Vol. 45, No. 2, 1973

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Vol. 323: 1  2
Vol. 322: 1  2
Vol. 321: 1  2
Online Archive
The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author index
To appear
Other MSP journals
On the rest points of a nonlinear nonexpansive semigroup

Chi-Lin Yen

Vol. 45 (1973), No. 2, 699–706

Let X be a reflexive Banach space and T a nonlinear nonexpansive semigroup on X. The results which we shall prove are the following:

Theorem 1. Suppose that for any closed convex set M with the property that T(t)M M for all t 0,M contains a precompact orbit. Then T has a rest point. Moreover, the set of all rest points of T is connected.

Theorem 2. Suppose that X is strictly convex and T has a bounded orbit. If there is an unbounded increasing sequence {ui} of positive numbers and point x such that limi→∞T(ui)x exists then T has a rest point. Moreover, if {ti} is an unbounded increasing sequence of positive numbers such that

-1  ti
y = w − ili→m∞ ti 0 T(t)x dt

exists, then y F.

Mathematical Subject Classification 2000
Primary: 47H99
Received: 24 November 1971
Published: 1 April 1973
Chi-Lin Yen