×

zbMATH — the first resource for mathematics

Classification of second order linear differential equations with respect to oscillation. (English) Zbl 0188.40101

PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Adamov, N.V., On certain transformations not changing the integral curves of a differential equation of the first order, Mat. sb., 23, 65, 187-228, (1948), no. 2
[2] Barrett, J.H., Behavior of solutions of second order selfadjoint differential equations, (), 247-251 · Zbl 0064.33201
[3] Barrett, J.H., Disconjugacy of 2nd order linear differential equations with nonnegative coefficients, (), 552-561 · Zbl 0092.08103
[4] Bôcher, M.; Bôcher, M., The theorems of oscillation of Sturm and Klein, I & II, Bull. am. math. soc., Bull. am. math. soc., 4, 365-376, (1897 1998)
[5] Bôcher, M., Non oscillatory linear differential equations of the second order, Bull. am. math. soc., 7, 333-340, (1900 1901)
[6] Bôcher, M., Application of a method of d’Alembert to the proof of Sturm’s theorem of comparison, Trans. amer. math. soc., 1, 414-420, (1900) · JFM 31.0352.03
[7] Bôcher, M., An elementary proof of a theorem of Sturm, Trans. amer. math. soc., 2, 150-151, (1901) · JFM 32.0342.01
[8] Bohl, P., Über eine differentialgleichung der störungstheorie, J. reine angew. math., 131, 268-321, (1906) · JFM 37.0343.02
[9] Bor\( u̦\)vka, O., Théorie analytique et constructive des transformations différentielles linéaires du second ordre, Bull. math. soc. roum. sci., 1, (49), no. 2, 125-130, (1957) · Zbl 0082.07501
[10] Bor\( u̦\)vka, O., Transformation of ordinary second-order linear differential equations, (), 27-38
[11] Bor\( u̦\)vka, O., Lineare differentialtransformationen 2, (1967), Ordnung, Veb Deutscher Verlag der Wissenschaften Berlin · Zbl 0153.11201
[12] Coles, W.J., An oscillation criterion for second-order linear differential equations, (), 755-759 · Zbl 0172.11702
[13] Coles, W.J.; Willett, D., Summability criteria for oscillation of second order linear differential equations, Annali di mat. pura ed app., 79, 391-398, (1968) · Zbl 0183.37001
[14] De Castro Brzezicki, A.; De Castro Brzezicki, A., Integrales oscilantes de las ecuaciones diferenciales lineales con segundo miembro, Las ciencias, Las ciencias, 79, 391-398, (1968)
[15] Drahlin, M.E., A comparison principle for differential equations of the second order on an infinite interval, Izv. vysš. Učebn. zaved. mat., no. 9 (64), 26-30, (1967), (Russian)
[16] El’šin, M.I., Qualitative problems on the linear differential equation of the second order, Dokl. adad. nauk SSSR (N. S.), 68, no. 2, 221-224, (1949), (Russian) · Zbl 0041.41803
[17] El’šin, M.I., The phase method and the classical method of comparison, Dokl. adad. nauk SSSR (N. S.), 68, 813-816, (1949), (Russian)
[18] El’šin, M.I., Qualitative investigation of a system of two linear homogeneous equations of the first order, Dokl. adad. nauk SSSR (N. S.), 94, 5-8, (1954), (Russian) · Zbl 0058.07601
[19] El’šin, M.I., On a solution of a classical oscillation problem, Dokl. adad. nauk SSSR (N. S.), Soviet math. dokl., 3, 1752-1755, (1962), (Russian) · Zbl 0136.07701
[20] El’šin, M.I., Qualitative solution of a linear differential equation of the second order, Uspehi mat. nauk. (N. S.), 5, no. 2 (36), 155-158, (1950), (Russian) · Zbl 0041.05703
[21] Fink, A.M.; St. Mary, D.F., A generalized Sturm comparison theorem and oscillation coefficients, Monat. math., 73, 207-212, (1969) · Zbl 0182.12301
[22] Fite, W.B., Concerning the zeros of the solutions of certain differential equations, Trans. amer. math. soc., 19, 341-352, (1918) · JFM 46.0702.02
[23] Fort, T., Some theorems of comparison and oscillation, Bull. amer. math. soc., 24, 330-334, (1917 1918)
[24] Fubini, G., Su un teorema di confronto per le equazioni del secondo ordine alle derivate ordinarie, Ann. scuola norm. super. Pisa, 2, 283-284, (1933), S. II · JFM 59.0443.05
[25] Gagliardo, E., Sui criteri di oscillazione per gli integrali di un’equazione differenziale lineare del secondo ordine, Boll. unione mat. ital., 9, 177-189, (1954) · Zbl 0056.08302
[26] Hartman, P., On the linear logarithmico-exponential differential equation of the second order, Amer. J. math., 70, 764-779, (1948) · Zbl 0035.18302
[27] Hartman, P., On linear second order differential equations with small coefficients, Amer. J. math., 73, 955-962, (1951) · Zbl 0045.36401
[28] Hartman, P., On non-oscillatory linear differential equations of second order, Amer. J. math., 74, 389-400, (1952) · Zbl 0048.06602
[29] Hartman, P.; Wintner, A., On non-conservatory linear oscillators of low frequency, Amer. J. math., 70, 529-539, (1948) · Zbl 0037.19002
[30] Hartman, P.; Wintner, A., On nonoscillatory linear differential equations, Amer. J. math., 75, 717-730, (1953) · Zbl 0051.06503
[31] Hille, E., Non-oscillation theorems, Trans. amer. math. soc., 64, 234-252, (1948) · Zbl 0031.35402
[32] Ielchin, M., Sur le problème d’oscillation pour l’équation différentielle linéaire du deuxième ordre, C. R. (doklady) acad. sci. URSS, 18, no. 3, 141-145, (1938) · JFM 64.0442.02
[33] Jelchin, M., Sur une méthode d’évaluation de la phase d’une équation différentielle linéaire du second ordre, Uchenye zapiski moskov. GoS. univ. mat., 45, 97-108, (1940) · JFM 66.0403.04
[34] Kamke, E., A new proof of Sturm’s comparison theorems, Amer. math. monthly, 46, 417-421, (1939) · JFM 65.1263.05
[35] Kamke, E., Über sturms vergleichssätze fur homogene lineare differentialgleichungen zweiter ordnung und systeme von zwei differentialgleichungen erster ordnung, Math. zeit., 47, 788-795, (1940) · JFM 68.0190.02
[36] Kamke, E., Bemerkungen zu einigen trennungssätzen von M. nicolesco und S. Takahashi fur die nullstellen der Lösungen von differentialgleichungen, Math. cluj, 15, 201-203, (1939) · Zbl 0020.35801
[37] Kneser, A., Untersuchungen über die reelen nullstellen der integrale linearer differentialgleichungen, Math. ann., 42, 409-435, (1893) · JFM 25.0522.01
[38] Knobloch, Hans-Wilhelm, Wachstum und oszillatorisches verhalten von Lösungen linearer differentialgleichungen zweiter ordnung, Jber. Deutsch. math. verein., 66, 138-152, (1963 1964), Abt. 1
[39] Kondratév, V.A., Elementary derivation of a necessary and sufficient condition for nonoscillation of the solutions of a linear differential equation of second order, Uspehi mat. nauk (N.S.), 12, no. 3 (75), 159-160, (1957), (Russian)
[40] Kondratév, V.A., Sufficient conditions for nonoscillatory or oscillatory nature of solutions of the second order equation y″ + p(x)y = 0, Dokl. akad. nauk SSSR, 113, 742-745, (1957), (Russian) · Zbl 0088.06104
[41] Kummer, E.E., De generali quadam aequatione differentiali tertii ordinis, J. reine angew. math., 100, 1-9, (1887), (Abdruck aus dem Programm des evangelischen Königl. und Stadtgym. in Liegnitz vom Jahre 1834) · JFM 18.0297.01
[42] Laitoch, M., Sur une théorie des critères comparatifs sur l’oscillation des intégrales de l’équation différentielle u″ = P(x)u, Spisy Přir. fak. M U Brno, 365, 255-267, (1955) · Zbl 0068.07002
[43] Landolino, G., Sul teorema di confronto di Sturm, Boll. un. mat. ital., 2, 16-19, (1947), Ser. III · Zbl 0031.21401
[44] Leighton, W., On self-adjoint differential equations of second order, (), 656-657 · Zbl 0034.35401
[45] Leighton, W., The detection of the oscillation of solutions of a second-order linear differential equation, Duke J. math., 17, 57-62, (1950) · Zbl 0036.06101
[46] Leighton, W., On self-adjoint differential equations of second-order, J. London math. soc., 27, 37-47, (1952) · Zbl 0048.06503
[47] Leighton, W., Comparison theorems for linear differential equations of second order, (), 603-610 · Zbl 0118.08202
[48] Levin, A.Ju., A comparison principle for second-order differential equations, Dokl. akad. nauk SSSR, Soviet math. dokl., 1, 1313-1316, (1961), (Russian) · Zbl 0102.07801
[49] Levin, A.Ju., On linear second-order differential equations, Dokl. akad. nauk SSSR, Soviet math. dokl., 4, 1814-1817, (1963), (Russian)
[50] Levin, A.Ju., Integral criteria for the equation \(ẍ + q(t)x = 0\) to be nonoscillatory, Uspehi mat. nauk, 20, no. 2 (122), 244-246, (1965), (Russian)
[51] Lie, S., Zur theorie der transformationsgruppen, (), Heft III, 386-395, (1894), Part 1, No. XVII · JFM 25.0626.01
[52] Magnus, W.; Winkler, S., Hill’s equation, (), 56-78
[53] Manacorda, T., Sul comportamento asintotico degli integrali dell’equazione: y″(x) + p(x)y′(x) + q(x)y(x) = 0 quando limx→∞q(x) = ∞, I, Atti accad. naz. lincei rend. cl. sci. fis. mat. nat., 2, 8, 537-541, (1947) · Zbl 0029.21203
[54] Markus, L.; Moore, R.A., Oscillation and disconjugacy for linear differential equations with almost periodic coefficients, Acta math., 96, 99-123, (1956) · Zbl 0071.08302
[55] Miller, J.C.P., On a criterion for oscillatory solutions of a linear differential equation of the second order, (), 283-287 · Zbl 0025.26401
[56] Milos, R., Kriterien für die oszillation der Zösungen der differentialgleichung [p(x)y′]′ + q(x)y = 0, Casopis pro pestovani mat., 84, 3, (1959)
[57] Min, I.M., Sur LES zéros des solutions des certaines équations différentielles, Bull. inst. politehn. bucuresti, 27, no. 6, 17-23, (1967) · Zbl 0219.34033
[58] Moore, R.A., The behavior of solutions of a linear differential equation of second order, Pacific J. math., 5, 125-145, (1955) · Zbl 0064.08401
[59] Moore, R.A., The least eigenvalue of Hill’s equation, J. D’analyse math., 5, 183-196, (1956 1957)
[60] Nakano, H., Über die verteilung der nullstellen von dem Lösungen der differentialgleichung \((d\^{}\{2\}w)(dz\^{}\{2\}) + G(z)(w) = 0\) einer komplexen veränderlichen, (), 337-339 · JFM 58.0459.02
[61] Nehari, Z., Oscillation criteria for second-order linear differential equations, Trans. amer. math. soc., 85, 428-445, (1957) · Zbl 0078.07602
[62] Nicolesco, M., Sur le théorème de Sturm, Math. cluj, 1, 111-114, (1929) · JFM 55.0254.03
[63] Nikolenko, L.D., On oscillation of solutions of the differential equation y″ + p(x)y = 0, Ukrain. mat. Z, 7, 124-127, (1955) · Zbl 0064.33204
[64] Oakley, C.O., A note on the methods of Sturm, Ann. math., 31, 660-662, (1930), ser. II · JFM 56.0387.01
[65] Olech, C.; Opial, Z.; Wazewski, T., Sur le problème d’oscillation des intégrales de l’équation y″ + q(t)y = 0, Bull. acad. polonaise sci., 5, 621-626, (1957), cl. III · Zbl 0078.07701
[66] Opial, Z., Sur LES intégrales oscillantes de l’équation différentielle u″ + f(t)u = 0, Ann. polon. math., 4, 308-313, (1958) · Zbl 0083.07701
[67] Opial, Z., Sur une critère d’oscillation des intégrales de l’équation différentielle (Q(t)x′)′ + f(t)x = 0, Ann. polon. math., 6, 99-104, (1959 1960)
[68] Petropavlovskaya, R.V., On oscillation of solutions of the equation u″ + p(x)u = 0, Dokl. akad. nauk SSSR (N. S.), 105, 29-31, (1955), (Russian)
[69] Potter, R.L., On self-adjoint differential equations of 2nd order, Pacific J. math., 3, 467-491, (1953) · Zbl 0051.06502
[70] Putnam, C.R., An oscillation criterion involving a minimuḿ principle, Duke math. J., 16, 633-636, (1949) · Zbl 0037.34302
[71] Putnam, C.R., Note on some oscillation criteria, (), 950-952 · Zbl 0067.06202
[72] Ráb, M., Eine bemerkung zu der frage über die oszillatorischen eigenschaften der Lösungen der differentialgleichungen y″ + A(x)y = 0, C̆as. Pěst. mat., 82, 342-348, (1957) · Zbl 0105.06702
[73] Ráb, M., Kriterien fur die oszillation der Lösungen der differentialgleichung [p(x)y′]′ + q(x)y = 0, C̆as. Pěst. mat., Erratum, C̆as. Pěst. mat., 85, 91-370, (1960) · Zbl 0087.29505
[74] Reid, W.T., A comparison theorem for self-adjoint differential equations of second order, Ann. math., 65, 197-202, (1957) · Zbl 0077.08604
[75] Richard, U., Sulla risoluzione asintoticonumerica dell’equazione differenziale (py′)′ + qy = 0 nel caso oscillante, Atti accad. sci. Torino, 97, 857-890, (1962 1963)
[76] Riemann, B.; Weber, H., Die partiellen differentialgleichungen der mathematischen physik, vol. II, 53-72, (1912), 5th ed., Braunschweig
[77] Mary, D.F.St., Some oscillation and comparison theorems for (r(t)y′)′ + p(t)y = 0, () · Zbl 0613.34006
[78] Sansone, G., ()
[79] Sobol, I.M., Investigation with the aid of polar coordinates of the asymptotic behavior of a linear differential equation of the second order, Mat. sb., 28, 70, 707-714, (1951), no. 3
[80] Stäckel, P., Ueber transformationen von differentialgleichungen, J. reine angew. math., 111, 290-302, (1893) · JFM 25.0167.01
[81] Sturm, C., Sur LES équations différentielles linéaires du second ordre, J. math. pures appl., 1, 106-186, (1836)
[82] Choy-tak, Taam, Non-oscillatory differential equations, Duke math. J., 19, 493-497, (1952) · Zbl 0049.34403
[83] Shin-ichi, Takahashi, On the zero points of an integral of a linear differential equation, Jap. J. math., 7, 335-346, (1931) · Zbl 0001.39401
[84] Tonolo, A., Sopra un teorema di confronto di Fubini, Boll. un. mat. Italy, 14, 67-70, (1935) · JFM 61.0496.03
[85] Vessoit, E., Encyklopedia math. wissenschaften, iia 4b, (), 271-285, Leipzig
[86] Willett, D., On the oscillatory behavior of the solutions of second order linear differential equations, Ann. polon. math., 21, 175-194, (1969) · Zbl 0174.13701
[87] Wintner, A., A norm criterion for non-oscillatory differential equations, Quart. appl. math., 6, 183-185, (1948) · Zbl 0034.35402
[88] Wintner, A., A criterion of oscillatory stability, Quart. appl. math., 7, 115-117, (1949) · Zbl 0032.34801
[89] Wintner, A., Asymptotic integrations of the adiabatic oscillator in its hyperbolic range, Duke math. J., 15, 55-67, (1948) · Zbl 0034.35501
[90] Wintner, A., On almost free linear motions, Amer. J. math., 71, 595-602, (1949) · Zbl 0033.06002
[91] Wintner, A., On the non-existence of conjugate points, Amer. J. math., 73, 368-380, (1951) · Zbl 0043.08703
[92] Wintner, A., On the comparison theorem of Kneser-Hille, Math. scand., 5, 2, 255-260, (1957) · Zbl 0080.29801
[93] Yelchin, M., Sur LES conditions pour qu’une solution d’un système linéaire du second ordre possède deux zéros, C. R. (doklady) acad. sci. URSS, 51, no. 8, 573-576, (1946) · Zbl 0063.09068
[94] Yosida, K., On the distribution of α-points of solutions for linear differential equations of the second order, (), 335-336 · JFM 58.1127.05
[95] Zlámal, M., Oscillation criterions, Cas. pěst. mat. a fis., 75, 213-217, (1950) · Zbl 0040.19502
[96] Zubova, A.F., Concerning oscillation of the solutions of an equation of the second order, Vestnik leningrad univ., 12, 211, 168-174, (1957), no. 1 · Zbl 0091.08104
[97] Zubova, A.F., Oscillations and stability of solutions of second-order equations, Sibirsk. mat. ž, 4, 1060-1070, (1963), (Russian)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.