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Oscillation theory of ordinary linear differential equations. (English) Zbl 0213.10801

This is a good and long survey article on oscillation theory, divided into the following themes: 1. Second order equations. 2. A third order equation. 3. Fourth order equations. 4. Self-adjoint fourth order equations. 5. Second order matrix equations and related first order systems.
There is an extensive bibliography of 112 items. The paper should be a must for people in the field of oscillations.
Reviewer: C. Imaz

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
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[1] Appel, P., Sur la transformation des equations differentielles lineares, Compt. Rend. (Paris), 91, 211-214 (1880)
[2] Atkinson, F. V., Discrete and Continuous Boundary Problems (1964), Academic Press: Academic Press New York · Zbl 0117.05806
[3] AMS transl., 42, 233-245 (1964), [English Translation: · Zbl 0158.08804
[4] Barrett, J. H., Behavior of solutions of second-order self-adjoint differential equations, (Proc. AMS, 6 (1955)), 247-251 · Zbl 0064.33201
[5] Barrett, J. H., Matrix systems of second order differential equations, Portugal. Math., 14, 79-89 (1955) · Zbl 0066.33703
[6] Barrett, J. H., A Prufer transformation for matrix differential equations, (Proc. Am. Math. Soc., 8 (1957)), 510-518 · Zbl 0079.10603
[7] Barrett, J. H., Second order complex differential equations with a real independent variable, Pacific J. Math., 8, 187-200 (1958) · Zbl 0085.30501
[8] Barrett, J. H., Disconjugacy of second order linear differential equations with nonnegative coefficients, (Proc. Am. Math. Soc., 10 (1959)), 552-561 · Zbl 0092.08103
[9] Barrett, J. H., Systems-disconjugacy of a fourth-order differential equation, (Proc. Am. Math. Soc., 12 (1961)), 205-213 · Zbl 0098.05902
[10] Barrett, J. H., Fourth-order boundary value problems and comparison theorems, Canadian J. Math., 13, 625-638 (1961) · Zbl 0101.06603
[11] Barrett, J. H., Two-point boundary problems for linear self-adjoint differential equations of the fourth order with middle term, Duke Math. J., 29, 543-554 (1962) · Zbl 0108.08204
[12] Barrett, J. H., Disconjugacy of generalized linear differential equations of third and fourth order, Notices AMS, 9, 469 (1962), (Abstract No. 594-36)
[13] Barrett, J. H., Canonical forms for third-order linear differential equations, Ann. Mat. Pura Appl., 65, 253-274 (1964) · Zbl 0125.32502
[14] Barrett, J. H., Third-order equations with nonnegative coefficients, J. Math. Anal. Appl., 24, 212-224 (1968) · Zbl 0174.39903
[15] Bellman, R., Introduction to Matrix Analysis (1960), McGraw-Hill: McGraw-Hill New York · Zbl 0124.01001
[16] Birkhoff, G. D., On the solutions of ordinary linear homogeneous differential equations of the third order, Ann. Math., 12, 103-127 (1910-1911) · JFM 42.0344.01
[17] Bliss, G. A., Lectures on the Calculus of Variations (1946), The University of Chicago Press: The University of Chicago Press Chicago · Zbl 0063.00459
[18] Coddington, E. A.; Levinson, N., Theory of Ordinary Differential Equations (1955), McGraw-Hill: McGraw-Hill New York · Zbl 0042.32602
[19] Coles, W. J., A general Wirtinger-type inequality, Duke Math. J., 27, 133-138 (1960) · Zbl 0096.28601
[21] Coppel, W. A., Comparison theorems for canonical systems of differential equations, J. Math. Anal. Appl., 12, 306-315 (1965) · Zbl 0137.27902
[22] Courant, R.; Hilbert, D., (Methods of Mathematical Physics, Vol. I (1953), Interscience: Interscience New York) · Zbl 0729.00007
[23] Dobrohotova, M. A., On boundedness of solutions of linear differential equations of third order, Yaroslavl. Gosudarstvennyi Pedagogichoskii Institut Uchenye Zapiski, 34, 19-34 (1960), (in Russian)
[24] Dolan, J. M., Oscillatory Behavior of Solutions of Linear Ordinary Differential Equations of Third Order (1967), University of Tennessee, Unpublished doctoral dissertation
[25] Etgen, G. J., Oscillatory properties of certain nonlinear matrix differential systems of second order, Trans. Am. Math. Soc., 122, 289-310 (1966) · Zbl 0151.12302
[26] Etgen, G. J., A note on trigonometric matrices, (Proc. Am. Math. Soc., 17 (1966)), 1226-1232 · Zbl 0152.08403
[27] Etgen, G. J., On the determinants of solutions of second order matrix differential systems, J. Math. Anal. Appl., 18, 585-598 (1967) · Zbl 0173.10101
[28] Frobenius, G., Über adjugierte lineare Differentialausdrücke, Journal für Mathematik, 85, 185-213 (1878) · JFM 10.0247.01
[29] Giuliano, Landolino, On linear self-adjoint equations of third order, Boll. Univ. Mat. Ital., 12, 16-18 (1957)
[30] Greguš, M., On certain new properties of solutions of the differential equation \(y\)‴ + Qy′ + \(Q\)′\(y = 0\), Publ. Fac. Sci. Université Masaryk, Bruo, Czech., 362, 237-251 (1955), (in Czech.)
[31] Greguš, M., On a new boundary-value problem for differential equations of third order, Czech. Math. J., 7, 41-47 (1957), (in Russian)
[32] Greguš, M., On the linear differential equations of third order with constant coefficients, Acta Fac. Rerum. Natur. Univ. Comenian. Math., 2, 61-65 (1957), (in Slovak)
[33] Greguš, M., A remark about the dispersions and transformations of a differential equation of third-order, Acta Fac. Rerum Natur. Univ. Comenian. Math., 4, 205-211 (1959), (in Czech.)
[34] Greguš, M., Oxzillatorische Eigenschatten der Lösungen der linearen Differentialgleichung dritter Ordnung \(y\)‴ + 2Ay′ + \((A\)′ + \(b)y = 0\), wo \(A = A(x)\) ⩽ 0 ist, Czech. Mat. J., 9, 416-427 (1959) · Zbl 0093.08801
[35] Greguš, M., Über einige Eigenschaften der Lösungen der Differentialgleichung \(y\)‴ + 2Ay′ + \((A\)′ + \(b)y = 0, A\) ⩽ 0, Czech. Mat. J., 11, 86, 106-115 (1961) · Zbl 0099.06702
[36] Greguš, M., Oscillatory properties of the solutions of the third-order differential equation of the type \(y\)‴ + \(2A(x)y\)′ + \([A\)′\((x) + b(x)]y = 0\), Acta Fac. Rerum natur. Univ. Comenian. Math., 6, 275-300 (1961)
[37] Greguš, M., Bemerkungen zu der unlösharen Randwert problemen dritter Ordnung, Acta Fac. Rerum Natur. Univ. Comenian. Math., 7, 639-647 (1963)
[38] Greguš, M., Über einige Eigenschatten der Lösungen der Differentialgleichung dritter Ordnung, Acta Fac. Rerum Natur. Univ. Comenian. Math., 7, 585-595 (1963) · Zbl 0142.34604
[39] Greguš, M., Über einige Radwertprobleme Dritter Ordnung, Czech. Math. J., 13, 551-560 (1963) · Zbl 0117.06005
[40] Greguš, M., Über die lineare homogene Differentialgleichung dritter Ordnung, Wiss. Zeit. Martin-Luther Univ. Halle-Wittenberg, XII/3, 265-286 (1963) · Zbl 0118.30501
[41] Greguš, M., Über die asymptotischen Eigenschatten der Lösungen der linearen Differentialgleichung dritter Ordnung, Ann. Mat. Pura Appl., 63, 1-10 (1963) · Zbl 0123.27801
[42] Greguš, M., Über die Eigenschaften der Lösungen einiger quasilinearer Gleichungen 3. Ordnung, Acta Fac. Rerum Natur. Univ. Comenian. Math., 10, 11-22 (1965) · Zbl 0145.10602
[43] Hahn, W., Über Orthogonalpolynome mit drei Parametern, Deutsche Math., 5, 273-278 (1940) · JFM 66.0314.02
[44] Hallam, T. G., Asymptotic behavior of the solutions of an \(n\) th-order nonhomogeneous ordinary differential equation, Trans. Am. Math. Soc., 122, 177-194 (1966) · Zbl 0138.33001
[45] Hanan, M., Oscillation criteria for third-order linear differential equations, Pacific J. Math., 11, 919-944 (1961) · Zbl 0104.30901
[46] Handleman, G.; Tu, Yih-O, Lateral vibrations of a beam under initial linear axial stress, J. SIAM, 9, 455-473 (1961) · Zbl 0108.37703
[47] Hartman, P., Ordinary Differential Equations (1964), Wiley: Wiley New York · Zbl 0125.32102
[48] Hartman, P.; Wintner, A., On disconjugate differential systems, Canadian J. Math., 8, 72-81 (1956) · Zbl 0075.07701
[49] Heidel, J., Qualitative behavior of solutions of a third order nonlinear differential equation, Pacific J. Math., 27, 507-526 (1968) · Zbl 0172.11703
[50] Hille, E., Nonoscillation theorems, Trans. Am. Math. Soc., 64, 234-252 (1948) · Zbl 0031.35402
[51] Hinton, D. B., Disconjugate properties of a system of differential equations, J. Differential Eqs., 2, 420-437 (1966) · Zbl 0161.27904
[52] Hinton, D. B., Clamped end boundary conditions for fourth-order self-adjoint differential equations, Duke Math. J., 34, 131-138 (1967) · Zbl 0148.06402
[53] Hinton, D. B., A Criterion for \(n-n\) oscillations in differential equations of order \(2n\), (Proc. Am. Math. Soc., 19 (1968)), 511-518 · Zbl 0157.15001
[54] Howard, H. C., Oscillation criteria for fourth-order linear differential equations, Trans. Am. Math. Soc., 96, 296-311 (1960) · Zbl 0094.28503
[55] Howard, H. C., Oscillation criteria for matrix differential equations, Canadian J. Math., 19, 184-199 (1967) · Zbl 0148.07203
[56] Hunt, R. W., The behavior of solutions of ordinary self-adjoint differential equations of arbitrary even order, Pacific J. Math., 12, 945-961 (1962) · Zbl 0118.08701
[57] Hunt, R. W., Oscillation properties of even-order linear differential equations, Trans. Am. Math. Soc., 115, 54-61 (1965) · Zbl 0142.34702
[58] Ince, E. L., Ordinary Differential Equations (1944), Dover Publications: Dover Publications New York · Zbl 0063.02971
[59] (Am. Math. Soc. AMS Transl., 42 (1964)), 247-288
[60] Kamke, E., Differentialgleichungen Lösungsmethoden und Lösungen, I (1943), Becker and Erler: Becker and Erler Leipzig, J. W. Edwards, Ann Arbor, 1945 · Zbl 0028.22702
[61] Kneser, A., Untersuchen über die reelen Nullstellen der Integrale linearer Differentialgleichungen, Math. Ann., 42, 409-435 (1893) · JFM 25.0522.01
[62] Dokl. Akad. Nauk. USSR, 118, 22-24 (1958) · Zbl 0080.06902
[63] Dolk. Akad. Nauk USSR, 120, 1180-1182 (1958) · Zbl 0085.30402
[64] Lasota, A., Sur la distance entre les zéros de l’équation differentielle lineaire du troisième ordre, Ann. Polon. Math., 13, 129-132 (1963) · Zbl 0117.05004
[65] Lazer, A. C., The behavior of solutions of the differential equation \(y\)‴ + \(p(x)y\)′ + \(q(x)y = 0\), Pacific J. Math., 17, 435-466 (1966) · Zbl 0143.31501
[66] Leighton, W., Principal quadratic functionals, Trans. Am. Math. Soc., 67, 253-274 (1949) · Zbl 0041.22404
[67] Leighton, W., The detection of the oscillation of solutions of a second order linear differential equation, Duke Math. J., 17, 57-62 (1950) · Zbl 0036.06101
[68] Leighton, W., On self-adjoint differential equations of second-order, J. London Math. Soc., 27, 37-47 (1952) · Zbl 0048.06503
[69] Leighton, W., Comparison theorems for linear differential equations of second-order, (Proc. Am. Math. Soc., 13 (1962)), 603-610 · Zbl 0118.08202
[70] Leighton, W.; Nehari, Z., On the oscillation of solutions of self-adjoint linear differential equations of the fourth order, Trans. Am. Math. Soc., 89, 325-377 (1958), (Contains an extensive bibliography) · Zbl 0084.08104
[71] Soviet Math., 4, 121-124 (1963) · Zbl 0125.32306
[72] Soviet Math., 5, 818-821 (1964) · Zbl 0117.05003
[73] Mammana, G., Decomposizione delle espressioni differenziali lineari omogenee in prodotti di fattori simbolici e applicazione relativa allo studio delle equazioni differenziali lineari, Math. Z., 33, 186-231 (1931) · JFM 57.0510.02
[74] Morse, M., The Calculus of Variations in the Large (1934), American Mathematical Society Colloquium Publications: American Mathematical Society Colloquium Publications New York · JFM 60.0450.01
[75] Nehari, Z., Oscillation criteria for second-order linear differential equations, Trans. Am. Math. Soc., 85, 428-445 (1958) · Zbl 0078.07602
[76] Nehari, Z., On the zeros of solutions of \(n\) th order linear differential equations, J. London Math. Soc., 39, 327-332 (1964) · Zbl 0125.04504
[77] Nehari, Z., Nonoscillation criteria for \(n\) th order linear differential equations, Duke Math. J., 32, 607-616 (1965) · Zbl 0134.07301
[78] Nehari, Z., Disconjugate Linear Operators, Trans. Am. Math. Soc. AMS, 129, 500-516 (1967) · Zbl 0183.09101
[79] Polya, G., On the mean-value theorem corresponding to a given linear homogeneous differential equation, Trans. Am. Math. Soc., 24, 312-324 (1922) · JFM 50.0299.02
[80] Prüfer, H., Neue Herleitung der Sturm-Liouvilleschen Reihenentwicklung stetiger Funktionen, Math. Ann., 95, 499-518 (1926) · JFM 52.0455.01
[81] Rab, M., Asymptotische Eigenschatten der Lösungen Linearer Differentialgleichung dritter Ordnung, Spisy Publ. Fac. Sci. Masaryk, 374, 177-184 (1956/1954)
[82] Am. J. Math., 70, 460 (1948), [See also
[83] Reid, W. T., Oscillation criteria for linear differential equations with complex coefficients, Pacific J. Math., 6, 733-751 (1956) · Zbl 0071.30203
[84] Reid, W. T., A comparison theorem for self-adjoint differential equations of second order, Ann. Math., 65, 197-202 (1957) · Zbl 0077.08604
[85] Reid, W. T., Oscillation criteria for linear differential systems with complex coefficients, Pacific J. Math., 6, 733-751 (1956) · Zbl 0071.30203
[86] Reid, W. T., Principal solutions of nonoscillatory self-adjoint linear differential systems, Pacific J. Math., 8, 147-169 (1958) · Zbl 0102.30005
[87] Reid, W. T., A Prüfer transformation for differential systems, Pacific J. Math., 8, 575-584 (1958) · Zbl 0102.30004
[88] Reid, W. T., Oscillation criteria for self-adjoint differential systems, Trans. Am. Math. Soc., 101, 91-106 (1961) · Zbl 0114.29202
[89] Reid, W. T., Riccati matrix differential equations and nonoscillation criteria for associated linear systems, Pacific J. Math., 13, 664-685 (1963) · Zbl 0119.07401
[90] Reid, W. T., Variational methods and boundary problems for ordinary linear differential systems, (Harris, W. A.; Sibuya, Y., The Proceedings of the Japan-United States Seminar on Functional and Differential Equations (1967), Benjamin: Benjamin New York), 267-299 · Zbl 0178.09102
[91] Sansone, G., Studi sulle equazioni differenziali lineari omogenee di terzo ordine nel campo real e, Revista Mat. Fis. Teor. (Tucuman), 6, 195-253 (1948) · Zbl 0041.41903
[92] Sansone, G., Equazioni Differenziali nel Campo Reale (1956), Zanichelli: Zanichelli Bologna · JFM 67.0306.01
[93] Schröder, J., Randwertantgaben vieter Ordnung mit positiver Greenscher Funktion, Math. Z., 90, 429-440 (1965) · Zbl 0135.29703
[94] Schröder, J., Hinreichende Bedingungen bei Differentialgleichungen vierter Ordnung, Math. Z., 92, 75-94 (1966) · Zbl 0135.29704
[95] Schwarz, B., Disconjugacy of Complex Differential Systems, Trans. Am. Math. Soc., 125, 482-496 (1966) · Zbl 0189.37901
[96] Seifert, G., A third order boundary value problem arising in aeroelastic wing theory, Quart. Appl. Math., 9, 210-218 (1951) · Zbl 0043.09102
[97] Seifert, G., A third order irregular boundary value problem and the associated series, Pacific J. Math., 2, 395-406 (1952) · Zbl 0047.32804
[98] Sherman, T. L., Properties of solutions of \(n\) th order linear differential equations, Pacific J. Math., 15, 1045-1060 (1965) · Zbl 0132.31204
[99] Shinn, D., Existence theorems for the quasi-differential equation of the \(n\) th order, Compt. Rend. Acad. Sci. URSS, 18, 515-518 (1938) · Zbl 0019.21402
[100] Sternberg, R. L., Variational methods and nonoscillation theorems for systems of differential equations, Duke Math. J., 19, 311-322 (1952) · Zbl 0049.18704
[101] Švec, M., Sur une propriété des integrales de l’equation \(y^{(n)} + Q(x)y = 0, n = 3, 4\), Czech. Math. J., 7, 450-461 (1957) · Zbl 0089.06604
[102] Švec, M., Some remarks on a third-order linear differential equation, Czech. Math. J., 15, 42-49 (1965), (in Russian) · Zbl 0129.06203
[103] Švec, M., Einige asymptotische und oszillatorische Eigenschaften der Differentialgleichung \(y\)‴\(p + A(x)y′ + B(x)y = 0\), Czech. Mat. J., 15, 378-393 (1965) · Zbl 0143.11202
[104] Tomastik, E. C., Singular quadratic functionals of \(n\) dependent variables, Trans. Am. Math. Soc., 124, 60-76 (1966) · Zbl 0161.09503
[105] Waltman, P., Oscillation criteria for third order nonlinear differential equations, Pacific J. Math., 18, 385-389 (1966) · Zbl 0144.11403
[106] Whyburn, W. M., On self-adjoint ordinary differential equations of the fourth order, Am. J. Math., 52, 171-196 (1930) · JFM 56.0399.01
[107] Willet, D., On the Oscillatory Behavior of the Solutions of Second-Order Linear Differential Equations, (AWU Differential Equations Symposium (1967)), unpublished
[108] Wintner, A., A criterion of oscillatory stability, Quart. Appl. Math., 7, 115-117 (1949) · Zbl 0032.34801
[109] Zedek, M., Cayley’s decomposition and Polya’s \(W\)-property of ordinary linear differential equations, Israel J. Math., 3, 81-86 (1965) · Zbl 0139.03705
[110] Zettl, A. J., Adjoint linear differential operators, (Proc. Am. Math. Soc., 16 (1965)), 1239-1241 · Zbl 0139.03801
[111] Zettl, A. J.; Coddington, E. A., Hermitian and anti-Hermitian properties of Green’s matrices, Pacific J. Math., 18, 451-454 (1966) · Zbl 0152.08501
[112] Zettl, A. J., Some identities related to Polya’s property \(W\) for linear differential equations, (Proc. Am. Math. Soc., 18 (1967)), 992-994 · Zbl 0204.09902
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