| چکیده انگلیسی مقاله |
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 , 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 of a Banach space 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 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 is a nonempty, weakly compact and convex subset of a Banach space which has the normal structure and 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 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_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 is called relatively nonexpansive whenever 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 may not be continuous, necessarily. For the noncyclic mapping , 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 is a best proximity pair for 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 which estimates the distance between the sets 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 is a nonempty, bounded, closed and convex pair in a uniformly convex Banach space and if is a noncyclic relatively nonexpansive mapping, then 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 be a nonempty and convex pair of subsets of a normed linear space and 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 -uniformly semi-normal structure defined on 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 -uniformly semi-normal structure under some sufficient conditions.
Conclusion
The following conclusions were drawn from this research.
We introduce a geometric notion of -uniformly semi-normal structure and prove that: Let be a nonempty, bounded, closed and convex pair in a strictly convex Banach space 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 be a noncyclic strongly relatively C-nonexpansive mapping. If has the -uniformly semi-normal structure, then has a best proximity pair.
In the setting of uniformly convex in every direction Banach space , we also prove that: Let be a nonempty, weakly compact and convex pair in and be a noncyclic mapping such that ![""]("file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image046.gif") for all 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 has -semi-normal structure.
We present some examples showing the useability of our main conclusions.
./files/site1/files/42/8Abstract.pdf |
