Lemma 3.1 Let be a complete bounded metric space with uniform normal structure, and let be a semigroup on with property (∗). Then, for each , each and for any constant , the normal structure coefficient with respect to the given convexity structure μ, there exist some , and satisfying the properties:
-
(I)
where
-
(II)
Proof For each integer , let , and . Then , and are decreasing sequences of admissible subsets of X, hence , and by Proposition 2.1. From Proposition 2.1, it is not difficult to see that , and . Indeed, observe that
Similarly, one can obtain
On the other hand, for any and any , we have
Therefore,
Also, one can deduce that for any , and any , we have
from which (ii) follows.
We now claim that for each , there exist , and such that
(3)
(4)
(5)
Indeed, if , then , we conclude that (3) holds. Without loss of generality, we may assume that . Then, for , we choose so small satisfying the following:
(6)
By the definition of , one can find such that
This shows that (3) holds. Obviously, it follows from (3) that for each ,
which implies
(7)
where . Noticing
we know that property (∗) yields a point such that
(8)
Since , and satisfies
similarly one can obtain that
where and .
Therefore (i) holds. □
Theorem 3.1 Let be a complete bounded metric space with uniform normal structure, and let be an asymptotically regular semigroup on with property (∗) and satisfying
Then there exist some such that , and for all .
Proof First, we write and . Choose a constant such that and . We can select a sequence such that and , where .
Observe that
for each and .
Now fix . Then, by Lemma 3.1, we can inductively construct sequences such that
(III)
where
(IV)
Let
Observe that for each , using (IV) we have
(9)
By the asymptotic regularity of on , we see that
which implies
(10)
Similarly, one can show that
(11)
(12)
Then it follows from (10), (11) and (12) that for each ,
which implies that for each ,
(13)
Hence, by using (III) and (9), we have
(14)
Hence, by the asymptotic regularity of ℑ on , we have, for each integer ,
(15)
which implies
(16)
It follows from (14) that
Similarly, one can deduce that
(17)
(18)
Thus, we have , and . Consequently, , and are Cauchy and hence convergent as X is complete. Let , and . Then we have
On the other hand, from (15) we have actually proven the following inequalities:
Since , it follows that
Similarly, one can obtain that
i.e., , and . Hence, for each , by the continuity of , we deduce
Similarly, we get that
Then we have , and , i.e., , and for each . □
From Remark 2.1 and Theorem 3.1, we immediately obtain the following results.
Corollary 3.1 Let be a complete bounded metric space with property (P) and uniform normal structure, and let be an asymptotically regular semigroup on satisfying
Then there exist some such that , and for all .
Remark 3.1 It will be interesting to establish Theorem 3.1 for representative on of a left amenable semigroup S as a complete bounded metric space with uniform normal structure as in Holmes and Lau [16], Lau and Takahashi [17] and Lau [18].