Convergence in the part metric for discrete dynamical systems in ordered topological cones

Nonlinear Analysis, Theory, Methods and Applications

Volumn 26, Issue 11, 1996, Pages 1753-1777

  Takáč, Peter ^a  

^a WASHINGTON STATE UNIVERSITY   (United States)

Author keywords

Cones without operation of addition; Kre n Rutman theorem; Nonlinear eigenvalue problem; Stable fixed point; Weakly noncooperative elliptic system

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; DYNAMICS; EIGENVALUES AND EIGENFUNCTIONS; EQUIVALENCE CLASSES; MATHEMATICAL TECHNIQUES; MATRIX ALGEBRA; NUMERICAL ANALYSIS; SET THEORY; THEOREM PROVING;

BANACH SPACES; CONES WITHOUT OPERATION OF ADDITION; DISCRETE DYNAMICAL SYSTEMS; KREIN-RUTMAN THEOREM; NONLINEAR EIGENVALUE PROBLEM; NONLINEAR FUNCTIONAL ANALYSIS; ORDERED TOPOLOGICAL CONES; PERRON-FROBENIUS THEOREM; STABLE FIXED POINT; WEAKLY NONCOOPERATIVE ELLIPTIC SYSTEMS;

TOPOLOGY;

EID: 0030174748     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/0362-546X(95)00015-N     Document Type: Article

References (25)

1. BIRKHOFF G., Extensions of Jentzsch's theorem, Trans. Am. math. Soc. 85(1), 219-227 (1957).
2. THOMPSON A. C., On certain contraction mappings in a partially ordered vector space, Proc. Am. math. Soc. 14, 438-443 (1963).
3. NUSSBAUM R. D., Hubert's projective metric and iterated nonlinear maps, Mem. Am. math. Soc. 391, Providence, R.I. (1988).
4. HIRSCH M. W., The dynamical systems approach to differential equations, Bull. Am. math. Soc. 114, 1-64 (1984).
5. KRAUSE U., A nonlinear extension of the Birkhoff-Jentzsch theorem, J. Math. Analysis Applic. 11, 522-568 (1986).
6. KRAUSE U. & NUSSBAUM R. D., A limit set trichotomy for self-mappings of normal cones in Banach spaces, Nonlinear Analysis 20(7), 855-870 (1993).
7. KRAUSE U. & RANFT P., A limit set trichotomy for monotone nonlinear dynamical systems, Nonlinear Analysis 19(4), 375-392 (1992).
8. NUSSBAUM R. D., Hubert's projective metric and iterated nonlinear maps II, Mem. Am. math. Soc. 401, Providence, R.I. (1989).
9. SMITH H. L., Cooperative systems of differential equations with concave nonlinearities, Nonlinear Analysis 10, 1037-1052 (1986).
10. TAKÁČ P., Asymptotic behavior of discrete-time semigroups of sublinear, strongly increasing mappings with applications in biology, Nonlinear Analysis 14(1), 35-42 (1990).
11. THOMPSON A. C., On the eigenvectors of some not-necessarily-linear transformations, Proc. London Math. Soc. 15(3), 577-598 (1965).
12. ALIKAKOS N. D. & FUSCO G., A dynamical systems proof of the Kreǐn-Rutman theorem and an extension of the Perron theorem, Proc. R. Soc. Edinb. 177A, 209-214 (1991).
13. JENTZSCH P., Über Integralgleichungen mit positivem Kern, J. reine angew. Math. 141, 235-244 (1912).
14. KRASNOSEL'SKIJ M. A., LIFSHITS JE. A. & SOBOLEV A. V., Positive Linear Systems: The Method of Positive Operators. Heldermann, Berlin (1989).
15. KREǏN M. G. & RUTMAN A., Linear operators leaving invariant a cone in a Banach space, Upsekhi Mat. Nauk (N.S.) 3, no. 1 (23), 3-95 (1948). English transl. in: Am. math. Soc. Transl. 26, 1-128 (1950).
16. KREǏN M. G. & RUTMAN A., Linear operators leaving invariant a cone in a Banach space, Upsekhi Mat. Nauk (N.S.) 3, no. 1 (23), 3-95 (1948). English transl. in: Am. math. Soc. Transl. 26, 1-128 (1950).
17. SCHAEFER H. H., Topological Vector Spaces. Springer, New York-Berlin-Heidelberg-Tokyo (1971) (5-th print., 1986).
18. SCHAEFER H. H., Banach Lattices and Positive Operators, Springer, New York-Berlin-Heidelberg-Tokyo (1974).
19. TAKÁČ P., A short elementary proof of the Kreǐn-Rutman theorem, Houston J. Math. 20(1), 93-98 (1994).
20. DEIMLING K., Nonlinear Functional Analysis. Springer, New York-Berlin-Heidelberg (1985).
21. BAUER H. & BEAR H. S., The part metric in convex sets, Pacif. J. Math. 30(1), 15-33 (1969).
22. HALE J. K., "Asymptotic Behavior of Dissipative Systems". Math. Surveys and Monogr. 25, Amer. Math. Soc., Providence, R.I. (1988).
23. TAKÁČ P., The local stability of positive solutions to the Hammerstein equation with a nonmonotonic Nemytskii operator, Mh. Math. 106, 313-335 (1988).
24. GILBARG D. & TRUDINGER N. S., Elliptic Partial Differential Equations of Second Order. Springer, New York-Berlin-Heidelberg (1977).
25. SELGRADE J. F., Asymptotic behavior of solutions to single loop positive feedback systems, J. Diff. Eqns. 38, 80-103 (1980).