Theorem 3.1.
Let be a weakly compact convex subset of a Banach space and let be a bounded sequence in regular with respect to . Then for every ,
Proof.
Denote and . We can assume that . By passing to a subsequence if necessary, we can also assume that is weakly convergent to a point . Since is regular with respect to , then passing through a subsequence does not have any effect to the asymptotic radius of the whole sequence. Let , then we have that
Denote . By the definition of we have that
Convexity of implies that and thus, we obtain
On the other hand, by the weak lower semicontinuity of the norm, we have that
For every there exists such that
(1),
(2)
(3)
(4)
Now, put , and . Using the above estimates, we obtain , and
Thus,
By the weak lower semicontinuity of the norm again, we conclude that and hence,
Therefore Since the above inequality is true for every and every , we obtain
and therefore,
Corollary 3.2.
Let be a nonempty bounded closed convex subset of a Banach space such that and let be a nonexpansive mapping. Then has a fixed point.
Proof.
When , then satisfies the DL condition by Theorem 3.1. So has a fixed point by Theorem 2.3.
Remark 3.3.
In particular, when , we get the result of Kaewkhao; a Banach space with
satisfies the DL condition.
Theorem 3.4.
Let be a weakly compact convex subset of a Banach space and let be a bounded sequence in regular with respect to . Then for every ,
Proof.
Let , and be as in the proof of the previous theorem. Thus,
Since and by the definition of , we obtain
On the other hand, by the weak lower semicontinuity of the norm, we have that
For every , there exists such that
(1)
(2)
(3)
(4)
Now, put , and and use the above estimates to obtain , , , and , so that
By the definition of , we get
Let ; we obtain that . Then we have
This holds for arbitrary ; hence, we have that
Corollary 3.5.
Let be a nonempty bounded closed convex subset of a Banach space such that and let be a nonexpansive mapping. Then has a fixed point.
Proof.
When , then satisfies the DL condition by Theorem 3.4. So has a fixed point by Theorem 2.3.
Remark 3.6.
In particular, when , we get the result of Kaewkhao; a Banach space with
satisfies the DL condition.
Repeating the arguments in the proof of Theorem 3.4, we can easily get the following conclusion.
Theorem 3.7.
Let be a nonempty bounded closed convex subset of a Banach space such that and let be a nonexpansive mapping. Then has a fixed point.
Remark 3.8.
In particular, when , we get that
satisfies the DL condition which improves the result of Kaewkhao; a Banach space with
satisfies the DL condition.
Theorem 3.9.
A Banach space has property (D) whenever .
Proof.
Let be a nonempty weakly compact convex subset of . Suppose that and are regular asymptotically uniforms relative to . Passing to a subsequence, we may assume that is weakly convergent to a point and exists. Let . Again, passing to a subsequence of , still denoted by , we assume in addition that
for all . Now, put
It is easy to see that , , , and . This implies that or Now we estimate as follows:
Hence,
Remark 3.10.

(1)
Theorem 3.9 strengthens the result of Saejung [7] and has property (D) whenever .

(2)
Theorem 3.9 also improves the result implying that the Banach space has normal structure from Theorem 2.2.