Elec onic Jou nal o Di e en ial Equa ions, Vol. 2023 (2023), No. 65, pp. 1–23.
ISSN: 1072-6691. URL: h ps://ejde.ma h. xs a e.edu, h ps://ejde.ma h.un .edu
DOI: 10.58997/ejde.2023.65
PARAMETER-DEPENDENT PERIODIC PROBLEMS FOR
NON-AUTONOMOUS DUFFING EQUATIONS WITH
SIGN-CHANGING FORCING TERM
JIˇ
R´
Iˇ
SREMR
Abs ac . We s udy he exis ence, exac mul iplici y, and s uc u e o he
se o posi i e solu ions o he pe iodic p oblem
u00 =p( )u+h( )|u|λsgn u+µ ( ); u(0) = u(ω), u0(0) = u0(ω),
whe e µ∈Ris a pa ame e . We assume ha p, h, ∈L([0, ω]), λ > 1, and
he unc ion his non-nega i e. The esul s ob ained ex end he esul s known
in he exis ing li e a u e. We do no equi e ha he G een’s unc ion o he
co esponding linea p oblem be posi i e and we allow he o cing e m o
change i s sign.
1. S a emen o he p oblem
We conside he pe iodic p oblem
u00 =p( )u+h( )|u|λsgn u+µ ( ); u(0) = u(ω), u0(0) = u0(ω),(1.1)
whe e p, h, ∈L([0, ω]), h≥0 a.e. on [0, ω], λ > 1, and µ∈Ris a pa ame e .
By a solu ion o p oblem (1.1), as usual, we unde s and a unc ion u: [0, ω]→R
which is absolu ely con inuous oge he wi h i s i s de i a i e, sa is ies he gi en
equa ion almos e e ywhe e, and mee s he pe iodic condi ions.
In [11], we conside ed p oblem (1.1) wi h µ= 0 and we showed, among o he
hings, ha o he exis ence o a posi i e solu ion i is necessa y ha p6∈ V−(ω)∪
V0(ω). Using a echnique de eloped in [11], we p o ided in [15] e ec i e condi ions
o he exis ence and exac mul iplici y o posi i e solu ions o he pe iodic p oblem
o a non-au onomous Du ing equa ion wi h a sign-changing o cing e m, i.e.,
p oblem (1.1) wi h µ= 1. In he p esen pape , we conclude ou s udies and show,
in he case o p6∈ V−(ω)∪V0(ω), he exis ence/non-exis ence as well as he exac
mul iplici y o sign-cons an solu ions o p oblem (1.1) depending on he choice o
he pa ame e µ. The esul s ob ained a e compa ed wi h he esul s known o
he au onomous case and he esul s a ailable in he exis ing li e a u e.
Fo he esul s co e ing he mul iplici y and local/global bi u ca ions o pe iodic
solu ions o supe -linea equa ions (and hei sys ems), we e e he eade s, o
ins ance, o [1, 2, 3, 4, 6, 8, 12, 13] (see also he e e ences he ein). We s udied
2020 Ma hema ics Subjec Classi ica ion. 34B08, 34C23, 34C25, 34B18.
Key wo ds and ph ases. Pe iodic solu ion; second-o de di e en ial equa ion; exis ence;
Du ing equa ion; mul iplici y; bi u ca ion; posi i e solu ion.
©2023. This wo k is licensed unde a CC BY 4.0 license.
Submi ed No embe 22, 2022. Published Oc obe 5, 2023.
1
2 J. ˇ
SREMR EJDE-2023/65
a bi u ca ion o posi i e solu ions o p oblem (1.1), wi h he non-posi i e unc ion
h, in [14].
In [2], he au ho s s udy he pa ame e -dependen p oblem
x00 +cx0+a( )x−b( )x3=λd( ); x(0) = x(T), x0(0) = x0(T),(1.2)
whe e c > 0, λ∈Ris a pa ame e , and a, b, d: [0, T ]→Ra e con inuous unc ions
such ha
a( )≤π2
T2+c2
4 o ∈[0, T],ZT
0
a(s) ds > 0,(1.3)
and
b( )>0, d( )>0 o ∈[0, T].(1.4)
Theo em 1.1 ([2, Theo em 1.1]).Assume ha (1.3) and (1.4) hold. Then, all
solu ions o (1.2) a e o one sign and he e is λ0>0such ha
(1) p oblem (1.2) has a unique solu ion which is nega i e (posi i e) and uns able
o λ>λ0(λ < −λ0),
(2) p oblem (1.2) has exac ly h ee o de ed solu ions o |λ|<|λ0|. Mo eo e ,
he middle solu ion is asymp o ically s able and he emaining wo a e un-
s able. When −λ0< λ < 0, he maximal solu ion is posi i e and he o he
wo a e nega i e. When λ= 0, p oblem (1.2) has one posi i e, one 0, and
one nega i e solu ion. When 0< λ < λ0, he minimal solu ion is nega i e
and he o he wo a e posi i e.
(3) p oblem (1.2) has exac ly wo one-signed solu ions o λ=±λ0; bo h o
hem a e uns able.
Recen ly, Liang [8] p o ed he conclusion o Theo em 1.1 unde he posi i i y
o a, b, d and he hypo hesis kakp≤(1 + c2)K(2p∗) wi h some p≥1. I seems
om he p oo o Theo em 1.1 ha i s conclusions, which conce n he exis ence
and mul iplici y o solu ions, emain ue e en in he case o c= 0.
In Sec ion 3, we ex end he conclusions o Theo em 1.1 o he case o undamped
Du ing equa ion (i.e., o c= 0). Mo eo e , we weaken hypo heses (1.3) and
(1.4). In pa icula , (1.3) is eplaced by a weake assump ion −a∈ V+(T) (see
De ini ion 2.1), bmay be equal o ze o on a se o posi i e measu e, and dmay
change i s sign so ha (−a, d)∈ U(T) (see De ini ion 2.7). Fu he mo e, we p o e
he exis ence/non-exis ence o solu ions o p oblem (1.2), wi h c= 0, depending on
he choice o he pa ame e λin he case o a( )>π2
T2on a se o posi i e measu e.
A he end o his sec ion, we show, as a mo i a ion, wha happens in he au-
onomous case o (1.1). I p( ) := −a, hen p6∈ V−(ω)∪V0(ω) i and only i a > 0
(see Rema k 2.4). The e o e, we conside he equa ion
x00 =−ax +b|x|λsgn x+µ, (1.5)
whe e a > 0 and b, µ ∈R. In his pape , we a e in e es ed in he equa ion in
(1.1) wi h a non-nega i e hand, hus, we assume ha b > 0 in (1.5). By di ec
calcula ion, he phase po ai s o his equa ion can be elabo a ed depending on
he choice o he pa ame e µand, hus, one can p o e he ollowing p oposi ion
conce ning pe iodic solu ions o equa ion (1.5).
P oposi ion 1.2. Le λ > 1and a, b > 0. Then, he ollowing conclusions hold:
(i) I µ > (λ−1)a
λa
λb 1
λ−1, hen equa ion (1.5) has a unique nega i e equilib-
ium (saddle) and no o he pe iodic solu ions occu .
EJDE-2023/65 PARAMETER-DEPENDENT PERIODIC PROBLEMS 3
(ii) I µ=(λ−1)a
λa
λb 1
λ−1, hen equa ion (1.5) has a unique posi i e equilib-
ium (cusp), a unique nega i e equilib ium (saddle), and no o he pe iodic
solu ions occu .
(iii) I 0< µ < (λ−1)a
λa
λb 1
λ−1, hen equa ion (1.5) possesses exac ly wo posi-
i e equilib ia x1> x2(x1is a saddle and x2is a cen e ), a unique nega i e
equilib ium x3(saddle), and non-cons an (bo h posi i e and sign-changing)
pe iodic solu ions wi h di e en pe iods. Mo eo e , all non-cons an pe i-
odic solu ions oscilla e a ound x2be ween x3and x1.
(i ) I µ= 0, hen equa ion (1.5) possesses a unique posi i e equilib ium x0
(saddle), a i ial equilib ium (cen e ), a unique nega i e equilib ium −x0,
and non-cons an sign-changing pe iodic solu ions wi h di e en pe iods.
Mo eo e , all non-cons an pe iodic solu ions oscilla e a ound 0be ween
−x0and x0.
( ) I −(λ−1)a
λa
λb 1
λ−1< µ < 0, hen equa ion (1.5) possesses exac ly wo neg-
a i e equilib ia x1< x2(x1is a saddle and x2is a cen e ), a unique posi i e
equilib ium x3(saddle), and non-cons an (bo h posi i e and sign-changing)
pe iodic solu ions wi h di e en pe iods. Mo eo e , all non-cons an pe i-
odic solu ions oscilla e a ound x2be ween x1and x3.
( i) I µ=−(λ−1)a
λa
λb 1
λ−1, hen equa ion (1.5) has a unique nega i e equilib-
ium (cusp), a unique posi i e equilib ium (saddle), and no o he pe iodic
solu ions occu .
( ii) I µ < −(λ−1)a
λa
λb 1
λ−1, hen equa ion (1.5) has a unique posi i e equilib-
ium (saddle) and no o he pe iodic solu ions occu .
2. No a ion and de ini ions
The ollowing no a ion is used h oughou his a icle:
•Ris he se o eal numbe s. Fo x∈R, we pu [x]+=1
2(|x|+x) and
[x]−=1
2(|x|−x).
•C(I) deno es he se o con inuous eal unc ions de ined on he in e al
I⊆R. Fo u∈C([a, b]), we pu kukC= max{|u( )|: ∈[a, b]}.
•AC1([a, b]) is he se o unc ions u: [a, b]→Rwhich a e absolu ely con-
inuous oge he wi h hei i s de i a i es.
•AC`([a, b]) ( esp. ACu([a, b])) is he se o absolu ely con inuous unc ions
u: [a, b]→Rsuch ha u0admi s he ep esen a ion u0( ) = γ( ) + σ( ) o
a. e. ∈[a, b], whe e γ: [a, b]→Ris absolu ely con inuous and σ: [a, b]→R
is a non-dec easing ( esp. non-inc easing) unc ion whose de i a i e is equal
o ze o almos e e ywhe e on [a, b].
•L([a, b]) is he Banach space o Lebesgue in eg able unc ions p: [a, b]→R
equipped wi h he no m kpkL=Rb
a|p(s)|ds. The symbol In As ands o
he in e io o he se A⊂L([a, b]).
De ini ion 2.1 ([10, De ini ion 0.1]).We say ha a unc ion p∈L([0, ω]) belongs
o he se V+(ω) ( esp. V−(ω)) i , o any unc ion u∈AC1([0, ω]) sa is ying
u00( )≥p( )u( ) o a.e. ∈[0, ω], u(0) = u(ω), u0(0) = u0(ω),
he inequali y
u( )≥0 o ∈[0, ω] esp. u( )≤0 o ∈[0, ω]
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holds.
Rema k 2.2. In an al e na i e e minology, p∈ V−(ω) ( esp. p∈ V+(ω)) means
ha he maximum p inciple ( esp. he an i-maximum p inciple) holds o he linea
pe iodic p oblem
u00 =p( )u;u(0) = u(ω), u0(0) = u0(ω).(2.1)
De ini ion 2.3 ([10, De ini ion 0.2]).We say ha a unc ion p∈L([0, ω]) belongs
o he se V0(ω)i p oblem (2.1) has a posi i e solu ion.
Rema k 2.4. Le ω > 0. I p( ) := p0 o ∈[0, ω], hen one can show by di ec
calcula ion ha :
Bp∈ V−(ω)i and only i p0>0,
Bp∈ V0(ω)i and only i p0= 0,
Bp∈ V+(ω)i and only i p0∈−π2
ω2,0,
Bp∈In V+(ω)i and only i p0∈−π2
ω2,0.
When he unc ion p∈L([0, ω]) is no cons an , e icien condi ions o p o belong
o each o he se s V+(ω)and V−(ω)a e p o ided in [10] (see also [1, 16]).
Rema k 2.5. I is well known ha , i he homogeneous p oblem (2.1) has only
he i ial solu ion, hen, o any ∈L([0, ω]), he p oblem
u00 =p( )u+ ( ); u(0) = u(ω), u0(0) = u0(ω) (2.2)
possesses a unique solu ion uand his solu ion sa is ies
|u( )| ≤ ∆(p)Zω
0| (s)|ds o ∈[0, ω],
whe e ∆(p), depending only on p, deno es a no m o he G een’s ope a o o p oblem
(2.1). Clea ly, ∆(p)>0.
Assuming ha p∈In V+(ω), we ex end he unc ion ppe iodically o he whole
eal axis deno ing i by he same symbol. I is p o ed in [10, Sec ion 6] ha , o
any a∈R, he p oblem
u00 =p( )u;u(a)=1, u(a+ω)=1
has a unique solu ion uaand ua( )>0 o ∈[0, ω]. We pu
Γ(p) := sup kuakC:a∈[0, ω]eRω
0[p(s)]+ds.(2.3)
I is clea ha Γ(p)≥1.
Rema k 2.6. I p∈ V+(ω), hen he numbe ∆(p) de ined in Rema k 2.5 can be
es ima ed, o example, by he a maximal alue o he G een’s unc ion o p oblem
(2.1) (see, e.g., [16]). On he o he hand, assuming p∈In V+(ω), some es ima es
o he numbe Γ(p) a e p o ided in [10, Sec ion 6].
Fo ins ance, i p( ) := p0 o ∈[0, ω] and p0∈[−π2
ω2,0[ , esp. p0∈−π2
ω2,0,
hen
∆(p)≤2p|p0|sin ωp|p0|
2−1
, esp. Γ(p) = cos ωp|p0|
2−1
.
De ini ion 2.7 ([10, De ini ion 16.1]).Le p, ∈L([0, ω]). We say ha a pai
(p, ) belongs o he se U(ω), i p oblem (2.1) has a unique solu ion which is
posi i e.
EJDE-2023/65 PARAMETER-DEPENDENT PERIODIC PROBLEMS 5
3. Main esul s
This sec ion con ains o mula ions o all he main esul s o he pape . Thei
p oo s a e p esen ed in de ail in Sec ion 5.
We s a wi h he mos gene al s a emen o he pape , which p o ides he
exis ence/non-exis ence esul s in he case o p6∈ V−(ω)∪V0(ω). This condi ion is
sa is ied, o ins ance, i
Zω
0
p(s) ds≤0, p( )6≡ 0
(see Lemma 4.15). No e also ha , o he Du ing equa ion wi h he cons an
coe icien s
x00 +ax −bx3=µ ( ),
he abo e-men ioned condi ion is sa is ied i and only i a > 0.
Theo em 3.1. Le λ > 1,p6∈ V−(ω)∪V0(ω), ( )6≡ 0, and
h( )>0 o a.e. ∈[0, ω].(3.1)
Then, he e exis −∞ ≤ µ∗<0and 0< µ∗≤+∞such ha he ollowing conclu-
sions hold:
(1) Fo any µ∈]µ∗, µ∗[, p oblem (1.1) has a posi i e solu ion u∗such ha
e e y solu ion u o p oblem (1.1) sa is ies
ei he u( )< u∗( ) o ∈[0, ω],o u( )≡u∗( ).(3.2)
Mo eo e , o any couple o dis inc posi i e solu ions u1,u2 o (1.1) sa -
is ying
u1( )6≡ u∗( ), u2( )6≡ u∗( ),(3.3)
he condi ions
min{u1( )−u2( ) : ∈[0, ω]}<0,
max{u1( )−u2( ) : ∈[0, ω]}>0(3.4)
hold.
(2) I µ∗<+∞, hen
(a) o µ>µ∗, p oblem (1.1) has no posi i e solu ion,
(b) o µ=µ∗, p oblem (1.1) has a unique non-nega i e solu ion u∗and
e e y solu ion u o (1.1) sa is ies (3.2).
(3) I µ∗>−∞, hen
(a) o µ<µ∗, p oblem (1.1) has no posi i e solu ion,
(b) o µ=µ∗, p oblem (1.1) has a unique non-nega i e solu ion u∗and
e e y solu ion u o (1.1) sa is ies (3.2).
(4) I Rω
0 (s) ds > 0( esp. Rω
0 (s) ds < 0), hen µ∗<+∞( esp. µ∗>−∞).
Co olla y 3.2. Le λ > 1,p6∈ V−(ω)∪V0(ω), ( )6≡ 0, and condi ion (3.1) hold.
Then, he e exis s 0< µ0<+∞such ha , o any µ∈]−µ0, µ0[, p oblem (1.1)
has a nega i e solu ion u∗and a posi i e solu ion u∗such ha e e y solu ion u o
p oblem (1.1) di e en om u∗,u∗sa is ies
u∗( )< u( )< u∗( ) o ∈[0, ω].(3.5)

6 J. ˇ
SREMR EJDE-2023/65
Rema k 3.3. The conclusions o Theo em 3.1 and Co olla y 3.2 ex end he conclu-
sions o P oposi ion 1.2 o non-au onomous Du ing equa ions wi h a sign-changing
o cing e m. Indeed, le ω > 0 and
p( ) := −a, h( ) := b, ( ) := 1 o ∈[0, ω],
whe e a, b > 0. Then, condi ion (3.1) holds and, by Rema k 2.4, we ob ain p6∈
V−(ω)∪V0(ω). We emphasize, in pa icula , he conclusion o Co olla y 3.2 which
claims ha he e exis s 0 < µ0<+∞such ha , o any µ∈]−µ0, µ0[ , equa ion
(1.5) has a maximal ( esp. a minimal) ω-pe iodic solu ion which is posi i e ( esp.
nega i e); compa e i wi h conclusions (iii), (i ), ( ) o P oposi ion 1.2.
We now p o ide a lowe ( esp. an uppe ) es ima e o he numbe µ∗( esp. µ∗)
appea ing in he conclusion o Theo em 3.1.
P oposi ion 3.4. Le λ > 1,p6∈ V−(ω)∪V0(ω), ( )6≡ 0,hsa is y (3.1), and µ∗,
µ∗be he numbe s appea ing in he conclusion o Theo em 3.1. I [ ( )]+6≡ 0, hen
µ∗≥1
Rω
0[ (s)]+dssup n
∆p+ λ−1h: > 0, p + λ−1h∈ V+(ω)o,(3.6)
and, i [ ( )]−6≡ 0, hen
µ∗≤ − 1
Rω
0[ (s)]−dssup n
∆p+ λ−1h: > 0, p + λ−1h∈ V+(ω)o,(3.7)
whe e ∆is de ined in Rema k 2.5.
Rema k 3.5. Le λ > 1, ω > 0, and
p( ) := −a, h( ) := b o ∈[0, ω],(3.8)
whe e a, b > 0, and
Φ(a, b, λ, ω) := (2ω
π
(λ−1)a
λ(a
λb )1
λ−1i a < λ
λ−1π
ω2,
2π
ω[1
b(a−π2
ω2)] 1
λ−1i a≥λ
λ−1(π
ω)2.
I ollows om he p oo o [15, Co olla y 3.19] ha , i [ ( )]+6≡ 0 and [ ( )]−6≡ 0,
hen
µ∗≥Φ(a, b, λ, ω)
Rω
0[ (s)]+ds, µ∗≤ − Φ(a, b, λ, ω)
Rω
0[ (s)]−ds.
I
( )≥0 o ∈[0, ω], ( )6≡ 0,(3.9)
hen i ollows om [15, Theo em 3.15(3)] ha , o any µ > 0, p oblem (1.1) has a
unique nega i e solu ion. The e o e, he conclusions o Theo em 3.1 can be e ined
as ollows.
Theo em 3.6. Le λ > 1,p6∈ V−(ω)∪ V0(ω)and condi ions (3.1) and (3.9) be
ul illed. Then, he e exis s 0< µ0<+∞such ha he ollowing conclusions hold:
(1) Fo any µ>µ0, p oblem (1.1) has a unique nega i e solu ion u∗and no
posi i e solu ion. Mo eo e , e e y solu ion u o (1.1) sa is ies
ei he u( )> u∗( ) o ∈[0, ω],o u( )≡u∗( ).(3.10)
(2) Fo µ=µ0, p oblem (1.1) has a unique nega i e solu ion u∗and a unique
non-nega i e solu ion u∗. Mo eo e , e e y solu ion u o p oblem (1.1) di -
e en om u∗,u∗sa is ies (3.5).
EJDE-2023/65 PARAMETER-DEPENDENT PERIODIC PROBLEMS 7
(3) Fo µ∈]0, µ0[, p oblem (1.1) has a unique nega i e solu ion u∗and a
posi i e solu ion u∗such ha e e y solu ion u o p oblem (1.1) di e en
om u∗,u∗sa is ies (3.5).
(4) Fo µ= 0, p oblem (1.1) has a unique posi i e solu ion u0, he i ial
solu ion, and a unique nega i e solu ion −u0. Mo eo e , e e y solu ion u
o p oblem (1.1) di e en om u∗,u∗changes i s sign and sa is ies (3.5).
(5) Fo µ∈]−µ0,0[ , p oblem (1.1) has a unique nega i e solu ion u∗and a
posi i e solu ion u∗such ha e e y solu ion u o p oblem (1.1) di e en
om u∗,u∗sa is ies (3.5).
(6) Fo µ=−µ0, p oblem (1.1) has a unique non-posi i e solu ion u∗and a
unique posi i e solu ion u∗. Mo eo e , e e y solu ion u o p oblem (1.1)
di e en om u∗,u∗sa is ies (3.5).
(7) Fo any µ < −µ0, p oblem (1.1) has a unique posi i e solu ion u∗an no
nega i e solu ion. Mo eo e , e e y solu ion u o (1.1) sa is ies (3.2).
Rema k 3.7. I ollows om Theo em 3.1(1) ha , in Theo em 3.6(3,5), i u1,u2
a e dis inc posi i e ( esp. nega i e) solu ions o p oblem (1.1) di e en om u∗
( esp. u∗), hen condi ions (3.4) hold.
Rema k 3.8. Le ω > 0 and
p( ) := −a, h( ) := b, ( ) := 1 o ∈[0, ω],
whe e a, b > 0. Then, condi ions (3.1) and (3.9) hold, p6∈ V−(ω)∪ V0(ω) (see
Rema k 2.4), and all he conclusions o Theo em 3.6 a e in compliance wi h hose
in P oposi ion 1.2.
We showed in [11, Example 2.8] ha assuming p6∈ V−(ω)∪ V0(ω), hypo hesis
(3.1) in Theo ems 3.1 and 3.6 (i.e. he posi i i y o ha. e. on [0, ω]) is essen ial
o he exis ence o a posi i e solu ion o p oblem (1.1) wi h µ= 0 and canno be
weakened o he non-nega i i y o h. Howe e , unde a s onge assump ion on he
coe icien p, namely, p∈ V+(ω), hypo hesis (3.1) o Theo ems 3.1 and 3.6 can be
elaxed o
h( )≥0 o a.e. ∈[0, ω], h( )6≡ 0.(3.11)
Theo em 3.9. Le λ > 1,p∈ V+(ω),hsa is y (3.11), and
(p, )∈ U(ω),Zω
0
(s) ds > 0.(3.12)
Then, he e exis −∞ ≤ µ∗<0and 0< µ∗<+∞such ha he ollowing conclu-
sions hold:
(1) Fo any µ > µ∗, p oblem (1.1) has no posi i e solu ion.
(2) Fo µ=µ∗, p oblem (1.1) has a unique posi i e solu ion u∗and, mo eo e ,
e e y solu ion u o p oblem (1.1) sa is ies (3.2).
(3) Fo µ∈]0, µ∗[, p oblem (1.1) has exac ly wo posi i e solu ions u1,u2and
hese solu ions sa is y
u1( )> u2( )>0 o ∈[0, ω].(3.13)
Mo eo e , e e y solu ion u o p oblem (1.1) di e en om u1is such ha
u( )< u1( ) o ∈[0, ω].(3.14)
(4) Fo µ= 0, p oblem (1.1) has exac ly h ee solu ions: a posi i e solu ion u0,
he i ial solu ion, a nega i e solu ion −u0.
8 J. ˇ
SREMR EJDE-2023/65
(5) Fo µ∈]µ∗,0[ , p oblem (1.1) has ei he one o wo posi i e solu ions.
Mo eo e , (1.1) has a posi i e solu ion u∗such ha e e y solu ion o p ob-
lem (1.1) sa is ies (3.2).
(6) I µ∗>−∞, hen, o any µ<µ∗, p oblem (1.1) has no posi i e solu ion.
Rema k 3.10. Assume ha hypo heses o Theo em 3.9 hold and µ∗>−∞. I ,
mo eo e , h( )>0 o a. e. ∈[0, ω], hen i ollows om Theo em 3.1(3b) ha
p oblem (1.1) wi h µ=µ∗has a unique non-nega i e solu ion u∗and, mo eo e ,
e e y solu ion o (1.1) wi h µ=µ∗sa is ies (3.2).
Open ques ions. The ollowing wo ques ions emain open in Theo em 3.9:
(1) Does he inequali y µ∗>−∞ hold wi hou any addi ional assump ion?
(2) Wha happens in he case o µ=µ∗, i µ∗>−∞ and h( ) = 0 on a se o
posi i e measu e?
Rema k 3.11. I is p o ed in [10, Theo em 16.4] ha , i p∈In V+(ω), hen he
inclusion (p, )∈ U(ω) holds o e e y unc ion ∈L([0, ω]) sa is ying ( )6≡ 0
and
Zω
0
[ (s)]+ds≥Γ(p)Zω
0
[ (s)]−ds,
whe e Γ is gi en by (2.3).
On he o he hand, i p∈ V+(ω) and sa is ies (3.9), hen (p, )∈ U(ω) as well
(see [10, Rema k 9.2]).
Rema k 3.12. In [1], o show a possible use o he main esul s, he au ho s
conside he pa ame e -dependen pe iodic p oblem o he o ced Ma hieu-Du ing
equa ion
z00 =−(e+bcos( ))z+νz3+c( ); z(0) = z(2π), z0(0) = z0(2π),(3.15)
whe e e≥0 and b∈Ra e such ha e+|b|>0 and
k[e+bcos(·)]+kLα≤max K(2α∗,2π) : α≥1,
Kis he so-called bes Sobole cons an , csa is ies −(e+bcos(·)), c∈ U(2π), and
ν∈Ris a pa ame e . I is p o ed in [1, Co olla y 45] ha he e exi s ν0>0such
ha p oblem (3.15) has a leas wo posi i e solu ions p o ided ha 0< ν < ν0.
Pu ing u( ) := √ν z( ), p oblem (3.15) is equi alen , in some sense, wi h p oblem
(1.1) in which p( ) := −(e+bcos( )), h( ) := 1, ( ) := c( ), λ:= 3, and µ:= √ν.
Since −(e+bcos(·)) ∈ V+(ω) in he case conside ed, Theo em 3.9 complemen s
he conclusion o [1, Co olla y 45] as ollows: The e exis s ν0>0such ha p oblem
(3.15) has exac ly wo posi i e solu ions p o ided ha 0< ν < ν0, a unique posi i e
solu ion p o ided ha ν=ν0, and no posi i e solu ion p o ided ha ν > ν0.
Theo em 3.9 gua an ees he exis ence o ce ain “c i ical” alues µ∗,µ∗o he
pa ame e µsuch ha c ossing hese alues, a bi u ca ion o posi i e solu ions o
p oblem (1.1) occu s. F om an applica ion poin o iew, he es ima es o hese
numbe s a e also needed.
P oposi ion 3.13. Le λ > 1,p∈In V+(ω),hsa is y (3.11), and
Zω
0
[ (s)]+ds > Γ(p)Zω
0
[ (s)]−ds > 0,(3.16)
EJDE-2023/65 PARAMETER-DEPENDENT PERIODIC PROBLEMS 9
whe e Γis gi en by (2.3). Then, he numbe s µ∗,µ∗appea ing in he conclusion
o Theo em 3.9 sa is y
µ∗≤ − (λ−1) [∆(p)]−λ
λ−1
λλRω
0h(s) ds1
λ−1Rω
0[ (s)]−ds
,(3.17)
µ∗≥(λ−1) [∆(p)]−λ
λ−1
λλRω
0h(s) ds1
λ−1Rω
0[ (s)]+ds
,(3.18)
whe e ∆is de ined in Rema k 2.5, and
µ∗<(λ−1)[Γ(p)Rω
0[p(s)]−ds−Rω
0[p(s)]+ds]λ
λ−1
λ[λRω
0h(s) ds]1
λ−1Rω
0[ (s)]+ds−Γ(p)Rω
0[ (s)]−ds.(3.19)
I he o cing e m is non-nega i e, hen, simila ly as in Theo em 3.6, he
conclusions o Theo em 3.9 can be ex ended as ollows.
Theo em 3.14. Le λ > 1,p∈ V+(ω), and condi ions (3.9) and (3.11) be ul illed.
Then, he e exis s 0< µ0<+∞such ha he ollowing conclusions hold:
(1) Fo any µ > µ0, p oblem (1.1) has a unique solu ion which is nega i e.
(2) Fo µ=µ0, p oblem (1.1) has exac ly wo solu ions: one posi i e and one
nega i e.
(3) Fo µ∈]0, µ0[, p oblem (1.1) has exac ly h ee solu ions u1,u2,u3and
hese solu ions sa is y
u1( )> u2( )>0, u3( )<0 o ∈[0, ω].
(4) Fo µ= 0, p oblem (1.1) has exac ly h ee solu ions: a posi i e solu ion u0,
he i ial solu ion, a nega i e solu ion −u0.
(5) Fo µ∈]−µ0,0[ , p oblem (1.1) has exac ly h ee solu ions u1,u2,u3and
hese solu ions sa is y
u1( )< u2( )<0, u3( )>0 o ∈[0, ω].
(6) Fo µ=−µ0, p oblem (1.1) has exac ly wo solu ions: one posi i e and one
nega i e.
(7) Fo any µ < −µ0, p oblem (1.1) has a unique solu ion which is posi i e.
Rema k 3.15. Theo em 3.14 ex ends he conclusions o Theo em 1.1 o he case
o c= 0 and con i ms a conjec u e o mula ed in [2, Rema k 3, p. 2502] because,
a leas in case o c= 0, he conclusions o Theo em 1.1 (excep o he asymp o ic
s abili y) a e s ill ue o dwhich changes i s sign (and belongs o a ce ain class
o unc ions).
We inally p o ide he uppe and lowe es ima es o he numbe µ0appea ing in
Theo em 3.14, which ollow immedia ely om P oposi ion 3.13.
P oposi ion 3.16. Le λ > 1,p∈In V+(ω), and condi ions (3.9) and (3.11)
hold. Then, he numbe µ0appea ing in he conclusion o Theo em 3.14 sa is ies
µ0≥(λ−1) [∆(p)]−λ
λ−1
λλRω
0h(s) ds1
λ−1Rω
0 (s) ds
,
16 J. ˇ
SREMR EJDE-2023/65
α2(0) = α2(ω), α0
2(0) = α0
2(ω),(4.33)
α00
2( ) = p( )α2( ) + µ∗
µλ−1h( )αλ
2( ) + µ ( )
≥p( )α2( ) + h( )αλ
2( ) + µ ( ) o a.e. ∈[0, ω],
(4.34)
meas  ∈[0, ω] : α00
2( )> p( )α2( ) + h( )αλ
2( ) + µ ( )>0,(4.35)
because 0 < µ < µ∗and hsa is ies (3.11).
The e o e, Lemma 4.3(2) (wi h α( ) := α2( )), p oblem (1.1) has a solu ion α1
such ha
α1( )≥α2( ) o ∈[0, ω].(4.36)
Consequen ly, he unc ions α1,α2sa is y condi ions (4.5) and (4.6). We inally
show ha (4.4) is ul illed as well. Suppose on he con a y ha (4.4) does no
hold. Ex end he unc ions p,h, ,α1,α2pe iodically o he whole eal axis
deno ing hem by he same symbols. Then, in iew o (4.32) and (4.36), he e
exis s a∈[0, ω[ such ha
α1(a) = α2(a), α0
1(a) = α0
2(a).(4.37)
Pu
w( ) := α1( )−α2( ) o ∈[a, a +ω],
ϕ( ) := gα1( ), α2( ) o ∈[a, a +ω],
whe e
g(x, y) := (xλ−yλ
x−y o x, y ∈R, x 6=y,
λ|x|λ−1sgn x o x, y ∈R, x =y.
I is no di icul o e i y ha g:R2→Ris a con inuous unc ion and, hus, he
unc ion ϕis con inuous and non-nega i e on [a, a +ω]. By (4.36) and (4.37), w
sa is ies (4.31) wi h b=a+ω. Since α1is a solu ion o p oblem (1.1) and α2
sa is ies (4.34), we ha e
w00( )≤p( )w( ) + h( )αλ
1( )−αλ
2( )
≤|p( )|+h( )ϕ( )w( ) o a.e. ∈[a, a +ω].
The e o e, Lemma 4.11 (wi h `( ) := |p( )|+h( )ϕ( ) and b:= a+ω) yields w( )≡0,
i., e., α1( )≡α2( ). Howe e , his con adic s condi ion (4.35), because α1is a
solu ion o p oblem (1.1). 
Lemma 4.13. Le λ > 1,µ∗>0,p, h, ∈L([0, ω]),hsa is y (3.11), and he e
exis unc ions α1, α2∈AC1([0, ω]) such ha (4.5) wi h µ=µ∗and (4.6) hold and
0≤α2( )< α1( ) o ∈[0, ω].(4.38)
Then, he e exis µ>µ∗and a posi i e unc ion α∈AC1([0, ω]) sa is ying (4.1)
wi h µ=µ∗and (4.2).
The p oo o he abo e lemma is simila o he p oo o [14, Lemma 4.9] and
hus, i is omi ed.
Lemma 4.14. Le λ > 1,µ∗∈R,p6∈ V−(ω)∪V0(ω), and hsa is y (3.1). Then,
o any c > 0, he e exis s a unc ion β∈AC 1([0, ω]) such ha
β00( )≤p( )β( ) + h( )βλ( ) + µ∗ ( ) o a.e. ∈[0, ω],(4.39)

EJDE-2023/65 PARAMETER-DEPENDENT PERIODIC PROBLEMS 17
β(0) = β(ω), β0(0) = β0(ω),(4.40)
β( )≥c o ∈[0, ω].(4.41)
P oo . Pu
q0( , x) := h( )|x|λ−1 o a.e. ∈[0, ω] and all x∈R.
Since limx→+∞xλ−1REh(s) ds= +∞ o e e y E⊆[0, ω], meas E > 0, i ollows
om [15, Lemma 4.15] ha he e exis s R > 0 such ha p+q0(·, R)∈ V−(ω).
The e o e, he conclusion o he lemma ollows om [15, P oposi ion 4.21] (wi h
q( , x) := q0( , x) and x0:= 0). 
Lemma 4.15 ([10, P oposi ion 10.8, Rema k 0.7]).I p∈ V−(ω)∪ V0(ω), hen
ei he Rω
0p(s) ds > 0o p( )≡0.
5. P oo s o main esul s
P oo o Theo em 3.1. Pu
A:= µ∈R: p oblem (1.1) has a posi i e solu ion.(5.1)
In iew o Lemmas 4.1(1) and 4.2, he e exis s ε > 0 such ha ]−ε, ε[∩A 6=∅. Le
µ∗:= in A, µ∗:= sup A.(5.2)
Then, −∞ ≤ µ∗<0 and 0 < µ∗≤+∞.
Conclusion (1):Le µ0∈ A {0}be a bi a y and µ∈Rbe such ha 0 <|µ| ≤ |µ0|
and sgn µ= sgn µ0. Le , mo eo e , u0be a posi i e solu ion o p oblem (1.1) wi h
µ=µ0. Pu
α( ) := µ
µ0
u0( ) o ∈[0, ω].(5.3)
Clea ly, α( )>0 o ∈[0, ω]. I ollows om (1.1) wi h µ=µ0 ha αsa is ies
(4.2) and
α00( ) = p( )α( ) + µ0
µλ−1h( )αλ( ) + µ ( )
≥p( )α( ) + h( )αλ( ) + µ ( ) o a.e. ∈[0, ω],
(5.4)
because |µ0|≥|µ|>0 and (3.1) holds. The e o e, Lemma 4.1(1) yields µ∈ A.
Consequen ly, ]µ∗, µ∗[⊆ A and, hus, conclusion (1) o he heo em ollows om
Lemma 4.1(1).
Conclusion (2):Assume ha µ∗<+∞. Then, i ollows immedia ely om (5.1)
and (5.2) ha conclusion (2a) o he heo em holds.
Le {µn}∞
n=1 be a sequence o posi i e numbe s such ha
µn∈ A o n∈N,lim
n→+∞µn=µ∗(5.5)
and, o any n∈N, le unbe a solu ion o p oblem (1.1) wi h µ=µn. Lemma 4.10
yields (4.13). By he s anda d a gumen s using in he p oo o a well-possedness
o he pe iodic p oblems o second-o de ODEs, one can show ha he e exis s a
subsequence {unk}∞
k=1 o {un}∞
n=1 such ha
lim
k→+∞u(i)
nk( ) = (u∗)(i)( ) uni o mly on [0, ω], i = 0,1,(5.6)
18 J. ˇ
SREMR EJDE-2023/65
whe e u∗∈AC1([0, ω]) is a solu ion o p oblem (1.1) wi h µ=µ∗. All he unc ions
unka e posi i e and, hus, i is clea ha
u∗( )≥0 o ∈[0, ω].
We now p o e ha u∗is a unique non-nega i e solu ion o p oblem (1.1) wi h
µ=µ∗. Suppose on he con a y ha u∗is a non-nega i e solu ion o (1.1) wi h
µ=µ∗such ha
u∗(ξ)6=u∗(ξ) o some ξ∈[0, ω].(5.7)
Pu
α( ) := max u∗( ), u∗( ) o ∈[0, ω].
I is no di icul o e i y ha α∈AC`([0, ω]), condi ion (4.2) wi h µ=µ∗holds,
and
α(a) = α(ω), α0(a)≥α0(ω).(5.8)
Le us show ha
α( )>0 o ∈[0, ω].(5.9)
I his condi ion does no hold, hen, in iew o he non-nega i i y o u∗,u∗, he e
exis s 0∈[0, ω] such ha
u∗( 0)=0, u∗( 0)=0.(5.10)
Ex end he unc ions p,h, ,u∗,u∗pe iodically o he whole eal axis deno ing
hem by he same symbols. Then, using (5.10) and he non-nega i i y o u∗,u∗,
we ob ain
u0
∗( 0)=0,(u∗)0( 0)=0.(5.11)
Since he unc ion x7→ |x|λsgn xis Lipschi z on e e y compac in e al, o any
c1, c2∈R, he Cauchy p oblem
u00 =p( )u+h( )|u|λsgn u+µ∗ ( ); u( 0) = c1, u0( 0) = c2(5.12)
is uniquely sol able. The e o e, (5.10) and (5.11) yield u∗( )≡u∗( ), which con-
adic s (5.7). Hence, (5.9) holds. Now, in iew o (4.2) wi h µ=µ∗, (5.8), and
(5.9), i ollows om Lemma 4.1(1) ha p oblem (1.1) wi h µ=µ∗has a posi i e
solu ion ˜u∗such ha
0≤u∗( )<˜u∗( ) o ∈[0, ω] o 0 ≤u∗( )<˜u∗( ) o ∈[0, ω].
The e o e, Lemma 4.13 gua an ees ha he e exis ˜µ > µ∗and a posi i e unc ion
˜α∈AC1([0, ω]) sa is ying
˜α00( )≥p( )˜α( ) + h( )˜αλ( ) + ˜µ ( ) o a.e. ∈[0, ω],(5.13)
˜α(0) = ˜α(ω),˜α0(0) = ˜α0(ω).(5.14)
Consequen ly, i ollows om Lemma 4.1 (wi h α( ) := ˜α( ) and µ:= ˜µ) ha
p oblem (1.1) wi h µ= ˜µhas a leas one posi i e solu ion, which con adic s
he abo e-p o ed conclusion (2a). The con adic ion ob ained p o es ha u∗is a
unique non-nega i e solu ion o p oblem (1.1) wi h µ=µ∗.
I emains o show ha e e y solu ion u o p oblem (1.1) wi h µ=µ∗sa is ies
(3.2). Indeed, suppose on he con a y ha uis a solu ion o p oblem (1.1) wi h
µ=µ∗such ha (3.2) does no hold. We ha e men ioned abo e ha , o any
0∈[0, ω] and c1, c2∈R, he Cauchy p oblem (5.12) is uniquely sol able and, hus,
he solu ion usa is ies
max u( )−u∗( ) : ∈[0, ω]>0.(5.15)
EJDE-2023/65 PARAMETER-DEPENDENT PERIODIC PROBLEMS 19
Pu
α( ) := max u( ), u∗( ) o ∈[0, ω].(5.16)
I is no di icul o e i y ha α∈AC`([0, ω]) and condi ions (4.2) wi h µ=µ∗and
(5.8) hold. Mo eo e , i ollows om Lemma 4.14 ha he e exis s β∈AC 1([0, ω])
sa is ying (4.39), (4.40), and
β( )≥α( ) o ∈[0, ω].(5.17)
The e o e, by (4.2) wi h µ=µ∗, (4.39), (4.40), (5.8), and (5.17), we conclude
ha αand β o m a well-o de ed pai o lowe and uppe unc ions and, hus,
p oblem (1.1) wi h µ=µ∗has a solu ion ˆusuch ha
α( )≤ˆu( )≤β( ) o ∈[0, ω].
Howe e , his condi ion, oge he wi h (5.15) and (5.16), implies ha ˆuis a non-
nega i e solu ion o p oblem (1.1) wi h µ=µ∗di e en om u∗, which con adic s
he abo e-p o ed ac conce ning he uniqueness o he non-nega i e solu ion u∗.
Conclusion (3):I can be p o ed in much he same way as conclusion (2) consid-
e ing −µand − ins ead o µand .
Conclusion (4):I ollows immedia ely om Lemma 4.9. 
P oo o Co olla y 3.2. I is clea ha uis a solu ion o p oblem (1.1) i and only
i −uis a solu ion o p oblem (4.10). The e o e, he conclusion o he co olla y
ollows om Theo em 3.1(1). 
P oo o P oposi ion 3.4. Le µ∗,µ∗be he numbe s appea ing in he conclusion o
Theo em 3.1.
Assume ha [ ( )]+6≡ 0 and suppose on he con a y ha (3.6) does no hold,
i.e.,
µ∗<1
Rω
0[ (s)]+dssup 
∆p+ λ−1h: > 0, p + λ−1h∈ V+(ω),
whe e ∆ is de ined by Rema k 2.5. Then, µ∗∈]0,+∞[ and he e exis s ε > 1 such
ha
Zω
0
[εµ∗ (s)]+ds < sup 
∆p+ λ−1h: > 0, p + λ−1h∈ V+(ω).
The e o e, om Lemmas 4.2 and 4.1(1) ha p oblem (1.1) wi h µ=εµ∗has a
leas one posi i e solu ion, which con adic s conclusion (2a) o Theo em 3.1.
Assuming [ ( )]−6≡ 0, es ima e (3.7) can be p o ed analogously o (3.6). 
P oo o Theo em 3.6. I ollows om Theo em 3.1 and Lemmas 4.6 and 4.9 ha
he e exis s µ0∈]0,∞[ such ha conclusions (1), (2), and (3) o he heo em hold.
Since uis a solu ion o p oblem (1.1) i and only i −uis a solu ion o p oblem (4.10),
conclusions (5), (6), and (7) o he heo em hold as well. Finally, conclusion (4) o
he heo em ollows om Lemma 4.6 and he abo e-men ioned equi alence. 
P oo o Theo em 3.9. Le he se Abe gi en by o mula (5.1). In iew o Lem-
mas 4.3(2,3) and 4.4(1), he e exis s ε > 0 such ha ] −ε, ε[∩A 6=∅. De ine he
numbe s µ∗and µ∗by (5.2). Then, −∞ ≤ µ∗<0, µ∗>0, and Lemma 4.8 implies
ha µ∗<+∞.
Conclusion (1):I ollows om (5.1), (5.2), and he condi ion µ∗∈]0,+∞[ .
20 J. ˇ
SREMR EJDE-2023/65
Conclusion (2):We i s show ha
µ∗∈ A.(5.18)
Indeed, le {µn}∞
n=1 be a non-dec easing sequence o posi i e numbe s such ha
µn∈ A o n∈N,lim
n→+∞µn=µ∗.
Mo eo e , o any n∈N, le unbe a posi i e solu ion o p oblem (1.1) wi h µ=µn.
I ollows om Lemma 4.8 ha condi ion (4.13) holds. By he s anda d a gumen s
using in he p oo o a well-possedness o he pe iodic p oblems o second-o de
ODEs, one can show ha he e exis s a subsequence {unk}∞
k=1 o {un}∞
n=1 such
ha (5.6) is sa is ied, whe e u∗∈AC1([0, ω]) is a solu ion o p oblem (1.1) wi h
µ=µ∗. Since he unc ions unk,k∈N, a e posi i e, i is clea ha
u∗( )≥0 o ∈[0, ω].(5.19)
In iew o he hypo hesis (p, )∈ U(ω) and he posi i i y o µ∗, p oblem (4.23) has
a unique solu ion , which is posi i e. By (1.1) wi h µ=µ∗, (3.11), (4.23), and
(5.19), we ob ain
z00( )≥p( )z( ) o a.e. ∈[0, ω], z(0) = z(ω), z0(0) = z0(ω),
whe e z( ) := u∗( )−µ∗ ( ) o ∈[0, ω]. The e o e, he hypo hesis p∈ V+(ω)
yields z( )≥0 o ∈[0, ω]. Hence, we ha e
u∗( )≥µ∗ ( )>0 o ∈[0, ω]
and, hus condi ion (5.18) holds.
Since u∗is a posi i e solu ion o p oblem (1.1) wi h µ=µ∗, in iew o Lemma
4.3(2), o p o e conclusion (2) o he heo em, i is su icien o show ha p oblem
(1.1) wi h µ=µ∗does no ha e mo e han one posi i e solu ion. Suppose on he
con a y ha p oblem (1.1) wi h µ=µ∗has a posi i e solu ion di e en om u∗.
Then, i ollows om Lemma 4.3(2) (wi h α( ) := u∗( ) and µ:= µ∗) ha p oblem
(1.1) wi h µ=µ∗possesses solu ions ˜u∗, ˜u∗such ha
˜u∗( )>˜u∗( )>0 o ∈[0, ω].
The e o e, Lemma 4.13 (wi h α1( ) := ˜u∗( ) and α2( ) := ˜u∗( )) gua an ees ha
he e exis ˜µ > µ∗and a posi i e unc ion ˜α∈AC1([0, ω]) sa is ying (5.13) and
(5.14). Consequen ly, by Lemma 4.1(1) (wi h α( ) := ˜α( ) and µ:= ˜µ), we conclude
ha p oblem (1.1) wi h µ= ˜µhas a leas one posi i e solu ion, which con adic s
he abo e-p o ed conclusion (1).
Conclusions (3):Ha ing a posi i e solu ion u∗ o p oblem (1.1) wi h µ=µ∗, i
is clea ha all he hypo heses o Lemma 4.12 (wi h α( ) := u∗( )) a e ul illed.
Consequen ly, o any µ∈]0, µ∗[ , (p, µ )∈ U(ω) and he e exis unc ions α1, α2∈
AC1([0, ω]) sa is ying condi ions (4.4), (4.5), and (4.6) and, he e o e, conclusion
(3) o he heo em ollows om Lemma 4.3(3).
Conclusion (4):I ollows immedia ely om [11, Co olla y 2.31(2)].
Conclusion (5):Le µ0∈ A∩]−∞,0[ and µ∈[µ0,0[ be a bi a y and le u0be
a posi i e solu ion o p oblem (1.1) wig h µ=µ0. De ine he unc ion αby (5.3).
Clea ly, α( )>0 o ∈[0, ω]. I ollows om (1.1) wi h µ=µ0 ha αsa is ies
(4.2) and (5.4), because µ0≤µ < 0 and (3.11) holds. The e o e, Lemma 4.3(2)
yields µ∈ A. Consequen ly, ]µ∗,0[ ⊆ A and, hus, conclusion (5) o he heo em
ollows om Lemma 4.3(1,2).
EJDE-2023/65 PARAMETER-DEPENDENT PERIODIC PROBLEMS 21
Conclusion (6):Assume ha µ∗>−∞. Then, i ollows immedia ely om (5.1)
and (5.2) ha , o any µ < µ∗, p oblem (1.1) has no posi i e solu ion. 
P oo o P oposi ion 3.13. By Rema k 3.11, i ollows om (3.16) ha condi ion
(3.12) holds. Le µ∗,µ∗be he numbe s appea ing in he conclusion o Theo em 3.9.
We i s show ha µ∗sa is ies (3.17), whe e ∆ is de ined in Rema k 2.5. Suppose
on he con a y ha (3.17) does no hold, i.e.,
µ∗>−(λ−1) [∆(p)]−λ
λ−1
λλRω
0h(s) ds1
λ−1Rω
0[ (s)]−ds
.
Then, µ∗∈]−∞,0[ and he e exis s ε > 1 such ha
0<Zω
0
[εµ∗ (s)]+ds=−εµ∗Zω
0
[ (s)]−ds≤(λ−1) [∆(p)]−λ
λ−1
λλRω
0h(s) ds1
λ−1
.
The e o e, i ollows om Lemmas 4.4(1) and 4.3(2) ha p oblem (1.1) wi h
µ=εµ∗has a posi i e solu ion, which con adic s conclusion (6) o Theo em 3.9.
Now we show ha µ∗sa is ies (3.18), whe e ∆ is de ined in Rema k 2.5. Suppose
on he con a y ha (3.18) does no hold, i.e.,
µ∗<(λ−1) [∆(p)]−λ
λ−1
λλRω
0h(s) ds1
λ−1Rω
0[ (s)]+ds
.(5.20)
By he condi ions (p, )∈ U(ω) and µ∗>0, we ob ain (p, µ∗ )∈ U(ω). The e o e,
in iew o (5.20), i ollows om Lemmas 4.4(2) and 4.3(3) ha p oblem (1.1)
wi h µ=µ∗has exac ly wo posi i e solu ions, which con adic s conclusion (2) o
Theo em 3.9.
We inally show ha µ∗sa is ies (3.19), whe e Γ is gi en by (2.3). Suppose on
he con a y ha (3.19) does no hold, i.e.,
µ∗≥(λ−1)[Γ(p)Rω
0[p(s)]−ds−Rω
0[p(s)]+ds]λ
λ−1
λλRω
0h(s) ds1
λ−1[Rω
0[ (s)]+ds−Γ(p)Rω
0[ (s)]−ds]
.
Then, i ollows om Lemma 4.5 ha p oblem (1.1) wi h µ=µ∗has no posi i e
solu ion, which con adic s conclusion (2) o Theo em 3.9. 
P oo o Theo em 3.14. We i s no e ha , by Rema k 3.11, condi ion (3.12) holds.
The e o e, i ollows om Theo em 3.9(1,2,3) and Lemmas 4.6 and 4.7 ha he e
exis s µ0∈]0,+∞[ such ha conclusions (1), (2), and (3) o he heo em hold.
Since uis a solu ion o p oblem (1.1) i and only i −uis a solu ion o p oblem
(4.10), conclusions (5), (6), and (7) o he heo em hold as well. Finally, he alidi y
o conclusion (4) o he heo em ollows immedia ely om Theo em 3.9(4). 
6. Conclusions
The exis ence and exac mul iplici y o solu ions o p oblem (1.1) was s udied
depending on he choice o he pa ame e µ. We ex ended he conclusions s a ed in
[2, Theo em 1.1] o he case o undamped Du ing equa ion (1.2) wi h c:= 0 and
weakened hypo heses (1.3) and (1.4). Ou esul s con i m a conjec u e o mula ed
in [2, Rema k 3, p. 2502] because, a leas in he case o c= 0, he conclusions o
Theo em 1.1 (excep o he asymp o ic s abili y) a e s ill ue o dwhich changes
i s sign (and belongs o a ce ain class o unc ions). We also p o ided bo h lowe

22 J. ˇ
SREMR EJDE-2023/65
and uppe es ima es o he “c i ical” alues µ∗,µ∗( esp. µ0) o he pa ame e µ
appea ing in he conclusions o Theo ems 3.1 and 3.9 ( esp. Theo ems 3.6 and 3.14).
The app oach used in [2] employs iden i ying he old poin on bi u ca ion cu es
and he con inua ion me hod combined wi h he S u m’s compa ison heo em, opo-
logical deg ee, and he maximum p inciple. We used a sligh ly di e en app oach;
we p o ed ou esul s by using he me hod o lowe and uppe unc ions only, which
was combined wi h he he maximum and an i-maximum p inciples. The esul s
ob ained subs an ially gene alize he esul s a ailable in he li e a u e because hey
a e no only speci ic su icien condi ions. Ou gene al esul s hold o all he equa-
ions o he ype s udied whose coe icien in he linea pa belongs o a ce ain
su icien ly wide class o unc ions. Such a class is desc ibed in e ms o he beha -
io o he co esponding linea pe iodic p oblem and does no exclude he so-called
esonan cases.
Finally, i is wo h men ioning ha i he esul s conce ning he maximum and
an i-maximum p inciples a e known o he pe iodic linea p oblem
u00 =p( )u+g( )u0;u(0) = u(ω), u0(0) = u0(ω)
wi h p, g ∈L([0, ω]), he pa ame e -dependen p oblem
u00 =p( )u+g( )u0+h( )|u|λsgn u+µ ( ); u(0) = u(ω), u0(0) = u0(ω)
migh be also s udied in a simila way as (1.1). The i s s eps a e al eady done o
he Du ing equa ion wi h a cons an damping coe icien g(see, e. g., [2, 8]).
Acknowledgmen s. This esea ch was suppo ed by he in e nal g an FSI-S-20-
6187 o FME BUT.
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F ench.
Jiˇ
´
ıˇ
S em
Ins i u e o Ma hema ics, Facul y o Mechanical Enginee ing, B no Uni e si y o
Technology, Technick´
a 2, 616 69 B no, Czech Republic
Email add ess:[email p o ec ed]