Kamenev type theorems for second-order matrix differential systems

Authors:
Lynn H. Erbe, Qingkai Kong and Shi Gui Ruan

Journal:
Proc. Amer. Math. Soc. **117** (1993), 957-962

MSC:
Primary 34C10; Secondary 34A30

MathSciNet review:
1154244

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the second order matrix differential systems (1) and (2) where and are real continuous matrix functions with symmetric and positive definite for . We establish sufficient conditions in order that all prepared solutions of (1) and (2) are oscillatory. The results obtained can be regarded as generalizing well-known results of Kamenev in the scalar case.

**[1]**W. Allegretto and L. Erbe,*Oscillation criteria for matrix differential inequalities*, Canad. Math. Bull.**16**(1973), 5–10. MR**0322263****[2]**F. V. Atkinson, Hans G. Kaper, and Man Kam Kwong,*An oscillation criterion for linear second-order differential systems*, Proc. Amer. Math. Soc.**94**(1985), no. 1, 91–96. MR**781063**, 10.1090/S0002-9939-1985-0781063-2**[3]**G. J. Butler and L. H. Erbe,*Oscillation results for second order differential systems*, SIAM J. Math. Anal.**17**(1986), no. 1, 19–29. MR**819208**, 10.1137/0517003**[4]**G. J. Butler and L. H. Erbe,*Oscillation results for selfadjoint differential systems*, J. Math. Anal. Appl.**115**(1986), no. 2, 470–481. MR**836240**, 10.1016/0022-247X(86)90009-0**[5]**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**, 10.1090/S0002-9947-1987-0896022-5**[6]**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**, 10.1016/0022-0396(86)90117-8**[7]**Garret J. Etgen and Roger T. Lewis,*Positive functionals and oscillation criteria for second order differential systems*, Proc. Edinburgh Math. Soc. (2)**22**(1979), no. 3, 277–290. MR**560991**, 10.1017/S001309150001645X**[8]**William Benjamin Fite,*Concerning the zeros of the solutions of certain differential equations*, Trans. Amer. Math. Soc.**19**(1918), no. 4, 341–352. MR**1501107**, 10.1090/S0002-9947-1918-1501107-2**[9]**Philip Hartman,*Ordinary differential equations*, 2nd ed., Birkhäuser, Boston, Mass., 1982. MR**658490****[10]**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**, 10.1016/0022-0396(79)90013-5**[11]**Don B. Hinton and Roger T. Lewis,*Oscillation theory for generalized second-order differential equations*, Rocky Mountain J. Math.**10**(1980), no. 4, 751–766. MR**595103**, 10.1216/RMJ-1980-10-4-751**[12]**I. V. Kamenev,*An integral test for conjugacy for second order linear differential equations*, Mat. Zametki**23**(1978), no. 2, 249–251 (Russian). MR**0486798****[13]**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**, 10.1016/0022-0396(85)90104-4**[14]**Man Kam Kwong, Hans G. Kaper, Kazuo Akiyama, and Angelo B. Mingarelli,*Oscillation of linear second-order differential systems*, Proc. Amer. Math. Soc.**91**(1984), no. 1, 85–91. MR**735570**, 10.1090/S0002-9939-1984-0735570-8**[15]**Angelo B. Mingarelli,*On a conjecture for oscillation of second-order ordinary differential systems*, Proc. Amer. Math. Soc.**82**(1981), no. 4, 593–598. MR**614884**, 10.1090/S0002-9939-1981-0614884-3**[16]**Ch. G. Philos,*Oscillation theorems for linear differential equations of second order*, Arch. Math. (Basel)**53**(1989), no. 5, 482–492. MR**1019162**, 10.1007/BF01324723**[17]**E. C. Tomastik,*Oscillation of systems of second order differential equations*, J. Differential Equations**9**(1971), 436–442. MR**0274863****[18]**Terry Walters,*A characterization of positive linear functionals and oscillation criteria for matrix differential equations*, Proc. Amer. Math. Soc.**78**(1980), no. 2, 198–202. MR**550493**, 10.1090/S0002-9939-1980-0550493-1**[19]**Aurel Wintner,*A criterion of oscillatory stability*, Quart. Appl. Math.**7**(1949), 115–117. MR**0028499****[20]**Ju Rang Yan,*Oscillation theorems for second order linear differential equations with damping*, Proc. Amer. Math. Soc.**98**(1986), no. 2, 276–282. MR**854033**, 10.1090/S0002-9939-1986-0854033-4

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
34C10,
34A30

Retrieve articles in all journals with MSC: 34C10, 34A30

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1993-1154244-0

Keywords:
Matrix differential system,
oscillation theory,
Riccati equation

Article copyright:
© Copyright 1993
American Mathematical Society