A nonlinear Perron-Frobenius theorem

Proceedings of the American Mathematical Society - Tập 109 Số 2 - Trang 331-336
Robert Sine

Tóm tắt

If T T is a nonexpansive map on a domain in a finite-dimensional sup norm space then there is a universal bound on the periods of periodic points. This yields the same result for T T nonexpansive on a domain in a finite-dimensional Banach space which has a polyhedral unit ball. Similar results are obtained for certain nonexpansive maps defined on all of an infinite-dimensional L p {L_p} space with 1 > p > 1 > p > \infty .

Từ khóa


Tài liệu tham khảo

Akcoglu, M. A., 1986, Finite orbits in finite-dimensional 𝑙₁ spaces, C. R. Math. Rep. Acad. Sci. Canada, 8, 75

Akcoglu, M. A., 1987, Nonlinear models of diffusion on a finite space, Probab. Theory Related Fields, 76, 411, 10.1007/BF00960065

Aronszajn, N., 1956, Extension of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math., 6, 405, 10.2140/pjm.1956.6.405

Bruck, Ronald E., Jr., 1973, Nonexpansive projections on subsets of Banach spaces, Pacific J. Math., 47, 341, 10.2140/pjm.1973.47.341

DeMarr, Ralph, 1963, Common fixed points for commuting contraction mappings, Pacific J. Math., 13, 1139, 10.2140/pjm.1963.13.1139

Edelstein, Michael, 1964, On non-expansive mappings of Banach spaces, Proc. Cambridge Philos. Soc., 60, 439, 10.1017/S0305004100037956

Krengel, Ulrich, 1987, Nonlinear ergodic theorems, 99

Lacey, H. Elton, 1974, The isometric theory of classical Banach spaces, 10.1007/978-3-642-65762-7

Mankiewicz, Piotr, 1972, On extension of isometries in normed linear spaces, Bull. Acad. Polon. Sci. S\'{e}r. Sci. Math. Astronom. Phys., 20, 367

Nussbaum, Roger D., 1990, Omega limit sets of nonexpansive maps: finiteness and cardinality estimates, Differential Integral Equations, 3, 523

\bysame, Hilbert’s projective metric and iterated nonlinear maps, Mem. Amer. Math. Soc., no. 391, 1988.

Roehrig, Stephen F., 1981, The structure of 𝜔-limit sets of nonexpansive maps, Proc. Amer. Math. Soc., 81, 398, 10.2307/2043473

Royden, H. L., 1963, Real analysis

M. Scheutzow, Periods of nonexpansive operators on finite ℓ₁-spaces, preprint.

Sine, R. C., 1979, On nonlinear contraction semigroups in sup norm spaces, Nonlinear Anal., 3, 885, 10.1016/0362-546X(79)90055-5

Sine, Robert, 1983, Recurrence of nonexpansive mappings in Banach spaces, 175, 10.1090/conm/018/728600