Kamenev type theorems for secondorder matrix differential systems
Authors:
Lynn H. Erbe, Qingkai Kong and Shi Gui Ruan
Journal:
Proc. Amer. Math. Soc. 117 (1993), 957962
MSC:
Primary 34C10; Secondary 34A30
MathSciNet review:
1154244
Fulltext 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 wellknown 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 (48 #625)
 [2]
F.
V. Atkinson, Hans
G. Kaper, and Man
Kam Kwong, An oscillation criterion for linear
secondorder differential systems, Proc. Amer.
Math. Soc. 94 (1985), no. 1, 91–96. MR 781063
(86h:34028), http://dx.doi.org/10.1090/S00029939198507810632
 [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 (87h:34045), http://dx.doi.org/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 (87f:34032), http://dx.doi.org/10.1016/0022247X(86)900090
 [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
(88h:34023), http://dx.doi.org/10.1090/S00029947198708960225
 [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
(87f:34033), http://dx.doi.org/10.1016/00220396(86)901178
 [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
(81b:34051), http://dx.doi.org/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, http://dx.doi.org/10.1090/S00029947191815011072
 [9]
Philip
Hartman, Ordinary differential equations, 2nd ed.,
Birkhäuser, Boston, Mass., 1982. MR 658490
(83e:34002)
 [10]
Philip
Hartman, Oscillation criteria for selfadjoint secondorder
differential systems and “principal sectional curvatures”,
J. Differential Equations 34 (1979), no. 2,
326–338. MR
550049 (81a:34034), http://dx.doi.org/10.1016/00220396(79)900135
 [11]
Don
B. Hinton and Roger
T. Lewis, Oscillation theory for generalized secondorder
differential equations, Rocky Mountain J. Math. 10
(1980), no. 4, 751–766. MR 595103
(82c:34039), http://dx.doi.org/10.1216/RMJ1980104751
 [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
(58 #6497)
 [13]
Man
Kam Kwong and Hans
G. Kaper, Oscillation of twodimensional linear secondorder
differential systems, J. Differential Equations 56
(1985), no. 2, 195–205. MR 774162
(86j:34032), http://dx.doi.org/10.1016/00220396(85)901044
 [14]
Man
Kam Kwong, Hans
G. Kaper, Kazuo
Akiyama, and Angelo
B. Mingarelli, Oscillation of linear secondorder
differential systems, Proc. Amer. Math.
Soc. 91 (1984), no. 1, 85–91. MR 735570
(85g:34027), http://dx.doi.org/10.1090/S00029939198407355708
 [15]
Angelo
B. Mingarelli, On a conjecture for oscillation of
secondorder ordinary differential systems, Proc. Amer. Math. Soc. 82 (1981), no. 4, 593–598. MR 614884
(82j:34028), http://dx.doi.org/10.1090/S00029939198106148843
 [16]
Ch.
G. Philos, Oscillation theorems for linear differential equations
of second order, Arch. Math. (Basel) 53 (1989),
no. 5, 482–492. MR 1019162
(90m:34080), http://dx.doi.org/10.1007/BF01324723
 [17]
E.
C. Tomastik, Oscillation of systems of second order differential
equations, J. Differential Equations 9 (1971),
436–442. MR 0274863
(43 #621)
 [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 (81a:34037), http://dx.doi.org/10.1090/S00029939198005504931
 [19]
Aurel
Wintner, A criterion of oscillatory stability, Quart. Appl.
Math. 7 (1949), 115–117. MR 0028499
(10,456a)
 [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
(87k:34052), http://dx.doi.org/10.1090/S00029939198608540334
 [1]
 W. Allegretto and L. Erbe, Oscillation criteria for matrix differential inequalities, Canad. Math. Bull. 16 (1973), 510. MR 0322263 (48:625)
 [2]
 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)
 [3]
 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)
 [4]
 , Oscillation results for selfadjoint differential systems, J. Math. Anal. Appl. 115 (1986), 470481. MR 836240 (87f:34032)
 [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), 263282. MR 896022 (88h:34023)
 [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]
 G. J. Etgen and R. T. Lewis, Positive functionals and oscillation criteria for second order differential systems, Proc. Edinburgh Math. Soc. (2) 22 (1979), 277290. MR 560991 (81b:34051)
 [8]
 B. Fite, Concerning the zeros of the solutions of certain differential equations, Trans. Amer. Math. Soc. 19 (1918), 341352. MR 1501107
 [9]
 P. Hartman, Ordinary differential equations, Wiley, New York, 1982. MR 658490 (83e:34002)
 [10]
 , Oscillation criteria for selfadjoint second order differential systems and "principal sectional curvatures", J. Differential Equations 34 (1979), 326338. MR 550049 (81a:34034)
 [11]
 D. B. Hinton and R. T. Lewis, Oscillation theory of generalized second order differential equations, Rocky Mountain J. Math. 10 (1980), 751761. MR 595103 (82c:34039)
 [12]
 I. V. Kamenev, Integral criterion for oscillations of linear differential equations of second order, Mat. Zametki 23 (1978), 249251. MR 0486798 (58:6497)
 [13]
 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)
 [14]
 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)
 [15]
 A. B. Mingarelli, On a conjecture for oscillation of second order differential systems, Proc. Amer. Math. Soc. 82 (1981), 593598. MR 614884 (82j:34028)
 [16]
 Ch. G. Philos, Oscillation theorems for linear differential equations of second order, Arch. Math. (Basel) 53 (1989), 482492. MR 1019162 (90m:34080)
 [17]
 E. E. Tomastik, Oscillation of systems of second order differential equations, J. Differential Equations 9 (1971), 436442. MR 0274863 (43:621)
 [18]
 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)
 [19]
 A. Wintner, A criterion of oscillatory stability, Quart. Appl. Math. 7 (1949), 115117. MR 0028499 (10:456a)
 [20]
 J. Yan, Oscillation theorems for second order linear differential equations with damping, Proc. Amer. Math. Soc. 98 (1986), 276282. MR 854033 (87k:34052)
Similar Articles
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/S00029939199311542440
PII:
S 00029939(1993)11542440
Keywords:
Matrix differential system,
oscillation theory,
Riccati equation
Article copyright:
© Copyright 1993
American Mathematical Society
