Hndw Publshng Corporton Interntonl Journl of Dfferentl Equtons Volume 0, Artcle ID 7703, pges do:055/0/7703 Reserch Artcle On the Upper Bounds of Egenvlues for Clss of Systems of Ordnry Dfferentl Equtons wth Hgher Order Go J, L-N Hung, nd We Lu College of Scence, Unversty of Shngh for Scence nd Technology, Shngh 00093, Chn Correspondence should be ddressed to Go J, goj79@yhoocomcn Receved 4 My 0; Revsed 6 July 0; Accepted 9 July 0 Acdemc Edtor: Bshr Ahmd Copyrght q 0 Go J et l Ths s n open ccess rtcle dstrbuted under the Cretve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, nd reproducton n ny medum, provded the orgnl work s properly cted The estmte of the upper bounds of egenvlues for clss of systems of ordnry dfferentl equtons wth hgher order s consdered by usng the clculus theory Severl results bout the upper bound nequltes of the n th egenvlue re obtned by the frst n egenvlues The estmte coeffcents do not hve ny relton to the geometrc mesure of the domn Ths knd of problem s nterestng nd sgnfcnt both n theory of systems of dfferentl equtons nd n pplctons to mechncs nd physcs Introducton In mny physcl settngs, such s the vbrtons of the generl homogeneous or nonhomogeneous strng, rod nd plte cn yeld the Sturm-Louvlle egenvlue problems or other egenvlue problems However, t s not esy to get the ccurte vlues by the nlytc method Sometmes, t s necessry to consder the estmtons of the egenvlues And snce 960s, the problems of the egenvlue estmtes hd become one of the hotspots of the dfferentl equtons There re lots of chevements bout the upper bounds of rbtrry egenvlues for the dfferentl equtons nd unformly ellptc opertors wth hgher orders 9 However, there re few chevements ssocted wth the estmtes of the egenvlues for systems of dfferentl equtons wth hgher order In the followng, we wll obtn some nequltes concernng the egenvlue λ n n terms of λ,λ,,λ n n the systems of ordnry dfferentl equtons wth hgher order In fct, the egenvlue problems hve gret strong prctcl bckgrounds nd mportnt theoretcl vlues 0,
Interntonl Journl of Dfferentl Equtons Let, b R be bounded domn nd t be n nteger The followng egenvlue problems re studed: t D t D t y D t y n D t y n λs x y, t D t D t y D t y n D t y n λs x y, t D t n D t y n D t y nn D t y n λs x yn, D k y D k y b 0,,,n, k 0,,,,t, where D d/, D k condtons: d k / k, j x, j,,,n nd s x stsfes the followng j x C t, b, j x j x,, j,,,n; for the rbtrry ξ ξ,ξ,,ξ n R n, we hve μ ξ j x ξ ξ j μ ξ, x, b,,j where μ μ > 0, μ,μ re both constnts; 3 s x C, b, nd there re constnts ν ν, such tht 0 <ν s x ν Accordng to the theores of the dfferentl equtons,, the egenvlues of re ll postve rel numbers, nd they re dscrete We chnge to the form of mtrx Let y T y y y n D t y x x n x D t y x x n x, D t y T, A x 3 D t y n n x n x nn x By vrtue of j x j x, therefore A T x A x, cn be chnged nto the followng form: t D t A x D t y T λs x y T, 4 y k y k b 0, k 0,,,,t 5 Obvously, 4-5 s equvlent to
Interntonl Journl of Dfferentl Equtons 3 Suppose tht 0 < λ λ λ n re egenvlues of 4-5, y, y,,y n, re the correspondng egenfunctons nd stsfy the followng weghted orthogonl condtons:, j, s x y y T j δ j 0, / j,, j,, 6 Multplyng y n sdes of 4, byusng 5 nd ntegrton by prts, we hve λ D t y A x D t y T,,, 7 From, we hve b D t y D t y D t y T λ,,, 8 μ For fxed n,let Φ xy b j y j,,,,n, 9 j where b j xs x y y T j Obvously, b j b j, nd Φ re weghted orthogonl to y, y,,y n Furthermore, Φ Φ b 0,, j,,,n We cn use the well-known Rylegh theorem, to obtn t Φ D t A x D t Φ T λ n 0 s x Φ It s esy to see tht t D t A x D t Φ T t td t A x D t y T t td t A x D t y T t xd t A x D t y T t n b j D t A x D t y T j t td t A x D t y T j t td t A x D t y T λ xs x y T s x λ j b j y T j j
4 Interntonl Journl of Dfferentl Equtons We hve Φ t D t A x D t Φ T λ xs x Φ y T t t t t Φ D t A x D t y T s x Φ λ j b j y T j j Φ D t A x D t y T In ddton, usng the fct tht Φ re weghted orthogonl to y, y,,y n nd s x Φ xs x Φ y T, 3 we know tht the lst term of s equl to zero Thus, we hve Φ t D t A x D t Φ T λ s x Φ t t t t Φ D t A x D t y T Φ D t A x D t y T 4 Set I t t J t t Φ D t A x D t y T Φ D t A x D t y T,, I J I, J 5 From 4, we hve Φ t D t A x D t Φ T λ s x Φ I J 6 By usng 0 nd 6, one cn gve λ n s x Φ λ s x Φ I J 7
Interntonl Journl of Dfferentl Equtons 5 Substtutng λ n for λ n 7, weget λ n λ n s x Φ I J 8 In order to get the estmtons of the egenvlues, we only need to show the estmtes bout I,J, nd n b s x Φ Lemms Lemm Suppose tht the egenfunctons y of 4-5 correspond to the egenvlues λ Then one hs Dp y ν / p Dp y p/ p, p,,,t ; Dy ν /t λ /μ /t Proof By nducton If p, usng ntegrton by prts nd the Schwrz nequlty, we hve b b Dy Dy Dy Dy T y D y T / / y D y T b / ν / D y T Therefore, when p, s true If for p k, s true, tht s, D k b k/ k / k y ν D k y For p k, usng ntegrton by prts, the Schwrz nequlty nd the result when p k, one cn gve b D k b y D k y D k y D k y T / D k b y D k y T / ν / k D k b y / k/ k D k y 3
6 Interntonl Journl of Dfferentl Equtons By further clcultng, one cn gve D k b k / k / k y ν D k y 4 Therefore, when p k, s true Usng nd the nductve method, we hve D p y ν / p p/ p D p y ν / p p/ p D p y 5 ν p/t p/t D t y From 8 nd 5, weget D p y ν p/t D t y p/t ν p/t λ μ p/t, p,,,t 6 Tkng p, we hve /t Dy ν /t λ 7 μ So Lemm s true Lemm Let λ,λ,,λ n be the egenvlues of 4-5 Then one hs I J t t μ /t ν /t 8 μ λ /t
Interntonl Journl of Dfferentl Equtons 7 Proof Snce I t t t t Φ D t A x D t y T xy b j y j D t A x D t y T j t t t t xy D t A x D t y T j b j y j D t A x D t y T 9 t D t y A x D t y T t xd t y A x D t y T t J t t j b j D t y j A x D t y T Φ D t A x D t y T, t t D t y A x D t y T t xd t y A x D t y T t j b j D t y j A x D t y T, 0 we hve I J I J t t td t y A x D t y T t D t y A x D t y T,j b j D t y j A x D t y T D t y j A x D t y T By j x j x, the lst term of s zero Then we cn get I J t td t y A x D t y T t D t y A x D t y T
8 Interntonl Journl of Dfferentl Equtons Usng, Lemm, nd 6, we hve D t y A x D t y T μ D t y μ ν /t λ μ /t 3 Usng, the Schwrz nequlty, Lemm, nd 6, one cn gve b D t y A x D t y T μ D t y / /t μ ν /t λ μ D t y T / 4 Therefore, we obtn I J t t μ /t ν /t μ λ /t 5 Lemm 3 If Φ nd λ,,,n s bove, then one hs Proof By the defnton of Φ, one hs s x Φ μ/t ν /t n λ /t 4ν 6 Φ Dy T xy Dy T,j b j y j Dy T 7 Usng b j b j nd y jdy T y Dy T j, t s esy to see tht the lst term of 7 s zero Then we hve Φ Dy T xy Dy T 8 Usng ntegrton by prts, one cn gve xy Dy T y xy Dy T xy Dy T, 9 y 0
Interntonl Journl of Dfferentl Equtons 9 By /ν y /ν, we hve xy Dy T y ν From 8 nd, we cn get Φ Dy T n ν Usng the Schwrz nequlty, Lemm, nd 3, we hve n 4ν s x Φ s x Φ ν /t μ /t Dy s x λ /t 3 By further clcultng, we cn esly get Lemm 3 3 Mn Results Theorem 3 If λ,,,n re the egenvlues of 4-5,then λ n λ n 4t t μ ν μ ν n λ n 4t t μ ν λ μ ν n λ /t λ /t ; 3 3 Proof From 8, we cn get λ n λ n I J s x Φ 33 Usng Lemms nd 3, we cn esly get 3 In 3, Replcng λ wth λ n, by further clcultng, we cn get 3 Theorem 3 For n, one hs λ /t μ ν n λ n λ 4t t μ ν λ /t 34
0 Interntonl Journl of Dfferentl Equtons Proof Choosng the prmeter σ>λ n,usng 7, one cn gve λ n s x Φ σ s x Φ λ σ s x Φ I J 35 By nd the Young nequlty, we obtn n ν δ σ λ s x Φ δ σ λ Dy s x, 36 where δ>0 s constnt to be determned Set V s x Φ, T σ λ s x Φ 37 Usng Lemm, 35,nd 36, we cn get the followng results, respectvely, λ n σ V T I J, n δt ν δ μ /t ν /t 38 σ λ λ /t 39 In order to get the mnmum of the rght of 39, we cn tke / δ T / μ /t ν /t σ λ λ /t 30 By 39,nd 30, we cn esly get T μ/t ν /t n 4ν λ /t σ λ 3 Usng Lemm, 38,nd 3, we hve λ n σ V μ/t ν /t n 4ν λ /t σ λ t t μ /t ν /t μ λ /t, 3 tht s, λ n σ V t t μ /t ν /t μ λ /t μ/t ν /t n 4ν λ /t σ λ 33
Interntonl Journl of Dfferentl Equtons Let the rght term of 33 be f σ It s esy to see tht lm f σ, σ lm f σ t t μ /t σ λ ν /t n μ λ /t > 0 34 Hence, there s σ 0 λ n,, such tht λ /t μ ν n σ 0 λ 4t t μ ν λ /t 35 On the other hnd, lettng g σ λ /t, σ λ 36 we hve g λ /t σ σ λ 0 37 It mples tht g σ s the monotone decresng nd contnuous functon, nd ts vlue rnge s 0, Therefore, there exts exctly one σ 0 to stsfy 35 From 33, we know tht σ 0 >λ n Replcng σ 0 wth λ n n 35, we cn get the result References L E Pyne, G Poly, nd H F Wenberger, Sur le quotent de deux frquences propres conscutves, Comptes Rendus de l AcdémedesScencesdePrs, vol 4, pp 97 99, 955 L E Pyne, G Poly, nd H F Wenberger, On the rto of consecutve egenvlues, Journl of Mthemtcs nd Physcs, vol 35, pp 89 98, 956 3 G N Hle nd M H Protter, Inequltes for egenvlues of the Lplcn, Indn Unversty Mthemtcs Journl, vol 9, no 4, pp 53 538, 980 4 G N Hle nd R Z Yeb, Inequltes for egenvlues of the bhrmonc opertor, Pcfc Journl of Mthemtcs, vol, no, pp 5 33, 984 5 S M Hook, Domn ndependent upper bounds for egenvlues of ellptc opertors, Trnsctons of the Amercn Mthemtcl Socety, vol 38, no, pp 65 64, 990 6 Z C Chen nd C L Qn, On the dfference of consecutve egenvlues of unformly ellptc opertors of hgher orders, Chnese Annls of Mthemtcs B, vol 4, no 4, pp 435 44, 993 7 Z C Chen nd C L Qn, On the upper bound of egenvlues for ellptc equtons wth hgher orders, Journl of Mthemtcl Anlyss nd Applctons, vol 86, no 3, pp 8 834, 994 8 G J, X P Yng, nd C L Qn, On the upper bound of second egenvlues for unformly ellptc opertors of ny orders, Act Mthemtce Applcte Snc, vol 9, no, pp 07 6, 003 9 G J nd X P Yng, The upper bounds of rbtrry egenvlues for unformly ellptc opertors wth hgher orders, Act Mthemtce Applcte Snc, vol, no 4, pp 589 598, 006 0 M H Protter, Cn one her the shpe of drum? SIAM Revew, vol 9, no, pp 85 97, 987
Interntonl Journl of Dfferentl Equtons R Cournt nd D Hlbert, Methods of Mthemtcl Physcs, Interscence Publshers, New York, NY, USA, 989 H Weyl, Rmfctons, old nd new, of the egenvlue problem, Bulletn of the Amercn Mthemtcl Socety, vol 56, pp 5 39, 950
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