Nonoscillation of the Mathieu-type half-linear differential equation and its application to the generalized Whittaker–Hill-type equation
The nonoscillation of Mathieu-type half-linear differential equations was investigated. The particular equation under consideration is an extension of the Mathieu equation, which has been widely applied in mechanical and electrical engineering. The investigation led to the main finding that all nont...
Saved in:
| Published in | Monatshefte für Mathematik Vol. 198; no. 4; pp. 741 - 756 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Vienna
Springer Vienna
01.08.2022
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
Cover
Loading…
| Abstract | The nonoscillation of Mathieu-type half-linear differential equations was investigated. The particular equation under consideration is an extension of the Mathieu equation, which has been widely applied in mechanical and electrical engineering. The investigation led to the main finding that all nontrivial solutions of the Mathieu-type half-linear differential equations are nonoscillatory under simple parametric conditions. Proving the finding requires a simple nonoscillation theorem to compare the two equations. As another application of the findings, by using a simple nonoscillation comparison theorem, we propose that all nontrivial solutions of the half-linear Whittaker–Hill-type equation do not oscillate. |
|---|---|
| AbstractList | The nonoscillation of Mathieu-type half-linear differential equations was investigated. The particular equation under consideration is an extension of the Mathieu equation, which has been widely applied in mechanical and electrical engineering. The investigation led to the main finding that all nontrivial solutions of the Mathieu-type half-linear differential equations are nonoscillatory under simple parametric conditions. Proving the finding requires a simple nonoscillation theorem to compare the two equations. As another application of the findings, by using a simple nonoscillation comparison theorem, we propose that all nontrivial solutions of the half-linear Whittaker–Hill-type equation do not oscillate. |
| Author | Ishibashi, Kazuki |
| Author_xml | – sequence: 1 givenname: Kazuki orcidid: 0000-0003-1812-9980 surname: Ishibashi fullname: Ishibashi, Kazuki email: ishibashi_kazuaoi@yahoo.co.jp organization: Department of Electronic Control Engineering, National Institute of Technology (KOSEN), Hiroshima College |
| BookMark | eNp9kLtKBTEQhoMoeLy8gFXAOjpJNnspRbyBl0axDDm7E090TdYkp9DK2tY39ElcXcXOaQaG__sG_g2y6oNHQnY47HGAaj8BlKAYCMGAVwKYWCEzXsiSKaj5KpkBiJI1Qql1spHSPQBwWTYz8nYZfEit63uTXfA0WJoXSC9MXjhcsvw8IF2Y3rLeeTSRds5ajOizMz3Fp-VEGd9RlxM1w9C7drrl8G26Q4_R9O4FO3q7cDmbB4wfr--n48tJ_2vZImvW9Am3f_YmuTk-uj48ZedXJ2eHB-eslbXKTJa2EYXgOG8KtNVcdbZRQpZczYtxbFFBbYxURtWqkqK1puNQF3OluC1rVchNsjt5hxielpiyvg_L6MeXWpT1CHGlYEyJKdXGkFJEq4foHk181hz0V-d66lyPnevvzrUYITlBaQz7O4x_6n-oT6aUiQ0 |
| CitedBy_id | crossref_primary_10_14232_ejqtde_2024_1_19 |
| Cites_doi | 10.1017/S0013091500002297 10.1002/9783527617586 10.1016/j.jfa.2006.06.013 10.21136/CMJ.2018.0645-16 10.1017/S0308210500000287 10.1002/qua.22255 10.1016/j.aml.2017.12.012 10.1063/1.3455367 10.1007/978-94-017-2515-6 10.1016/j.jat.2005.03.004 10.1016/j.jmaa.2018.05.037 10.1063/1.1696228 10.1016/j.jmaa.2016.07.013 |
| ContentType | Journal Article |
| Copyright | The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2022 The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2022. |
| Copyright_xml | – notice: The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2022 – notice: The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2022. |
| DBID | AAYXX CITATION |
| DOI | 10.1007/s00605-022-01720-2 |
| DatabaseName | CrossRef |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics |
| EISSN | 1436-5081 |
| EndPage | 756 |
| ExternalDocumentID | 10_1007_s00605_022_01720_2 |
| GrantInformation_xml | – fundername: JSPS KAKENHI Grant-in-Aid for Early-Career Scientists grantid: JP22K13933 |
| GroupedDBID | --Z -52 -5D -5G -BR -EM -Y2 -~C -~X .86 .VR 06D 0R~ 0VY 199 1N0 1SB 2.D 203 29M 2J2 2JN 2JY 2KG 2KM 2LR 2P1 2VQ 2WC 2~H 30V 406 408 409 40D 40E 5QI 5VS 67Z 692 6NX 6TJ 78A 8UJ 95- 95. 95~ 96X AAAVM AABHQ AABYN AAFGU AAHNG AAIAL AAJKR AANZL AARHV AARTL AATNV AATVU AAUYE AAYFA AAYIU AAYQN AAYTO ABBBX ABBXA ABDZT ABECU ABFGW ABFTV ABHLI ABHQN ABJNI ABJOX ABKAS ABKCH ABKTR ABMNI ABMQK ABNWP ABQBU ABSXP ABTEG ABTHY ABTKH ABTMW ABULA ABWNU ABXPI ACBMV ACBRV ACBXY ACBYP ACGFS ACHSB ACHXU ACIGE ACIPQ ACIWK ACKNC ACMDZ ACMLO ACNCT ACOKC ACOMO ACTTH ACVWB ACWMK ADHHG ADHIR ADIMF ADINQ ADKNI ADKPE ADMDM ADOXG ADTPH ADURQ ADYFF ADZKW AEBTG AEEQQ AEFIE AEFTE AEGAL AEGNC AEJHL AEJRE AEKMD AENEX AEOHA AEPOP AEPYU AESKC AESTI AETLH AEVLU AEVTX AEXYK AFDYV AFEXP AFFNX AFLOW AFMKY AFNRJ AFQWF AFWTZ AFZKB AGDGC AGGBP AGGDS AGJBK AGMZJ AGPAZ AGQMX AGWIL AGWZB AGYKE AHAVH AHBYD AHKAY AHSBF AHYZX AIAKS AIIXL AILAN AIMYW AITGF AJBLW AJDOV AJRNO AJZVZ AKQUC ALMA_UNASSIGNED_HOLDINGS ALWAN AMKLP AMXSW AMYLF AMYQR AOCGG ARMRJ ASPBG AVWKF AXYYD AYJHY AZFZN B-. BA0 BAPOH BBWZM BDATZ BGNMA CAG COF CSCUP DDRTE DL5 DNIVK DPUIP DU5 EBLON EBS EIOEI EJD ESBYG FERAY FFXSO FIGPU FINBP FNLPD FRRFC FSGXE FWDCC GGCAI GGRSB GJIRD GNWQR GQ6 GQ7 GQ8 GXS HG5 HG6 HMJXF HQYDN HRMNR HZ~ H~9 IHE IJ- IKXTQ ITM IWAJR IXC IZIGR IZQ I~X I~Z J-C J0Z JBSCW JCJTX JZLTJ KDC KOV KOW LAS LLZTM M4Y MA- N2Q N9A NB0 NDZJH NPVJJ NQJWS NU0 O9- O93 O9G O9I O9J OAM OK1 P19 P2P P9R PF0 PT4 PT5 QOK QOS R4E R89 R9I REI RHV RNI ROL RPX RSV RYB RZK RZZ S16 S1Z S26 S27 S28 S3B SAP SCLPG SDD SDH SDM SHX SJYHP SMT SNE SNPRN SNX SOHCF SOJ SPISZ SRMVM SSLCW STPWE SZN T13 T16 TSG TSK TSV TUC TWZ U2A UG4 UNUBA UOJIU UTJUX UZXMN VC2 VFIZW W23 W48 WK8 Y6R YLTOR YNT YQT Z45 Z7U ZMTXR ZWQNP ~EX AACDK AAEOY AAJBT AASML AAYXX ABAKF ACAOD ACDTI ACZOJ AEARS AEFQL AEMSY AFBBN AGQEE AGRTI AIGIU CITATION H13 |
| ID | FETCH-LOGICAL-c385t-36f92421eb94ef7b5df9523615b4444f4708aa35a585732cfad1084b551f68543 |
| IEDL.DBID | AGYKE |
| ISSN | 0026-9255 |
| IngestDate | Thu Oct 10 18:11:43 EDT 2024 Thu Sep 12 18:16:58 EDT 2024 Sat Dec 16 12:10:18 EST 2023 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 4 |
| Keywords | Nonoscillation 34C10 Half-linear differential equation Mathieu’s equation 34B30 Riccati technique Comparison theorem Whittaker–Hill’s equation |
| Language | English |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c385t-36f92421eb94ef7b5df9523615b4444f4708aa35a585732cfad1084b551f68543 |
| ORCID | 0000-0003-1812-9980 |
| PQID | 2688571550 |
| PQPubID | 2043621 |
| PageCount | 16 |
| ParticipantIDs | proquest_journals_2688571550 crossref_primary_10_1007_s00605_022_01720_2 springer_journals_10_1007_s00605_022_01720_2 |
| PublicationCentury | 2000 |
| PublicationDate | 2022-08-01 |
| PublicationDateYYYYMMDD | 2022-08-01 |
| PublicationDate_xml | – month: 08 year: 2022 text: 2022-08-01 day: 01 |
| PublicationDecade | 2020 |
| PublicationPlace | Vienna |
| PublicationPlace_xml | – name: Vienna – name: Heidelberg |
| PublicationTitle | Monatshefte für Mathematik |
| PublicationTitleAbbrev | Monatsh Math |
| PublicationYear | 2022 |
| Publisher | Springer Vienna Springer Nature B.V |
| Publisher_xml | – name: Springer Vienna – name: Springer Nature B.V |
| References | MagnusWWinklerSHill’s equation1979New YorkDover1141.34002 EnakaYOnitsukaMIntegral averaging technique for oscillation of damped half-linear oscillatorsCzechoslov. Math. J.201868755770385188910.21136/CMJ.2018.0645-16 MathieuÉMémoire sur le mouvement vibratoire d’une membrane de forme elliptiqueJ. Math. Pure. Appl.18681313720301.0354.02 KusanoTMarićVNonoscillatory linear and half-linear differential equations having regularly varying solutionsAdv. Math. Sci. Appl.20041435135720836331069.34050 RoncarattiLFAquilantiVWhittaker–Hill equation, Ince polynomials, and molecular torsional modesInt. J. Quant. Chem.201011071673010.1002/qua.22255 WhittakerETOn a class of differential equations whose solutions satisfy integral equationsProc. Edinburgh Math. Soc.191433142310.1017/S0013091500002297 AgarwalRPGraceSRO’ReganDOscillation theory for second order linear, half-linear, superlinear, and sublinear dynamic equations2002Dordrecht, Boston, LondonKluwer Academic Publishers10.1007/978-94-017-2515-6 SugieJIshibashiKIntegral condition for oscillation of half-linear differential equations with dampingAppl. Math. Lett.201879146154374862410.1016/j.aml.2017.12.012 DošlýOŘehákPHalf-linear differential equations, North-Holland Math. Stud2005AmsterdamElsevier Science B.V.1090.34001 IshibashiKSugieJSimple conditions for parametrically excited oscillations of generalized Mathieu equationsJ. Math. Anal. Appl.2017446233247355472410.1016/j.jmaa.2016.07.013 DjakovPMityaginBAsymptotics of instability zones of the Hill operator with a two term potentialJ. Funct. Anal.2007242157194227401910.1016/j.jfa.2006.06.013 HuntRHLeacockRAPetersCWHechtKTInternal-rotation in hydrogen peroxide: the far-infrared spectrum and the determination of the hindering potentialJ. Chem. Phys.1965421931194610.1063/1.1696228 HemeryADVeselovAPWhittaker–Hill equation and semifinite-gap Schrödinger operatorsJ. Math. Phys.20105170721081710.1063/1.3455367 NayfehAHMookDTNonlinear oscillations1995New YorkJohn Wiley & Sons10.1002/9783527617586 SugieJIshibashiKOscillation problems for Hill’s equation with periodic dampingJ. Math. Anal. Appl.20184465670381810510.1016/j.jmaa.2018.05.037 DjakovPMityaginBSimple and double eigenvalues of the Hill operator with a two-term potentialJ. Approx. Theory200513570104215125010.1016/j.jat.2005.03.004 DošlýOElbertÁConjugacy of half-linear second-order differential equationsProc. Roy. Soc. Edinburgh Sect. A2000130517525176924010.1017/S0308210500000287 SugieJMatsumuraKA nonoscillation theorem for half-linear differential equations with periodic coefficientsAppl. Math. Comput.200819944745524205741217.34056 McLachlanNWTheory and application of Mathieu functions1964New YorkDover0128.29603 É Mathieu (1720_CR13) 1868; 13 NW McLachlan (1720_CR14) 1964 O Došlý (1720_CR5) 2005 J Sugie (1720_CR16) 2008; 199 W Magnus (1720_CR12) 1979 Y Enaka (1720_CR6) 2018; 68 AD Hemery (1720_CR7) 2010; 51 J Sugie (1720_CR17) 2018; 79 ET Whittaker (1720_CR19) 1914; 33 P Djakov (1720_CR3) 2007; 242 K Ishibashi (1720_CR9) 2017; 446 T Kusano (1720_CR10) 2004; 14 P Djakov (1720_CR2) 2005; 135 RH Hunt (1720_CR8) 1965; 42 AH Nayfeh (1720_CR11) 1995 O Došlý (1720_CR4) 2000; 130 J Sugie (1720_CR18) 2018; 446 LF Roncaratti (1720_CR15) 2010; 110 RP Agarwal (1720_CR1) 2002 |
| References_xml | – volume: 14 start-page: 351 year: 2004 ident: 1720_CR10 publication-title: Adv. Math. Sci. Appl. contributor: fullname: T Kusano – volume: 199 start-page: 447 year: 2008 ident: 1720_CR16 publication-title: Appl. Math. Comput. contributor: fullname: J Sugie – volume-title: Half-linear differential equations, North-Holland Math. Stud year: 2005 ident: 1720_CR5 contributor: fullname: O Došlý – volume-title: Hill’s equation year: 1979 ident: 1720_CR12 contributor: fullname: W Magnus – volume: 33 start-page: 14 year: 1914 ident: 1720_CR19 publication-title: Proc. Edinburgh Math. Soc. doi: 10.1017/S0013091500002297 contributor: fullname: ET Whittaker – volume: 13 start-page: 137 year: 1868 ident: 1720_CR13 publication-title: J. Math. Pure. Appl. contributor: fullname: É Mathieu – volume-title: Nonlinear oscillations year: 1995 ident: 1720_CR11 doi: 10.1002/9783527617586 contributor: fullname: AH Nayfeh – volume: 242 start-page: 157 year: 2007 ident: 1720_CR3 publication-title: J. Funct. Anal. doi: 10.1016/j.jfa.2006.06.013 contributor: fullname: P Djakov – volume: 68 start-page: 755 year: 2018 ident: 1720_CR6 publication-title: Czechoslov. Math. J. doi: 10.21136/CMJ.2018.0645-16 contributor: fullname: Y Enaka – volume: 130 start-page: 517 year: 2000 ident: 1720_CR4 publication-title: Proc. Roy. Soc. Edinburgh Sect. A doi: 10.1017/S0308210500000287 contributor: fullname: O Došlý – volume: 110 start-page: 716 year: 2010 ident: 1720_CR15 publication-title: Int. J. Quant. Chem. doi: 10.1002/qua.22255 contributor: fullname: LF Roncaratti – volume: 79 start-page: 146 year: 2018 ident: 1720_CR17 publication-title: Appl. Math. Lett. doi: 10.1016/j.aml.2017.12.012 contributor: fullname: J Sugie – volume: 51 start-page: 07210817 issue: 7 year: 2010 ident: 1720_CR7 publication-title: J. Math. Phys. doi: 10.1063/1.3455367 contributor: fullname: AD Hemery – volume-title: Oscillation theory for second order linear, half-linear, superlinear, and sublinear dynamic equations year: 2002 ident: 1720_CR1 doi: 10.1007/978-94-017-2515-6 contributor: fullname: RP Agarwal – volume: 135 start-page: 70 year: 2005 ident: 1720_CR2 publication-title: J. Approx. Theory doi: 10.1016/j.jat.2005.03.004 contributor: fullname: P Djakov – volume-title: Theory and application of Mathieu functions year: 1964 ident: 1720_CR14 contributor: fullname: NW McLachlan – volume: 446 start-page: 56 year: 2018 ident: 1720_CR18 publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2018.05.037 contributor: fullname: J Sugie – volume: 42 start-page: 1931 year: 1965 ident: 1720_CR8 publication-title: J. Chem. Phys. doi: 10.1063/1.1696228 contributor: fullname: RH Hunt – volume: 446 start-page: 233 year: 2017 ident: 1720_CR9 publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2016.07.013 contributor: fullname: K Ishibashi |
| SSID | ssj0001369 |
| Score | 2.325326 |
| Snippet | The nonoscillation of Mathieu-type half-linear differential equations was investigated. The particular equation under consideration is an extension of the... |
| SourceID | proquest crossref springer |
| SourceType | Aggregation Database Publisher |
| StartPage | 741 |
| SubjectTerms | Differential equations Mathematical analysis Mathematics Mathematics and Statistics Theorems |
| Title | Nonoscillation of the Mathieu-type half-linear differential equation and its application to the generalized Whittaker–Hill-type equation |
| URI | https://link.springer.com/article/10.1007/s00605-022-01720-2 https://www.proquest.com/docview/2688571550 |
| Volume | 198 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| link | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV07T8MwED5BWWDgjSiPygMbGCWO4yRjQZQKRCcqwRTZiQ1VqwTadOnEzMo_5Jdg51WeAxlj65LcXc53vrvPAEeBxXwhfIo9IgWmng5YeUQcrIjgyibM8_IN_Zse6_bp1Z17N-_jzovdq4xkbqjrXjeDHGKaiU0lgadjHm13l0zjqduApfbl_fVFbYBth5WVHQwH2mUue2V-p_J1PZo7md_yovly01mD26ppp6gyGZ5OM3EazX5iOP7nS9ZhtXQ_UbvQlw1YkMkmrNzU2K2TLXjtpYlBuBwVVXIoVUiPIjNnIKfY7NmiRz5S2DiofIyqI1a0qRgh-VxAhyOexGiQTdCnBDnK0pzSQ4F0PZjJGJnz-TI-lOP3l7eufmRBvqKyDf3Oxe15F5cnNuDI8d0MO0wFJscsRUCl8oQbq8A18C6uoPpSWhd8zh2X6yDFc0ikeGxbPhXabVPMd6mzA40kTeQuoFhZ5payPGlRSSPflpYKOAsCqkMkizbhuJJb-FQAc4Q1BHPO4VBzOMw5HJImHFSiDcufdBIS5uvXMDFaE04qUc2H_6a297_p-7BMcmmbssEDaGTjqTzUrkwmWlp1O2dnvVapwi1Y7JP2B0B37O8 |
| link.rule.ids | 315,783,787,27938,27939,41095,41537,42164,42606,52125,52248 |
| linkProvider | Springer Nature |
| linkToHtml | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV07T8MwELYQDMCAeIpCAQ9sYClxHCceK0QVoO3USt0iO7GhokqgTRcmZlb-Ib8E23kUEAxkjK1L5O9s3_nuPgNwzhwaChESFGApEAm0w8oT7CGFBVcupkFgD_T7AxqNyO3YH1dFYfM6270OSdqVuil2M9QhpprYpBIE2unRC--a4Vc3ej3CnWb9dT1aJXZQxLTFXJXK_C7j-3a0tDF_hEXtbtPdBluVmQg7Ja47YEVmu2Cz33CszvfA2yDPDBPltMxmg7mCuhWaPhO5QOZsFT7wqULGkOQzWF-Foqf0FMrnkuIb8iyFk2IOvwSyYZFbSfclI_XkRabQ3KNX8Ec5-3h9j_QnS_G1lH0w6l4PryJU3ayAEi_0C-RRxUwsWApGpAqEnyrmGxoWXxD9KI1ZyLnnc-1MBB5OFE9dJyRCm1eKhj7xDsBqlmfyEMBUOeaVcgLpEEmS0JWOYpwyRrQr45AWuKgHOH4qCTTihirZwhFrOGILR4xboF1jEFeTaR5jGurfML5UC1zWuCyb_5Z29L_uZ2A9GvZ7ce9mcHcMNrBVE5Pq1warxWwhT7T5UYhTq22fbarRew |
| linkToPdf | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV07T8MwED6hIiEYeCPK0wMbuOThOMmIgFJeFQNIZYrsxIaqVVLadOnEzMo_5Jdg59EWBAMiY2xdbJ9jf-e7-wxw4BvU49wj2LUEx8RVBisLLRtLizNpWtR1swP92yZtPJCrltOayuLPot1Ll2Se06BZmuL0uBfJ43Him6YR0ZnFOqzAVQaQWoRniangQgVmTy4er8_Hq7Fp0yLMg2Jf4eciceZnKV83pwni_OYkzfae-hKwstV5yEmnNkx5LRx9I3T8T7eWYbEApugkn0krMCPiVVi4HbO6DtbgrZnEmvuym8fPoUQiVYp0nbYYYn2ai55ZV2INXVkflZevqEWki8RLTiqOWByhdjpAU65zlCaZpKecA7s9EhHSN_elrCP6H6_vDfXJXHwpZR0e6uf3pw1c3OWAQ9tzUmxT6Wvvs-A-EdLlTiR9RxO_OJyoR6pZ4jFmO0yZL65thZJFpuERrgCdpJ5D7A2oxEksNgFF0tCvpOEKgwgSeqYwpM-o7xNlPBmkCoelEoNeTtkRjMmZsxEO1AgH2QgHVhV2Sj0Hxe87CCzqqWZo660KR6XaJsW_S9v6W_V9mLs7qwc3l83rbZi3MsXr2MIdqKT9odhVeCfle8WU_gS-_PdC |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Nonoscillation+of+the+Mathieu-type+half-linear+differential+equation+and+its+application+to+the+generalized+Whittaker%E2%80%93Hill-type+equation&rft.jtitle=Monatshefte+f%C3%BCr+Mathematik&rft.au=Ishibashi%2C+Kazuki&rft.date=2022-08-01&rft.issn=0026-9255&rft.eissn=1436-5081&rft.volume=198&rft.issue=4&rft.spage=741&rft.epage=756&rft_id=info:doi/10.1007%2Fs00605-022-01720-2&rft.externalDBID=n%2Fa&rft.externalDocID=10_1007_s00605_022_01720_2 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0026-9255&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0026-9255&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0026-9255&client=summon |