Riccati techniques and variational principles in oscillation theory for linear systems
Authors:
G. J. Butler, L. H. Erbe and A. B. Mingarelli
Journal:
Trans. Amer. Math. Soc. 303 (1987), 263282
MSC:
Primary 34C10; Secondary 34A30, 34C29
MathSciNet review:
896022
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Abstract: We consider the seond order differential system , where , are matrices with a continuous symmetric matrixvalued function, . We obtain a number of sufficient conditions in order that all prepared solutions of are oscillatory. Two approaches are considered, one based on Riccati techniques and the other on variational techniques, and involve assumptions on the behavior of the eigenvalues of (or of its integral). These results extend some wellknown averaging techniques for scalar equations to system .
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K. Akiyama, On the maximum eigenvalue conjecture for the oscillation of second order differential systems, M. Sc. Thesis, Univ. of Ottawa, Canada, 1983.
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W. Allegretto and L. Erbe, Oscillation criteria for matrix differential inequalities, Canad. Math. Bull. 16 (1973), 510. MR 0322263 (48:625)
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F. V. Atkinson, H. G. Kaper and M. K. Kwong, An oscillation criterion for linear secondorder differential systems, Proc. Amer. Math. Soc. 94 (1985), 9196. MR 781063 (86h:34028)
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G. J. Butler and L. H. Erbe, Oscillation results for second order differential systems, SIAM J. Math. Anal. 17 (1986), 1929. MR 819208 (87h:34045)
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, Oscillation results for selfadjoint differential systems, J. Math. Anal. Appl. 115 (1986), 470481. MR 836240 (87f:34032)
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R. Byers, B. J. Harris and M. K. Kwong, Weighted means and oscillation conditions for second order matrix differential equations, J. Differential Equations 61 (1986), 164177. MR 823400 (87f:34033)
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W. A. Coppel, Stability and asymptotic behavior of differential equations, Heath, Boston, Mass., 1965. MR 0190463 (32:7875)
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G. J. Etgen and R. T. Lewis, Positive functionals and oscillation criteria for second order differential systems, Proc. Edinburgh Math. Soc. 22 (1979), 277290. MR 560991 (81b:34051)
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W. B. Fite, Concerning the zeros of the solutions of certain differential equations, Trans. Amer. Math. Soc. 19 (1918), 341352. MR 1501107
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S. Halvorsen and A. B. Mingarelli, preprint, 1984.
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P. Hartman, Oscillation criteria for selfadjoint second order differential systems and "Principal sectional curvatures", J. Differential Equations 34 (1979), 326338. MR 550049 (81a:34034)
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, Ordinary differential equations, Wiley, New York, 1964. MR 0171038 (30:1270)
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D. B. Hinton and R. T. Lewis, Oscillation theory of generalized second order differential equations, Rocky Mountain J. Math 10 (1980), 751766. MR 595103 (82c:34039)
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M. K. Kwong and H. G. Kaper, Oscillation of twodimensional linear second order differential systems, J. Differential Equations 56 (1985), 195205. MR 774162 (86j:34032)
 [15]
M. K. Kwong and A. Zettl, A new approach to second order linear oscillation theory, Proc. Conf. Ordinary Differential Equations and operators (Dundee, 1982), Lecture Notes in Math., vol. 1032, Springer, Berlin and New York, 1983, pp. 328345. MR 742647 (85k:34074)
 [16]
, Integral inequalities and second order linear oscillation, J. Differential Equations 45 (1982), 1633. MR 662484 (83i:34034)
 [17]
M. K. Kwong, H. G. Kaper, K. Akiyama and A. B. Mingarelli, Oscillation of second order differential systems, Proc. Amer. Math. Soc. 91 (1984), 8591. MR 735570 (85g:34027)
 [18]
A. B. Mingarelli, On a conjecture for oscillation of second order differential systems, Proc. Amer. Math. Soc. 82 (1981), 593598. MR 614884 (82j:34028)
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, An oscillation criterion for second order selfadjoint differential systems, C. R. Math. Rep. Acad. Sci. Canada 2 (1980), 287290. MR 600563 (82b:34042)
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R. A. Moore, The behavior of solutions of a linear differential equation of second order, Pacific J. Math. 5 (1955), 125145. MR 0068690 (16:925b)
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C. Olech, Z. Opial and T. Ważewski, Sur le problème d'oscillation des intégrales de l'équation , Bull. Acad. Polon. Sci. Cl. III 5 (1957), 621626. MR 0089312 (19:650e)
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B. N. Parlett, The symmetric eigenvalue problem, PrenticeHall, Englewood Cliffs, N.J., 1980. MR 570116 (81j:65063)
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M. Rab, Kriterien fur die Oszillation der Lösungen der Differentialgleichung , Časobis. Pěst. Mat. 84 (1959), 335370. MR 0114964 (22:5773)
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C. A. Swanson, Comparison and oscillation theory of linear differential equations, Academic Press, New York, 1968. MR 0463570 (57:3515)
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E. C. Tomastik, Oscillation of systems of second order differential equations, J. Differential Equations 9 (1971), 436442. MR 0274863 (43:621)
 [26]
T. Walters, A characterization of positive linear functionals and oscillation criteria for matrix differential equations, Proc. Amer. Math. Soc. 78 (1980), 198202. MR 550493 (81a:34037)
 [27]
D. Willett, Classification of second order linear differential equations with respect to oscillation, Adv. in Math. 3 (1969), 594623. MR 0280800 (43:6519)
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, On the oscillatory behavior of the solutions of second order linear differential equations, Ann. Polon. Math. 21 (1969), 175194. MR 0249723 (40:2964)
 [29]
A. Wintner, A criterion of oscillatory stability, Quart. Appl. Math. 7 (1949), 115117. MR 0028499 (10:456a)
 [30]
J. S. W. Wong, Oscillation and nonoscillation of solutions of second order linear differential equations with integrable coefficients, Trans. Amer. Math. Soc. 144 (1969), 197215. MR 0251305 (40:4536)
 [1]
 K. Akiyama, On the maximum eigenvalue conjecture for the oscillation of second order differential systems, M. Sc. Thesis, Univ. of Ottawa, Canada, 1983.
 [2]
 W. Allegretto and L. Erbe, Oscillation criteria for matrix differential inequalities, Canad. Math. Bull. 16 (1973), 510. MR 0322263 (48:625)
 [3]
 F. V. Atkinson, H. G. Kaper and M. K. Kwong, An oscillation criterion for linear secondorder differential systems, Proc. Amer. Math. Soc. 94 (1985), 9196. MR 781063 (86h:34028)
 [4]
 G. J. Butler and L. H. Erbe, Oscillation results for second order differential systems, SIAM J. Math. Anal. 17 (1986), 1929. MR 819208 (87h:34045)
 [5]
 , Oscillation results for selfadjoint differential systems, J. Math. Anal. Appl. 115 (1986), 470481. MR 836240 (87f:34032)
 [6]
 R. Byers, B. J. Harris and M. K. Kwong, Weighted means and oscillation conditions for second order matrix differential equations, J. Differential Equations 61 (1986), 164177. MR 823400 (87f:34033)
 [7]
 W. A. Coppel, Stability and asymptotic behavior of differential equations, Heath, Boston, Mass., 1965. MR 0190463 (32:7875)
 [8]
 G. J. Etgen and R. T. Lewis, Positive functionals and oscillation criteria for second order differential systems, Proc. Edinburgh Math. Soc. 22 (1979), 277290. MR 560991 (81b:34051)
 [9]
 W. B. Fite, Concerning the zeros of the solutions of certain differential equations, Trans. Amer. Math. Soc. 19 (1918), 341352. MR 1501107
 [10]
 S. Halvorsen and A. B. Mingarelli, preprint, 1984.
 [11]
 P. Hartman, Oscillation criteria for selfadjoint second order differential systems and "Principal sectional curvatures", J. Differential Equations 34 (1979), 326338. MR 550049 (81a:34034)
 [12]
 , Ordinary differential equations, Wiley, New York, 1964. MR 0171038 (30:1270)
 [13]
 D. B. Hinton and R. T. Lewis, Oscillation theory of generalized second order differential equations, Rocky Mountain J. Math 10 (1980), 751766. MR 595103 (82c:34039)
 [14]
 M. K. Kwong and H. G. Kaper, Oscillation of twodimensional linear second order differential systems, J. Differential Equations 56 (1985), 195205. MR 774162 (86j:34032)
 [15]
 M. K. Kwong and A. Zettl, A new approach to second order linear oscillation theory, Proc. Conf. Ordinary Differential Equations and operators (Dundee, 1982), Lecture Notes in Math., vol. 1032, Springer, Berlin and New York, 1983, pp. 328345. MR 742647 (85k:34074)
 [16]
 , Integral inequalities and second order linear oscillation, J. Differential Equations 45 (1982), 1633. MR 662484 (83i:34034)
 [17]
 M. K. Kwong, H. G. Kaper, K. Akiyama and A. B. Mingarelli, Oscillation of second order differential systems, Proc. Amer. Math. Soc. 91 (1984), 8591. MR 735570 (85g:34027)
 [18]
 A. B. Mingarelli, On a conjecture for oscillation of second order differential systems, Proc. Amer. Math. Soc. 82 (1981), 593598. MR 614884 (82j:34028)
 [19]
 , An oscillation criterion for second order selfadjoint differential systems, C. R. Math. Rep. Acad. Sci. Canada 2 (1980), 287290. MR 600563 (82b:34042)
 [20]
 R. A. Moore, The behavior of solutions of a linear differential equation of second order, Pacific J. Math. 5 (1955), 125145. MR 0068690 (16:925b)
 [21]
 C. Olech, Z. Opial and T. Ważewski, Sur le problème d'oscillation des intégrales de l'équation , Bull. Acad. Polon. Sci. Cl. III 5 (1957), 621626. MR 0089312 (19:650e)
 [22]
 B. N. Parlett, The symmetric eigenvalue problem, PrenticeHall, Englewood Cliffs, N.J., 1980. MR 570116 (81j:65063)
 [23]
 M. Rab, Kriterien fur die Oszillation der Lösungen der Differentialgleichung , Časobis. Pěst. Mat. 84 (1959), 335370. MR 0114964 (22:5773)
 [24]
 C. A. Swanson, Comparison and oscillation theory of linear differential equations, Academic Press, New York, 1968. MR 0463570 (57:3515)
 [25]
 E. C. Tomastik, Oscillation of systems of second order differential equations, J. Differential Equations 9 (1971), 436442. MR 0274863 (43:621)
 [26]
 T. Walters, A characterization of positive linear functionals and oscillation criteria for matrix differential equations, Proc. Amer. Math. Soc. 78 (1980), 198202. MR 550493 (81a:34037)
 [27]
 D. Willett, Classification of second order linear differential equations with respect to oscillation, Adv. in Math. 3 (1969), 594623. MR 0280800 (43:6519)
 [28]
 , On the oscillatory behavior of the solutions of second order linear differential equations, Ann. Polon. Math. 21 (1969), 175194. MR 0249723 (40:2964)
 [29]
 A. Wintner, A criterion of oscillatory stability, Quart. Appl. Math. 7 (1949), 115117. MR 0028499 (10:456a)
 [30]
 J. S. W. Wong, Oscillation and nonoscillation of solutions of second order linear differential equations with integrable coefficients, Trans. Amer. Math. Soc. 144 (1969), 197215. MR 0251305 (40:4536)
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DOI:
http://dx.doi.org/10.1090/S00029947198708960225
PII:
S 00029947(1987)08960225
Keywords:
Oscillation,
Riccati equation,
variational techniques
Article copyright:
© Copyright 1987
American Mathematical Society
