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.