On the spectrum of a quasidifferential boundary value problem of the second order

M. Yu. Vatolkin

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2023), pp. 3-24.

Résumé

The structure of the spectrum of a quasidifferential boundary value problem of the second order is studied, i.e. $(_{P}^{ 2}x)(t)=-\lambda( _{P}^{ 0}x)(t) (t\in[a,b], \lambda \in {\mathbb R})$ (the coefficients in the equation are real valued functions) with given homogeneous boundary conditions at the ends of this segment, i.e. ${ }_P^{ 0}x(a)={ }_P^{ 0}x(b)=0.$ First, we consider the auxiliary Cauchy problem with the real parameter $\beta$ in the coefficient $p_{20}(t)$ of the equation, namely, $p_{22}(t) \left(p_{11}(t)v'(t) \right)' +(p_{20}(t)+\beta)v(t)=0,$ $ v(a)=0, p_{11}(a)v'(a)=1.$ In terms of the solution to this problem, a fundamental theorem on the continuous or discrete real spectrum of the boundary value problem is formulated (see Theorem 1). Examples are given to illustrate the cases of the continuous spectrum and discrete spectrum for the boundary value problem under consideration.

Détail

Keywords: quasiderivatives, quasidifferential equation, two-point quasidifferential boundary value problem, Cauchy problem, real parameter, sequence of solutions, sum of series, eigenvalue, eigenfunction, spectrum of boundary value problem, discrete spectrum, continuous spectrum, root of equation, Taylor series.

Bibliographie

[1] Everitt W.N., Marcus L., Boundary value problems and symplectic algebra for ordinary differential and quasi-differential operators, Mathematical Surveys Monographs, 61, Amer. Math. Soc., 1999 | MR

[2] Eckhardt J., Gestezy F., Nichols R., Teschl G., “Weyl–Titchmarsh theory for Sturm–Liuville operators with distributional potentials”, Opuscula Math., 33:3 (2013), 467–563 | DOI | MR

[3] Everitt W.N., Race D., “The regular representation of singular second order differential expressions using quasi-derivatives”, Proc. London Math. Soc. (3), 65:2 (1992), 383–404 | DOI | MR

[4] Xiao xia Lv, Ji-jun Ao, Zettl A., “Dependence of eigenvalues of fourth-order differential equations with discontinuous boundary conditions on the problem”, J. Math. Anal. Appl., 456:1 (2017), 671–685 | DOI | MR

[5] Qinglan Bao, Jiong Sun, Xiaoling Hao, Zettl A., “Characterization of self-adjoint domains for regular even order C-symmetric differential operators”, Electronic J. Qualitative Theory Diff. Equat., 62 (2019), 1–17 | MR

[6] Zettl A., Sturm–Liouville Theory, Amer. Math. Soc., 2005 | MR

[7] Zettl A., Recent Developments in Sturm-Liouville Theory, De Gruyter, Berlin–Boston, 2021 | MR

[8] Jianfang Qin, Kun Li, Zhaowen Zheng, Jinming Cai, “Dependence of eigenvalues of discontinuous fourth-order differential operators with eigenparameter dependent boundary conditions”, J. Nonlinear Math. Phys., 29:4 (2022), 776–793 | DOI | MR

[9] Mirzoev K.A., “Operatory Shturma–Liuvillya”, Tr. MMO, 75, no. 2, 2014, 335–359

[10] Vladykina V.E., Shkalikov A.A., “Asimptotika reshenii uravneniya Shturma–Liuvillya s singulyarnymi koeffitsientami”, Matem. zametki, 98:6 (2015), 832–841 | MR

[11] Konechnaya N.N., Mirzoev K.A., “Ob asimptotike reshenii lineinykh differentsialnykh uravnenii nechetnogo poryadka”, Vestn. Moskovsk. un-ta. Ser. 1. Matem., Mekhan., 2020, no. 1, 23–28 | MR

[12] Savchuk A.M., Shkalikov A.A., “Asimptoticheskii analiz reshenii obyknovennykh differentsialnykh uravnenii s koeffitsientami-raspredeleniyami”, Matem. sb., 211:11 (2020), 129–166 | MR

[13] Savchuk A.M., Sadovnichaya I.V., “O suschestvovanii operatornoi gruppy, porozhdennoi odnomernoi sistemoi Diraka”, Tr. MMO, 80, no. 2, 2019, 275–294

[14] Savchuk A.M., Sadovnichaya I.V., “Spektralnyi analiz odnomernoi sistemy Diraka s summiruemym potentsialom i operatora Shturma–Liuvillya s koeffitsientami-raspredeleniyami”, Sovr. matem. Fundament. naprav., 66, no. 3, 2020, 373–530

[15] Savchuk A.M., Sadovnichaya I.V., “Ravnoskhodimost spektralnykh razlozhenii operatora Shturma–Liuvillya s potentsialom-raspredeleniem v shkalakh prostranstv”, Dokl. RAN. Matem., Inform., Prots. uprav., 496 (2021), 56–58

[16] Derr V.Ya., “Neostsillyatsiya reshenii lineinogo kvazidifferentsialnogo uravneniya”, Izv. in-ta matem. i inform. UdGU, 1999, no. 1(16), 3–105

[17] Derr V.Ya., “Ob adekvatnom opisanii sopryazhennogo operatora”, Vestn. Udmurtsk. un-ta. Matem. Mekhan. Kompyut. nauki, 2011, no. 3, 43–63

[18] Shin D.Yu., “O resheniyakh lineinogo kvazidifferentsialnogo uravneniya $n$-go poryadka”, Matem. sb., 7(49):3 (1940), 479–532

[19] Shin D.Yu., “O kvazidifferentsialnykh operatorakh v gilbertovom prostranstve”, Matem. sb., 13(55):1 (1943), 39–70

[20] Naimark M.A., Lineinye differentsialnye operatory, Nauka, M., 1969

[21] Nehary Z., “Disconjugate linear differential operators”, Trans. Amer. Math. Soc., 129 (1969), 500–516 | DOI | MR

[22] Elias U., “The extremal solutions of the equation $Ly~+~p(x)y=0$, II”, J. Math. Anal. Appl., 55 (1976), 253–265 | DOI | MR

[23] Everitt W.N., “Fourth order singular differential equation”, J. Math. Anal., 149 (1963), 320–340 | DOI | MR

[24] Kusano T., Naito M., “Oscillation criteria of general linear ordinary differential equations”, Pacific J. Math., 92:2 (1981), 345–355 | DOI | MR

[25] Trench W.F., “Canonical forms and principal systems for general disconjugate equations”, Trans. Amer. Math. Soc., 189 (1974), 319–327 | DOI | MR

[26] Trench W.F., “Asymptotic theory perturbed general disconjugate equations”, Hiroshima Math. J., 12 (1982), 43–58 | DOI | MR

[27] Everitte W.N., Key J.D., “On some properties of matrices associated with linear ordinary quasi-differential equations”, Proc. of the Royal Soc. of Edinburgh, 96A (1984), 211–219 | DOI | MR

[28] Everitte W.N., “A note of linear ordinary quasi-differential equations”, Proc. of the Royal Soc. of Edinburgh, 101A (1985), 1–14 | DOI | MR

[29] Nicolson L.S., “Disconjugate systems of linear differential equation”, J. Diff. Equat., 1970, no. 7, 570–583 | DOI | MR

[30] Derr V.Ya., “O preobrazovanii nekotorykh mnogotochechnykh kraevykh zadach v zadachu Valle–Pussena i usloviyakh razreshimosti”, Differents. uravneniya, 23:4 (1987), 598–608 | MR

[31] Derr V.Ya., “K obobschennoi zadache Valle–Pussena”, Differents. uravneniya, 23:11 (1987), 1861–1872 | MR

[32] Mukhtarov O.Sh., Kadakal M., “Spektralnye svoistva odnoi zadachi tipa Shturma–Liuvillya s razryvnym vesom”, Sib. matem. zhurn., 46:4 (2005), 860–875 | MR

[33] Vladimirov A.A., Sheipak I.A., “Asimptotika sobstvennykh znachenii zadachi Shturma–Liuvillya s diskretnym samopodobnym vesom”, Matem. zametki, 88:5 (2010), 662–672 | MR

[34] Vladimirov A.A., Sheipak I.A., “Asimptotika sobstvennykh znachenii zadachi vysshego chetnogo poryadka s diskretnym samopodobnym vesom”, Algebra i analiz, 24:2 (2012), 104–119

[35] Vladimirov A.A., Karulina E.S., “Ob odnoi apriornoi mazhorante naimenshikh sobstvennykh znachenii zadachi Shturma–Liuvillya”, Itogi nauki i tekhn. Ser. Sovr. matem. i ee pril. Temat. obz., 193, 2021, 25–27

[36] Vatolkin M.Yu., Derr V.Ya., “O predstavlenii reshenii kvazidifferentsialnogo uravneniya”, Izv. vuz. Matem., 1995, no. 10, 27–34 | MR

[37] Vatolkin M.Yu., “O strukture spektra odnoi kraevoi zadachi vtorogo poryadka”, Teoriya upravleniya i matem. modelirovanie, Mater. Vserossiisk. konf. s mezhdunar. uchast., posvyasch. pamyati prof. N.V. Azbeleva i prof. E.L. Tonkova, izd. 2, dop., Izd. tsentr “Udmurtsk. un-t”, Izhevsk, 2020, 56–58

[38] Vatolkin M.Yu., “Primery na issledovanie spektra odnoi kraevoi zadachi vtorogo poryadka”, Teoriya upravleniya i matem. modelirovanie, Mater. Vserossiisk. konf. s mezhdunar. uchast., posvyasch. pamyati prof. N.V. Azbeleva i prof. E.L. Tonkova, izd. 2, dop., Izd. tsentr “Udmurtsk. un-t”, Izhevsk, 2020, 58–60

[39] Kamke E., Spravochnik po obyknovennym differentsialnym uravneniyam, Gl. red. fiz.-matem. lit., M., 1971