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Ahlbrandt, Calvin D.; Hinton, Don B.

The effect of variable change on oscillation and disconjugacy criteria with applications to spectral theory and asymptotic theory. (English) Zbl 0459.34018

J. Math. Anal. Appl. 81, 234-277 (1981).
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Cited in 30 Documents

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34L99 Ordinary differential operators
47A10 Spectrum, resolvent
34E99 Asymptotic theory for ordinary differential equations
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References:

[1] Ahlbrandt, C. D., Disconjugacy criteria for self-adjoint differential systems, J. Differential Equations, 6, 271-295 (1969) · Zbl 0175.09301
[2] Ahlbrandt, C. D., Equivalent boundary value problems for self-adjoint differential systems, J. Differential Equations, 9, 420-435 (1971) · Zbl 0218.34020
[4] Akhiezer, N. I., The Calculus of Variations (1962), Blaisdell: Blaisdell New York · Zbl 0119.05604
[5] Atkinson, F. V., Discrete and Continuous Boundary Problems (1964), Academic Press: Academic Press New York · Zbl 0117.05806
[6] Barrett, J. H., Disconjugacy of second-order linear differential equations with non-negative coefficients, (Proc. Amer. Math. Soc., 10 (1959)), 552-561 · Zbl 0092.08103
[7] Barrett, J. H., Disconjugacy of a self-adjoint differential equation of the fourth order, Pacific J. Math., 11, 25-37 (1961) · Zbl 0105.06701
[8] Barrett, J. H., Oscillation theory of ordinary linear differential equations, Adv. in Math., 3, 415-509 (1969) · Zbl 0213.10801
[9] Birkhoff, G.; Rota, G.-C, Ordinary Differential Equations (1978), Wiley: Wiley New York · Zbl 0183.35601
[10] Bliss, G. A., A boundary value problem for a system of ordinary linear differential equations of the first order, Trans. Amer. Math. Soc., 28, 561-584 (1926) · JFM 52.0453.13
[11] Bolza, O., Lecture on the Calculus of Variations (1904), Univ. of Chicago Press: Univ. of Chicago Press Chicago · JFM 35.0373.01
[12] Boyce, W. E.; DiPrima, R. C., Elementary Differential Equations and Boundary Value Problems (1977), Wiley: Wiley New York · Zbl 0807.34002
[13] Coppel, W. A., Stability and Asymptotic Behavior of Differential Equations (1965), Heath: Heath Boston · Zbl 0154.09301
[14] Davis, D. R., The inverse problem of the calculus of variations in higher space, Trans. Amer. Math. Soc., 30, 710-736 (1928) · JFM 54.0533.01
[15] Devinatz, A., The deficiency index of a certain class of ordinary self-adjoint differential operators, Adv. in Math., 8, 434-473 (1972) · Zbl 0259.34019
[16] Dunford, N.; Schwartz, J., Linear Operators (1963), Interscience: Interscience New York, Part II · Zbl 0128.34803
[17] Eastham, M. S.D; El-Deberky, A. A., The spectrum of differential operators with large coefficients, J. London Math. Soc. (2), 2, 257-266 (1970) · Zbl 0188.46402
[18] 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
[19] Friedrichs, K. O., Über die ausgezeichnete Randbedingung in der Spektraltheorie der halbbeschränkten gewöhnlichen Differentialoperatoren zweiter Ordnung, Math. Ann., 112, 1-23 (1935) · JFM 61.1198.02
[20] Gantmacher, F. R., (The Theory of Matrices, Vol. 2 (1964), Chelsea: Chelsea New York) · Zbl 0085.01001
[21] Glazman, I. M., Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators (1965), Israel Program for Sci. Transl: Israel Program for Sci. Transl Jerusalem · Zbl 0143.36505
[22] Hadass, R., On the zeros of the solutions of the differential equation \(y^{(n)}(z) + p(z) y(z) = 0\), Pacific J. Math., 31, 33-46 (1969) · Zbl 0266.34037
[23] Harris, W. A.; Lutz, D. A., On the asymptotic integration of linear differential systems, J. Math. Anal. Appl., 48, 1-16 (1974) · Zbl 0304.34043
[24] Hartman, P., Ordinary Differential Equations (1964), Wiley: Wiley New York · Zbl 0125.32102
[25] Hartman, P., On disconjugate differential equations, Trans. Amer. Math. Soc., 134, 53-70 (1968) · Zbl 0164.38704
[26] Hartman, P., Principal solutions of disconjugate \(n\)-th order linear differential equations, Amer. J. Math., 91, 306-362 (1969) · Zbl 0184.11603
[27] Hartman, P., On nonoscillatory linear differential equations of second order, (Proc. Amer. Math. Soc., 64 (1977)), 251-259 · Zbl 0412.34017
[28] Hartman, P., Oscillation criteria for self-adjoint second-order differential systems and “principal section curvatures”, J. Differential Equations, 34, 326-338 (1979) · Zbl 0443.34029
[29] Hawking, S. W.; Ellis, G. F.R, The Large Scale Structure of Space-Time (1973), Cambridge Univ. Press: Cambridge Univ. Press London/New York · Zbl 0265.53054
[30] Hawking, S. W.; Penrose, R., The singularities of gravity collapse and cosmology, (Proc. Roy. Soc. London Ser. A, 314 (1970)), 529-548 · Zbl 0954.83012
[31] Hille, E., Nonoscillation theorems, Trans. Amer. Math. Soc., 64, 234-252 (1948) · Zbl 0031.35402
[32] Hinton, D. B., Asymptotic behavior of solutions of \(( ry^{(m)})^{(k)\) · Zbl 0175.38101
[33] Hinton, D. B., Limit point criteria for differential equations, II, Canad. J. Math., 26, 340-351 (1974) · Zbl 0249.34010
[34] Hinton, D. B., Continuous spectra of an even order differential operator, Illinois J. Math., 18, 444-450 (1974) · Zbl 0285.34016
[35] Hinton, D. B., Principal solutions of positive hamiltonian systems, J. Austral. Math. Soc. Ser. A, 22, 411-420 (1976) · Zbl 0347.34008
[36] Hinton, D. B.; Lewis, R. T., Discrete spectra criteria for singular differential operators with middle terms, (Math. Proc. Cambridge Philos. Soc., 77 (1975)), 337-347 · Zbl 0298.34018
[37] Hinton, D. B.; Lewis, R. T., Oscillation theory at a finite singularity, J. Differential Equations, 30, 235-247 (1978) · Zbl 0362.34025
[38] Hinton, D. B.; Lewis, R. T., Singular differential operators with spectra discrete and bounded below, (Proc. Roy. Soc. Edinburgh Sect. A, 84 (1979)), 117-134 · Zbl 0418.47021
[39] Kauffman, R. M., On the limit \(n\) classification of ordinary differential operators with positive coefficients, (Proc. London Math. Soc. (3), 35 (1977)), 496-526 · Zbl 0382.47025
[40] Knowles, I. W., On second order differential operators of limit-circle type, (Proceedings, Conference on Ordinary and Partial Differential Equations. Proceedings, Conference on Ordinary and Partial Differential Equations, Dundee, Scotland, 1974. Proceedings, Conference on Ordinary and Partial Differential Equations. Proceedings, Conference on Ordinary and Partial Differential Equations, Dundee, Scotland, 1974, Lecture Notes in Mathematics No. 415 (1974), Springer-Verlag: Springer-Verlag Berlin), 184-187 · Zbl 0337.34018
[41] Knowles, I. W., On stability conditions for second order linear differential equations, J. Differential Equations, 34, 179-203 (1979) · Zbl 0388.34029
[42] Komkov, V., A technique for the detection of oscillation of second order differential equations, Pacific J. Math., 42, 105-115 (1972) · Zbl 0241.34038
[43] Kummer, E. E., De generali quadam equatione differentialit tertii ordinis, J. Reine Angew. Math., 100, 1-9 (1887), [a reprint of an article first published in 1834] · JFM 18.0297.01
[44] Kurss, H., A limit-point criterion for non-oscillartory Sturm-Liouville differential operators, (Proc. Amer. Math. Soc., 18 (1967)), 445-449 · Zbl 0155.13101
[45] Kwong, M. K.; Zettl, A., Discreteness conditions for the spectrum of ordinary differential operators, J. Differential Equations, 40, 53-70 (1981) · Zbl 0486.34018
[46] 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
[47] Leighton, W.; Nehäri, Z., On the oscillation of solutions of self-adjoint linear differential equations of the fourth order, Trans. Amer. Math. Soc., 89, 325-377 (1958) · Zbl 0084.08104
[48] Lewis, R. T., Oscillation and nonoscillation criteria for some self-adjoint even order linear differential operators, Pacific J. Math., 51, 221-234 (1974) · Zbl 0281.34027
[49] Lewis, R. T., The discreteness of the spectrum of self-adjoint, even order, one term, differential operators, (Proc. Amer. Math. Soc., 42 (1974)), 480-482 · Zbl 0254.34027
[50] Lewis, R. T., The existence of conjugate points for self-adjoint differential equations of even order, (Proc. Amer. Math. Soc., 56 (1976)), 162-166 · Zbl 0294.34004
[51] Lewis, R. T., Conjugate points of vector-matrix differential equations, Trans. Amer. Math. Soc., 231, 167-178 (1977) · Zbl 0366.34022
[52] Liouville, J., Second Mémoire sur le développement des fonctions ou parties de fonctions en séries dont les divers termes sont assujettis à satisfaire à une même équation différentielle du second ordre contenant un paramètre variable, J. Math. Pures Appl., 2, 16-35 (1837)
[53] Macki, J. W., Applications of change of variables in the qualitative theory of second order linear ordinary differential equations, SIAM Rev., 18, 269-274 (1976) · Zbl 0332.34039
[54] 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
[55] Morse, M., The Calculus of Variations in the Large, (Colloquium Publications, Vol. 18 (1934), Amer. Math. Soc: Amer. Math. Soc Providence, R.I) · JFM 58.0537.01
[56] Müller-Pfeiffer, E., Spektraleigenschaften Singularer Gewöhlicher Differentialoperatoren, Teubner-text zur mathematik (1977), Leipzig · Zbl 0373.47022
[58] Naimark, M. A., Linear differential operators (1968), Ungar: Ungar New York, Part II · Zbl 0227.34020
[59] Nehäri, Z., Disconjugate differential equations, Trans. Amer. Math. Soc., 129, 500-516 (1967) · Zbl 0183.09101
[60] Pfeiffer, G. W., Asymptotic solutions of \(y\)‴ + qy′ + ry = 0, J. Differential Equations, 11, 145-155 (1972) · Zbl 0263.34057
[61] Radon, J., Zum Problem von Lagrange, (Abh. Math. Sem. Univ. Hamburg, 6 (1928)), 273-299 · JFM 54.0530.01
[62] Rankin, S. M., Oscillation results for a nonhomogeneous equation, Pacific J. Math., 80, 237-243 (1979) · Zbl 0407.34029
[63] Rasmussen, C. H., Oscillation and asymptotic behavior of systems of ordinary linear differential equations, Trans. Amer. Math. Soc., 256, 1-47 (1979) · Zbl 0416.34036
[64] Read, T. T., Factorization and discretre spectra for second-order differential expressions, J. Differential Equations, 35, 388-406 (1980) · Zbl 0447.34026
[65] Reid, W. T., Principal solutions of nonoscillatory linear differential systems, Pacific J. Math., 8, 147-169 (1958) · Zbl 0102.30005
[66] Reid, W. T., A Prufer transformation for differential systems, Pacific J. Math., 8, 575-584 (1958) · Zbl 0102.30004
[67] Reid, W. T., Monotoneity properties of solutions of Hermitian-Riccati matrix differential equations, SIAM J. Math. Anal., 1, 195-213 (1970) · Zbl 0209.11601
[68] Reid, W. T., Some remarks on special disconjugacy criteria for differential systems, Pacific J. Math., 35, 763-772 (1970) · Zbl 0208.11102
[69] Reid, W. T., Ordinary Differential Equations (1971), Wiley: Wiley New York · Zbl 0212.10901
[70] Reid, W. T., Sturmian Theory for Ordinary Differential Equations, (Applied Mathematical Sciences, Vol. 31 (1980), Springer-Verlag: Springer-Verlag New York) · Zbl 0212.10901
[71] Selby, S. M., (CRC Handbook of Tables for Mathematics (1970), Chem. Rubber Co: Chem. Rubber Co Cleveland)
[72] Siegel, C. L.; Moser, J. K., Lectures on Celestial Mechanics, (Die Grundlehren der mathematischen Wissenschaften, Vol. 187 (1971), Springer-Verlag: Springer-Verlag New York) · Zbl 0312.70017
[73] Tipler, F. J., General relativity and conjugate ordinary differential equations, J. Differential Equations, 30, 165-174 (1978) · Zbl 0362.34023
[74] Walker, P. W., Asymptotics of the solutions to [(ry″)′ − py′]′ + qy = σy, J. Differential Equations, 9, 108-132 (1971) · Zbl 0211.11202
[75] Walker, P. W., Asymptotics for a class of fourth order differential equations, J. Differential Equations, 11, 321-334 (1972) · Zbl 0278.34049
[76] Weinstock, R., Calculus of Variations with Applications to Physics and Engineering (1952), McGraw-Hill: McGraw-Hill New York · Zbl 0049.19503
[77] Wilczynski, E. J., Projective Differential Geometry of Curves and Ruled Surfaces (1906), Teubner: Teubner Leipzig · JFM 37.0620.02
[78] Willett, D., Classification of second order linear differential equations with respect to oscillation, Adv. in Math., 3, 594-623 (1969) · Zbl 0188.40101
[79] Willett, D., A necessary and sufficient condition for the oscillation of some linear second order differential equations, Rocky Mountain J. Math., 1, 357-365 (1971) · Zbl 0231.34032
[80] Wintner, A., A criterion of oscillatory stability, Quart. Appl. Math., 7, 115-117 (1949) · Zbl 0032.34801
[81] Wong, J. S.W, Oscillation theorems for second order nonlinear differential equations, Bull. Inst. Math. Acad. Sinica, 3, No. 2, 283-309 (Dec. 1975) · Zbl 0316.34035
[82] Zlǎmal, M., Oscillation criterions, Časopis Pěst. Mat. Fys. Roč., 75, 213-217 (1950) · Zbl 0040.19502
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