EXISTENCE AND UNIQUENESS OF SOLUTIONS OF
HIGHER-ORDER ANTIPERIODIC DYNAMIC EQUATIONS
ALBERTO CABADA AND DOLORES R. VIVERO
Recei ed 8 Oc obe 2003 and in e ised o m 9 Feb ua y 2004
We p o e exis ence and uniqueness esul s in he p esence o coupled lowe and uppe
solu ions o he gene al n h p oblem in ime scales wi h linea dependence on he i h ∆-
de i a i es o i=1,2,...,n, oge he wi h an ipe iodic bounda y alue condi ions. He e
he nonlinea igh -hand side o he equa ion is de ined by a unc ion ( ,x) which is
d-con inuous in and con inuous in xuni o mly in . To do ha , we ob ain he exp es-
sion o he G een’s unc ion o a ela ed linea ope a o in he space o he an ipe iodic
unc ions.
1. In oduc ion
The heo y o dynamic equa ions has been in oduced by S e an Hilge in his Ph.D. hesis
[12]. This new heo y uni ies diffe ence and diffe en ial equa ions and has expe ienced
an impo an g ow h in he las yea s. Recen ly, many pape s de o ed o he s udy o
his kind o p oblems ha e been p esen ed. In he monog aphs o Bohne and Pe e son
[5,6] he e a e he undamen al ools o wo k wi h his ype o equa ions. Su eys on his
heo y gi en by Aga wal e al. [2]andAga wale al.[1]gi eusanideao heimpo ance
o his new ield.
In his pape , we s udy he exis ence and uniqueness o solu ions o he ollowing n h-
o de dynamic equa ion wi h an ipe iodic bounda y alue condi ions:
(Ln)
u∆n( )+
n−1

j=1
Mju∆j( )=  ,u( ),∀ ∈I=[a,b],
u∆i(a)=−u∆iσ(b),0≤i≤n−1.
(1.1)
He e, n≥1, Mj∈Ra e gi en cons an s o j∈{1,...,n−1},[a,b]=Tκn,wi hT⊂Ran
a bi a y bounded ime scale and :I×R→Rsa is ies he ollowing condi ion:
Copy igh ©2004 Hindawi Publishing Co po a ion
Ad ances in Diffe ence Equa ions 2004:4 (2004) 291–310
2000 Ma hema ics Subjec Classi ica ion: 39A10
URL: h p://dx.doi.o g/10.1155/S1687183904310022
292 Highe -o de an ipe iodic dynamic equa ions
(H ) o allx∈R, (·,x)∈C d(I)and ( ,·)∈C(R)uni o mlya ∈I, ha is, o
all >0, he e exis s δ>0such ha
|x−y|<δ=⇒ 
 ( ,x)− ( ,y)
<,∀ ∈I. (1.2)
Asolu iono p oblem(Ln) will be a unc ion u:T→Rsuch ha u∈Cn
d(I) and sa -
is ies bo h equali ies. He e, we deno e by Cn
d(I) he se o all unc ions u:T→Rsuch
ha he i h de i a i e is con inuous in Tκi,i=0,...,n−1, and he n h de i a i e is d-
con inuous in I.
I is clea ha o any gi en cons an M∈R,p oblem(Ln)canbe ew i enas
u∆n( )+
n−1

j=1
Mju∆j( )+Mu( )=  ,u( )+Mu( ), ∀ ∈I,
u∆i(a)=−u∆iσ(b),0≤i≤n−1.
(1.3)
De ining he linea ope a o Tn[M]:Cn
d(I)→C d(I) o e e yu∈Cn
d(I)as
Tn[M]u( ):=u∆n( )+
n−1

j=1
Mju∆j( )+Mu( ), o e e y ∈I, (1.4)
and he se
Wn:=u∈Cn
d(I):u∆i(a)=−u∆iσ(b),0≤i≤n−1, (1.5)
we can ew i e he dynamic equa ion (Ln)as
Tn[M]u( )=  ,u( )+Mu( ), ∈I,u∈Wn.(1.6)
F om his ac , we deduce ha o ensu e he exis ence and uniqueness o solu ions o
he dynamic equa ion (Ln), we mus de e mine he eal alues M,M1,...,Mn−1 o which
he ope a o Tn[M] is in e ible on he se Wn, ha is, he alues o which G een’s unc-
ion associa ed wi h he ope a o T−1
n[M]inWncan be de ined. In Sec ion 2,wep esen
he exp ession o G een’s unc ion associa ed o he ope a o T−1in Wn,whe eTis a
gene al n h-o de linea ope a o ha is in e ible on ha se . This o mula is analogous
o he one gi en in [9] o n h-o de dynamic equa ions wi h pe iodic bounda y alue
condi ions.
In Sec ion 3,wep o easufficien condi ion o he exis ence and uniqueness o solu-
ions o he dynamic equa ion (Ln). Fo his, we ake as e e ence he esul s ob ained in
[3,4], whe e he exis ence and uniqueness o solu ions o p oblem (Ln)iss udiedin he
pa icula case T={0,1,...,P+n}and so (Ln)isadiffe ence equa ion wi h an ipe iodic
bounda y condi ions. In his case, he classical i e a i e me hods based on he exis ence
o a lowe and an uppe solu ion and on compa ison p inciples o some adequa e linea
A. Cabada and D. R. Vi e o 293
ope a o s, canno be applied and, as a consequence, ex emal solu ions do no exis in a
gi en unc ion’s se . Hence, o s udy he exis ence and uniqueness o solu ions o p ob-
lem (Ln) in an a bi a y bounded ime scale T⊂R, we use he echnique de eloped in
[3,4], based on he concep o coupled lowe and uppe solu ions, simila o he de i-
ni ion gi en in [10] o ope a o s de ined in abs ac spaces and in [11] o an ipe iodic
bounda y i s -o de diffe en ial equa ions. A su ey o hose esul s o diffe ence equa-
ions can be ounded in [8].
Using he esul s p o ed in Sec ions 2and 3,wewillob aininSec ions4and 5 he
exp ession o G een’s unc ion and a sufficien condi ion o he exis ence and uniqueness
o solu ions o he dynamic equa ions o i s - and second-o de , espec i ely; likewise,
we will gi e de ails abou he con inuous case whe e a dynamic equa ion is a diffe en ial
equa ion and he disc e e case, in which ei he a diffe ence equa ion o a q-diffe ence
equa ion a e ea ed.
2. Exp ession o G een’s unc ion
In his sec ion, we ob ain he exp ession o G een’s unc ion associa ed wi h he ope a o
T−1in Wn,whe eTis a gene al linea ope a o o n h-o de ha is in e ible on he
men ioned se .
Fi s , we in oduce he concep o n h-o de eg essi e ope a o , see [5, De ini ion 5.89
and Theo em 5.91].
De ini ion 2.1. Le Mi∈R,0≤i≤n−1 be gi en cons an s, he ope a o T:Cn
d(I)→
C d(I), de ined o e e y u∈Cn
d(I)as
Tu( ):=u∆n( )+
n−1

i=0
Miu∆i( ), o e e y ∈I, (2.1)
is eg essi e on Ii and only i 1 + n
i=1(−µ( ))iMn−i= 0 o all ∈I.
Theo em 2.2. Le Mi∈R,0≤i≤n−1be gi en cons an s such ha he ope a o Tde ined
in (2.1) is eg essi e on I(see De ini ion 2.1). I he ope a o Tis in e ible on Wn, hen
G een’s unc ion associa ed o he ope a o T−1in Wn,G:T×I→Ris gi en by he ollowing
exp ession:
G( ,s)=


u( ,s)+ ( ,s), i a≤σ(s)≤ ≤σn(b),
u( ,s), i a≤ <σ(s)≤σ(b), (2.2)
whe e, o e e y s∈[a,b] ixed, (·,s)is he unique solu ion o he p oblem
(Qs)
Txs( )=0, ∈σ(s),b,
x∆i
sσ(s)=0, i=0,1,...,n−2,
x∆n−1
sσ(s)=1,
(2.3)
and o e e y s∈[a,b] ixed, u(·,s)is gi en as he unique solu ion o he p oblem
294 Highe -o de an ipe iodic dynamic equa ions
(Rs)
Tys( )=0, ∈[a,b],
y∆i
s(a)+y∆i
sσ(b)=−
∆iσ(b),s,i=0,1,...,n−1.(2.4)
P oo . Fi s , we see ha he unc ion Gis well de ined, ha is, o e e y s∈[a,b] ixed,
p oblems (Qs)and(Rs) ha e a unique solu ion.
Since he ope a o Tis eg essi e on I,weha e,see[5, Co olla y 5.90 and Theo em
5.91], ha o e e y s∈[a,b] ixed, he ini ial alue p oblem (Qs) has a unique solu ion.
To e i y ha he pe iodic bounda y p oblem (Rs) is uniquely sol able, we conside
he ollowing bounda y alue p oblem:
(Pλ)
w∆n( )+
n−1

i=0
Miw∆i( )=h( ), ∈I,
w∆i(a)+w∆iσ(b)=λi,i=0,1,...,n−1,
(2.5)
wi h h∈C d(I)andλi∈R,0≤i≤n−1 ixed.
We know ha w∈Cn
d(I)isasolu iono p oblem(Pλ)i andonlyi W( )=
(w( ),w∆( ),...,w∆n−1( ))Tis a solu ion o he ma ix equa ion
W∆( )=AW( )+H( ), ∈I,W(a)+Wσ(b)=λ, (2.6)
whe e H( )=(0,...,0,h( ))T,λ=(λ0,...,λn−1)T,and
A=












010··· 0
001··· 0
.
.
..
.
..
.
.....
.
.
000
··· 1
−M0−M1−M2··· −Mn−1












.(2.7)
Since he ope a o Tis eg essi e on I,weha e,by[5, De ini ions 5.5 and 5.89], ha
he ma ix Ais eg essi e on I ooandso,i ollows om[5, Theo em 5.24] ha he
ini ial alue p oblem
W∆( )=AW( )+H( ), ∈I,W(a)=Wa, (2.8)
has a unique solu ion ha is gi en by he ollowing exp ession:
W( )=eA( ,a)Wa+
aeA ,σ(s)H(s)∆s. (2.9)
A. Cabada and D. R. Vi e o 295
I we deno e he n×niden i y ma ix by In, hen we ob ain, om he bounda y con-
di ions, ha p oblem (2.6) has a unique solu ion i and only i he e exis s a unique
Wa=W(a)∈Rnsuch ha
In+eAσ(b),aWa=λ−σ(b)
aeAσ(b),σ(s)H(s)∆s, (2.10)
o equi alen ly, i and only i he ma ix In+eA(σ(b),a)isin e ible.
Now, since he ope a o Tis in e ible on Wn,weha e ha p oblem(P0)hasaunique
solu ion and hen he e exis s he in e se o such ma ix. As a consequence, p oblem (Rs)
has a unique solu ion.
Now, le z:T→Rbe de ined o e e y ∈Tas
z( )=σ(b)
aG( ,s)h(s)∆s. (2.11)
I is no difficul o p o e, by using [5, Theo em 1.117], ha zis he unique solu ion
o he p oblem (P0). 
Now, we p o e he ollowing p ope ies o G een’s unc ion associa ed o he ope a o
T−1in Wn.
P oposi ion 2.3. Le Mi∈R,0≤i≤n−1be gi en cons an s such ha he ope a o Tde-
ined in (2.1) is eg essi e on I.I G:T×I→Ris G een’s unc ion associa ed o he ope a o
T−1in Wn,de inedin(2.2), hen he ollowing condi ions a e sa is ied.
(1) The e exis s k>0such ha |G( ,s)|≤k o all ( ,s)∈T×I.
(2) I n=1, hen o e e ys∈I, he unc ionG(·,s)is con inuous a ∈Texcep a
=s=σ(s).
(3) I n>1, hen o e e ys∈I, he unc ionG(·,s)is con inuous in T.
(4) I n=1, hen o e e y ∈T, he unc ionG( ,·)is d-con inuous a s∈Iexcep
when s= =σ( ).
(5) I n>1, hen o e e y ∈T, he unc ionG( ,·)is d-con inuous in I.
P oo . As we ha e seen in he p oo o Theo em 2.2, we know ha G een’s unc ion asso-
cia ed o he ope a o T−1in Wnis gi en as he 1 ×n e m o he ma ix unc ion
F( ,s)=




eA ,σ(s)−eA( ,a)In+eAσ(b),a−1eAσ(b),σ(s),σ(s)≤ ,
−eA( ,a)In+eAσ(b),a−1eAσ(b),σ(s), <σ(s), (2.12)
whe e Ais he ma ix gi en in (2.7).
F om [5, De ini ion 5.18 and Theo em 5.23], we know ha he ma ix exponen ial
unc ion is con inuous in bo h a iables and so he unc ion Gis bounded in he compac
se T×I.
Now, since eA( , )=In,i =σ(s)=s, hen he diagonal e ms o F(·,s) a e no con-
inuous a .
I is clea ha in any o he si ua ion, he unc ion F(·,s)iscon inuousa .

296 Highe -o de an ipe iodic dynamic equa ions
On he o he hand, gi en 0∈T, o e e ys0∈Isuch ha s0= 0,i ollows, om
he con inui y o he exponen ial unc ion, ha i s→s0and σ(s)→σ(s0), hen F( 0,s)→
F( 0,s0).
Hence, since G( ,s)(≡F1,n( ,s)) belongs o he diagonal o F( ,s)onlywhenn=1, he
p ope ies (2), (3), (4), and (5) o he s a emen hold. 
3. Exis ence and uniqueness esul s
In his sec ion, we p o e exis ence and uniqueness esul s o he n h-o de nonlinea
dynamic equa ion wi h an ipe iodic bounda y condi ions (Ln).
Suppose ha he unc ion :I×R→Rsa is ies condi ion (H ), he ope a o Tn[M]
is eg essi e on Iand in e ible on Wnand Gis G een’s unc ion associa ed o he ope a o
Tn−1[M]inWn,de inedin(2.2).
We de ine he unc ions G+,G−:T×I→Ras
G+:=max{G,0}≥0, G−:=−min{G,0}≥0, (3.1)
and so,
G=G+−G−on T×I. (3.2)
Conside ing he ope a o s A+
n[M],A−
n[M]:C(T)→C(T)de ined o e e yη∈C(T)
as
A+
n[M]η( ):=σ(b)
aG+( ,s) s,η(s)+Mη(s)∆s, ∈T,
A−
n[M]η( ):=σ(b)
aG−( ,s) s,η(s)+Mη(s)∆s, ∈T,
(3.3)
he solu ions o he dynamic equa ion (Ln) a e he ixed poin s o he ope a o
An[M]:=A+
n[M]−A−
n[M].(3.4)
No e ha i condi ion (H ) holds, hen he ope a o s A+
n[M]andA−
n[M]a ewell
de ined.
To deduce he exis ence and uniqueness o solu ions o he dynamic equa ion (Ln), we
in oduce he concep o coupled lowe and uppe solu ions o such p oblem.
De ini ion 3.1. Gi en M∈Rsuch ha he ope a o Tn[M] is eg essi e on Iand in e ible
on Wn, a pai o unc ions α,β∈Cn
d(I)such ha α≤βin Tis a pai o coupled lowe
and uppe solu ions o he dynamic equa ion (Ln) i he inequali ies
α( )≤A+
n[M]α( )−A−
n[M]β( ), ∀ ∈T,
β( )≥A+
n[M]β( )−A−
n[M]α( ), ∀ ∈T,(3.5)
hold.
A. Cabada and D. R. Vi e o 297
Unde he condi ions o he p e ious de ini ion, i αand βa e a pai o coupled lowe
and uppe solu ions o he dynamic equa ion (Ln), hen de ining he ope a o
B[M]:[α,β]×[α,β]−→ C(T) (3.6)
as
B[M](η,ξ):=A+
n[M]η−A−
n[M]ξ, (3.7)
and conside ing he hypo hesis
(H) o e e y ∈Iand α( )≤u≤ ≤β( ), i is sa is ied ha
( ,u)+Mu≤ ( , )+M , (3.8)
we p o e he ollowing mono onici y p ope y.
Lemma 3.2. Suppose ha M∈Ris a gi en cons an such ha he ope a o Tn[M]is e-
g essi e on Iand in e ible on Wn,αand βa e a pai o coupled lowe and uppe solu-
ions o he dynamic equa ion (Ln)and he unc ion :I×R→Rsa is ies hypo heses (H )
and (H).Then,B[M](η,ξ)∈[α,β] o all η,ξ∈[α,β].Mo eo e ,i α≤η1≤η2≤βand
α≤ξ2≤ξ1≤β, hen
B[M]η1,ξ1≤B[M]η2,ξ2in T.(3.9)
P oo . Le α≤η1≤η2≤βand α≤ξ2≤ξ1≤β. I ollows, om he de ini ions o A+
n[M]
and A−
n[M], ha
A+
n[M]α≤A+
n[M]η1≤A+
n[M]η2≤A+
n[M]βin T,
A−
n[M]α≤A−
n[M]ξ2≤A−
n[M]ξ1≤A−
n[M]βin T.(3.10)
F om he de ini ions o αand β,weob ain ha
α≤A+
n[M]α−A−
n[M]β≤A+
n[M]η1−A−
n[M]ξ1
≤A+
n[M]η2−A−
n[M]ξ2≤A+
n[M]β−A−
n[M]α≤βin T.(3.11)
This comple es he p oo . 
Now, we ob ain a esul which gi es us a egion whe e all he solu ions in [α,β]o he
dynamic equa ion (Ln)lie.
P oposi ion 3.3. Suppose ha M∈Ris a gi en cons an such ha he ope a o Tn[M]is
eg essi e on Iand in e ible on Wn,αand βa e a pai o coupled lowe and uppe solu ions
o he dynamic equa ion (Ln)and he unc ion :I×R→Rsa is ies hypo heses (H )and
(H).
Then, he e exis wo mono one sequences in C(T),{ϕm}m∈N,and{ψm}m∈N,wi hα=
ϕ0≤ϕm≤ψl≤ψ0=βin T,m,l∈Nwhich con e ge uni o mly o he unc ions ϕand ψ
ha sa is y
ϕ=A+
n[M]ϕ−A−
n[M]ψ,ψ=A+
n[M]ψ−A−
n[M]ϕin T.(3.12)
298 Highe -o de an ipe iodic dynamic equa ions
Mo eo e , any solu ion u∈[α,β]o (Ln)belongs o he sec o [ϕ,ψ]. I , in addi ion,
ϕ=ψ, hen ϕis he unique solu ion o (Ln)in [α,β].
P oo . The sequences {ϕm}m∈Nand {ψm}m∈Na e ob ained ecu si ely as ϕ0:=α,ψ0:=β
and o e e y m≥1,
ϕm:=B[M]ϕm−1,ψm−1,ψm:=B[M]ψm−1,ϕm−1.(3.13)
F om Lemma 3.2,weknow ha α=:ϕ0≤ϕ1≤ψ1≤ψ0:=βin T.
By induc ion, we conclude ha he sequence {ϕm}m∈Nis mono one inc easing, he
sequence {ψm}m∈Nis mono one dec easing, and ϕm≤ψlin T o e e y m,l∈N.
As a consequence, o e e y ∈T, he e exis ϕ( ):=limm→∞ ϕm( )andψ( ):=
limm→∞ ψm( ).
F om hypo hesis (H )andP oposi ion 2.3, we know ha bo h sequences a e uni-
o mly equicon inuous on Iand so, Ascoli-A zel`
a’s heo em, (see [7, page 72], [14,page
735]), implies ha such con e gence is uni o m in T.Now,[13, Theo em 1.4.3] shows
ha
ϕ=A+
n[M]ϕ−A−
n[M]ψ,ψ=A+
n[M]ψ−A−
n[M]ϕin T.(3.14)
Le ube a solu ion o he dynamic equa ion (Ln)such ha u∈[α,β]. F om Lemma
3.2,weknow ha
ϕ1:=B[M](α,β)≤B[M](u,u)=u≤B[M](β,α)=:ψ1in T.(3.15)
By ecu ence, we a i e a ϕm≤u≤ψlin T o all m,l∈N. Thus, passing o he limi ,
we ob ain ha ϕ≤u≤ψin T.
Finally, i ϕ=ψ, henweha e ha ϕ=A+
n[M]ϕ−A−
n[M]ϕ=:An[M]ϕ, ha is, ϕ=ψ
is a solu ion o he dynamic equa ion (Ln)in[α,β]. Since all solu ions o (Ln) ha belong
o [α,β] lie in he sec o [ϕ,ψ], we conclude ha ϕis he unique solu ion o (Ln)in
[α,β]. 
Now, le ·be he sup emum no m in C(T).
We p o e he ollowing exis ence esul , ha gi es us a sufficien condi ion o assu e
ha he dynamic equa ion (Ln) has a unique solu ion lying be ween a pai o coupled
lowe and uppe solu ions o (Ln).
Theo em 3.4. Assume ha M∈Ris a gi en cons an such ha he ope a o Tn[M]is
eg essi e on Iand in e ible on Wn,αand βa e a pai o coupled lowe and uppe solu ions
o he dynamic equa ion (Ln)and he unc ion :I×R→Rsa is ies hypo hesis (H ).
I o e e y ∈Iand α( )≤u≤ ≤β( ) he inequali ies
−M( −u)≤ ( , )− ( ,u)≤(K−M)( −u) (3.16)
A. Cabada and D. R. Vi e o 299
a e sa is ied o some K≥0such ha
K·



σ(b)
a
G( ,s)
∆s




<1, (3.17)
hen he dynamic equa ion (Ln)has a unique solu ion in [α,β].
P oo . Since he i s pa o he inequali y (3.16) is hypo hesis (H), we know, by
P oposi ion 3.3, ha he e exis s a pai o unc ions ϕ,ψ∈C(T)such ha o e e y ∈T
we ha e
0≤(ψ−ϕ)( )
=A+
n[M]ψ( )−A−
n[M]ϕ( )−A+
n[M]ϕ( )+A−
n[M]ψ( )
=σ(b)
aG+( ,s) s,ψ(s)− s,ϕ(s)+Mψ(s)−ϕ(s)∆s
+σ(b)
aG−( ,s) s,ψ(s)− s,ϕ(s)+Mψ(s)−ϕ(s)∆s
=σ(b)
a
G( ,s)
 s,ψ(s)− s,ϕ(s)+Mψ(s)−ϕ(s)∆s
≤σ(b)
a
G( ,s)
·K·ψ(s)−ϕ(s)∆s
≤
ψ−ϕ
·K·



σ(b)
a
G( ,s)
∆s




.
(3.18)
Thus, i ollows om he inequali y (3.17) ha ϕ=ψin Tand P oposi ion 3.3 allows
us o conclude ha he dynamic equa ion (Ln) has a unique solu ion in [α,β]. 
Rema k 3.5. One can check, ollowing he p oo s gi en in hese sec ions, ha we can
de elop an analogous heo y o p oblem
(¯
Ln)
−u∆n( )+
n−1

j=1
Mju∆j( )=  ,u( ),∀ ∈I=[a,b],
u∆i(a)=−u∆iσ(b),0≤i≤n−1.
(3.19)
In his case, we mus s udy he ope a o
¯
Tn[M]u≡−u∆n+
n−1

j=1
Mju∆j+Mu (3.20)
in he space Wn.
The unc ions αand βa e gi en as in De ini ion 3.1,wi hGG een’s unc ion ela ed
wi h ope a o ¯
Tn[M]inWn.
306 Highe -o de an ipe iodic dynamic equa ions
One can e i y ha he ope a o
T2[M]u( ):=−u( )+M2u( ), ∈I, (5.12)
is eg essi e on Iand in e ible on W2 o all M= 0. Mo eo e , G een’s unc ion associ-
a ed o he ope a o T−1
2[M]inW2is gi en by he ollowing exp ession.
I M=0, hen
G( ,s)=








1
2T
2− +s,i 0≤s< ≤T,
1
2T
2−s+ ,i 0≤ ≤s≤T.
(5.13)
I M= 0, hen
G( ,s)=












1
2Me−M( −s)
1+e−MT −eM( −s)
1+eMT ,i 0≤s< ≤T,
1
2MeM(T+ −s)
1+eMT −e−M(T+ −s)
1+e−MT ,i 0≤ ≤s≤T.
(5.14)
F om he p e ious exp essions we ob ain ha i M=0, hen




T
0
G( ,s)
ds




≤5T2
4=:K0>0, (5.15)
and i M= 0, hen




T
0
G( ,s)
ds




≤1
2|M|MeMT −1
1+eMT +1−e−MT
1+e−MT =:KM>0.(5.16)
The e o e, om Co olla y 5.1 we ob ain he ollowing esul .
Co olla y 5.2. Assume ha M∈Ris a gi en cons an , αand βa e a pai o coupled
lowe anduppe solu ionso hediffe en ial equa ion (¯
L2),and :I×R→Ris a con inu-
ous unc ion. Assuming condi ion (3.16)(wi hM2ins ead o M) o someK≥0such ha
K<1/KM,wi hK0and KMde ined in (5.15)and(5.16), espec i ely, hen he diffe en ial
equa ion (¯
L2)has a unique solu ion in [α,β].
Diffe ence equa ions. Le h>0, P∈N;P≥2, and T={0,h,...,hP}⊂R.
Gi en u:T→R,i ollows om[5, Theo em 1.16] ha o e e y ∈Tκ2,uis wice
∆-diffe en iable a , mo eo e , we ha e ha
u∆2( )=u( +2h)−2u( +h)+u(h)
h2.(5.17)

A. Cabada and D. R. Vi e o 307
In his case, one can e i y ha o all M∈Rsuch ha |M| = 1/h,1+(1+Mh)P−1= 0,
and 1 + (1 −Mh)P−1= 0, he ope a o
¯
T2[M]u( ):=−u( +2h)+2u( +h)−1−M2h2u(h)
h2, ∈I, (5.18)
is eg essi e on Iand in e ible on W2. Now, he unc ion Gis gi en by he ollowing
exp ession.
I M=0, hen
G( ,s)=








1
2h(P−1)
2− +s+h,i 0≤s+h≤ ≤hP,
1
2h(P−1)
2−s−h+ ,i 0≤ ≤s≤h(P−2).
(5.19)
I M= 0, |M| = 1/h,1+(1+Mh)P−1= 0, and 1 + (1 −Mh)P−1= 0, hen
G( ,s)=










1
2M(1 −Mh)( −s−h)/h
1+(1−Mh)P−1−(1 + Mh)( −s−h)/h
1+(1+Mh)P−1,i 0≤s+h≤ ≤hP,
1
2M(1 + Mh)(Ph+ −s−2h)/h
1+(1+Mh)P−1−(1 −Mh)(Ph+ −s−2h)/h
1+(1−Mh)P−1,i 0≤ ≤s≤h(P−2).
(5.20)
F om hese exp essions, we deduce he ollowing es ima es.
I M=0, hen we ha e ha





h·
h(P−2)

s=0
G( ,s)





≤5h(P−1)2
4=:K0>0.(5.21)
I |M|<1/h,M= 0, hen we a i e a





h·
h(P−2)

s=0
G( ,s)





≤1
2|M|M(1 + Mh)P−1−1
1+(1+Mh)P−1+1−(1 −Mh)P−1
1+(1−Mh)P−1
=:KM>0.
(5.22)
As a consequence, we a i e a he ollowing esul .
Co olla y 5.3. Assume ha M∈Ris a gi en cons an such ha |M|<1/h,αand βa e
a pai o coupled lowe and uppe solu ions o he dynamic equa ion (¯
L2),and he unc ion
:I×R→R e i ies ha o e e y ∈I, ( ,·)∈C(R).I (3.16)(wi hM2ins ead o M)
holds o some K≥0such ha K<1/KM,wi hK0and KMde ined in (5.21)and(5.22),
espec i ely, hen he dynamic equa ion (¯
L2)has a unique solu ion in [α,β].
q-diffe ence equa ions. Gi en q∈R,q>1, and N∈N,N≥2, le T={1,q,...,qN}⊂R.
308 Highe -o de an ipe iodic dynamic equa ions
I u:T→R, henweknow,by[5, Theo em 1.16], ha o e e y ∈Tκ2,uis wice
∆-diffe en iable a and
u∆2( )=uq2 −(q+1)u(q )+qu( )
(q−1)2q 2.(5.23)
I =qiand s=qj, hen one can e i y ha G een’s unc ion is gi en by he ollowing
exp ession.
I M=0, hen
G( ,s)=








1
2qN−1
2−qi+qj+1,i 0≤j+1≤i≤N,
1
2qN−1
2−qj+1 +qi,i 0≤i≤j≤N−2.
(5.24)
I M= 0, |M| = 1/(q−1)qk o each k∈JN−2={0,1,...,N−2},1+
N−2
k=0(1 + M(q−
1)qk)= 0and1+N−2
k=0(1 −M(q−1)qk)= 0, hen
G( ,s)=





























1
2Mi−1
k=j+1 1−M(q−1)qk
1+N−2
k=01−M(q−1)qk−i−1
k=j+1 1+M(q−1)qk
1+N−2
k=01+M(q−1)qk,
i 0 ≤j+1≤i≤N,
1
2Mk∈Ii,j1−M(q−1)qk
1+N−2
k=01−M(q−1)qk−k∈Ii,j1+M(q−1)qk
1+N−2
k=01+M(q−1)qk,
i 0 ≤i≤j≤N−2,
(5.25)
wi h Ii,jde ined in (4.23).
F om he exp ession o G een’s unc ion, we ob ain he ollowing uppe bounds.
I M=0, hen
(q−1) ·max
0≤i≤NN−2

j=0
qj
Gqi,qj
≤maxK0,1,K0,2=:K0>0, (5.26)
wi h
K0,1 :=5qN−1−12
4,
K0,2 :=1
2qN−1qN−1−1+1
2qN−1−12.
(5.27)
I |M|<1/(q−1)qN−2,M= 0, hen
(q−1) ·max
0≤i≤NN−2

j=0
qj
Gqi,qj
≤maxKM,1,KM,2=:KM>0, (5.28)
A. Cabada and D. R. Vi e o 309
whe e
KM,1 :=1
2|M|MN−2
k=01+M(q−1)qk−1
1+N−2
k=01+M(q−1)qk+1−N−2
k=01−M(q−1)qk
1+N−2
k=01−M(q−1)qk,
KM,2 :=1
2|M|MN−1
k=01+M(q−1)qk−1−M(q−1)qN−1
1+N−2
k=01+M(q−1)qk
+1−M(q−1)qN−1−N−1
k=01−M(q−1)qk
1+N−2
k=01−M(q−1)qk.
(5.29)
F om he p e ious exp essions, we ha e he ollowing esul .
Co olla y 5.4. Assume ha M∈Ris such ha |M|<1/(q−1)qN−2,αand βa e a pai o
coupled lowe and uppe solu ions o he dynamic equa ion (¯
L2),and :I×R→R e i ies
ha o e e y ∈I, ( ,·)∈C(R).I (3.16)(wi hM2ins ead o M)is ue o someK≥0
such ha K<1/KM,wi hK0and KMde ined in (5.26)and(5.28), espec i ely, hen he
dynamic equa ion (¯
L2)has a unique solu ion in [α,β].
Acknowledgmen s
The au ho s hank he e e ees o he pape o hei in e es ing commen s and sugges-
ions. This esea ch was pa ially suppo ed by D. G. I. and F.E.D.E.R. p ojec BFM2001-
3884-C02-01, and by Xun a de Galicia and F.E.D.E.R. p ojec PGIDIT020XIC20703PN,
Spain.
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Albe o Cabada: Depa amen o de An´
alise Ma em´
a ica, Facul ade de Ma em´
a icas, Uni e sidade
de San iago de Compos ela, 15782 San iago de Compos ela, Galicia, Spain
E-mail add ess:[email p o ec ed]
Dolo es R. Vi e o: Depa amen o de An´
alise Ma em´
a ica, Facul ade de Ma em´
a icas, Uni e sidade
de San iago de Compos ela, 15782 San iago de Compos ela, Galicia, Spain
E-mail add ess:[email p o ec ed]