Oscillation of superlinear matrix differential equations

Ahlbrandt, Calvin D.; Ridenhour, Jerry; Thompson, Russell C.

doi:10.1090/S0002-9939-1989-0946622-6

Oscillation of superlinear matrix differential equations
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by Calvin D. Ahlbrandt, Jerry Ridenhour and Russell C. Thompson PDF

Proc. Amer. Math. Soc. 105 (1989), 141-148 Request permission

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

Shair Ahmad, On Sturmian theory for second order systems, Proc. Amer. Math. Soc. 87 (1983), no. 4, 661–665. MR 687636, DOI 10.1090/S0002-9939-1983-0687636-8
Shair 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. 9 (1978), no. 6, 1137–1150. MR 512517, DOI 10.1137/0509092
Shair Ahmad and C. C. Travis, Oscillation criteria for second-order differential systems, Proc. Amer. Math. Soc. 71 (1978), no. 2, 247–252. MR 486792, DOI 10.1090/S0002-9939-1978-0486792-2
F. V. Atkinson, On second-order non-linear oscillations, Pacific J. Math. 5 (1955), 643–647. MR 72316

Oscillatory solutions of certain non-linear differential equations of second order

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G. J. Butler and L. H. Erbe, Oscillation theory for second order differential systems with functional commutative coefficients, Differential and integral equations (Iowa City, Iowa, 1983/Argonne, Ill., 1984) Univ. Missouri-Rolla, Rolla, MO, 1985, pp. 15–18. MR 821763
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. 303 (1987), no. 1, 263–282. MR 896022, DOI 10.1090/S0002-9947-1987-0896022-5
Ralph Byers, B. J. Harris, and Man Kam Kwong, Weighted means and oscillation conditions for second order matrix differential equations, J. Differential Equations 61 (1986), no. 2, 164–177. MR 823400, DOI 10.1016/0022-0396(86)90117-8
J. Dieudonné, Sur un théorème de Schwerdtfeger, Ann. Polon. Math. 29 (1974), 87–88 (French). MR 344276, DOI 10.4064/ap-29-1-87-88
Garret J. Etgen and James F. Pawlowski, A comparison theorem and oscillation criteria for second order differential systems, Pacific J. Math. 72 (1977), no. 1, 59–69. MR 450675
Stuart Goff, Hermitian function matrices which commute with their derivative, Linear Algebra Appl. 36 (1981), 33–40. MR 604327, DOI 10.1016/0024-3795(81)90217-2
Philip Hartman, Oscillation criteria for selfadjoint second-order differential systems and “principal sectional curvatures”, J. Differential Equations 34 (1979), no. 2, 326–338. MR 550049, DOI 10.1016/0022-0396(79)90013-5
Stuart P. Hastings, Boundary value problems in one differential equation with a discontinuity, J. Differential Equations 1 (1965), 346–369. MR 180723, DOI 10.1016/0022-0396(65)90013-6
Athanassios 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 (Proc. Regional Conf., Mississippi State Univ., Mississippi State, Miss., 1975) Lecture Notes in Pure and Appl. Math., Vol. 28, Dekker, New York, 1977, pp. 17–72. MR 0594954
A. G. Kartsatos and T. Walters, Some oscillation results for matrix and vector differential equations with forcing term, J. Math. Anal. Appl. 73 (1980), no. 2, 506–513. MR 564000, DOI 10.1016/0022-247X(80)90295-4
M. S. Keener and C. C. Travis, Sturmian theory for a class of nonselfadjoint differential systems, Ann. Mat. Pura Appl. (4) 123 (1980), 247–266. MR 581932, DOI 10.1007/BF01796547
Kurt Kreith, Oscillation criteria for nonlinear matrix differential equations, Proc. Amer. Math. Soc. 26 (1970), 270–272. MR 264163, DOI 10.1090/S0002-9939-1970-0264163-4
Takeshi Kura, A matrix analogue of Atkinson’s oscillation theorem, Funkcial. Ekvac. 25 (1982), no. 2, 223–226. MR 694914
Man Kam Kwong and Hans G. Kaper, Oscillation of two-dimensional linear second-order differential systems, J. Differential Equations 56 (1985), no. 2, 195–205. MR 774162, DOI 10.1016/0022-0396(85)90104-4
E. S. Noussair and C. A. Swanson, Oscillation criteria for differential systems, J. Math. Anal. Appl. 36 (1971), 575–580. MR 296417, DOI 10.1016/0022-247X(71)90039-4
K. Schmitt and H. L. Smith, Positive solutions and conjugate points for systems of differential equations, Nonlinear Anal. 2 (1978), no. 1, 93–105. MR 512658, DOI 10.1016/0362-546X(78)90045-7
E. C. Tomastik, Oscillation of nonlinear matrix differential equations of second order, Proc. Amer. Math. Soc. 19 (1968), 1427–1431. MR 232046, DOI 10.1090/S0002-9939-1968-0232046-2