Two point boundary problems for second order matrix differential systems
Author:
Garret J. Etgen
Journal:
Trans. Amer. Math. Soc. 149 (1970), 119132
MSC:
Primary 34.30
MathSciNet review:
0273096
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Abstract: This paper is concerned with second order matrix differential systems involving a parameter together with boundary conditions specified at two points. The object of the paper is to establish sufficient conditions for the existence of eigenvalues for the system. Although such problems have been considered using the results of and techniques from the calculus of variations, the methods and results here are entirely in the context of ordinary differential equations. Use is made of the matrix generalization of the polar coordinate transformation introduced by J. H. Barrett and the unitary transformation suggested by F. V. Atkinson and V. A. Jakubovič. The sufficient conditions for the existence of eigenvalues obtained here represent certain extensions of W. M. Whyburn's work concerning linear and nonlinear boundary problems for second order differential systems.
 [1]
F.
V. Atkinson, Discrete and continuous boundary problems,
Mathematics in Science and Engineering, Vol. 8, Academic Press, New York,
1964. MR
0176141 (31 #416)
 [2]
John
H. Barrett, A Prüfer transformation for
matrix differential equations, Proc. Amer.
Math. Soc. 8
(1957), 510–518. MR 0087821
(19,415f), http://dx.doi.org/10.1090/S00029939195700878217
 [3]
Earl
A. Coddington and Norman
Levinson, Theory of ordinary differential equations,
McGrawHill Book Company, Inc., New YorkTorontoLondon, 1955. MR 0069338
(16,1022b)
 [4]
Garret
J. Etgen, Oscillatory properties of certain
nonlinear matrix differential systems of second order, Trans. Amer. Math. Soc. 122 (1966), 289–310. MR 0190421
(32 #7834), http://dx.doi.org/10.1090/S00029947196601904211
 [5]
Garret
J. Etgen, A note on trigonometric
matrices, Proc. Amer. Math. Soc. 17 (1966), 1226–1232. MR 0213646
(35 #4504), http://dx.doi.org/10.1090/S00029939196602136460
 [6]
Garret
J. Etgen, On the oscillation of solutions of second order
selfadjoint matrix differential equations, J. Differential Equations
6 (1969), 187–195. MR 0241752
(39 #3091)
 [7]
A.
K. Hinds and W.
M. Whyburn, A nonselfadjoint differential system of the second
order, J. Elisha Mitchell Sci. Soc. 68 (1952),
32–43. MR
0051987 (14,556g)
 [8]
E.
L. Ince, Ordinary Differential Equations, Dover Publications,
New York, 1944. MR 0010757
(6,65f)
 [9]
V.
A. Jakubovič, Oscillatory properties of solutions of
canonical equations, Mat. Sb. (N.S.) 56 (98) (1962),
3–42 (Russian). MR 0138863
(25 #2303)
 [10]
Marston
Morse, The calculus of variations in the large, American
Mathematical Society Colloquium Publications, vol. 18, American
Mathematical Society, Providence, RI, 1996. Reprint of the 1932 original.
MR
1451874 (98f:58070)
 [11]
W.
T. Reid, Boundary value problems of the
calculus of variations, Bull. Amer. Math.
Soc. 43 (1937), no. 10, 633–666. MR
1563613, http://dx.doi.org/10.1090/S00029904193706616X
 [12]
William
T. Reid, An IntegroDifferential Boundary Value Problem, Amer.
J. Math. 60 (1938), no. 2, 257–292. MR
1507311, http://dx.doi.org/10.2307/2371292
 [13]
William
T. Reid, A Prüfer transformation for differential
systems, Pacific J. Math. 8 (1958), 575–584. MR 0099474
(20 #5913)
 [14]
William
M. Whyburn, Existence and oscillation theorems for
nonlinear differential systems of the second order, Trans. Amer. Math. Soc. 30 (1928), no. 4, 848–854. MR
1501463, http://dx.doi.org/10.1090/S00029947192815014631
 [15]
William
M. Whyburn, A nonlinear boundary value problem for second order
differential systems, Pacific J. Math. 5 (1955),
147–160. MR 0069368
(16,1027d)
 [1]
 F. V. Atkinson, Discrete and continuous boundary problems, Academic Press, New York, 1964. MR 31 #416. MR 0176141 (31:416)
 [2]
 J. H. Barrett, A Prüfer transformation for matrix differential equations, Proc. Amer. Math. Soc. 8 (1957), 510518. MR 19, 415. MR 0087821 (19:415f)
 [3]
 E. A. Coddington and N. Levinson, Theory of ordinary differential equations, McGrawHill, New York, 1955. MR 16, 1022. MR 0069338 (16:1022b)
 [4]
 G. J. Etgen, Oscillatory properties of certain nonlinear matrix differential systems of second order, Trans. Amer. Math. Soc. 122 (1966), 289310. MR 32 #7834. MR 0190421 (32:7834)
 [5]
 , A note on trigonometric matrices, Proc. Amer. Math. Soc. 17 (1966), 12261232. MR 35 #4504. MR 0213646 (35:4504)
 [6]
 , On the oscillation of solutions of second order, selfadjoint matrix differential equations, J. Differential Equations 6 (1969). MR 0241752 (39:3091)
 [7]
 A. K. Hinds and W. M. Whyburn, A nonselfadjoint differential system of the second order, J. Elisha Mitchell Sci. Soc. 68 (1952), 3243. MR 14, 556. MR 0051987 (14:556g)
 [8]
 E. L. Ince, Ordinary differential equations, Longmans, Green & Co., 1926; reprint, Dover, New York, 1944. MR 6, 65. MR 0010757 (6:65f)
 [9]
 V. A. Jakubovič, Oscillatory properties of solutions of canonical equations, Mat. Sb. 56 (98) (1962), 342; English transl., Amer. Math. Soc. Transl. (2) 42 (1964), 247288. MR 25 #2303. MR 0138863 (25:2303)
 [10]
 M. Morse, The calculus of variations in the large, Amer. Math. Soc. Colloq. Publ., vol. 17, Amer. Math. Soc., Providence, R. I., 1934. MR 1451874 (98f:58070)
 [11]
 W. T. Reid, Boundary value problems of the calculus of variations, Bull. Amer. Math Soc. 43 (1937), 633666. MR 1563613
 [12]
 W. T. Reid, An integrodifferential boundary value problem, Amer. J. Math. 60 (1938), 257292. MR 1507311
 [13]
 , A Prüfer transformation for differential systems, Pacific J. Math. 8 (1958), 575584. MR 20 #5913. MR 0099474 (20:5913)
 [14]
 W. M. Whyburn, Existence and oscillation theorems for nonlinear differential equations of second order, Trans. Amer. Math. Soc. 30 (1928), 848854. MR 1501463
 [15]
 , A nonlinear boundary value problem for second order differential systems, Pacific J. Math. 5 (1955), 147160. MR 16, 1027. MR 0069368 (16:1027d)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197002730963
PII:
S 00029947(1970)02730963
Keywords:
Matrix differential equations,
second order matrix differential systems,
two point boundary problems,
existence of eigenvalues,
oscillatory behavior
Article copyright:
© Copyright 1970 American Mathematical Society
