Elec onic Jou nal o Di e en ial Equa ions, Vol. 2017 (2017), No. 245, pp. 1–17.
ISSN: 1072-6691. URL: h p://ejde.ma h. xs a e.edu o h p://ejde.ma h.un .edu
POSITIVE SOLUTIONS FOR SECOND-ORDER
BOUNDARY-VALUE PROBLEMS WITH SIGN CHANGING
GREEN’S FUNCTIONS
ALBERTO CABADA, RICARDO ENGUIC¸A, LUC´
IA L´
OPEZ-SOMOZA
Communica ed by Pa el D abek
Abs ac . In his a icle we analyze some possibili ies o inding posi i e
solu ions o second-o de bounda y- alue p oblems wi h he Di ichle and
pe iodic bounda y condi ions, o which he co esponding G een’s unc ions
change sign. The ob ained esul s can also be adap ed o Neumann and mixed
bounda y condi ions.
1. In oduc ion
In he li e a u e, he exis ence o posi i e solu ions o bounda y- alue p oblems
(BVP) has been widely s udied, in pa icula o second-o de BVP wi h pe iodic
and Di ichle bounda y condi ions. A s anda d echnique consis s in ob aining he
exis ence o posi i e solu ions h ough K asnoselskii’s ixed poin heo em on cones,
o o use ixed poin index heo y. In hese cases, he posi i i y o he associa ed
G een’s unc ions is usually undamen al o p o e such esul s. In his pape we
a e able o p o e exis ence o solu ions o se e al p oblems whe e he associa ed
G een’s unc ion changes sign.
Hill’s ope a o p ope ies ha e been desc ibed in se e al pape s, whe e exis ence
and mul iplici y esul s, compa ison p inciples, G een’s unc ions and spec al anal-
ysis we e s udied. Some o hese esul s can be o iginally ound in [4, 5, 6, 12, 15].
Posi i i y esul s o BVP whe e he G een’s unc ion can anish a e ea ed
o example in [8, 13]. G ae , Kong and Wang [8] s udied he pe iodic BVP (wi h
T= 1)
u00( ) + a( )u( ) = g( ) (u( )), ∈(0, T),
u(0) = u(T), u0(0) = u0(T),
wi h and gnonnega i e con inuous unc ions and gsa is ying he condi ion
min ∈[0,T ]g( )>0. They assumed he G een’s unc ion o be nonnega i e and
o sa is y he condi ion
min
0≤s≤TZT
0
G( , s)d > 0.(1.1)
2010 Ma hema ics Subjec Classi ica ion. 34B15, 34A40.
Key wo ds and ph ases. Second o de di e en ial equa ions; Di ichle bounda y condi ions;
pe iodic bounda y condi ions; sign changing G een’s unc ion.
c
2017 Texas S a e Uni e si y.
Submi ed June 19, 2017. Published Oc obe 6, 2017.
1
2 A. CABADA, R. ENGUIC¸ A, L. L´
OPEZ-SOMOZA EJDE-2017/245
Webb [13] conside ed weake assump ions o p o e he exis ence o posi i e solu ions
o he p e ious p oblem, bu he s ill assumed G een’s unc ion o be nonnega i e.
Despi e ou esul s do no equi e he G een’s unc ion o be nonnega i e, hey
could be applied o his pa icula case, ob aining posi i e solu ions assuming an
in eg al condi ion weake han (1.1) (see Rema ks 3.6 and 3.11).
On he o he hand, some exis ence esul s o BVP wi h sign-changing G een’s
unc ion ha e been conside ed in [7, 10], whe e he au ho s asked o he exis ence
o a subin e al [c, d]⊂[0, T], a unc ion φ∈L1([0, T]) and a cons an c∈(0,1]
such ha he G een’s unc ion Gsa is ies he condi ion
|G( , s)| ≤ φ(s) o all ∈[0, T] and almos e e y s∈[0, T],
G( , s)≥c φ(s) o all ∈[c, d] and almos e e y s∈[0, T].(1.2)
I mus be poin ed ou ha , i we conside a pe iodic p oblem wi h cons an
po en ial a( ) = ρ2 o which he ela ed G een’s unc ion changes i s sign (i.e.
ρ > π/T ,ρ6= 2kπ/T ,k= 1,2, . . .), condi ion (1.2) is ne e ul illed o any s ic ly
posi i e unc ion φ. This is due o he ac ha in such si ua ion he G een’s
unc ion is cons an along he s aigh lines o slope equals o one (see [2, 3] o
de ails). Meanwhile, as we will p o e on Sec ion 4, ou esul s can be applied
wi hou u he complica ions o his case.
Mo eo e , o he Di ichle BVP wi h cons an po en ial a( ) = ρ2wi h sign-
changing G een’s unc ion (i.e. ρ > π/T,ρ6=kπ/T ,k= 1,2, . . .), as a di ec
consequence o exp ession (5.1) below, i is immedia e o e i y ha condi ion (1.2)
holds i and only i ρ2lies be ween he i s and he second eigen alues o he
p oblem ( π
T< ρ < 2π
T) bu i is ne e sa is ied o ρ > 2π
T. Howe e , as we will
poin ou in Sec ion 5, ou esul s can be applied o any non esonan alue o
ρ > π/T . Despi e his, we mus no e ha he imposed es ic ions inc ease wi h ρ.
Fu he mo e, in [7, 10] he au ho s p o ed he exis ence o solu ions in he cone
K0=u∈ C[0, T] : min
∈[c,d]u( )≥ckuk,
ha is, hey ensu ed he posi i i y o he solu ions on he subin e al [c, d] bu such
solu ions we e allowed o change sign when conside ing he whole in e al [0, T].
As a as we know, posi i e solu ions o BVP wi h sign-changing G een’s unc-
ion can be acked only as back as 2011 in he pape s [11, 16]. In he i s o hese
pape s, Ma conside s he one-pa ame e amily o p oblems
u00( ) + a( )u( ) = λ g( ) (u( )), ∈(0, T),
u(0) = u(T), u0(0) = u0(T).(1.3)
By using he Schaude ’s ixed poin Theo em, he au ho ob ains he exis ence o
a posi i e solu ion o su icien ly small alues o λ. These exis ence esul s a e no
compa able wi h he ones we will ob ain in his pape . In he second pape , Zhong
and An [16] s udy he ollowing au onomous pe iodic BVP, wi h cons an po en ial
ρ∈(0,3π
2T]:
u00 +ρ2u= (u), ∈(0, T ), u(0) = u(T), u0(0) = u0(T).(1.4)
In his case, i is e y well known ha he ela ed G een’s unc ion GP( , s)≥0
o all ρ∈(0,π
T] and i changes sign o ρ∈(π
T,3π
2T] (see [2, 3]). Wi h his, i can
EJDE-2017/245 POSITIVE SOLUTIONS 3
be de ined he cons an
δ=


∞i ρ∈(0,π
T],
in ∈IRT
0G+
P( ,s)ds
RT
0G−
P( ,s)ds i ρ∈(π
T,3π
2T],
and using he K asnoselskii’s ixed poin Theo em, he au ho s p o e he ollowing
exis ence esul :
Theo em 1.1. [16, Theo em 3] Suppose ha he ollowing assump ions a e ul illed:
(1) : [0,∞)→[0,∞)is con inuous.
(2) 0 ≤m= in u≥0{ (u)}and M= supu≥0{ (u)} ≤ M≤ ∞.
(3) M/m ≤δ, wi h M/m =∞when m= 0.
Mo eo e , i δ=∞assume ha
lim
x→∞
(x)
x< ρ2<lim
x→0+
(x)
x.
Then p oblem (1.4) has a posi i e solu ion on [0, T].
Conce ning his speci ic case, along his pape we imp o e he ange o he alues
ρ o which he esul is s ill alid. Fu he mo e, we apply ou s udy o noncons an
po en ials and nonau onomous nonlinea pa s.
As we will see, some o he posi i i y condi ions imposed o he pe iodic BVP
canno be adap ed o he Di ichle BVP, so he app oach ha mus be used needs
o be conside ably modi ied, by using, in his case, a di e en ype o cones.
The es o his a icle is o ganized he ollowing way: In Sec ion 2 we s a e some
p elimina y esul s conside ing he Hill’s ope a o . In Sec ion 3 some new esul s
conce ning he exis ence o a posi i e solu ion o he Hill’s pe iodic BVP in he case
ha he G een’s unc ion may change sign a e p o ed. Mo eo e , in his sec ion,
such exis ence esul s a e gene alized o o he bounda y condi ions. In Sec ion 4
we imp o e Theo em 1.1 o he pe iodic p oblem wi h a cons an po en ial. In
Sec ion 5 we app oach he Di ichle BVP, also in he case o a cons an po en ial,
whe e as a as we know, no esul s o sign changing G een’s unc ion we e p o ed
be o e.
2. P elimina ies
Le L[a] be he Hill’s ope a o ela ed o he po en ial a
L[a]u( )≡u00( ) + a( )u( ), ∈[0, T]≡I,
whe e a:I→R,a∈Lα(I), α≥1.
Le X⊂W2,1(I) be a Banach space such ha he homogeneous p oblem
L[a]u( ) = 0, o a. e. ∈I, u ∈X(2.1)
has only he i ial solu ion. This condi ion is known as ope a o L[a] being non-
esonan in X. Mo eo e , i is e y well known ha i his condi ion is sa is ied
and σ∈L1(I), he nonhomogeneous p oblem
L[a]u( ) = σ( ), o a. e. ∈I, u ∈X
has a unique solu ion
u( ) = ZT
0
G( , s)σ(s)ds, ∈I,
4 A. CABADA, R. ENGUIC¸ A, L. L ´
OPEZ-SOMOZA EJDE-2017/245
whe e Gis he co esponding G een’s unc ion.
We deno e x0 on Ii x≥0 on Iand RT
0x(s)ds > 0. I is said ha ope a o
L[a] sa is ies a s ong maximum p inciple (MP) in Xi
u∈X, L[a]u0 on I⇒u < 0 in (0, T).
Analogously, L[a] sa is ies he an imaximum p inciple (AMP) in Xi
u∈X, L[a]u0 on I⇒u > 0 in (0, T).
The nex esul is a di ec consequence o [3, Co olla ies 1.6.6 and 1.6.12], and i
ensu es ha he maximum and an i-maximum p inciples o he pe iodic p oblem
a e equi alen o he cons an sign o he G een’s unc ion.
Lemma 2.1. The ollowing claims a e equi alen :
(1) The ela ed G een’s unc ion Go p oblem (2.1) sa is ies G( , s)≥0 (≤0)
on I×I.
(2) Ope a o L[a]sa is ies a s ong maximum (an imaximum) p inciple in X.
We will conside now he pe iodic bounda y- alue p oblem
u00( ) + a( )u( ) = 0, ∈I, u(0) = u(T), u0(0) = u0(T),(2.2)
and we will deno e i s ela ed G een’s unc ion as GP.
Now, le λPbe he smalles eigen alue o he pe iodic p oblem
u00( )+(a( ) + λ)u( ) = 0, o a. e. ∈I, u(0) = u(T), u0(0) = u0(T),
and le λAbe he smalles eigen alue o he an i-pe iodic p oblem
u00( )+(a( ) + λ)u( )=0, o a. e. ∈I, u(0) = −u(T), u0(0) = −u0(T).
In [15] i is p o ed ha λP< λA. The ollowing esul ela es he cons an sign o
he pe iodic G een’s unc ion wi h he sign o hese eigen alues:
Lemma 2.2. [15, Theo em 1.1] Suppose ha a∈L1(I), hen:
(1) GP( , s)≤0on I×Ii and only i λP>0.
(2) GP( , s)≥0on I×Ii and only i λP<0≤λA.
I we conside o he bounda y- alue p oblems, such as he Neumann p oblem
u00( ) + a( )u( )=0, ∈I, u0(0) = u0(T) = 0; (2.3)
he Di ichle p oblem
u00( ) + a( )u( )=0, ∈I, u(0) = u(T) = 0; (2.4)
and he mixed p oblems
u00( ) + a( )u( ) = 0, ∈I, u0(0) = u(T) = 0; (2.5)
u00( ) + a( )u( ) = 0, ∈I, u(0) = u0(T) = 0; (2.6)
deno ing by GN,GD,GM1and GM2 he ela ed G een’s unc ions and λN,λD,
λM1and λM2 he co esponding smalles eigen alue o each o he p oblems, we
know ha he ollowing esul s a e sa is ied (see [6]):
Lemma 2.3. (1) GN( , s)<0on I×Ii and only i λN>0.
(2) GN( , s)≥0on I×Ii and only i λN<0,λM1≥0and λM2≥0.
(3) GNchanges sign i and only i min{λM1, λM2}<0.
(4) GD( , s)<0on (0, T)×(0, T)i and only i λD>0.
EJDE-2017/245 POSITIVE SOLUTIONS 5
(5) GDchanges sign i and only i λD<0.
(6) GM1( , s)<0on [0, T)×[0, T)i and only i λM1>0.
(7) GM1changes sign i and only i λM1<0.
(8) GM2( , s)<0on (0, T]×(0, T]i and only i λM2>0.
(9) GM2changes sign i and only i λM2<0.
3. Pe iodic bounda y- alue p oblems
Conside now he nonlinea and nonau onomous pe iodic bounda y alue p ob-
lem
u00( ) + a( )u( ) = ( , u( )), ∈I, u(0) = u(T), u0(0) = u0(T).(3.1)
We will assume ha p oblem (2.2) is non esonan and λA<0. F om Lemma 2.2,
i is clea ha in his case he ela ed G een’s unc ion changes i s sign on I×I.
On he o he hand, i is well-known ha he e exis s P, a posi i e eigen unc ion
on I, unique up o a cons an , ela ed o λP; ha is, Pis such ha
00
P( ) + a( ) P( ) = −λP P( ),a. e. ∈I,
P(0) = P(T), 0
P(0) = 0
P(T).
The e o e,
P( ) = −λPZT
0
GP( , s) P(s)ds
and, since Pis posi i e and λP<0, we ha e ha
ZT
0
GP( , s) P(s)ds > 0∀ ∈I
and, consequen ly,
ZT
0
G+
P( , s) P(s)ds > ZT
0
G−
P( , s) P(s)ds ∀ ∈I,
whe e G+
Pand G−
Pa e he posi i e and nega i e pa s o GP.
Since he G een’s unc ion changes sign, i makes sense o de ine
γ= in
∈IRT
0G+
P( , s) P(s)ds
RT
0G−
P( , s) P(s)ds (>1).
Mo eo e , o ensu e he exis ence o solu ions o p oblem (3.1), we will make he
ollowing assump ions:
(H1) :I×[0,∞)→[0,∞) sa is ies L1-Ca a h´eodo y condi ions, ha is, (·, u)
is measu able o e e y u∈R, ( , ·) is con inuous o a. e. ∈Iand
o each > 0 he e exis s φ ∈L1(I) such ha ( , u)≤φ ( ) o all
u∈[− , ] and a. e. ∈I.
(H2) The e exis wo posi i e cons an s mand Msuch ha m P( )≤ ( , x)≤
M P( ) o e e y ∈Iand x≥0. Mo eo e , hese cons an s sa is y ha
M
m≤γ.
(H3) The e exis s [c, d]⊂Isuch ha Rd
cGP( , s)d ≥0, o all s∈Iand
Rd
cGP( , s)d > 0, o all s∈[c, d].

6 A. CABADA, R. ENGUIC¸ A, L. L ´
OPEZ-SOMOZA EJDE-2017/245
Rema k 3.1. We no e ha condi ion (H2) includes, as pa icula cases, hypo heses
(2) and (3) in Theo em 1.1 imposed in [16]. This is so because i a( ) = ρ2, as in
p oblem (1.4), we ha e ha λP=−ρ2and P( ) = 1 o all ∈I. Mo eo e , as we
will poin ou in Sec ion 4, we ha e ha i a( ) = ρ2 hen
ZT
0
GP( , s)ds =1
ρ2,
and condi ion (H3) is i ially ul illed o [c, d] = I.
Mo eo e , we no e ha in (H2) we a e no conside ing he possibili y o m= 0.
Theo em 1.1 includes his case, bu only when γ= +∞, which only happens when
he G een’s unc ion is nonnega i e. In [16] he au ho s conside his possibili y
because hey a e assuming ha ρ∈0,3π
2Tand when ρ∈0,π
T,GPis nonnega i e.
As we will see in Co olla y 3.5, hypo hesis (H2) is no necessa y in his case, so his
is he eason why we do no conside he possibili y m= 0.
We will conside he Banach space (C(I, R),k · k) coupled wi h he sup emum
no m kuk≡kuk∞, and de ine he cone
K=u∈ C(I, R) : u≥0 on I, ZT
0
u(s)ds ≥σkuk,
whe e
σ=η
max , s∈I{GP( , s)},
wi h
η= min
s∈[c,d]Zd
c
GP( , s)d .(3.2)
Now, i is clea ha uis a solu ion o he pe iodic p oblem (3.1) i and only i i is
a ixed poin o he ollowing ope a o :
Tu( ) = ZT
0
GP( , s) (s, u(s)) ds.
Lemma 3.2. Assume hypo hesis (H1)–(H3). Then T:C(I)→ C(I)is a comple ely
con inuous ope a o which maps he cone K o i sel .
P oo . The p oo ha ope a o Tis a comple ely con inuous ope a o ollows s an-
da d a gumen s and we omi i .
Le us see now ha Tmaps he cone o i sel . Conside ing u∈K, hen, o all
∈I, he ollowing inequali ies a e ul illed:
Tu( ) = ZT
0
GP( , s) (s, u(s)) ds
=ZT
0G+
P( , s)−G−
P( , s) (s, u(s)) ds
≥ZT
0m P(s)G+
P( , s)−M P(s)G−
P( , s)ds
≥mZT
0
G+
P( , s) P(s)ds −γZT
0
G−
P( , s) P(s)ds≥0.
EJDE-2017/245 POSITIVE SOLUTIONS 7
Mo eo e ,
ZT
0Tu( )d ≥Zd
cTu( )d =Zd
cZT
0
GP( , s) (s, u(s)) ds d
=ZT
0
(s, u(s)) Zd
c
GP( , s)d ds
≥ηZT
0
(s, u(s)) ds,
and since
Tu( )≤max
,s∈I{GP( , s)}ZT
0
(s, u(s)) ds,
we deduce ha RT
0Tu( )d ≥σTu( ) o all ∈I, ha is
ZT
0Tu( )d ≥σkTuk,
and he esul is concluded. 
Now, o p o e he exis ence o solu ions o p oblem (3.1), we use some classical
esul s ega ding he ixed poin index. We compile hem in he ollowing lemma.
Le Ω be an open bounded subse o C(I) and le us deno e ¯
Ω and ∂Ω i s closu e
and bounda y, espec i ely. Mo eo e , le us deno e ΩK= Ω ∩K.
Lemma 3.3. [1, Lemma 12.1] Le ΩKbe an open bounded se wi h 0∈ΩKand
¯
ΩK6=K. Assume ha F:¯
ΩK→Kis a comple ely con inuous map such ha
x6=Fx o all x∈∂ΩK. Then he ixed poin index iK(F, ΩK)has he ollowing
p ope ies:
(1) I he e exis s e∈K {0}such ha x6=Fx +λe o all x∈∂ΩKand all
λ > 0, hen iK(F, ΩK)=0.
(2) I x6=µ Fx o all x∈∂ΩKand o e e y µ≤1, hen iK(F, ΩK)=1.
(3) I iK(F, ΩK)6= 0, hen Fhas a ixed poin in ΩK.
(4) Le Ω1
Kbe an open se wi h ¯
Ω1
K⊂ΩK. I iK(F, ΩK)=1and iK(F, Ω1
K) =
0, hen Fhas a ixed poin in ΩK ¯
Ω1
K. The same esul holds i iK(F, ΩK) =
0and iK(F, Ω1
K)=1.
Now we a e in a posi ion o p o e he exis ence esul s conce ning he pe iodic
p oblem (3.1) as ollows. Fi s , we no e ha , as an immedia e consequence o
condi ion (H2), we deduce he ollowing p ope ies:
0= lim
x→0+min
∈[c,d]
( , x)
x=∞, ∞= lim
x→∞ max
∈I
( , x)
x= 0,
whe e he in e al [c, d] is gi en in (H3). These p ope ies will le us p o e he
ollowing heo em.
Theo em 3.4. Assume ha λA<0and hypo hesis (H1)–(H3) hold. Then he e
exis s a leas one posi i e solu ion o p oblem (3.1) in he cone K.
P oo . Taking in o accoun he de ini ion o 0, we know ha he e exis s δ1>0
such ha when kuk ≤ δ1, hen
( , u( )) >u( )
η,∀ ∈[c, d],
8 A. CABADA, R. ENGUIC¸ A, L. L ´
OPEZ-SOMOZA EJDE-2017/245
wi h ηde ined in (3.2). Le
Ω1={u∈K:kuk< δ1}
and choose u∈∂Ω1and e∈K {0}.
We will p o e ha u6=Tu+λ e o e e y λ > 0. Assume, on he con a y, ha
he e exis s some λ > 0 such ha u=Tu+λ e, ha is,
u( ) = Tu( ) + λ e( )≥ Tu( )∀ ∈I.
Then
Zd
c
u( )d ≥Zd
cTu( )d =Zd
cZT
0
GP( , s) (s, u(s)) ds d
=ZT
0Zd
c
GP( , s)d  (s, u(s)) ds
≥Zd
cZd
c
GP( , s)d  (s, u(s)) ds > Zd
c
u(s)ds,
which is a con adic ion. The e o e iK(T, Ω1) = 0.
P oceeding in an analogous way o [5, 8, 9], we de ine ˜
( , u) = max0≤z≤u ( , z).
Clea ly ˜
( , ·) is a nondec easing unc ion on [0,∞). Mo eo e , since ∞= 0 i is
ob ious ha
lim
x→∞ max
∈I
˜
( , x)
x= 0.
As a consequence, he e exis s δ2>0 such ha i kuk ≥ δ2 hen
˜
( , kuk)<σ2
T2ηkuk ∀ ∈I.
Le
Ω2={u∈K;kuk< δ2}
and choose u∈∂Ω2.
We will p o e ha u6=µTu o e e y µ≤1. Assume, on he con a y, ha
he e exis s some µ≤1 such ha u( ) = µTu( ) o all ∈I. Then
σkuk ≤ ZT
0
u( )d =µZT
0Tu( )d
=µZT
0ZT
0
GP( , s) (s, u(s)) ds d
=µZT
0ZT
0
GP( , s)d  (s, u(s)) ds
≤µT max
,s∈I{GP( , s)}ZT
0
(s, u(s)) ds
≤µT max
,s∈I{GP( , s)}ZT
0
˜
(s, u(s)) ds
≤µT max
,s∈I{GP( , s)}ZT
0
˜
(s, kuk)ds
< µT2η
σ
σ2
T2ηkuk ≤ σkuk,
EJDE-2017/245 POSITIVE SOLUTIONS 9
which is a con adic ion. As a consequence, iK(T, Ω2) = 1. We conclude ha
ope a o Thas a ixed poin , ha is, he e exis s a leas a non i ial solu ion o
p oblem (3.1). 
The p e ious heo em is also alid i he G een’s unc ion is nonnega i e. In
his case, hypo hesis (H3) would be i ially ul illed and hypo hesis (H2) is no
necessa y since i is only used o p o e ha Tmaps he cone o i sel , which is
ob ious (since is nonnega i e) when GPis nonnega i e. On he o he hand, we
would need o add he hypo hesis ha 0=∞and ∞= 0 (which can no be
deduced i we elimina e (H2)). The esul eads as ollows:
Co olla y 3.5. Assume ha λP<0≤λAand hypo hesis (H1) is ul illed. Then,
i 0=∞and ∞= 0 he e exis s a leas one posi i e solu ion o p oblem (3.1)
in he cone K.
Rema k 3.6. We no e ha o a nonnega i e G een’s unc ion, we gene alize he
esul s o G ae , Kong and Wang [8, 9] and Webb [13] since ou condi ion (H3) is
weake han condi ion (1.1) conside ed by hem.
Co olla y 3.7. I ( , x)≡ ( )∈L1(I)sa is ies (H2), hen he unique solu ion
o (3.1) is a nonnega i e unc ion on [0, T].
Rema k 3.8. We no e ha u( )≡1 is he unique solu ion o he pe iodic p oblem
u00( ) + a( )u( ) = a( ), ∈I,
u(0) = u(T), u0(0) = u0(T).
The e o e i is clea ha
ZT
0
GP( , s)a(s)ds = 1 >0 (3.3)
and so he p e ious easoning is also alid i a≥0, a > 0 on [c, d], and we change
he de ini ion o γby
γ∗= in
∈IRT
0G+
P( , s)a(s)ds
RT
0G−
P( , s)a(s)ds.
In his case, assump ion (H2) would be subs i u ed by
(H2’) The e exis wo posi i e cons an s mand Msuch ha ma( )≤ ( , u)≤
Ma( ) o e e y ∈I,u > 0. Mo eo e , hese cons an s sa is y ha
M
m≤γ∗.
3.1. Neumann, Di ichle and mixed bounda y alue p oblems. F om he
classical spec al heo y [14], i is e y well know ha , as in he pe iodic case, o
any o he bounda y condi ions in oduced in Lemma 2.3, he e exis s a posi i e
eigen unc ion on (0, T) ela ed o he co esponding smalles eigen alue. The e o e,
i we a e in he case in which L[a] ope a o coupled wi h he associa ed bounda y
condi ions is non esonan and he ela ed G een’s unc ion changes sign (di e en
cases a e cha ac e ized in Lemma 2.3), we could ollow he same a gumen as in
he p e ious sec ion o de ine γand we would ob ain analogous exis ence esul s.
Hypo hesis (H1)–(H3) would be he same wi h he sui able no a ion o each o he
p oblems ( ha is, conside ing in each case he app op ia e G een’s unc ion and
eigen unc ion).
16 A. CABADA, R. ENGUIC¸ A, L. L ´
OPEZ-SOMOZA EJDE-2017/245
Π
2Π
3Π
4Π
5Π
6Π
1
2
3
4
Figu e 3. G aph o γ o he Di ichle p oblem.
0.2
0.4
0.6
0.8
1.0
0.001
0.002
0.003
0.004
0.005
Figu e 4. Solu ion o p oblem (5.2)
.
0.2
0.4
0.6
0.8
1.0
-0.01
0.01
0.02
Figu e 5. Solu ion o p oblem (5.3).
Rema k 5.2. Analogous a gumen s and calcula ions can be done o he Neumann
and mixed p oblems.

EJDE-2017/245 POSITIVE SOLUTIONS 17
Acknowledgmen s. A. Cabada and L. L´opez-Somoza we e pa ially suppo ed by
Minis e io de Econom´ıa y Compe i i idad, Spain, and FEDER, p ojec MTM2013-
43014-P, and by he Agencia Es a al de In es igaci´on (AEI) o Spain unde g an
MTM2016-75140-P, co- inanced by he Eu opean Communi y und FEDER.
L. L´opez-Somoza was spa ially suppo ed by FPU schola ship, Minis e io de
Educaci´on, Cul u a y Depo e, Spain.
R. Engui¸ca was pa ially suppo ed by Funda¸cao pa a a Ciˆencia e a Tecnologia,
Po ugal, UID/MAT/04561/2013.
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[1] H. Amann; Fixed poin equa ions and Nonlinea P oblems in O de ed Banach Spaces, SIAM
Re iew, Vol. 18, 4 (1976), pp. 620–709.
[2] A. Cabada; The me hod o lowe and uppe solu ions o second, hi d, ou h, and highe
o de bounda y alue p oblems. J. Ma h. Anal. Appl., 185 (1994), 2, 302–320.
[3] A. Cabada; G een’s unc ions in he heo y o o dina y di e en ial equa ions. Sp inge B ie s
in Ma hema ics. Sp inge , New Yo k, 2014.
[4] A. Cabada, J. A. Cid; Exis ence and mul iplici y o solu ions o a pe iodic Hill’s equa ion
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[14] A. Ze l; S u m-Liou ille heo y. Ma hema ical Su eys and Monog aphs, 121. Ame ican
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[15] M. Zhang; Op imal condi ions o maximum and an i-maximum p inciples o he pe iodic
solu ion p oblem, Bound. Value P ob. 2010 (2010), 26 pages.
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Albe o Cabada
Ins i u o de Ma em´
a icas, Facul ade de Ma em´
a icas, Uni e sidade de San iago de Com-
pos ela, 15782, San iago de Compos ela, Galicia, Spain
E-mail add ess:[email p o ec ed]
Rica do Enguic¸a
Depa amen o de Ma em´
a ica, Ins i u o Poli ´
ecnico de Lisboa, Lisboa, Po ugal
E-mail add ess:[email p o ec ed]
Luc´
ıa L´
opez-Somoza
Ins i u o de Ma em´
a icas, Facul ade de Ma em´
a icas, Uni e sidade de San iago de Com-
pos ela, 15782, San iago de Compos ela, Galicia, Spain
E-mail add ess:[email p o ec ed]