Right-indefinite half-linear Sturm Liouville problems

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1 Computes nd Mthemtics with Applictions ) Right-indefinite hlf-line Stum Liouville poblems Lingju Kong, Qingki Kong b, Deptment of Mthemtics, The Univesity of Tennessee t Chttnoog, Chttnoog, TN 37403, USA b Deptment of Mthemtics, Nothen Illinois Univesity, DeKlb, IL 605, USA Received 5 June 2007; eceived in evised fom 4 Septembe 2007; ccepted 0 Octobe 2007 Abstct We study the egul hlf-line Stum Liouville eqution pφ y )) + qφ y) = λwφ y) on J =, b), whee φ u) = u u, > 0, p, q, w L, b), nd p > 0.e. on J. Let Nλ) denote the numbe of zeos in J of nontivil solution of the eqution. Asymptotic fomuls e found fo Nλ) when w 0.e. nd w chnges sign, espectively. As consequence, the existence nd symptotics of el eigenvlues e estblished fo the hlf-line Stum Liouville poblem consisting of the bove eqution nd septed boundy condition when w chnges sign. Ou esults cove the wok of Atkinson nd Mingelli on second-ode line equtions s specil cse. The genelized Püfe tnsfomtion plys key ole in the poofs. c 2007 Elsevie Ltd. All ights eseved. Keywods: Hlf-line; Asymptotics, Stum Liouville poblems; Eigenvlues; Eigenfunctions. Intoduction Fo ny > 0, let φ u) = u u, u R. Conside the hlf-line Stum Liouville poblem SLP) consisting of the eqution pt)φ y )) + qt)φ y) = λwt)φ y) on J :=, b),.) nd the septed boundy condition BC) y) 2 p y )) = 0, 2 yb) 22 p y )b) = 0..2) Coesponding utho. E-mil ddess: kong@mth.niu.edu Q. Kong) /$ - see font mtte c 2007 Elsevie Ltd. All ights eseved. doi:0.06/j.cmw

2 L. Kong, Q. Kong / Computes nd Mthemtics with Applictions ) Thoughout this ppe we ssume without futhe mention tht > 0, λ R, < < b <, i j R, i, j =, 2, such tht , , p, q, w : J R, p, q, w LJ), nd p > 0.e. on J..3) A function yt) is sid to be solution of Eq..) on J if y, pφ )y ) ACJ), nd y stisfy Eq..).e. on J. Hee ACJ) denotes the set of functions which e bsolutely continuous on J. We obseve tht Eq..) hs hlf-line popety, i.e., if yt) is solution of Eq..), then, fo ny c C, cyt) is lso solution of Eq..). If, fo some λ C, Eq..) hs nontivil solution yt) stisfying BC.2), then λ is clled n eigenvlue of SLP.) nd.2), nd yt) is sid to be n eigenfunction of the poblem ssocited with this λ. SLP.) nd.2) is sid to be ight-definite RD) if w > 0 o w < 0).e. on J; ight-semidefinite RSD) if w 0 o w 0).e. on J; nd it is sid to be ight-indefinite RID) if w chnges sign on J, i.e., both sets {t J wt) > 0} nd {t J wt) < 0} hve positive Lebesgue mesues. When =, SLP.) nd.2) educes to the line SLP consisting of the eqution pt)y ) + qt)y = λwt)y on J,.4) nd the septed BC y) 2 py )) = 0, 2 yb) 22 py )b) = 0. The line SLP.4) nd.5) is one of the most impotnt subjects in the clssicl theoy of second-ode line diffeentil equtions, nd hs been extensively studied, see, fo exmple, [ 3] nd the efeences theein. Hlf-line SLPs hve ecently been investigted in the litetue, fo exmple, see [4 2]. Elbet [7] systemticlly studied nd lid sevel foundtions in the theoy of hlf-line SLPs. He developed the genelized tigonometic functions nd the genelized Püfe tnsfomtion nd used them to pove the existence of el eigenvlues fo the RSD Diichlet hlf-line SLPs with q 0. Binding nd Dábek [4] futhe extended Elbet s esults to the genel RD poblems, nd Kong nd Kong [0] discussed the continuous nd discontinuous dependence of the eigenvlues of the RD poblems on the eqution nd on the BC. The zeo count is n impotnt concept in the theoy of Stum Liouville equtions nd SLPs. It is elted to oscilltion, disconjugcy, existence of eigenvlues, nd the symptotics of eigenvlues. Atkinson nd Mingelli [3] deived symptotic fomuls of the zeo counts fo line Stum Liouville equtions. Moe specificlly, they obtined the following: let Nλ) be the numbe of zeos of nontivil solution of Eq..4) in J. i) Assume tht w 0.e. on J. Then Nλ) b λπ ws) ds s λ. ps) ii) Assume tht w chnges sign on J, nd let f + = mx{ f, 0} nd f = mx{ f, 0}. Then Nλ) b ws) ) λ π ds s λ ±, esp. ps) ± Motivted by the wok in [3], in this ppe, we study the zeo counts fo the genel hlf-line Stum Liouville equtions. Let Nλ) be the numbe of zeos of nontivil solution of Eq..) in J. We find symptotic fomuls fo Nλ) when wt) 0.e. on J nd when wt) chnges sign on J, espectively. As pplictions, we estblish the existence nd detemine the symptotics of el eigenvlues of SLP.) nd.2) fo the RID cse. ou ppoch is in the sme diection s in [3] fo the line cse, but moe technicl guments e involved due to the nonline ntue of the poblem. This ppe is ognized s follows. Some peliminy knowledge on genelized tigonometic functions e intoduced in Section 2, nd the min esults e pesented in Section 3, followed by the poofs in Section 4..5)

3 2556 L. Kong, Q. Kong / Computes nd Mthemtics with Applictions ) Genelized tigonometic functions In this section, we summize the bsic knowledge on the genelized tigonometic functions intoduced by Elbet [7], which cn be egded s geneliztions of the stndd tigonometic functions. With thei help, the Püfe tnsfomtion fo the line equtions cn be extended to the hlf-line equtions in ntul wy. We efe the ede to [7] fo moe detils. Let S = Sθ) be the unique solution of the hlf-line diffeentil eqution d dθ φ ds dθ )) + φ S) = 0 stisfying the initil condition dsθ) S0) = 0, dθ =. θ=0 S = Sθ) is clled the genelized sine function nd hs popeties simil to tht of the clssicl sine function sin θ. Fo instnce, it is odd nd peiodic with peiod 2π, whee / π = sin 2π + π +, nd fo k Z, the set of integes, Skπ ) = 0, Sθ) > 0 fo θ 2kπ, 2k + )π ), nd Sθ) < 0 fo θ 2k + )π, 2k + 2)π ). The genelized cosine function Cθ) is defined by Cθ) = dsθ)/dθ. Clely, Cθ) is even nd peiodic with peiod 2π, nd fo k Z, Ck + /2)π ) = 0, Cθ) > 0 fo θ 2k /2)π, 2k + /2)π ), nd Cθ) < 0 fo θ 2k + /2)π, 2k + 3/2)π ). The functions Sθ) nd Cθ) stisfy the eltion tht Sθ) + Cθ) = fo θ R. The genelized tngent function T θ) is defined by T θ) = Sθ) Cθ) fo θ k + /2)π, k Z. It is peiodic function of peiod π nd stisfies T θ) = + T θ) fo θ k + /2)π, k Z. Fo k Z, T θ) is stictly incesing fo θ k /2)π, k + /2)π ) nd T θ) s θ k /2)π + nd T θ) s θ k + /2)π. 3. Min esults The following is esult on the existence of eigenvlues fo the RD hlf-line SLPs obtined by Binding nd Dábek in 2003, see [4, Theoem 3.]. Poposition 3.. Assume tht w > 0.e. on [, b]. Then SLP.) nd.2) hs countbly infinite numbe of el eigenvlues; they e bounded below nd unbounded bove, nd they e ll simple nd cn be odeed to stisfy < λ 0 < λ < λ 2 < < λ n < with λ n. Moeove, ny eigenfunction y n = y n t, λ n ) ssocited with λ n hs exctly n zeos in J. Remk 3.. The poblem whethe o not SLP.) nd.2) hs nonel eigenvlues is still open. Hee, we only conside the el eigenvlues of SLP.) nd.2) odeed s in Poposition 3..

4 L. Kong, Q. Kong / Computes nd Mthemtics with Applictions ) The min pupose of this ppe is to discuss the symptotics of the numbe of zeos of ny nontivil solution of Eq..) in J fo the RID cse. To this end, we fist pesent esult fo the RSD cse. Theoem 3.. Assume tht w 0.e. on J. Let Nλ) be the numbe of zeos of ny nontivil solution of Eq..) in J. Then Nλ) λ π ws) ds s λ. 3.) ps) Note tht when w > 0.e. on J, the eigenfunctions of SLP.) nd.2) ssocited with the nth eigenvlue λ n hs exctly n zeos in J. The following esult on the symptotics of eigenvlues, which hs been poved in [4], is n immedite consequence of Theoem 3.. Coolly 3.. Assume tht w > 0.e. on J. Let λ n be the nth eigenvlue of SLP.) nd.2). Then λ n n π ws) ps) ds ) s n. The next is the min esult in this ppe. Theoem 3.2. Assume tht w chnges sign on J. Let Nλ) be the numbe of zeos of ny nontivil solution of Eq..) in J. Then nd Nλ) λ π w+ s) ds s λ 3.2) ps) Nλ) λ π whee f + = mx{ f, 0} nd f = mx{ f, 0}. w s) ds s λ, 3.3) ps) Ou lst theoem is bout the existence of eigenvlues nd the zeo counts of the ssocited eigenfunctions of SLP.) nd.2) fo the RID cse. Theoem 3.3. Assume tht w chnges sign on J. Then thee exist i, j N 0 := {0,, 2,...} such tht SLP.) nd.2) hs sequence of negtive eigenvlues {λ m } m=i nd sequence of positive eigenvlues {λ+ n } n= j stisfying nd < λ m < < λ i+ < λ i < 0 < λ + j < λ + j+ < < λ+ n < lim m λ m = nd lim n λ+ n = +. Moeove, fo m = i, i +,..., the eigenfunctions y = yt, λ m ) ssocited with λ = λ m hve exctly m zeos in J, nd fo n = j, j +,..., the eigenfunctions y = yt, λ + n ) ssocited λ = λ+ n hve exctly n zeos in J. Futhemoe, λ ± n dmit the symptotic epesenttions λ ± n n π w± s) ps) ds ) s n. 3.4)

5 2558 L. Kong, Q. Kong / Computes nd Mthemtics with Applictions ) Remk 3.2. Ebehd nd Elbet [5] studied specil cse of the indefinite hlf-line SLP.) nd.2) with q 0 nd the Diichlet BC, i.e., the poblem consisting of the eqution nd the BC φ y )) = λwt)φ y) on J, y) = yb) = 0. Unde the condition tht w is continuous nd piecewise diffeentible on [, b] nd othe technicl ssumptions on w, they obtin esult on the existence nd symptotic behvio of el eigenvlues, see Theoem 2.4 in [5]. Clely, ou Theoem 3.3 is n extension of thei esult to the moe genel poblem.) nd.2), nd impoves it even fo the specil cse by eplcing ll technicl equiements fo w by w LJ). 4. Poofs of the min esults It is esy to see tht Eq..) is equivlent to n eqution of the sme fom with p. In fct, let τ = t ds. p s) Then Eq..) becomes d )) dyτ) φ + qτ)φ yτ)) = λ wτ)φ yτ)), dτ dτ 0 τ c, 4.2) whee qτ) = p tτ))qtτ)), wτ) = p tτ))wtτ)), nd c = /p s)ds. Clely, q, w L0, c) whee the integtion is with espect to τ. Convesely, the invese of the function τ defined in 4.) tnsfoms Eq. 4.2) to Eq..). Without loss of genelity, we pove Theoems 3. nd 3.2 fo the cse with p, nd the genel conclusions follow esily fom the tnsfomtion in 4.). We efe the ede to [4, Lemm 4.] fo the following esult nd its poof. Lemm 4.. Assume tht w 0.e. on J. Then fo ech ɛ > 0, thee exists positive function h ACJ) such tht ws) hs) ds < ɛ 4.3) 4.) nd ws)h s) hs) ds < ɛ. 4.4) Poof of Theoem 3.. Without loss of genelity, we ssume tht p. Fo ny ɛ > 0, let h be given in Lemm 4.. Let yt, λ) be nontivil solution of Eq..) nd define the genelized Püfe ngle θt, λ) of yt, λ) to be continuous function stisfying T θt, λ)) = yt, λ)ht)λ y t, λ) nd θ, λ) [0, π ). 4.5) Then, it is esy to see tht θt, λ) = nπ fo some n Z wheneve yt, λ) = 0, nd ) θ t, λ) = λ ht) + λ wt)h t) Sθt, λ)) λ qt)h t) Sθt, λ)) + h t)h t)sθt, λ))φ Cθt, λ))).e. on J. 4.6)

6 L. Kong, Q. Kong / Computes nd Mthemtics with Applictions ) It hs been shown in [4, Lemm 2.3] tht θ inceses t zeos of y. Then fo λ > 0 { π Nλ) = θb, λ) if yb, λ) 0 π θb, λ) if yb, λ) = 0, whee is the getest intege function. Fom 4.6) hs)ds λ θb, λ) = λ θ, λ) + λ qs)h s) Sθs, λ)) ds λ Thus, combining 4.3), 4.4) nd 4.8) we hve w ds λ θb, λ) < 2ɛ + λ ) ws)h s) hs) Sθs, λ)) ds 4.7) h s)h s)sθs, λ))φ Cθs, λ)))ds. 4.8) qs) h s)ds + λ Letting λ in the bove inequlity nd consideing Eq. 4.7), we ech the conclusion. h s) h s)ds + λ π. Poof of Theoem 3.2. Without loss of genelity, we ssume tht p. It suffices to show tht 3.2) holds becuse 3.3) is implied by 3.2) due to the fct tht λwt) = λ) wt)). Fo function h ACJ), which is to be detemined lte, we poceed s in the poof of Theoem 3., i.e., let yt, λ) be nontivil solution of Eq..) nd define the genelized Püfe ngle θt, λ) s in 4.5). Then 4.6) nd 4.7) hold. We need to show tht θb, λ) λ w+ s) ds s λ. By Theoem 3. nd the Stum compison theoem we see tht lim sup λ θb, λ) λ w+ s) Hence, it suffices to show tht fo ny ɛ > 0, thee exists h such tht lim inf λ λ θb, λ) w+ s) ds. 4.9) ds ɛ. 4.0) Fo δ > 0 to be chosen lte, by [4, Theoems.5.2. nd.5.3.], thee exist polynomils α +, α ACJ) such tht nd α + t) 0 nd α t) 0 fo t J, 4.) w+ s) α+ s) ds < b )δ, 4.2) w s) α s) Define the sets J, J 2, J 3, nd by J = {t [, b] : α + t) δ J 2 = ds < b )δ. 4.3) }, {t [, b] : α + t) > δ, α t) > δ, wt) 0 },

7 2560 L. Kong, Q. Kong / Computes nd Mthemtics with Applictions ) } J 3 = {t [, b] : α + t) > δ, α t) > δ, wt) < 0, } = {t [, b] : α + t) > δ, α t) δ. Let h = α + + δ. Since α ± cn be chosen in such wy so tht J nd consist of finite collection of intevls, we my pply 4.6) ove only nd gue tht, in ech of its intevls, θt, λ) cn decese by t most π since it is incesing t t wheneve θt, λ) = nπ fo n Z. Thus λ θb, λ) hs)ds ws)h s) hs) Hence, to pove 4.0), we only need to show tht fo given ɛ > 0 nd h with smll δ, w+ hs)ds ds < ɛ 2 nd ws)h s) hs) ds < ɛ 2. ds + Oλ ). 4.4) 4.5) 4.6) We fist estimte the left-hnd side of 4.5). Since h = α + + δ, by the Hölde inequlity nd fom 4.2), we hve hs) w+ s) ds w+ s) α+ s) ds + b )δ Thus hs)ds which implies tht hs)ds w+ s) w+ s) Fom the definition of J we see tht α + s)ds b )δ. J b ) w+ s) α+ s) ds + b )δ b ) b δ + b )δ = 2b )δ. 4.7) ds 2b )δ + ds 5b )δ + 3 i= 3 i= J i hs)ds, Since w t)/ α t) > δ fo t J 2, fom 4.3) we hve µj 2 ) b )δ, J i α + s)ds. 4.8) 4.9) 4.20) whee µ ) denotes the Lebesgue mesue. Similly, since w + t)/ α + t) > δ fo t J 3, fom 4.2) we hve µj 3 ) b )δ. By the Hölde inequlity nd fom 4.2) nd 4.20) α + s)ds b ) δ 2 α+ s)ds J 2 J 2 4.2)

8 L. Kong, Q. Kong / Computes nd Mthemtics with Applictions ) b ) δ 2 b ) δ 2 α w+ s) +s) 2 α w+ s) +s) + w+ s) b ) δ 2 2 b )δ + 2 w + ds ds + 2 w + s)ds = 2b ) δ b 2 )δ + w +, 4.22) whee denotes the nom in LJ). Similly, by the Hölde inequlity nd fom 4.2) nd 4.2) α + s)ds 2b ) δ b 2 )δ + w ) J 3 Now, combining 4.8), 4.9), 4.22) nd 4.23) we hve w+ s) hs)ds ds 5b )δ + b )δ + 4b ) δ b 2 )δ + w ) We then estimte the left-hnd side of 4.6). Since wt) = w + t) w t), we obtin tht ws)h s) hs) ds w +s)h s) hs) ds + w s)h s) ds w +s)h s) hs) ds + δ w s) ds. 4.25) Note tht w t) = Fom 4.3), w t) w s) ds 2 α t) + α t) w s) 2 b )δ + 2 δµ ) 2 w t) α t) α s) + 2 α s) + 2 α t). 2 b )δδ + ). 4.26) One cn veify tht w +t)h t) Mt) + w+ t), 4.27) whee w+ t) Mt) = h w+ t) t) h t). 4.28) )

9 2562 L. Kong, Q. Kong / Computes nd Mthemtics with Applictions ) It is esy to see tht Mt) h w+ t) t) By the men vlue theoem, Mt) h w+ t) t) w+ t) w+ t) ) ) + h t) + h w+ t) t) h t). 4.29) ) ) ) + h t) + ξt) w+ t), 4.30) whee ξt) is between w + t)/ nd ht). In wht follows, we discuss two cses when 0, ) nd [, ), espectively. Cse. 0, ). If w + t)/) ξt) ht), then fom 4.28) Mt) = w +t)/ ξt) ξt) ht) ) w+ t) If ht) < ξt) < w + t)/), then fom 4.30) Mt) h w+ t) w+ t) t) + h t) ξt) w+ t) h w+ t) t) ) ht) w+ t) + ξt) 2h w+ t) t) 2 w+ t) 2δ w+ t) w+ t) + 4 w+ t) + 4 Combining 4.3) nd 4.32) we see tht fo 0, ) Mt) 2δ w+ t) w+ t) + 5 w+ t) ) + ht) + h t). 4.3) + 2 h t) + h t) Cse 2. [, ). If w + t)/) ξt) ht), then s in Cse, 4.3) holds. If ht) < ξt) < w + t)/), then fom Eq. 4.30) Mt) = h t) ξt) w+ t) w+ t) ) h w+ t) w+ t) t). 4.32). 4.33)

10 = h t) h t) L. Kong, Q. Kong / Computes nd Mthemtics with Applictions ) w+ t) 2 w+ t) = 2 h w+ t) t) 2 δ w+ t) ) w+ t) + ht) + 2 h t) w+ t) Combining 4.3) nd 4.34) we see tht Mt) 2 δ w+ t) + 2 w+ t) + 2 w+ t) Fom both Cses nd 2 we obtin tht fo ny 0, ) Mt) 2 + 2)δ w+ t) Then fom 4.27) + 2 w+ t) + ). 4.34) + 2 w+ t) + + 5). 4.35). w +t)h t) 2 + 2)δ w+ t) + 2 w+ t) + + 6). Thus, since h = α + + δ, by the Minkowski inequlity nd fom Eqs. 4.2) nd 4.7), w +s)h s) hs) ds 2 + 2)δ w+ s) + 2 w+ s) + + 6) 2 + 2)δ w+ s) + δ ds hs) ds hs) ds α+ s) ds )b )δ ) 2 + 2)δ b ) δ + b ) δ )b )δ = )b )δ. 4.36) Combining 4.25), 4.26) nd 4.36) we obtin ws)h s) hs) ds )b )δ + 2 b )δ δ + ). 4.37)

11 2564 L. Kong, Q. Kong / Computes nd Mthemtics with Applictions ) Choose δ > 0 smll enough so tht nd 5b )δ + b )δ + 4b ) δ )b )δ + 2 b )δ δ + ) < ɛ 2. b )δ + w + < ɛ 2 Then, fom 4.24) nd 4.37), we obtin 4.5) nd 4.6). This completes the poof. Poof of Theoem 3.3. BC.2) cn be nomlized to Cα)y) Sα)p y )) = 0, Cβ)yb) Sβ)p y )b) = 0, whee 0 α < π nd 0 < β π. Let y = yt, λ) be the unique solution of the initil vlue poblem consisting of Eq..) nd the initil condition y, λ) = Sα), p )y, λ) = Cα). Define the genelized Püfe ngle θt, λ) of yt, λ) by yt, λ) T θt, λ)) =, θ, λ) = α. p y )t, λ) It is esy to see tht θ t, λ) = p t) Cθt, λ)) + λwt) qt)) Sθt, λ)).e. on J. By [4, Lemm 2.3], θt, λ) inceses t the zeos of y. Then Nλ) stisfies 4.7). By Theoem 3.2, Nλ) s λ. Consequently, so does θb, λ). Theefoe, fo sufficiently lge n, θb, λ) = β + nπ hs solution λ = λ + n nd hence λ + n is n eigenvlue of the SLP.) nd.2) nd ll the coesponding eigenfunctions must hve exctly n zeos in J. An nlogous gument holds fo λ. Finlly, 3.4) follows fom 3.2) nd 3.3). This completes the poof. Refeences [] F.V. Atkinson, Discete nd Continuous Boundy Poblems, Acdemic Pess, New Yok, 964. [2] J. Weidmnn, Spectl Theoy of Odiny Diffeentil Equtions, in: Lectue Notes in Mthemtics, Spinge-Velg, Belin, Heidelbeg, 987. [3] A. Zettl, Stum Liouville Theoy, in: Mthemticl Suveys nd Monogphs, vol. 2, Ameicn Mthemticl Society, [4] P.A. Binding, P. Dábek, Stum Liouville theoy fo the p-lplcin, Studi Sci. Mth. Hung ) [5] W. Ebehd, Á. Elbet, On the eigenvlues of hlf-line boundy vlue poblems, Mth. Nch ) [6] W. Ebehd, Á. Elbet, Hlf-line eigenvlue poblems, Mth. Nch ) [7] Á. Elbet, A hlf-line second ode diffeentil equtions, in: Qulittive theoy f diffeentil equtions, Vol. I, II Szeged, 979), Coll. Mth. Soc. J. Bolyi [8] Á. Elbet, On the hlf-line second ode diffeentil equtions, Act. Mth. Hung ) [9] M.F. Estbooks, J.W. Mcki, A nonline Stum Liouville poblem, J. Diffeentil Equtions [0] L. Kong, Q. Kong, Right-definite hlf-line Stum Liouville poblems, Poc. Royl. Soc. Edinbugh A ) [] T. Kusno, M. Nito, Stum Liouville eigenvlue poblems fo hlf-line odiny diffeentil equtions, Rocky Mountin J. Mth ) [2] H.J. Li, C.C. Yeh, Stumin compison theoem fo hlf-line second-ode diffeentil equtions, Poc. Roy. Soc. Edinbugh 25A 995) [3] F.V. Atkinson, A. Mingelli, Asymptotics of the numbe of zeos nd of the eigenvlues of genelized weighted Stum Liouville poblems, J. Reine Angew. Mth ) [4] G. Szegö, Othogonl Polynomils, vol. XXIII, Ameicn Mthemticl Society Colloquium Publictions, New Yok, 959.