Introduction
Let
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif)
be a nonempty subset of a normed linear space
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif)
. A self-mapping
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif)
is said to be nonexpansive provided that
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image008.gif)
for all
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif)
. In 1965, Browder showed that every nonexpansive self-mapping defined on a nonempty, bounded, closed and convex subset of a uniformly convex Banach space
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif)
, has a fixed point. In the same year, Kirk generalized this existence result by using a geometric notion of normal structure. We recall that a nonempty and convex subset
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif)
of a Banach space
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif)
is said to have normal structure if for any nonempty, bounded, closed and convex subset
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image012.gif)
of
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif)
with
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image014.gif)
, there exists a point
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image016.gif)
for which
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image018.gif)
. The well-known Kirk’s fixed point theorem states that if
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif)
is a nonempty, weakly compact and convex subset of a Banach space
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif)
which has the normal structure and
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif)
is a nonexpansive mapping, then
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif)
has at least one fixed point. In view of the fact that every nonempty, bounded, closed and convex subset of a uniformly convex Banach space
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif)
has the normal structure, the Browder’ fixed point result is an especial case of Kirk’s theorem.
Material and methods
Let
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image022.gif)
be a nonempty pair of subsets of a normed linear space
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif)
.
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image024.gif)
is said to be a noncyclic mapping if
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image026.gif)
. Also the noncyclic mapping
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif)
is called relatively nonexpansive whenever
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image008.gif)
for any
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image028.gif)
. Clearly, if
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image030.gif)
, then we get the class of nonexpppansive self-mappings. Moreover, we note the noncyclic relatively nonexpansive mapping
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif)
may not be continuous, necessarily. For the noncyclic mapping
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif)
, a point
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image032.gif)
is called a best proximity pair provided that
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image034.gif)
In the other words, the point
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image032.gif)
is a best proximity pair for
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif)
if
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image036.gif)
and
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image038.gif)
are two fixed points of
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif)
which estimates the distance between the sets
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif)
and
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image040.gif)
.
The first existence result about such points which is an interesting extension of Browder’s fixed point theorem states that if
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image022.gif)
is a nonempty, bounded, closed and convex pair in a uniformly convex Banach space
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif)
and if
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image024.gif)
is a noncyclic relatively nonexpansive mapping, then
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif)
has a best proximity pair. Furthermore, a real generalization of Kirk’s fixed point result for noncyclic relatively nonexpansive mappings was proved by using a geometric concept of proximal normal structure, defined on a nonempty and convex pair in a considered Banach space.
Results and discussion
Let
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image022.gif)
be a nonempty and convex pair of subsets of a normed linear space
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif)
and
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image024.gif)
be a noncyclic mapping. The main purpose of this article is to study of the existence of best proximity pairs for another class of noncyclic mappings, called noncyclic strongly relatively C-nonexpansive. To this end, we use a new geometric notion entitled
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif)
-uniformly semi-normal structure defined on
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image022.gif)
in a Banach space which is not reflexive, necessarily. To illustrate this geometric property, we show that every nonempty, bounded, closed and convex pair in uniformly convex Banach spaces has
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif)
-uniformly semi-normal structure under some sufficient conditions.
Conclusion
The following conclusions were drawn from this research.
We introduce a geometric notion of
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif)
-uniformly semi-normal structure and prove that: Let
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image022.gif)
be a nonempty, bounded, closed and convex pair in a strictly convex Banach space
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif)
such that
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image042.gif)
is nonempty and
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image044.gif)
. Let
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image024.gif)
be a noncyclic strongly relatively C-nonexpansive mapping. If
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image022.gif)
has the
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif)
-uniformly semi-normal structure, then
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif)
has a best proximity pair.
In the setting of uniformly convex in every direction Banach space
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif)
, we also prove that: Let
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image022.gif)
be a nonempty, weakly compact and convex pair in
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif)
and
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image024.gif)
be a noncyclic mapping such that
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image046.gif)
for all
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image028.gif)
with
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image048.gif)
. If
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image050.gif)
where
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image052.gif)
is a projection mapping defined on
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image054.gif)
then
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image022.gif)
has
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif)
-semi-normal structure.
We present some examples showing the useability of our main conclusions.
./files/site1/files/42/8Abstract.pdf