Vol. 45, No. 2, 1973

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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