Mathematical Researches

fa ﻧﺘﺎیج وﺟﻮدی ﺑﻬﺘﺮیﻦ زوجﻫﺎی نزدینی ﺑﺮای رده‌ای ﺧﺎص از ﻧﮕﺎﺷﺖﻫﺎی ﻏﯿﺮدوری در ﻓﻀﺎﻫﺎی ﺑﺎﻧﺎخ ﻏﯿﺮبازتابی Existence Results of best Proximity Pairs for a Certain Class of Noncyclic Mappings in Nonreflexive Banach Spaces Polynomials جبر alg مقاله مستقل Original Manuscript ﻓﺮض ﮐﻨﯿﺪ یﮏ زوج ﻧﺎﺗﻬﯽ از زیﺮﻣﺠﻤﻮﻋﻪﻫﺎی ﻓﻀﺎی ﻣﺘﺮیﮏ  ﺑﺎﺷﺪ.  یک نگاشت غیردوری نامیده می‌شود هرگاه . عضو  یک بهترین زوج نزدینی ﺑﺮای ﻧﮕﺎﺷﺖ ﻏﯿﺮدوری  ﻧﺎﻣﯿﺪه ﻣﯽﺷﻮد ﻫﺮﮔﺎه  ﻧﻘﺎط ﺛﺎﺑﺖ ﺑﻮده که ﻓﺎﺻﻠﻪ دو ﻣﺠﻤﻮﻋﻪ   و  را ﺗﻘﺮیﺐ ﺑﺰﻧﻨﺪ، ﺑﻪ ایﻦ ﻣﻌﻨﺎ ﮐﻪ . ﻫﺪف اﺻﻠﯽ ایﻦ ﻣﻘﺎﻟﻪ ﺑﺮرﺳﯽ وﺟﻮد ﭼﻨﯿﻦ ﻧﻘﺎﻃﯽ ﺑﺮای رده‌ای ﺧﺎص از ﻧﮕﺎﺷﺖﻫﺎی ﻏﯿﺮدوری ﺗﺤﺖ ﻋﻨﻮان ﻧﮕﺎﺷﺖﻫﺎیC - ﻏﯿﺮاﻧﺒﺴﺎﻃﯽ ﻧﺴﺒﯽ است ﮐﻪ اﺧﯿﺮاً در ﻣﺮﺟﻊ [1] ﻣﻌﺮﻓﯽ شده است. ﺑﺮای ایﻦﻣﻨﻈﻮر از یﮏ ﻣﻔﻬﻮم ﻫﻨﺪﺳﯽ ﺟﺪیﺪ ﺑﻪﻧﺎم  - ﺳﺎﺧﺘﺎر ﺷﺒﻪ ﻧﺮﻣﺎل یک‌نواﺧﺖ ﮐﻪ ﺑﺮ یﮏ زوج ﻧﺎﺗﻬﯽ و ﻣﺤﺪب از زیﺮ ﻣﺠﻤﻮﻋﻪﻫﺎی یﮏ ﻓﻀﺎی ﺑﺎﻧﺎخ ﮐﻪ ﻟﺰوﻣﺎً بازتابی نیست، اﺳﺘﻔﺎده ﺧﻮاﻫﺪ ﺷﺪ. ﺑﻪﻣﻨﻈﻮر ﺗﺒﯿﯿﻦ ﺑﻬﺘﺮ ایﻦ ﺧﺎﺻﯿﺖ ﻫﻨﺪﺳﯽ ﻧﺸﺎن داده ﻣﯽﺷﻮد ﮐﻪ ﻫﺮ زوج ﻧﺎﺗﻬﯽ، ﺑﺴﺘﻪ، ﮐﺮاﻧﺪار و ﻣﺤﺪب در ﻓﻀﺎﻫﺎی ﺑﺎﻧﺎخ ﺑﻪﻃﻮر یک‌نواﺧﺖ ﻣﺤﺪب ﺗﺤﺖ ﺷﺮایﻂ ﮐﺎﻓﯽ دارای ﺳﺎﺧﺘﺎر ﺷﺒﻪ ﻧﺮﻣﺎل - یک‌نواﺧﺖ اﺳﺖ. در ﻧﻬﺎیﺖ ﺑﺎ اراﺋﻪ ﭼﻨﺪ ﻣﺜﺎل ﮐﺎرﺑﺮدی ﺑﻪ ﺑﺮرﺳﯽ اﺛﺮﺑﺨﺶ ﺑﻮدن ﻧﺘﺎیﺞ ﺣﺎﺻل می‌پردازیم. Introduction Let <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" width="9" > be a nonempty subset of a normed linear space <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="10" >. A self-mapping <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif" width="55" > is said to be nonexpansive provided that <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image008.gif" width="142" > for all <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif" width="49" >. In 1965, Browder showed that every nonexpansive self-mapping defined on a nonempty, bounded, closed and convex subset of a uniformly convex Banach space <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="10" >, 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 <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" width="9" > of a Banach space <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="10" > is said to have normal structure if  for any nonempty, bounded, closed and convex subset <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image012.gif" width="11" > of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" width="9" > with <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image014.gif" width="84" >, there exists a point <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image016.gif" width="36" > for which <img alt="" height="21" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image018.gif" width="205" >. The well-known Kirk’s fixed point theorem states that if <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" width="9" > is a nonempty, weakly compact and convex subset of a Banach space <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="10" > which has the normal structure and <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif" width="55" > is a nonexpansive mapping, then <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif" width="9" > 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 <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="10" > has the normal structure, the Browder’ fixed point result is an especial case of Kirk’s theorem.  Material and methods Let <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image022.gif" width="37" > be a nonempty pair of subsets of a normed linear space <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="10" >. <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image024.gif" width="108" > is said to be a noncyclic mapping if <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image026.gif" width="130" >. Also the noncyclic mapping <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif" width="9" > is called relatively nonexpansive whenever <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image008.gif" width="142" > for any <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image028.gif" width="88" >. Clearly, if <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image030.gif" width="39" >, then we get the class of nonexpppansive self-mappings. Moreover, we note the  noncyclic relatively nonexpansive mapping <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif" width="9" > may not be continuous, necessarily. For the noncyclic mapping <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif" width="9" >, a point <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image032.gif" width="88" > is called a best proximity pair provided that <img alt="" height="21" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image034.gif" width="263" > In the other words, the point <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image032.gif" width="88" > is a best proximity pair for <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif" width="9" > if <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image036.gif" width="8" > and <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image038.gif" width="8" > are two fixed points of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif" width="9" > which estimates the distance between the sets <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" width="9" > and <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image040.gif" width="10" >. The first existence result about such points which is an interesting extension of Browder’s fixed point theorem states that if <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image022.gif" width="37" > is a nonempty, bounded, closed and convex pair in a uniformly convex Banach space <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="10" > and if <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image024.gif" width="108" > is a noncyclic relatively nonexpansive mapping, then <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif" width="9" > 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 <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image022.gif" width="37" > be a nonempty and convex pair of subsets of a normed linear space <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="10" > and  <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image024.gif" width="108" > 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 <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif" width="9" >-uniformly semi-normal structure defined on <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image022.gif" width="37" > 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 <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif" width="9" >-uniformly semi-normal structure under some sufficient conditions. Conclusion The following conclusions were drawn from this research. We introduce a geometric notion of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif" width="9" >-uniformly semi-normal structure and prove that: Let <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image022.gif" width="37" > be a nonempty, bounded, closed and convex pair in a strictly convex Banach space <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="10" > such that <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image042.gif" width="16" > is nonempty and <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image044.gif" width="88" >. Let <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image024.gif" width="108" > be a noncyclic strongly relatively C-nonexpansive mapping. If <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image022.gif" width="37" > has the <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif" width="9" >-uniformly semi-normal structure, then <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif" width="9" > has a best proximity pair. In the setting of uniformly convex in every direction Banach space <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="10" >, we also prove that: Let <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image022.gif" width="37" > be a nonempty, weakly compact and convex pair in <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="10" > and  <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image024.gif" width="108" > be a noncyclic mapping such that <img alt="" height="21" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image046.gif" width="145" > for all <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image028.gif" width="88" > with <img alt="" height="21" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image048.gif" width="132" >. If <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image050.gif" width="267" > where <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image052.gif" width="11" > is a projection mapping defined on <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image054.gif" width="52" > then <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image022.gif" width="37" > has <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif" width="9" >-semi-normal structure. We present some  examples showing the useability of our main conclusions. <a href="./files/site1/files/42/8Abstract.pdf">./files/site1/files/42/8Abstract.pdf</a> ﻧﮕﺎﺷﺖﻫﺎی ﺑﻪﻃﻮر ﻗﻮی C- ﻏﯿﺮاﻧﺒﺴﺎﻃﯽ ﻧﺴﺒﯽ, ﺑﻬﺘﺮیﻦ زوج نزدینی, ﻓﻀﺎی ﺑﻪﻃﻮر یک‌نواﺧﺖ ﻣﺤﺪب,T - ﺳﺎﺧﺘﺎر ﺷﺒﻪ ﻧﺮﻣﺎل یک‌نواﺧﺖ. Strongly relatively C-nonexpansive mapping, Best proximity pair, Uniformly convex space, T-Uniformly semi-normal structure. 229 240 http://mmr.khu.ac.ir/browse.php?a_code=A-10-143-1&slc_lang=fa&sid=1 Moosa Gabeleh موسی گابله gab.moo@gmail.com 10031947532846002315 10031947532846002315 Yes Ayatollah Boroujerdi University دانشگاه آیت‌ا...العظمی بروجردی، گروه ریاضی