Pacific Journal of Mathematics

Linear differential equations on cones in Banach spaces.

Charles V. Coffman

Article information

Pacific J. Math., Volume 12, Number 1 (1962), 69-75.

First available in Project Euclid: 14 December 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34.95


Coffman, Charles V. Linear differential equations on cones in Banach spaces. Pacific J. Math. 12 (1962), no. 1, 69--75.

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  • [4] Special form of Theorem 1. An analogue of the theorem of Perron-Frobenius is the following: Let X, K, H be as in Theorem 1, A a boundednonnegative"perator. Then A has an eigenvalue ^ 0 and a corresponding eigenvector x0 ^ 0. This is contained in a stronger theorem of Schaefer [4], pp. 1013- 1014. A very simple proof results from an application of Tychonoff s fixed point theorem to the map PA restricted to H, where Px = xlf(x)
  • [1] P. Hartman and A. Wintner, Linear differential and differenceequations with monotone solutions, Amer. J. Math., 75 (1953), 731-743.
  • [2] S. Karlin, Positive operators, J. Math and Mech, 8 (1959), 907-938.
  • [3] M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspehi Matem. Nauk (1948), 3-95 (Amer. Math. Soc. Trans. No. 26).
  • [4] H. Schaefer, Some spectral properties of positive linear operators, Pacific J. Math., 10 (I960), 1009-1019.