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Irregular boundary value problems revisited. (English) Zbl 1266.34001
The author gives a survey on results that have been obtained during the last century for strongly irregular boundary eigenvalue problems of the form \[ \begin{aligned} l(y)=y^{(n)}+\sum_{j=0}^{n-2}p_j(x)y^{(j)}=\lambda y, \;\;0\leq x\leq 1,\\ U_{\nu}(y)=U_{\nu 0}(y)+U_{\nu 1}(y)=0,\;\;1\leq \nu \leq n, \end{aligned} \] with complex-valued coefficients \(p_j\in L[0,1]\) and \[ U_{\nu 0}(y)=a_{\nu}y^{(k_{\nu})}(0)+\sum_{h=0}^{k_{\nu}-1}\alpha_{\nu h}y^{(h)}(0),\;\;U_{\nu 1}(y)=b_{\nu}y^{(k_{\nu})}(1)+\sum_{h=0}^{k_{\nu}-1}\beta_{\nu h}y^{(h)}(1), \] and \(a_{\nu}, b_{\nu}, \alpha_{\nu h}, \beta_{\nu h}\in \mathbb{C}\). He especially focuses on the difference concerning the convergence properties of the expansion of a given function into a series of eigenfunctions and associated functions of the above problems between the irregular case and the regular case. Also, several related results on some further classes of non-selfadjoint boundary value problems are introduced.

MSC:
34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B27 Green’s functions for ordinary differential equations
34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
34B09 Boundary eigenvalue problems for ordinary differential equations
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[1] Aigunov G.A., Gadzhieva T.Yu.: Asymptotics of eigenvalues and estimate for the kernel of the resolvent in an irregular boundary value problem generated by a 2n-th order differential equation on the interval [0, a]. Russ. Math. Surv. 63, 155–157 (2008) · Zbl 1181.34095 · doi:10.1070/RM2008v063n01ABEH004504
[2] de Alfaro, V., Regge, T.: Potential Scattering. North-Holland Publishing Co., Amsterdam, Interscience Publishers Division John Wiley & Sons, Inc., New York (1965) · Zbl 0141.23202
[3] Behrndt J.: Boundary value problems with eigenvalue depending boundary conditions. Math. Nachr. 282(5), 659–689 (2009) · Zbl 1177.47088 · doi:10.1002/mana.200610763
[4] Ben Amara J.: Oscillation properties for the equation of vibrating beam with irregular boundary conditions. J. Math. Anal. Appl. 360, 7–13 (2009) · Zbl 1179.34033 · doi:10.1016/j.jmaa.2009.05.042
[5] Benzinger H.E.: Greens function for ordinary differential operators. J. Differ. Equ. 7, 478–496 (1970) · Zbl 0198.12102 · doi:10.1016/0022-0396(70)90096-3
[6] Bergmann J.: Nichtreguläre Randwertaufgaben (German). Beitr. Anal. 7, 71–85 (1975) · Zbl 0307.34014
[7] Binding P., Curgus B.: Riesz bases of root vectors of indefinite Sturm-Liouville problems with eigenparameter dependent boundary conditions. II. Integral Equ. Oper. Theory 63, 473–499 (2009) · Zbl 1210.34038 · doi:10.1007/s00020-009-1659-0
[8] Birkhoff G.D.: On the asymptotic character of the solutions of certain linear differential equations containing a parameter. Trans. Am. Math. Soc. 9, 219–231 (1908) · JFM 39.0386.01 · doi:10.1090/S0002-9947-1908-1500810-1
[9] Birkhoff G.D.: Boundary value and expansion problems of ordinary linear differential equations. Am. Math. Soc. Trans. 9, 373–395 (1908) · JFM 39.0386.02 · doi:10.1090/S0002-9947-1908-1500818-6
[10] Birkhoff G.D., Langer R.E.: The boundary problems and developments associated with a system of ordinary linear differential equations of the first order. Am. Acad. Proc. 58, 49–128 (1923) · JFM 49.0723.01
[11] Bogomolova, E.P., Pechentsov, A.S.: On basis properties of the system of eigenfunctions of a boundary value problem with a multiple root of the characteristic polynomial (Russian). Vestn. Mosk. Univ., Ser. I, pp. 17–22 (1989) · Zbl 0687.34021
[12] Curgus B., Langer H.: A Krein space approach to symmetric ordinary differential operators with an indefnite weight function. J. Differ. Equ. 79, 31–61 (1989) · Zbl 0693.34020 · doi:10.1016/0022-0396(89)90112-5
[13] Dmitriev, O.Yu.: Eigenfunction expansion of an nth-order differential operator with irregular boundary-value conditions. Mathematics and Its Applications, Saratov State Univ., Saratov, pp. 70–72 (1991)
[14] Dmitriev, O.Yu.: Eigenfunction expansion of a tenth-order boundary-value problem, Mathematics. Mechanics. Part 4, Saratov State Univ., Saratov, pp. 45–48 (2002)
[15] Eberhard W.: Das asymptotische Verhalten der Greenschen Funktion irregulärer Eigenwertprobleme mit zerfallenden Randbedingungen. Math. Z. 86, 45–53 (1964) · Zbl 0127.04001 · doi:10.1007/BF01111277
[16] Eberhard W.: Die Entwicklungen nach Eigenfunktionen irregulärer Eigenwertprobleme mit zerfallenden Randbedingungen. Math. Z. 86, 205–214 (1964) · Zbl 0137.06003 · doi:10.1007/BF01110391
[17] Eberhard W.: Die Entwicklungen nach Eigenfunktionen irregulärer Eigenwertprobleme mit zerfallenden Randbedingungen. II. Math. Z. 90, 126–137 (1965) · Zbl 0235.34048 · doi:10.1007/BF01112237
[18] Eberhard W.: Zur Vollständigkeit des Biorthogonalsystems von Eigenfunktionen irregulärer Eigenwertprobleme. Math. Z. 146, 213–221 (1976) · Zbl 0302.34026 · doi:10.1007/BF01181881
[19] Eberhard W.: Irregular eigenvalue-problems on the half-axis. Results Math. 14, 48–63 (1988) · Zbl 0654.34019 · doi:10.1007/BF03323216
[20] Eberhard W., Freiling G.: Nicht-S-hermitesche Rand- und Eigenwertprobleme (German). Math. Z. 133, 187–202 (1973) · Zbl 0255.34007 · doi:10.1007/BF01238037
[21] Eberhard W., Freiling G.: Das Verhalten der Greenschen Matrix und der Entwicklungen nach Eigenfunktionen N-regulärer Eigenwertprobleme (German). Math. Z. 136, 13–30 (1974) · Zbl 0265.34033 · doi:10.1007/BF01189253
[22] Eberhard W., Freiling G.: Stone-reguläre Eigenwertprobleme. Math. Z. 160, 139–161 (1978) · Zbl 0379.40011 · doi:10.1007/BF01214265
[23] Eberhard W., Freiling G.: Necessary conditions for the convergence and summability of eigenfunction expansions in the complex domain. Differ. Integral Equ. 4, 1217–1232 (1991) · Zbl 0738.34042
[24] Eberhard W., Freiling G.: The distribution of the eigenvalues for second order eigenvalue problems in the presence of an arbitrary number of turning points. Result Math. 21, 24–41 (1992) · Zbl 0777.34055 · doi:10.1007/BF03323070
[25] Eberhard W., Freiling G.: An expansion theorem for eigenvalue problems with several turning points. Analysis 13, 301–308 (1993) · Zbl 0788.34080
[26] Eberhard W., Freiling G., Koch P.: Über eine Klasse von indefiniten Eigenwertproblemen mit stückweise stetiger Gewichtsfunktion. Schriftenreihe des Fachbereichs Mathematik 169, 69 (1989)
[27] Eberhard W., Freiling G., Schneider A.: On the distribution of the eigenvalues of a class of indefinite eigenvalue problems. Differ. Integral Equ. 3, 1167–1179 (1990) · Zbl 0736.34071
[28] Eberhard W., Freiling G., Schneider A.: Expansion theorems for a class of regular indefinite eigenvalue problems. (English) Differ. Integral Equ. 3, 1181–1200 (1990) · Zbl 0736.34072
[29] Eberhard W., Freiling G., Schneider A.: Connection formulas for second order differential equations with a complex parameter and having an arbitrary number of turning points. Math. Nachr. 165, 205–229 (1994) · Zbl 0829.34044 · doi:10.1002/mana.19941650114
[30] Eberhard W., Freiling G., Stoeber T.: Ein indefinites Eigenwertproblem mit einem Vorzeichenwechsel in der Gewichtsfunktion. Schriftenreihe des Fachbereichs Mathematik 148, 82 (1989)
[31] Eberhard W., Freiling G., Wilcken-Stoeber K.: Indefinite eigenvalue problems with several singular points and turning points. Math. Nachr. 229, 51–71 (2001) · Zbl 0996.34067 · doi:10.1002/1522-2616(200109)229:1<51::AID-MANA51>3.0.CO;2-4
[32] Fage M.K.: Operator-analytic functions of one independent variable (Russian). Tr. Mosc. Mat. Obshch. 7, 227–268 (1958) · Zbl 0084.05304
[33] Fage M.K.: Integral representation of operator-analytic functions of one independent variable Veränderlichen (Russian). Tr. Mosc. Mat. Obshch. 8, 3–48 (1959) · Zbl 0090.03702
[34] Faierman M., Markus A., Matsaev V., Möller M.: On n-fold expansions for ordinary differential operators. Math. Nachr. 238, 62–77 (2002) · Zbl 1012.34079 · doi:10.1002/1522-2616(200205)238:1<62::AID-MANA62>3.0.CO;2-Y
[35] Fiedler H.: Zur Regularität selbstadjungierter Randwertaufgaben. Manuskripta Math. 7, 185–196 (1972) · Zbl 0248.34034 · doi:10.1007/BF01679712
[36] Flax, A.H.: Aeroelastic problems at supersonic speed. In: Proceedings of the Second International Aeronautics Conference, International Aeronautics Society, New York, pp. 322–366 (1949)
[37] Freiling G.: Nichtselbstadjungierte Differentialoperatoren im nichtkompakten Fall (German). Math. Z. 149, 267–279 (1976) · Zbl 0323.34018 · doi:10.1007/BF01175589
[38] Freiling G.: Ein äquikonvergenzsatz fr eine Klasse von singulären Differentialoperatoren (German). Math. Z. 153, 255–265 (1977) · Zbl 0345.34013 · doi:10.1007/BF01214479
[39] Freiling G.: Reguläre Eigenwertprobleme mit Mehrpunkt-Integral-Randbedingungen. Math. Z. 171, 113–131 (1980) · Zbl 0432.34013 · doi:10.1007/BF01176703
[40] Freiling G.: Irregular multipoint eigenvalue problems. Math. Methods Appl. Sci. 3, 88–103 (1981) · Zbl 0447.34027 · doi:10.1002/mma.1670030107
[41] Freiling G.: Expansion problems with irregular multipoint boundary conditions. Proc. R. Soc. Edinb. Sect. A 88, 235–246 (1981) · Zbl 0498.34009 · doi:10.1017/S0308210500020084
[42] Freiling G.: On the completeness of eigenfunctions and associated functions of irregular multipoint eigenvalue-problems. J. Reine Angew. Math. 346, 36–47 (1984) · Zbl 0514.34014
[43] Freiling G.: Zur Vollständigkeit des Systems der Eigenfunktionen irregulärer Eigenwertprobleme mit \(\lambda\)-abhängigen Randbedingungen. [On the completeness of the system of eigenfunctions of irregular eigenvalue problems with \(\lambda\)-dependent boundary conditions, Z. Anal. Anwendungen 3, 263–269 (1984) · Zbl 0536.34016
[44] Freiling G.: Zur Vollständigkeit des Systems der Eigenfunktionen und Hauptfunktionen irregulärer Operatorbüschel (German). Math. Z. 188, 55–68 (1984) · Zbl 0552.47004 · doi:10.1007/BF01163872
[45] Freiling G.: Necessary conditions for the uniform convergence and Abel-summability of eigenfunction expansions with irregular ordinary differential bundles. Z. Anal. Anwend. 5, 543–553 (1986) · Zbl 0657.34024
[46] Freiling G.: Necessary conditions for the L 2-convergence of series in eigenfunctions of irregular eigenvalue problems. J. Math. Anal. Appl. 114, 503–511 (1986) · Zbl 0603.34019 · doi:10.1016/0022-247X(86)90103-4
[47] Freiling G.: On the behaviour of eigenfunction expansions in the complex domain. Proc. R. Soc. Edinb., Sect. A 104, 73–91 (1986) · Zbl 0626.34018 · doi:10.1017/S0308210500019077
[48] Freiling G.: General boundary eigenvalue problems for systems of differential equations with multiple roots of the characteristic equation. Proc. R. Soc. Edinb., Sect. A 108, 45–56 (1988) · Zbl 0643.34022 · doi:10.1017/S0308210500026500
[49] Freiling G.: Multiple expansions in series of eigen- and associated functions of boundary eigenvalue problems nonlinearly dependent on the spectral parameter. I. Result Math. 17, 83–99 (1990) · Zbl 0702.34069 · doi:10.1007/BF03322632
[50] Freiling G.: Multiple expansions in series of eigen- and associated functions of boundary eigenvalue problems nonlinearly dependent on the spectral parameter. II. Result Math. 17, 241–255 (1990) · Zbl 0727.34073 · doi:10.1007/BF03322461
[51] Freiling G., Kaufmann F.J.: On uniform and L p -convergence of eigenfunction expansions for indefinite eigenvalue problems. Integral Equ. Oper. Theory 13, 193–215 (1990) · Zbl 0692.34019 · doi:10.1007/BF01193756
[52] Freiling G., Trooshin I.Yu.: Abel-summability of eigenfunction expansions of three-point boundary value problems. Math. Nachr. 190, 129–148 (1998) · Zbl 0909.34071 · doi:10.1002/mana.19981900107
[53] Gurevich, A.P., Khromov, A.P.: First and second order differentiation operators with weight functions of variable sign. Math. Notes 56, 653–661 (1994) [translation from Mat. Zametki 56, 3–15] (1994) · Zbl 0835.34093
[54] Heisecke G.: Rand-Eigenwertprobleme N(y) = \(\lambda\) P(y) bei \(\lambda\)-abhängigen Randbedingungen (German). Mitt. Math. Semin. Gießen 145, 1–74 (1980)
[55] Hopkins J.W.: Some convergent developments associated with irregular boundary conditions. Am. Math. Soc. Trans. 20, 245–259 (1919) · JFM 47.0420.01 · doi:10.1090/S0002-9947-1919-1501125-5
[56] Iskenderova, M.B.: To the problem on multiple summability of expansions in eigen functions of irregular boundary value problems of fourth order. (English summary) Trans. Natl. Acad. Sci. Azerb. Ser. Phys. Tech. Math. Sci. 27, Math. Mech., 79–88 (2007) · Zbl 1156.34071
[57] Jackson D.: Expansion problems with irregular boundary conditions. Am. Acad. Proc. 51, 383–417 (1915) · JFM 47.0419.03 · doi:10.2307/20025584
[58] Kandemir M., Mukhtarov O.Sh.: A method on solving irregular boundary value problems with transmission conditions. Kuwait J. Sci. Eng. 36, 79–99 (2009) · Zbl 1207.34025
[59] Kandemir M., Mukhtarov O., Yakubov Ya.: Irregular boundary value problems with discontinuous coefficients and the eigenvalue parameter. Mediterr. J. Math. 6, 317–338 (2009) · Zbl 1191.34016 · doi:10.1007/s00009-009-0011-x
[60] Kaufmann F.-J.: Abgeleitete Birkhoff-Reihen bei Randeigenwertproblemen zu N(y) = \(\lambda\) P(y) mit \(\lambda\)-abhängigen Randbedingungen. Mitt. Math. Semin. Gießen 190, 107 (1989)
[61] Kesel’man G.M.: On the unconditional convergence of eigenfunction expansions of certain differential operators (Russian). Izv. Vysš. Učebn. Zaved. Matematika 39, 82–93 (1964)
[62] Khromov(=Hromov), A.P.: Expansion in eigenfunctions of ordinary differential operators with nonregular decomposing boundary conditions. Sov. Math., Dokl. 4, 1575–1578 (1963) [translation from Dokl. Akad. Nauk SSSR 152, 1324–1326] (1963) · Zbl 0197.41502
[63] Khromov, A.P.: The summability of expansions in the eigenfunctions of a boundary value problem for an ordinary differential equation with splitting boundary conditions, and a certain analogue of the Weierstrass theorem (Russian). Ordinary differential equations and expansions in Fourier series, No. 1 (Russian), 29–41. Izdat. Saratov. Univ., Saratov (1968)
[64] Khromov A.P.: Finite-Dimensional perturbations of Volterra operators. Matem. Zametki. (Math. Notes) 16, 669–680 (1974)
[65] Khromov A.P.: A differential operator with nonregular splitting boundary conditions. Math. Notes 19, 451–456 (1976) · Zbl 0369.34010 · doi:10.1007/BF01142570
[66] Khromov, A.P.: Expansion by eigenfunctions of a third-order boundary-value problem, Studies in operator theory, Ufa, pp. 182–193 (1988)
[67] Khromov, A.P.: Finite dimensional perturbation of Volterra operators (Russian) Ph. D. Thesis, Novosibirsk (1973) · Zbl 0278.47029
[68] Khromov, A.P.: On equiconvergence of spectral expansions of integral operators. J. Math. Sci., New York 144, 4277–4283 (2007) [translation from Sovrem. Mat. Prilozh.] (2005) · Zbl 1194.47056
[69] Khromov, A.P.: Finite-dimensional perturbations of Volterra operators (Russian) Sovrem. Mat. Fundam. Napravl. 10 (2004), 3–163 (electronic) [translation in J. Math. Sci. (N. Y.) 138, 5893–6066] (2006)
[70] Kogan B.L.: Double completeness of the system of eigenfunctions and associated functions of the Regge problem. Funct. Anal. Appl. 5, 229–232 (1972) · Zbl 0238.34039
[71] Kostyuchenko A.G., Shkalikov A.A.: Summability of eigenfunction expansions of differential operators and convolution operators. Funct. Anal. Appl. 12, 262–276 (1979) · Zbl 0469.34014 · doi:10.1007/BF01076380
[72] Kravitskij, A.O.: Double expansion into series of eigenfunctions of a certain nonself- adjoint boundary-value problem. Differ. Equations 4 (1968), 86–92 (1972) · Zbl 0238.34045
[73] Il’in V.A., Kritskov L.V.: Properties of spectral expansions corresponding to non-self-adjoint differential operators. J. Math. Sci., New York 116, 3489–3550 (2003) · Zbl 1066.47044 · doi:10.1023/A:1024180807502
[74] Langer R.E.: The expansion problem in the theory of ordinary linear differential systems of the second order. Trans. Am. Math. Soc. 31, 868–906 (1929) · JFM 55.0258.01 · doi:10.1090/S0002-9947-1929-1501520-0
[75] Langer R.E.: A theory for ordinary differential boundary problems of the second order and of the highly irregular type. Trans. Am. Math. Soc. 53, 292–361 (1943) · Zbl 0061.18201 · doi:10.1090/S0002-9947-1943-0007826-6
[76] Levin B.: Distribution of Zeros of Entire Functions. American Mathematical Society, Providence (1964) · Zbl 0152.06703
[77] Levin, B.Ya: Lectures on entire functions. In collab. with Yu. Lyubarskii, M. Sodin, V. Tkachenko. Translations of Mathematical Monographs. 150. American Mathematical Society, Providence(1997)
[78] Lidskii B.V.: On summability on series in the principal vectors of non-self-adjoint operators. Tr. Mosk. Matem. Ob-va. 11, 3–35 (1962)
[79] Locker, J.: Spectral Theory of Non-self-adjoint Two-point Differential Operators. Mathematical Surveys and Monographs, vol. 73, American Mathematical Society, Providence (2000) · Zbl 0945.47004
[80] Locker, J.: Eigenvalues and completeness for regular and simply irregular two-point differential operators. Mem. Am. Math. Soc. 911, 1–177 (2008) (an extended version is available as a file from the web) · Zbl 1162.34001
[81] Markus, A.S.: Introduction to the spectral theory of polynomial operator pencils. Translated from the Russian by H. H. McFaden. Translations of Mathematical Monographs, vol. 71. American Mathematical Society, Providence (1988) · Zbl 0678.47005
[82] Mennicken R., Möller M.: Non-self-adjoint boundary eingenvalue problems. North-Holland Mathematics Studies 192. North-Holland, Amsterdam (2003) · Zbl 1033.34001
[83] Minkler, H.: Über eine Erweiterung des Regularitätsbegriffes bei Randwertproblemen gewöhnlicher Differentialgleichungen (German) Dissertation, RRWTH Aachen (1978) · Zbl 0446.34021
[84] Mikhailov V.P.: On Riesz bases in L 2(0,1),. Soviet Math. Dokl. 3, 851–855 (1962) · Zbl 0133.37602
[85] Minkin, A.M.: Regularity of selfadjoint boundary conditions, Mat. Zametki 22 (1977) (6), pp. 835–846 [translation in Math. Notes 22, 958–965] (1978)
[86] Minkin A.M.: Odd and even cases of Birkhoff-regularity. Math. Nachr. 174, 219–230 (1995) · Zbl 0919.47041 · doi:10.1002/mana.19951740115
[87] Minkin A.M.: Equiconvergence theorems for differential operators. J. Math. Sci., New York 96, 3631–3715 (1999) · Zbl 0951.47046 · doi:10.1007/BF02172664
[88] Minkin A.M.: Resolvent growth and Birkhoff-regularity. J. Math. Anal. Appl. 323, 387–402 (2006) · Zbl 1114.34064 · doi:10.1016/j.jmaa.2005.10.058
[89] Möller M.: Expansion theorems for Birkhoff-regular differential boundary operators. Proc. R. Soc. Edinb. Sect. A 107, 349–374 (1987) · Zbl 0655.47042 · doi:10.1017/S0308210500031218
[90] Möller, M., Uschold, C.: Über Randeigenwertprobleme für Differentialgleichungen mit der charakteristischen Gleichung \(\lambda\) p (\(\lambda\) l ) = 0 (On boundary eigenvalue problems for differential equations with characteristic equation \(\lambda\) p (\(\lambda\) l ) = 0) (German) Regensburger Math. Schr. 17 (1988) · Zbl 0658.34012
[91] Naimark, M.A.: Linear differential operators. Part I: Elementary theory of linear differential operators with additional material by the author. Frederick Ungar Publishing Co. XIII, New York (1967) · Zbl 0219.34001
[92] Rasulov, M.L.: Methods of contour integration. North-Holland Series in Applied Mathematics and Mechanics, vol. 3. North-Holland Publishing Company, Amsterdam, XIV (1967) · Zbl 0143.13206
[93] Regge T.: On the analytic behaviour of the eigenvalue of the S-matrix in the complex plane of the energy. Nuovo Cimento X. Ser. 9, 295–302 (1958) · Zbl 0082.42101 · doi:10.1007/BF02724933
[94] Rykhlov, V.S.: On completeness of eigenfunctions of quadratic bundles of ordinary differential operators (Russian original) Russ. Math. 36, 33–42 (1992) [translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1992, 35–44] (1992)
[95] Rykhlov V.S.: On double completeness of eigenfunctions of a quadratic pencil of differential operators of second order (Russian). Zb. Pr. Inst. Mat. NAN Ukr. 6, 237–249 (2009) · Zbl 1199.34451
[96] Rykhlov, V.S.: Completeness of eigenfunctions of one class of pencils of differential operators with constant coefficients. Russ. Math. 53, 33–43 (2009) [translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2009, 42–53] (2009) · Zbl 1190.34109
[97] Rykhlov, V.S.: Completeness of root functions of the simplest strongly irregular differential operators with two-point two-term boundary conditions. Dokl. Math. 80, 762–764 (2009) [translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 428, 740–743] (2009) · Zbl 1190.34110
[98] Salaff S.: Regular boundary conditions for ordinary differential operators. Trans. Am. Math. Soc. 134, 355–373 (1968) · Zbl 0198.42704
[99] Schröder, M.: Spectrum and asymptotic distribution of eigenvalues of singular Sturm-Liouville-problems with indefinite weight function (German) Dissertation, Univ. Duisburg, Duisburg, Fachbereich Mathematik (1997) · Zbl 1040.34506
[100] Schultze B.: Strongly irregular boundary value problems. Proc. R. Soc. Edinb. Sect. A 82, 291–303 (1978/79) · Zbl 0404.34017 · doi:10.1017/S0308210500011252
[101] Seifert G.: A third order boundary value problem arising in aeroelastic wing theory. Q. Appl. Math. 9, 210–218 (1951) · Zbl 0043.09102
[102] Seifert G.: A third order irregular boundary value problem and the associated series. Pac. J. Math. 2, 395–406 (1952) · Zbl 0047.32804 · doi:10.2140/pjm.1952.2.395
[103] Shiryaev, E.A.: Birkhoff regularity in terms of the growth of the norm for the Green function. J. Math. Sci., New York 151, 2793–2799 (2008) [translation from Fundam. Prikl. Mat. 12, 231–239] (2006) · Zbl 1151.34310
[104] Shiryaev, E.A., Shkalikov, A.A.: Regular and completely regular differential operators. Math. Notes 81, 566–570 (2007) [translation from Mat. Zametki 81, 636–640] (2007) · Zbl 1156.34075
[105] Shkalikov, A.A.: The completeness of eigenfunctions and associated functions of an ordinary differential operator with irregular-separated boundary conditions. Funct. Anal. Appl. 10, 305–316 (1976) [translation from Funkts. Anal. Prilozh. 10, 69–80] (1976) · Zbl 0354.34023
[106] Shkalikov, A.A.: Boundary problems for ordinary differential equations with parameter in the boundary conditions. J. Sov. Math. 33, 1311–1342 (1986) [translation from Tr. Semin. Im. I. G. Petrovskogo 9, 190–229] (1983) · Zbl 0609.34019
[107] Shkalikov, A.A.: Spectral analysis of the Redge problem (Russian original) J. Math. Sci., New York 144, 4292–4300 (2007) [translation from Sovrem. Mat. Prilozh.] (2005) · Zbl 1282.34091
[108] Shkalikov A.A., Tretter C.: Spectral analysis for linear pencils N \(\lambda\) P of ordinary differential operators. Math. Nachr. 179, 275–305 (1996) · Zbl 0862.34058 · doi:10.1002/mana.19961790116
[109] Stone M.H.: An unusual type of expansion problem. Trans. Am. Math. Soc. 26, 335–355 (1924) · JFM 50.0313.01 · doi:10.1090/S0002-9947-1924-1501281-0
[110] Stone M.H.: A comparision of the series of Fourier and Birkhoff. Trans. Am. Math. Soc. 28, 695–761 (1926) · JFM 53.0419.03 · doi:10.1090/S0002-9947-1926-1501372-6
[111] Stone M.H.: Irregular differential systems of order two and the related expansion problems. Trans. Am. Math. Soc. 29, 23–53 (1927) · JFM 53.0429.01 · doi:10.1090/S0002-9947-1927-1501375-2
[112] Stone M.H.: The expansion problems associated with regular differential systems of the second order. Trans. Am. Math. Soc. 29, 826–844 (1927) · JFM 53.0420.01 · doi:10.1090/S0002-9947-1927-1501416-2
[113] Tamarkin Ya.D., Tamarkine J.: Sur quelques points de la theorie des equations differentielles lineaires ordinaires et sur la generalisation de la serie de Fourier. Palermo Rend. 34, 345–382 (1912) · JFM 43.0397.03 · doi:10.1007/BF03015024
[114] Tamarkin, Ya.D.: O nekotorykh obshchikh zadachakh teorii obyknovennykh diferentsial’nykh uravnenii (On Some General Problems in the Theory of Ordinary Differential Equations), Petrograd (1917)
[115] Tamarkin Ya.D.: Some general problems of the theory of ordinary linear differential equations and expansion of an arbitrary function in series of fundamental functions. Math. Z. 27, 1–54 (1927) · JFM 53.0419.02 · doi:10.1007/BF01171084
[116] Tretter, C.: On \(\lambda\)-nonlinear boundary eigenvalue problems. Mathematical Research, vol. 71. Akademie Verlag, Berlin (1993) · Zbl 0785.34054
[117] Tretter C.: Boundary eigenvalue problems for differential equations N\(\eta\) = \(\lambda\) P\(\eta\) with \(\lambda\)-polynomial boundary conditions. J. Differ. Equ. 170, 408–471 (2001) · Zbl 0984.34010 · doi:10.1006/jdeq.2000.3829
[118] Trooshin, I.Yu.: Abel summability of the series of eigen- and associated functions of the integral and differential operators. In: Operators Theory: Advances and Applications, vol. 57, pp. 307–309 (1992) · Zbl 0875.45001
[119] Trooshin I.Yu.: On summability of eigenfunction expansions of differential and integral operators. Matem. Zametki. (Math. Notes) 54, 114–122 (1993)
[120] Vagabov, A.I.: On the multiple completeness of eigenfunctions and adjoint functions for ordinary differential bundles with irregular boundary conditions (Russian). Vestn. Mosk. Univ., Ser. I, pp. 3–6 (1982) · Zbl 0495.34012
[121] Vagabov A.I.: The completeness of eigenfunctions of an ordinary differential pencil of non-Keldysh type (Russian). Differ. Uravn. 20, 375–382 (1984) · Zbl 0548.34023
[122] Vagabov, A.I.: Completeness of the eigenfunctions of irregular differential operators in a space of vector-valued functions. Math. Notes 40, 611–614 (1986) [translation from Mat. Zametki 40, 197–202] (1986) · Zbl 0621.34015
[123] Vagabov, A.I.: Conditions of multiple completeness of the eigenelements of an ordinary differential bundle. (Russian original) Sov. Math. 30, 16–24 (1986) [translation from Izv. Vyssh. Uchebn. Zaved., Mat., 13–20] (1986) · Zbl 0616.34018
[124] Vagabov A.I.: On the basis property of root elements of ordinary differential operators in the space L 2, n(a, b). Differ. Equ. 44, 1651–1658 (2008) · Zbl 1175.34108 · doi:10.1134/S001226610812001X
[125] Vagabov, A.I., Ragimkhanov, V.R.: (C, 1)-summability of Fourier series in root functions of ordinary linear differential operators in a space of vector functions. Differ. Equ. 41, 1087–1096 (2005) [translation from Differ. Uravn. 41, 1037–1045] (2005) · Zbl 1145.42004
[126] Vass J.I.: A class of boundary problems of highly irregular type. Duke Math. J. 2, 151–165 (1936) · JFM 62.0521.01 · doi:10.1215/S0012-7094-36-00214-4
[127] Ward L.E.: An irregular boundary value and expansion problem. Ann. Math. 26(2), 21–36 (1924) · JFM 50.0314.03 · doi:10.2307/1967740
[128] Ward L.E.: Some third-order irregular boundary value problems. Trans. Am. Math. Soc. 29, 716–745 (1927) · JFM 53.0428.03 · doi:10.1090/S0002-9947-1927-1501411-3
[129] Ward L.E.: Series associated with certain irregular third-order boundary value problems. Trans. Am. Math. Soc. 32, 544–557 (1930) · JFM 56.0396.01 · doi:10.1090/S0002-9947-1930-1501552-0
[130] Ward L.E.: A third-order irregular boundary value problem and the associated series. Trans. Am. Math. Soc. 34, 417–434 (1932) · JFM 58.0467.02 · doi:10.1090/S0002-9947-1932-1501646-1
[131] Ward L.E.: A third-order irregular boundary value problem and the associated series. Am. J. Math. 57, 345–362 (1935) · JFM 61.0497.01 · doi:10.2307/2371212
[132] Wermuth, E.: Konvergenzuntersuchungen bei Eigenfunktionsentwicklungen zu Randeigenwertproblemen n-ter Ordnung mit parameterabhängigen Randbedingungen (German) Dissertation, RWTH Aachen (1984) · Zbl 0563.34023
[133] Wermuth E.: A generalization of Lebesgues convergence criterion for Fourier series. Result Math. 15, 186–195 (1989) · Zbl 0667.42002 · doi:10.1007/BF03322455
[134] Wilder C.E.: Expansion problems of ordinary linear differential equations with auxiliary conditions at more than two points. Am. Math. Soc. Trans. 18, 415–442 (1917) · JFM 46.0697.02 · doi:10.1090/S0002-9947-1917-1501077-6
[135] Wolter M.: äber die Entwicklungen nach Eigenfunktionen N-irregulärer Eigenwertprobleme mit zerfallenden Randbedingungen (German). Math. Z. 178, 99–113 (1981) · Zbl 0459.34010 · doi:10.1007/BF01218374
[136] Wolter M.: Das asymptotische Verhalten der Greenschen Funktion N-irregulärer Eigenwertprobleme mit zerfallenden Randbedingungen (German English summary) [The asymptotic behavior of the Green function of N-irregular eigenvalue problems with splitting boundary conditions]. Math. Methods Appl. Sci. 5, 331–345 (1983) · Zbl 0588.34019 · doi:10.1002/mma.1670050122
[137] Yakubov S.: On a new method for solving irregular problems. J. Math. Anal. Appl. 220, 224–249 (1998) · Zbl 0914.34073 · doi:10.1006/jmaa.1997.5876
[138] Yakubov Ya.: Irregular boundary value problems for ordinary differential equations. Analysis (Munich) 18, 359–402 (1998) · Zbl 0919.34071
[139] Yakubov S., Yakubov, Ya.: Differential-operator equations: ordinary and partial differential equations. In: Monographs and Surveys in Pure and Applied Mathematics, vol. 103, Chapman &amp; Hall/CRC, Boca Raton (2000) · Zbl 0936.35002
[140] Zavgorodniij M.G.: The spectrum of a multipoint boundary value problem (Russian). Differ. Uravneniya 20, 1443–1444 (1984) · Zbl 0562.34009
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