equa ion x′′ = (σ( , y), x, x′)
wi h inc easing (in he la ge sense). The main esul s p o ed in his sec ion a e
Theo em 6.1 and Co olla y 6.2.
Finally, in Sec ion 7, we discuss some gene aliza ions o ou main esul s ( heo ems
5.4 and 6.1). One o his ype o esul s is es ablished in Theo em 7.1 (see also
co olla ies 7.2 and 7.3).
2. Boh /Le i an Almos Pe iodic and Almos Au omo phic Mo ions
o Dynamical Sys ems
We ecall now some no ions, ac s and cons uc ions om he heo y o dynamical
sys ems.
Al hough we could e e he eade s o o he publica ions o hese p elimina ies
(see, o ins ance, Ca aballo and Cheban [5, 6]), in o de o keep ou pape as much
sel -con ained as possible, we p e e o include he esul s he e.
2.1. Recu en , Boh Almos Pe iodic and Almos Au omo phic Mo-
ions. Le (X, ρ) be a comple e me ic space, Sbe one o he wo se s Ro
Z, and T⊆S(S+⊆T) be a sub-semig oup o he addi i e g oup S, whe e
S+:= {s∈S:s≥0}.
Le (X, T, π) be a dynamical sys em on X, i.e., le π:T×X→Xbe a con inuous
unc ion such ha π(0, x) = x o all x∈X, and π( 1+ 2, x) = π( 2, π( 1, x)), o
all x∈X, and 1, 2∈T.
Gi en ε > 0,a numbe τ∈Tis called an ε−shi ( espec i ely, an ε−almos pe iod)
o x, i ρ(π(τ, x), x)< ε ( espec i ely, ρ(π(τ+ , x), π( , x)) < ε o all ∈T).
A poin x∈Xis called almos ecu en ( espec i ely, Boh almos pe iodic), i
o any ε > 0 he e exis s a posi i e numbe lsuch ha in any segmen o leng h l
he e is an ε−shi ( espec i ely, an ε−almos pe iod) o he poin x∈X.
I he poin x∈Xis almos ecu en and he se H(x) := {π( , x)| ∈T}is
compac , hen xis called ecu en , whe e he ba deno es he closu e in X.
Deno e by Nx:= {{ n} ⊂ T: such ha {π( n, x)} → xand { n} → ∞} and
Mx:= {{ n} ⊂ T: such ha {π( n, x)}is con e gen and { n} → ∞}.
A poin x∈Xis called Poisson s able in he posi i e di ec ion i he e exis s a
sequence { n} ∈ Nxsuch ha n→+∞as n→ ∞.
Le (X, T, π) be a wo-sided dynamical sys em (i.e., T=S). A poin x∈Xis
called Poisson s able in he nega i e di ec ion i he e exis s a sequence { n} ∈ Nx
such ha n→ −∞ as n→ ∞. The poin x∈Xis called Poisson s able i i is
Poisson s able in bo h di ec ions.
LEVITAN/BOHR ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS ... 5
A dynamical sys em (X, T, π) is said o be
(i) ansi i e, i he e exis s a poin x0∈Xsuch ha H(x0) = X, whe e
H(x0) := {π( , x0) : ∈T};
(ii) pseudo ecu en i Xis compac , he dynamical sys em (X, T, π) is an-
si i e, and e e y poin x∈Xis Poisson s able.
A poin x∈Xis called [26, 28] pseudo ecu en i he dynamical sys em (H(x),T, π)
is pseudo ecu en .
Rema k 2.1. E e y ecu en poin is pseudo ecu en , bu he e exis pseudo
ecu en poin s which a e no ecu en [26, 28].
An m-dimensional o us is deno ed by Tm:= Rm/2πZ.Le (Tm,T, σ) be an i a-
ional winding o Tm, i.e., σ( , ν) := (ν1 , ν2 ,...,νm ) o all ∈Sand ν∈ T m.
A poin x∈Xis called quasi-pe iodic wi h he equency ν:= (ν1, ν2,...,νm)∈
Tm, i he e exis s a con inuous unc ion Φ : Tm→Xsuch ha π( , x) :=
Φ(σ( , ω)) o all ∈T,whe e (Tm,T, σ) is an i a ional winding o he o us
Tmand ω∈ T m.
A poin x∈Xo he dynamical sys em (X, T, π) is called Le i an almos pe iodic
[18], i he e exis s a dynamical sys em (Y, T, σ) and a Boh almos pe iodic poin
y∈Ysuch ha Ny⊆Nx.
Rema k 2.2. Le xi∈Xi(i= 1,2,...,m) be a Le i an almos pe iodic poin o
he dynamical sys em (Xi,T, πi).Then he poin x:= (x1, x2,...,xm)∈X:=
X1×X2×... ×Xmis also Le i an almos pe iodic in he p oduc dynamical
sys em (X, T, π),whe e π:T×X→Xis de ined by he equali y π( , x) :=
(π1( , x1), π2( , x2), . . . , πm( , xm)) o all ∈Tand x:= (x1, x2,...,xm)∈X.
A poin x∈Xis called s able in he sense o Lag ange (s .L) ( espec i ely, s able in
he sense o Lag ange in he posi i e di ec ion (s .L+)), i i s ajec o y {π( , x) :
∈T}( espec i ely, i s posi i e semi- ajec o y {π( , x) : ∈T+}) is ela i ely
compac , whe e T+; = { ∈T: ≥0}.
A poin x∈Xis called almos au omo phic [18, 24] in he dynamical sys em
(X, T, π),i he ollowing condi ions hold:
(i) xis s .L;
(ii) he e exis s a dynamical sys em (Y, T, σ),a homomo phism h om (X, T, π)
on o (Y, T, σ), and a poin y∈Ywhich is almos pe iodic, in he sense o
Boh , such ha h−1(y) = {x}.
Rema k 2.3. No ice he ollowing well-known ac s.
1. E e y almos au omo phic poin is Le i an almos pe iodic.
2. A Le i an almos pe iodic poin is almos au omo phic i and only i is s able in
he sense o Lag ange.
2.2. Shi Dynamical Sys ems, Le i an/Boh Almos Pe iodic and Al-
mos Au omo phic Func ions. Below we ecall a gene al me hod o cons uc-
ion o dynamical sys ems on spaces o con inuous unc ions. In his way, we will

6 TOM´
AS CARABALLO AND DAVID CHEBAN
ob ain many well-known dynamical sys ems on some unc ional spaces (see, o
example, [2, 23, 26]).
Le (X, T, π) be a dynamical sys em on X, Y a comple e pseudo me ic space, and P
a amily o pseudo me ics on Y. We deno e by C(X, Y ) he amily o all con inuous
unc ions :X→Yequipped wi h he compac -open opology. This opology is
gi en by he ollowing amily o pseudo me ics {dp
K}(p∈ P, K ∈ C(X)),whe e
dp
K( , g) := sup
x∈K
p( (x), g(x))
and C(X) deno es he amily o all compac subse s o X. Fo all τ∈Twe
de ine a mapping στ:C(X, Y )→C(X, Y ) by he ollowing equali y: (στ )(x) :=
(π(τ, x)), x ∈X. We no e ha he amily o mappings {στ:τ∈T}possesses
he nex p ope ies:
a. σ0=idC(X,Y );
b. στ1◦στ2=στ1+τ2, o all τ1, τ2∈T;
c. στis con inuous o all τ∈T.
Lemma 2.4. [7] The mapping σ:T×C(X, Y )→C(X, Y ),de ined by he equali y
σ(τ, ) := στ ( ∈C(X, Y ), τ ∈T),is con inuous, and he iple (C(X, Y ),T, σ)
is a dynamical sys em on C(X, Y ).
Conside now some examples o dynamical sys ems o he o m (C(X, Y ),T, σ),
which a e use ul in he applica ions.
Example 2.5. Le X=T, and deno e by (X, T, π) a dynamical sys em on T,
whe e π( , x) := x+ . The dynamical sys em (C(T, Y ),T, σ) is called Bebu o ’s
dynamical sys em [2, 23, 26] (dynamical sys em o ansla ions, o shi s dynamical
sys em).
I is said ha he unc ion ϕ∈C(T, Y )possesses a p ope y (A), i he mo ion
σ(·, ϕ) : T→C(T, Y ), gene a ed by his unc ion, possesses his p ope y in he
Bebu o dynamical sys em (C(T, Y ),T, σ). As p ope y (A) we can ake pe iodici y,
quasi-pe iodici y, Boh /Le i an almos pe iodici y, almos au omo phy, ecu ence,
pseudo ecu ence, Poisson s abili y, e c.
Example 2.6. Le X:= T×W, whe e Wis a me ic space, and le (X, T, π)
deno e a dynamical sys em on Xde ined in he ollowing way: π( , (s, w)) :=
(s+ , w). Using he gene al me hod p oposed abo e, we can de ine on C(T×W, Y )
a dynamical sys em o ansla ions (C(T×W, Y ),T, σ).
The unc ion ∈C(T×W, Y ) is called Boh /Le i an almos pe iodic (quasi-
pe iodic, ecu en , almos au omo phic, e c) wi h espec o ∈T, uni o mly in
won e e y compac om W, i he mo ion σ(·, ) is Boh /Le i an almos pe i-
odic (quasi-pe iodic, ecu en , almos au omo phic, e c.) in he dynamical sys em
(C(T×W, Y ),T, σ).
Rema k 2.7. Recall ha o a compac me ic space W, he opology on C(T×
W, Y )is me izable. Fo example, he equali y
d( , g) :=
∞
X
k=1
1
2k
dk( , g)
1 + dk( , g)
LEVITAN/BOHR ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS ... 7
de ines a comple e me ic on he space C(T×W, X)which is compa ible wi h he
compac -open opology on C(T×W, X), whe e dk( , g) := max
| |≤k, x∈Wρ( ( , x), g( , x)).
The space C(T×W, Y )is opologically isomo phic o C(T, C(W, Y )) (see [26]), and
also he shi s dynamical sys ems (C(T×W, Y ),T, σ)and (C(T, C(W, Y )),T, σ)a e
dynamically isomo phic.
2.3. Cocycles, Skew-P oduc Dynamical Sys ems and Non-Au onomous
Dynamical Sys ems. Le T1⊆T2be wo sub-semig oups o he g oup S(S+⊆
T1).
A iple h(X, T1, π),(Y, T2, σ), hi, whe e his a homomo phism om (X, T1, π)
on o (Y, T2, σ) (i.e., his con inuous and h(π( , x)) = σ( , h(x)) o all ∈T1and
x∈X), is called a non-au onomous dynamical sys em.
Le (Y, T2, σ) be a dynamical sys em, Wa comple e me ic space, and ϕa con in-
uous mapping om T1×W×Yin o W, possessing he ollowing p ope ies:
a. ϕ(0, u, y) = u(u∈W, y ∈Y);
b. ϕ( +τ, u, y) = ϕ(τ, ϕ( , u, y), σ( , y)) ( , τ ∈T1, u ∈W, y ∈Y).
Then, he iple hW, ϕ, (Y, T2, σ)i(o sho ly ϕ) is called [23] a cocycle on (Y, T2, σ)
wi h ibe W.
Le X:= W×Yand le us de ine a mapping π:X×T1→Xas ollows:
π((u, y), ) := (ϕ( , u, y), σ( , y)) (i.e., π= (ϕ, σ)). Then, i is easy o see ha
(X, T1, π) is a dynamical sys em on X, which is called a skew-p oduc dynamical
sys em [23] and h=p 2:X→Yis a homomo phism om (X, T1, π) on o (Y, T2, σ)
and, hence, h(X, T1, π),(Y, T2, σ), hiis a non-au onomous dynamical sys em.
Thus, i we ha e a cocycle hW, ϕ, (Y, T2, σ)ion he dynamical sys em (Y, T2, σ)
wi h ibe W, hen i gene a es a non-au onomous dynamical sys em h(X, T1, π),
(Y, T2, σ), hi(X:= W×Y), called a non-au onomous dynamical sys em gene a ed
by he cocycle hW, ϕ, (Y, T2, σ)ion (Y, T2, σ).
Non-au onomous dynamical sys ems (cocycles) play a e y impo an ole in he
s udy o non-au onomous e olu iona y di e en ial equa ions. Unde app op ia e
assump ions, e e y non-au onomous di e en ial equa ion gene a es a cocycle (a
non-au onomous dynamical sys em). Below we gi e some examples o his ype.
Example 2.8. Conside he sys em o di e en ial equa ions
(6) u′=F(y, u)
y′=G(y),
whe e Y⊆Em( o example, Y=Tmis an m– o us), G∈C(Y, En) and F∈
C(Y×En, En). Suppose ha , o he sys em (6), he condi ions ensu ing exis ence,
uniqueness and ex endabili y o solu ions o R+a e ul illed. Deno e by (Y, R+, σ)
a dynamical sys em on Ygene a ed by he second equa ion o he sys em (6) and
by ϕ( , u, y) we deno e he solu ion o he equa ion
u′=F(σ( , y), u)
passing h ough he poin u∈Ena = 0. Then, he mapping ϕ:R+×En×Y→
Ensa is ies condi ions a. and b. om he de ini ion o cocycle and, consequen ly,
8 TOM´
AS CARABALLO AND DAVID CHEBAN
sys em (6) gene a es a non-au onomous dynamical sys em h(X, R+, π),(Y, R+, σ), hi
(whe e X:= En×Y,π:= (ϕ, σ) and h:= p 2:X→Y).
Example 2.9. Le (Y, R, σ) be a dynamical sys em on he me ic space Y. We
conside he equa ion
(7) u′=F(σ(y, ), u) (y∈Y),
whe e F∈C(Y×Rn,Rn). Suppose again ha , o equa ion (7), he condi ions
o he exis ence, uniqueness and ex endabili y o solu ions o R+a e ul illed.
The non-au onomous dynamical sys em h(X, R+, π),(Y, R, σ), hi( espec i ely, he
cocycle hE, ϕ, (Y, R, σ)i), whe e X:= Rn×Y,π:= (ϕ, σ), ϕ(·, x, y) is he solu ion
o (7) passing h ough he poin xa ime = 0, and h:= p 2:X→Yis gene a ed
by equa ion (7).
Example 2.10. We conside he equa ion
(8) u′= ( , u),
whe e ∈C(R×Rn,Rn). Along wi h equa ion (8), conside he amily o equa ions
(9) u′=g( , u),
whe e g∈H( ) := { τ:τ∈R}and τis he τ-shi o wi h espec o he ime
a iable , i.e., τ( , u) := ( +τ, u) o all ( , u)∈R×Rn. Suppose ha he unc ion
is egula [23], i.e., o all g∈H( ) and u∈Rn he e exis s a unique solu ion
ϕ( , u, g) o equa ion (9). Deno e by Y=H( ) and (Y, R, σ) a shi dynamical
sys em on Yinduced by he Bebu o dynamical sys em (C(R×Rn,Rn),R, σ).
Now he amily o equa ions (9) can be w i en as (7) i we ake he mapping
F∈C(Y×Rn,Rn) de ined by F(g, u) := g(0, u), o all g∈H( ) and u∈Rn.
A solu ion ϕ( , u, y) o equa ion (7) is called [26, 28] compa ible ( espec i ely, uni-
o mly compa ible) by he cha ac e o ecu ence i Ny⊆Nϕ( espec i ely, My⊆
Mϕ), whe e Nϕ( espec i ely, Mϕ) is he se o all sequences { n} ⊂ Rsuch ha
{ϕ( + n, u, y}con e ges o ϕ( , u, y) ( espec i ely, {ϕ( + n, u, y}con e ges) in
he space C(T,Rn).
Rema k 2.11. The sequence {ϕ( + n, u, y)}con e ges o he unc ion ψin he
space C(T,Rn)i and only i {ϕ( n, u, y)}con e ges o ψ(0).
Theo em 2.12. [26, 28] The ollowing s a emen s hold:
1. Le y∈Ybe a s a iona y ( espec i ely, τ-pe iodic, Le i an almos pe iodic,
almos ecu en , Poisson s able) poin . I ϕ( , u, y)is a compa ible solu ion o
equa ion (7), hen so is ϕ( , u, y).
2. Le y∈Ybe a s a iona y ( espec i ely, τ-pe iodic, Boh almos pe iodic, al-
mos au omo phic, ecu en , pseudo ecu en ) poin . I ϕ( , u, y)is a uni o mly
compa ible solu ion o equa ion (7), hen so is ϕ( , u, y).
Example 2.13. Le us conside a second o de di e en ial equa ion
(10) x′′ = (σ( , y), x, x′),(y∈Y)
whe e ∈C(Y×Rn×Rn,Rn), and s a e a c i e ion o he exis ence o Le i an
almos pe iodic and almos au omo phic solu ions o his equa ion. Below we will
suppose ha he unc ion is egula , i.e., o all y∈Yand x, x′∈Rn he equa ion
LEVITAN/BOHR ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS ... 9
(10) admi s a unique solu ion ϕ( , x, x′, y) de ined on R+wi h he ini ial condi ions
ϕ(0, x, x′, y) = xand ϕ′(0, x, x′, y) = x′.
As i is well-known, we can educe equa ion (10) o he ollowing equi alen sys em
(11) u′=
′= (σ( , y), u, ),
(y∈Y) o o he equa ion
z′=F(σ( , y), z)
on he p oduc space Rn×Rn,whe e z:= (u, ) and F∈C(Y×Rn×Rn,Rn×Rn)
is he unc ion de ined by he equali y F(y, z) := ( , (y, u, )) o all y∈Yand
z:= (u, )∈Rn×Rn.
Theo em 2.14. [26] Le ϕ∈C(R,Rn)be a con inuously di e en iable unc ion.
I i s de i a i e ϕ′∈C(R,Rn)is uni o mly con inuous on R, hen ϕ′is uni o mly
compa able by he cha ac e o ecu ence wi h ϕ, i.e., Mϕ⊆Mϕ′.
We can now p o e he ollowing esul .
Lemma 2.15. Suppose ha he ollowing condi ions hold:
(i) Yis compac ;
(ii) ∈C(Y×Rn×Rn,Rn)is egula ;
(iii) ϕ( , x0, x′
0, y)is a solu ion o equa ion (10) de ined and bounded on Rand
such ha i s de i a i e ϕ′( , x0, x′
0, y)is also bounded on R.
Then, he ollowing wo s a emen s a e equi alen :
a. The solu ion ϕ( , x0, x′
0, y)o equa ion (10) is compa ible ( espec i ely, uni-
o mly compa ible) by he cha ac e o ecu ence wi h he igh -hand side;
b. The solu ion (ϕ( , x0, x′
0, y), ϕ′( , x0, x′
0, y)) o equa ion (11) is compa ible
( espec i ely, uni o mly compa ible) by he cha ac e o ecu ence wi h he
igh -hand side.
P oo . The implica ion b.=⇒a. is e iden . Thus, o p o e he lemma, i is su -
icien o es ablish he con e se implica ion. Le ϕ( , x0, x′
0, y) be a solu ion o
equa ion (10) such ha ϕ( , x0, x′
0, y) and ϕ′( , x0, x′
0, y) a e de ined and bounded
on R. Then Ny⊆Nϕ( espec i ely, My⊆Mϕ). We need o show ha he in-
clusion Ny⊆Nϕ′( espec i ely, My⊆Mϕ′) also holds. Indeed, le { n} ∈ Ny
( espec i ely, { n} ∈ My), hen he sequence {σ( n, y)}con e ges o y( espec-
i ely, he sequence {σ( n, y)}con e ges o some poin ˜y∈Y). Consequen ly, he
unc ional sequence { (σ( + n, y), u, )}con e ges o (σ( , y), u, ) ( espec i ely,
o (σ( , ˜y), u, )) uni o mly wi h espec o on e e y compac subse om R
and u, ∈Q:= ϕ(R, x0, x′
0, y)×ϕ′(R, x0, x′
0, y). Since Ny⊆Nϕ( espec i ely,
My⊆Mϕ), he sequence {ϕ( n, x0, x′
0, y)}con e ges o x0( espec i ely, o some
poin ˜x0∈R). Since he unc ion ∈C(Y×Rn×Rn,Rn) is egula , hen he
unc ional sequence {ϕ( + n, x0, x′
0, y)}con e ges o he unc ion ϕ( , x0, x′
0, y)
( espec i ely, o ϕ( , ˜x0,˜x0′,˜y)) uni o mly wi h espec o on e e y compac sub-
se om R. No e ha , unde he condi ions o Lemma, he second de i a i e
ϕ′′( , x0, x′
0, y) o he unc ion ϕ( , x0, x′
0, y) is bounded on Rand, consequen ly,
he i s de i a i e ϕ′( , x0, x′
0, y) is uni o mly con inuous in ∈R. Thus, acco ding
16 TOM´
AS CARABALLO AND DAVID CHEBAN
I (16) admi s a solu ion which is bounded on R+, hen i possesses a unique s a-
iona y ( espec i ely, τ–pe iodic, quasi-pe iodic, Boh almos pe iodic, almos au o-
mo phic, ecu en , pseudo ecu en ) solu ion which is globally uni o mly asymp-
o ically s able.
P oo . This s a emen ollows di ec ly om Theo em 4.2 and Rema k 3.1 (i em 2
(ii)). 
Rema k 4.4. 1. The analog o Theo em 4.2 (as well as Co olla y 4.3) holds i we
eplace he condi ion “ is s ic ly dec easing” by “ is s ic ly inc easing”. This
case can be educed o he conside ed one by he ime subs i u ion → − .
2. No e ha Theo em 4.2 and Co olla y 4.3 emain ue also o a ec o ial equa ion
(i.e., o sys ems o equa ions). Indeed, o his end, we assume ha ∈C(Y×
Rn,Rn), and eplace he condi ion “ is s ic ly dec easing” by he condi ion
h (y, u1)− (y, u2), u1−u2i<0
o all y∈Yand u1, u2∈Rn(u16=u2), whe e h,iis he scala p oduc on he
space Rn.
3. I he unc ion ∈C(Y×R,R)is con inuously di e en iable wi h espec o
x∈Rand
(18) ∂
∂x(y, x)≤ −k < 0
o all y∈Yand x∈R, hen Theo em 4.2 and Co olla y 4.3 also hold wi hou
he equi emen ha equa ion (16) admi s a leas one solu ion which is bounded on
R+. Owing o condi ion (18), i ollows
(19) h (y, u1)− (y, u2), u1−u2i ≤ −k|u1−u2|2
o all y∈Yand u1, u2∈R. Bu condi ion (19) gua an ees (see [11]) ha equa ion
(16) is con e gen .
4. We plan o s udy in mo e de ails he mul i-dimensional case in one o ou nex
publica ions.
5. Le i an almos pe iodic and almos au omo phic solu ions o
second o de di e en ial equa ions
In his sec ion we conside a scala di e en ial equa ion o he ype (10), i.e., n= 1.
In he sequel, we suppose ha he unc ion ∈C(Y×R2,R) is egula and
inc easing (in he la ge sense) wi h espec o he a iable x, i.e., i u1≤u2 hen
(y, u1, )≤ (y, u2, ) o all y∈Yand ∈R.
Lemma 5.1. [22] Le u( ), ( )be wo solu ions o equa ion (10) de ined on R.
Then, only one o he ollowing h ee cases is possible:
(i) The unc ion u( )− ( )is mono one on he eal axis R;
(ii) u( )− ( )is posi i e on R, and he e exis s a numbe 0∈Rsuch ha his
unc ion is non-dec easing on he in e al ( 0,+∞), and non-inc easing on
(−∞, 0);

LEVITAN/BOHR ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS ... 17
(iii) The unc ion u( )− ( )is nega i e on R, and he e exis s a numbe 0∈R
such ha i is non-inc easing on he in e al ( 0,+∞), and non-dec easing
on (−∞, 0).
Le ϕ∈C(R,R), and deno e by aϕ:= in {ϕ( )| ∈R}and bϕ:= sup{ϕ( )| ∈R}.
Rema k 5.2. No ice ha he ollowing ac s ake place:
1. aϕ≤bϕ o all ϕ∈C(R,R).
2. The inequali ies
(20) aϕ≤aψ≤bψ≤bϕ
hold o all ψ∈H(ϕ).
3. I he unc ion ϕis ecu en , hen
aϕ=aψand bψ=bϕ
o all ψ∈H(ϕ).
Theo em 5.3. [25, 29] Le ∈C(R×Rn,Rn)be Poisson s able wi h espec o
he ime a iable ∈R. I he equa ion
x′= ( , x)
admi s a solu ion ϕwhich is bounded on R, hen i admi s a leas a Poisson s able
solu ion ψ∈H(ϕ).
Le us now es ablish ou i s main esul in his sec ion.
Theo em 5.4. Suppose ha ∈C(Y×R2,R)is egula and inc easing (in he
la ge sense) wi h espec o he a iable x∈R, and assume ha he poin y∈Yis
Poisson s able. Then, he ollowing s a emen s hold:
(i) I (10) admi s a solu ion φsuch ha φand φ′a e bounded on R, hen i has
a leas one compa ible (by he cha ac e o ecu ence wi h he igh -hand
side) solu ion;
(ii) I u( )and ( )a e wo compa ible solu ions o equa ion (10), hen u( )−
( ) = c o all ∈R, whe e c∈Ris some cons an ;
(iii) I he unc ion is s ic ly inc easing wi h espec o he a iable x∈R,
hen equa ion (10) admi s a mos one compa ible solu ion which is bounded
on R.
P oo . Le φ∈C(R,R) be a solu ion o equa ion (10) such ha φand φ′a e
bounded on R. To p o e he i s s a emen , on accoun o Lemma 2.15, i is
su icien o show ha he unc ion φis compa able wi h yby he cha ac e o
ecu ence, i.e., he unc ional sequence {φ( + n)}con e ges o φ( ) uni o mly
on e e y compac subse om R, o e e y sequence { n} ∈ Ny. Conside he
mo ion σ( , φ) in he shi dynamical sys em (Bebu o ’s sys em) (C(R,R),R, σ).
Acco ding o Theo em 5.3, he se H(φ) := {σ(τ, φ)|τ∈R}con ains a leas one
Poisson s able solu ion ϕ∈H(φ) o equa ion (10) (in ac , he unc ion ϕand
he poin ya e join ly Poisson s able) . We will p o e ha he solu ion ϕis
compa ible. To his end, we will show ha equa ion (10) possesses a mos one
18 TOM´
AS CARABALLO AND DAVID CHEBAN
solu ion om H(ϕ)⊆H(φ). Indeed, i ψ∈H(ϕ) is a solu ion o equa ion (10) and
( ) := ψ( )−ϕ( ) o all ∈R, hen, by Lemma 5.1, he e exis he limi s
lim
→+∞ ( ) = c+,lim
→−∞ ( ) = c−
and
|c+|+|c−|>0.
Suppose, o example, ha c+>0. Then, by he join Poisson s abili y o he poin
yand he solu ion ϕ, he e exis s a sequence { n} ∈ Ny∩Nϕsuch ha n→+∞as
n→ ∞. Wi hou loss o gene ali y, we can suppose ha he sequence {ψ( + n)}
is con e gen in he space C(R,R). Le ¯
ψbe i s limi , i.e., ¯
ψ( ) = lim
→+∞ψ( + n).
Then,
(21) ¯
ψ( ) = ϕ( ) + c+ o all ∈R.
F om (20) and he ac ha ¯
ψ∈H(ψ)⊆H(ϕ), we ha e
(22) aϕ≤aψ≤a¯
ψ≤b¯
ψ≤bψ≤bϕ.
On he o he hand, om (21) we ha e b¯
ψ=bϕ+c+. F om he las equali y and
(22) we ob ain c+≤0.This con adic ion p o es ou s a emen . The o he cases
can be ea ed simila ly.
Le now u( ) and ( ) be wo compa ible solu ions o equa ion (10). Then, hanks o
Lemma 5.1, he e exis s a numbe 0∈Rsuch ha he unc ion ( ) := u( )− ( ) is
mono one on one o he wo in e als: (−∞, 0) o ( 0,+∞). Conside , o example,
he case when ( ) is mono one on he in e al (−∞, 0). Since he solu ions uand
a e compa ible, and he poin yis Poisson s able, he unc ion ( ) is Poisson
s able oo. In pa icula , i is Poisson s able in he nega i e di ec ion. On he
o he hand, his unc ion is mono one on he in e al (−∞, 0) and, consequen ly,
is a cons an . Thus u( )− ( ) = c o all ∈R, whe e c∈Ris some cons an .
Finally, we p o e he hi d s a emen o ou heo em. Suppose ha he unc ion
is s ic ly inc easing wi h espec o he a iable x∈R. I we suppose ha
equa ion (10) admi s wo di e en solu ions uand which a e bounded on R, hen
he unc ion
(23) ( ) := u( )− ( ) ( ∈R)
possesses he limi s c±:= lim
→±∞ ( ) and |c−|+|c+|>0. Suppose, o example,
ha c+>0. Then, we ake a sequence { n} ∈ Nysuch ha n→+∞and he
unc ional sequences {u( + n)}and { ( + n)}a e con e gen (since he unc ions
uand a e solu ions o (10) which a e bounded on R. Deno e by ¯u( espec i ely,
¯ ) he limi o he sequence {u( + n)}( espec i ely, { ( + n)})). F om equali y
(23) we ha e
¯u( ) := ¯ ( ) + c+ o all ∈R
and, consequen ly, we ob ain (σ( , y),¯ ( ),¯ ′( )) = (σ( , y),¯ ( ) + c+,¯ ′( )) o
all ∈R. The las iden i y con adic s he s ic mono ony o he unc ion wi h
espec o he a iable x. This con adic ion comple es he p oo . 
As a consequence o Theo em 5.4 and Theo em 2.12, we ha e he second main
esul in his sec ion.
LEVITAN/BOHR ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS ... 19
Co olla y 5.5. Suppose ha ∈C(Y×R2,R)is egula and mono one inc easing
(in he la ge sense) wi h espec o he a iable x∈R, and he poin y∈Yis
s a iona y ( espec i ely, τ–pe iodic, Le i an almos pe iodic, almos au omo phic,
almos ecu en , Poisson s able). Then, he ollowing s a emen s hold:
(i) I (10) admi s a solu ion φsuch ha φand φ′a e bounded on R, hen i has
a leas one s a iona y ( espec i ely, τ–pe iodic, Le i an almos pe iodic,
almos au omo phic, almos ecu en , Poisson s able) solu ion;
(ii) I u( )and ( )a e wo s a iona y ( espec i ely, τ–pe iodic, Le i an almos
pe iodic, almos au omo phic, almos ecu en , Poisson s able) solu ions
o equa ion (10), hen u( )− ( ) = c o all ∈R, whe e c∈Ris some
cons an ;
(iii) I he unc ion is s ic ly inc easing wi h espec o he a iable x∈R,
hen equa ion (10) admi s a mos one s a iona y solu ion ( espec i ely, τ–
pe iodic, Le i an almos pe iodic, almos au omo phic, almos ecu en ,
Poisson s able) which is bounded on R.
6. Quasi-pe iodic, Boh almos pe iodic, almos au omo phic and
ecu en solu ions
In his sec ion we analyze p oblem (10) in he scala case, i.e., n= 1, and suppose
ha Yis compac and (Y, R, σ) is a minimal dynamical sys em, i.e., Ydoes no
con ain a p ope compac in a ian subse .
Ou main esul below ensu es ha compa ibili y is now uni o m.
Theo em 6.1. Suppose ha ∈C(Y×R2,R)is egula and mono one inc eas-
ing (in he la ge sense) wi h espec o he a iable x∈R. Then, he ollowing
s a emen s hold:
(i) I (10) admi s a solu ion ϕsuch ha ϕand ϕ′a e bounded on R hen i
has a leas one uni o mly compa ible (by he cha ac e o ecu ence wi h
he igh -hand side) solu ion;
(ii) I u( )and ( )a e wo uni o mly compa ible solu ions o equa ion (10),
hen u( )− ( ) = c o all ∈R, whe e c∈Ris some cons an ;
(iii) I he unc ion is s ic ly inc easing wi h espec o he a iable x∈R,
hen equa ion (10) admi s a mos one uni o mly compa ible solu ion.
P oo . Le ϕ∈C(R,R) be a solu ion such ha ϕand a φ′a e bounded on
R. To p o e he i s s a emen , aking in o accoun Lemma 2.15, i is su icien
o show ha ϕis uni o mly compa able wi h yby he cha ac e o ecu ence,
i.e., he unc ional sequence {ϕ( + n)}is con e gen uni o mly on e e y compac
subse om R, o e e y sequence { n} ∈ My. Deno e by X:= C(R,R)×Y
and (X, R, π) he p oduc dynamical sys em, i.e., π(τ, (ϕ, y)) := (ϕτ, σ(τ, y)) o
all (ϕ, y)∈C(R,R)×Yand τ∈R, whe e ϕτis a τ–shi o he unc ion ϕ
(ϕτ( ) := ϕ( +τ) o all ∈R). Conside he mo ion π( , (ϕ, y)) in he p oduc
dynamical sys em (X, R, π). Unde he condi ions o ou heo em, his mo ion
is s able in he sense o Lag ange, i.e., he se H(ϕ, y) := {π(τ, (ϕ, y))|τ∈R}
is compac . Acco ding o Bi kho ’s heo em, he se H(ϕ, y) con ains a leas
20 TOM´
AS CARABALLO AND DAVID CHEBAN
one minimal se M ⊆ H(ϕ, y). No e ha he mapping h:= p 2:M 7→ Y
is an homomo phism o he dynamical sys em (H(ϕ, y),R, π) on o (Y, R, σ) and,
consequen ly, My:= {(ψ, y) : (ψ, y)∈H(ϕ, y)}is a nonemp y compac subse
o H(ϕ, y). Now we will show ha he se Myconsis s o a single poin o e e y
y∈Y. Indeed, i we assume he opposi e, hen he e exis s a poin y0∈Ysuch ha
My0con ains a leas wo di e en poin s ( i, y0) (i= 1,2 and 16= 2). Acco ding
o Theo em 5.4, wi hou loss o gene ali y we may suppose, o example, ha 1
is compa able by he cha ac e o ecu ence wi h he poin y0, i.e., Ny0⊆N 1.
On he o he hand, by Lemma 5.1, he e exis he limi s lim
→±∞ ( ) = c±and
|c−|+|c+|>0, whe e ( ) := 2( )− 1( ) o all ∈R. Suppose, o example,
ha c−>0. Then, aking in o accoun he ac ha he poin ( 1, y0) is nega i ely
Poisson s able, we ha e a sequence { n} ∈ N 1∩Ny0such ha n→ −∞ as n→ ∞.
We can suppose ha he sequence { 2( + n)}is con e gen . Deno e by ¯ 2i s limi .
Then, we ha e ¯ 2( ) = 1( ) + c− o all ∈Rand, consequen ly, we ha e
(24) a¯ 2=a 1+c−.
Bu he unc ions 1,¯ 2∈H( 1), and he unc ion 1is ecu en and, consequen ly,
we ha e
(25) a¯ 2=a 1=a 2.
F om (24) and (25) i ollows ha c−= 0. This con adic ion p o es ou s a emen .
The o he cases can be conside ed in a simila way. Thus, we ha e es ablished
ha he se Myconsis s o a single poin o all y∈Y. Le φbe a solu ion o
equa ion (10) such ha {(φ, y)}=My. Now, i is easy o show ha he solu ion
φis uni o mly compa ible. Indeed, le { n} ∈ My. Then, he sequence {σ( n, y)}
con e ges. Deno e by ˜yi s limi . We will show ha he unc ional sequence {φ( +
n)}is also con e gen in he space C(R,R). I i is no ue, hen he e exis a
leas wo poin s o accumula ion ψi(i= 1,2 and ψ16=ψ2) o his sequence. On
he o he hand, i is easy o see ha (ψi,˜y)∈ M˜y(i= 1,2). The las inclusion
con adic s he ac ha e e y subse s My⊆ M consis s o a single poin o all
y∈Y. This con adic ion p o es he i s s a emen o ou heo em.
The second and hi d s a emen s ollow om Theo em 5.4. 
Co olla y 6.2. Suppose ha ∈C(Y×R2,R)is egula and mono one inc easing
(in he la ge sense) wi h espec o he a iable x∈R, and he poin y∈Yis s a-
iona y ( espec i ely, τ–pe iodic, quasi-pe iodic, Boh almos pe iodic, ecu en ).
Then, he ollowing s a emen s hold:
(i) I (10) admi s a solu ion ϕsuch ha ϕand ϕ′a e bounded on R, hen i
has a leas one s a iona y ( espec i ely, τ–pe iodic, quasi-pe iodic, Boh
almos pe iodic, ecu en ) solu ion;
(ii) I u( )and ( )a e wo s a iona y ( espec i ely, τ–pe iodic, quasi-pe iodic,
Boh almos pe iodic, ecu en ) solu ions o equa ion (10), hen u( )−
( ) = c o all ∈R, whe e c∈Ris some cons an ;
(iii) I he unc ion is s ic ly inc easing wi h espec o he a iable x∈R,
hen equa ion (10) admi s a mos one s a iona y ( espec i ely, τ–pe iodic,
quasi-pe iodic, Boh almos pe iodic, ecu en ) solu ion.
P oo . These s a emen s ollow om Theo em 6.1 and Theo em 2.12. 
LEVITAN/BOHR ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS ... 21
Rema k 6.3. In he pa icula case in which Yis a Boh almos pe iodic minimal
se , hen Co olla y 6.2 coincides wi h he esul p o ed by Opial in [22].
7. Some gene aliza ions
Le now I:= (a, b),whe e a, b ∈[−∞,+∞]. Fo example, I=R, I = (0,+∞),
I= (a, b) and a, b ∈R, e c. Conside equa ion (10) when ∈C(Y×I×R,R). Fo
example, o he equa ion
x′′ +cx′+1
xα= (σ( , y))
we ha e (y, x, x′) := −cx′−1/xα+ (y) and I= (0,+∞), whe e α > 0.
In his con ex , we will now highligh an ex ended meaning o he concep o bound-
edness on R. To his espec , a solu ion ϕ∈C(R,R) o equa ion (10) is said o be
bounded on R( espec i ely, on R+) i Q:= ϕ(R) is a compac subse om I, i.e., i
he e exis wo eal numbe s αand βsuch ha a < α ≤ϕ( )≤β < b o all ∈R
( espec i ely, ∈R+).
All o ou esul s abou ou second o de equa ion (10) (especially, heo ems 5.4,
6.1 and Co olla ies 5.5 and 6.2) emain ue also when ∈C(Y×I×R,R). We
will o mula e o example he ollowing s a emen s.
Theo em 7.1. Suppose ha ∈C(Y×I×R,R)is egula and inc easing (in he
la ge sense) wi h espec o he a iable x∈I. Then, he ollowing s a emen s hold:
(i) I (10) admi s a solu ion ϕsuch ha ϕand ϕ′a e bounded on R, hen i
has a leas one uni o mly compa ible (by he cha ac e o ecu ence wi h
he igh -hand side) solu ion;
(ii) I u( )and ( )a e wo uni o mly compa ible solu ions o equa ion (10),
hen u( )− ( ) = c o all ∈R, whe e c∈Ris some cons an ;
(iii) I he unc ion is s ic ly inc easing wi h espec o he a iable x∈I,
hen equa ion (10) admi s a mos one uni o mly compa ible solu ion.
P oo . We omi he p oo because is comple ely simila o he p oo o Theo em
6.1. 
Co olla y 7.2. Suppose ha he unc ion ∈C(Y×I×R,R)is egula and
inc easing (in he la ge sense) wi h espec o he a iable x∈I, and he poin
y∈Yis s a iona y ( espec i ely, τ–pe iodic, quasi-pe iodic, Boh almos pe iodic,
ecu en ). Then, he ollowing s a emen s hold:
(i) I (10) admi s a solu ion ϕsuch ha ϕand ϕ′a e bounded on R, hen i
has a leas one s a iona y ( espec i ely, τ–pe iodic, quasi-pe iodic, Boh
almos pe iodic, ecu en ) solu ion;
(ii) I u( )and ( )a e wo s a iona y ( espec i ely, τ–pe iodic, quasi-pe iodic,
Boh almos pe iodic, ecu en ) solu ions o equa ion (10), hen u( )−
( ) = c o all ∈R, whe e c∈Ris some cons an ;
(iii) I he unc ion is s ic ly inc easing wi h espec o he a iable x∈I,
hen equa ion (10) admi s a mos one s a iona y ( espec i ely, τ–pe iodic,
quasi-pe iodic, Boh almos pe iodic, ecu en ) solu ion.

22 TOM´
AS CARABALLO AND DAVID CHEBAN
P oo . This esul ollows om Theo em 7.1 and Theo em 2.12. 
Co olla y 7.3. Suppose ha he ollowing condi ions a e ul illed:
(i) ∈C(Y×I×R,R)and he e exis s a cons an C > 0such ha
(26) | (y, x, x′)| ≤ C(1 + |x′|2)
o all (y, x, x′)∈Y×I×R;
(ii) The unc ion is egula and mono one inc easing (in he la ge sense)
wi h espec o he a iable x∈R;
(iii) The poin y∈Yis s a iona y ( espec i ely, τ–pe iodic, quasi-pe iodic,
Boh almos pe iodic, ecu en ).
Then, he ollowing s a emen s hold:
(i) I (10) admi s a solu ion which is bounded on R, hen i has a leas one
s a iona y ( espec i ely, τ–pe iodic, quasi-pe iodic, Boh almos pe iodic,
ecu en ) solu ion;
(ii) I u( )and ( )a e wo s a iona y ( espec i ely, τ–pe iodic, quasi-pe iodic,
Boh almos pe iodic, ecu en ) solu ions o equa ion (10), hen u( )−
( ) = c o all ∈R, whe e c∈Ris some cons an ;
(iii) I he unc ion is s ic ly inc easing wi h espec o he a iable x∈I,
hen equa ion (10) admi s a mos one s a iona y ( espec i ely, τ–pe iodic,
quasi-pe iodic, Boh almos pe iodic, ecu en ) solu ion.
P oo . These s a emen s ollow om Co olla y 7.2. To his end, i is su icien o
no e ha unde condi ion (26), i ϕ∈C(R,R) is a solu ion o equa ion (10) which
is bounded on R, hen i s de i a i e ϕ′is also bounded on R(see Lemma 2.1 [19]
and also Lemma 5.1 om [17, Ch.XII]). 
Rema k 7.4. 1. Co olla y 7.3 (i em (iii)) imp o es and gene alizes some o he
esul s om [4, 12, 14, 19] when he unc ion is s ic ly inc easing wi h espec o
he second a iable.
2. We plan o s udy in mo e de ail his case ( is s ic ly inc easing wi h espec o
second a iable) in one o ou u u e publica ion.
Acknowledgemen s. We would like o hank he e e ee o in e es ing sugges-
ions which allowed us o imp o e he p esen a ion o his pape .
This pape was w i en while he second au ho was isi ing he Uni e si y o Se illa
(Feb ua y–Sep embe 2010) unde he P og ama de Mo ilidad de P o eso es Uni-
e si a ios y Ex anje os (Minis e io de Educaci´on, Spain) g an SAB2009-0078.
He would like o hank people o his uni e si y o hei e y kind hospi ali y. He
also g a e ully acknowledges he inancial suppo o he Minis e io de Educaci´on
(Spain). The i s au ho is pa ially suppo ed by g an MTM2008-00088 (Minis-
e io de Ciencia e Inno aci´on, Spain) and P oyec o de Excelencia P07-FQM02468
(Jun a de Andaluc´ıa, Spain).
LEVITAN/BOHR ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS ... 23
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E-mail add ess, T. Ca aballo: ca [email protected]
E-mail add ess, D. Cheban: [email protected]
(T. Ca aballo) Depa amen o de Ecuaciones Di e enciales y An´
alisis Num´
e ico, Uni e -
sidad de Se illa, Apdo. Co eos 1160, 41080-Se illa (Spain)
(D. Cheban) S a e Uni e si y o Moldo a, Depa men o Ma hema ics and In o ma ics,
A. Ma ee ich S ee 60, MD–2009 Chis¸in˘
au, Moldo a