Parameter-dependent periodic problems for non-autonomous Duffing equations with sign-changing forcing term

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] 4 J. ˇ SREMR EJDE-2023/65 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. Re e ences [1] A. Cabada. J. ´ A. Cid, L. L´opez-Somoza; Maximum p inciples o he Hill’s equa ion, Aca- demic P ess, London, 2018. [2] H. Chen, Y. 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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]