Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Oscillation of superlinear matrix differential equations


Authors: Calvin D. Ahlbrandt, Jerry Ridenhour and Russell C. Thompson
Journal: Proc. Amer. Math. Soc. 105 (1989), 141-148
MSC: Primary 34C10
DOI: https://doi.org/10.1090/S0002-9939-1989-0946622-6
MathSciNet review: 946622
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The main theorems extend to matrix differential equations, Atkinson's classic theorem giving necessary and sufficient conditions for the oscillation of superlinear second-order scalar differential equations. The theorems improve recent results of Kura and of Butler and Erbe by removing a very restrictive hypothesis that solutions be symmetric.


References [Enhancements On Off] (What's this?)

  • [1] S. Ahmad, On Sturmian theory for second order systems, Proc. Amer. Math. Soc. 87 (1983), 661-665. MR 687636 (84k:34042)
  • [2] S. Ahmad and A. C. Lazer, An $ n$-dimensional extension of the Sturm separation and comparison theory to a class of nonselfadjoint systems, SIAM J. Math. Anal. 6 (1978), 1137-1150. MR 512517 (80a:34035)
  • [3] S. Ahmad and C. C. Travis, Oscillation criteria for second-order differential systems, Proc. Amer. Math. Soc. 71 (1978), 247-252. MR 0486792 (58:6492)
  • [4] F. V. Atkinson, On second-order non-linear oscillations, Pacific J. Math. 5 (1955), 643-647. MR 0072316 (17:264e)
  • [5] S. Belohorec, Oscillatory solutions of certain non-linear differential equations of second order, Mat.-Fyz. Casopis Sloven. Akad. Vied. 11 (1961), 250-255.
  • [6] G. J. Butler and L. H. Erbe, Oscillation theory for second order differential systems with functional commutative coefficients, Differential and Integral Equations (Proc. Conf. Twelfth and Thirteenth Midwest; J. L. Henderson, ed.) Institute of Applied Mathematics, University of Missouri-Rolla, 1985, pp. 15-18. MR 821763
  • [7] G. J. Butler, L. H. Erbe and A. B. Mingarelli, Riccati techniques and variational principles in oscillation theory for linear systems, Trans. Amer. Math. Soc. 302 (1987), 263-282. MR 896022 (88h:34023)
  • [8] 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), 164-177. MR 823400 (87f:34033)
  • [9] J. Dieudonné, Sur un Theorèm de Schwerdtfeger, Ann. Polon. Math. 24 (1974), 87-88. MR 0344276 (49:9015)
  • [10] G. T. Etgen and J. F. Pawlowski, A comparison theorem and oscillation criteria for second order differential systems, Pacific J. Math. 72 (1977), 59-69. MR 0450675 (56:8968)
  • [11] S. Goff, Hermitian function matrices which commute with their derivative, Linear Algebra and Appl. 36 (1981), 33-40. MR 604327 (83j:15018)
  • [12] P. Hartman, Oscillation criteria for self-adjoint second-order differential systems and "principal sectional curvatures", J. Differential Equations 34 (1979), 326-338. MR 550049 (81a:34034)
  • [13] S. P. Hastings, Boundary value problems in one differential equation with a discontinuity, J. Differential Equations 1 (1965), 346-369. MR 0180723 (31:4954)
  • [14] A. G. Kartsatos, Recent results on oscillation of solutions of forced and perturbed nonlinear differential equations of even order, Stability of Dynamical Systems, Theory and Applications (John R. Graef, ed.) Marcel Dekker, New York, 1977, pp. 17-72. MR 0594954 (58:28853)
  • [15] A. G. Kartsatos and T. Walters, Some oscillation results for matrix and vector differential equations with forcing term, J. Math. Anal. Appl. 73 (1980), 506-513. MR 564000 (81f:34039)
  • [16] M. S. Keener and C. C. Travis, Sturmian theory for a class of nonselfadjoint differential systems, Ann. Mat. Pura Appl. 123 (1980), 247-266. MR 581932 (81g:34035)
  • [17] K. Kreith, Oscillation criteria for nonlinear matrix differential equations, Proc. Amer. Math. Soc. 26 (1970), 270-272. MR 0264163 (41:8759)
  • [18] T. Kura, A matrix analogue of Atkinson's oscillation theorem, Funkcialaj Ekvacioj 25 (1982), 223-226. MR 694914 (84i:34037)
  • [19] M. K. Kwong and H. G. Kaper, Oscillation of two-dimensional linear second order differential systems, J. Differential Equations 56 (1985), 195-205. MR 774162 (86j:34032)
  • [20] E. S. Noussair and C. A. Swanson, Oscillation criteria for differential systems, J. Math. Anal. Appl. 36 (1971), 575-580. MR 0296417 (45:5477)
  • [21] K. Schmitt and H. L. Smith, Positive solutions and conjugate points for systems of differential equations, Nonlinear Anal. 2 (1978), 93-105. MR 512658 (80a:34033)
  • [22] E. C. Tomastik, Oscillation of nonlinear matrix differential equations of second order, Proc. Amer. Math. Soc. 19 (1968), 1427-1431. MR 0232046 (38:372)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 34C10

Retrieve articles in all journals with MSC: 34C10


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0946622-6
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society