Mathematical Researches

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Introduction
Let  be a nonempty subset of a normed linear space . A self-mapping  is said to be nonexpansive provided that  for all . 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  of  with , there exists a point  for which . 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  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  be a nonempty pair of subsets of a normed linear space .  is said to be a noncyclic mapping if . Also the noncyclic mapping  is called relatively nonexpansive whenever  for any . Clearly, if , 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  is called a best proximity pair provided that

In the other words, the point  is a best proximity pair for  if  and  are two fixed points of  which estimates the distance between the sets  and .
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  is nonempty and . 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  for all  with . If

where  is a projection mapping defined on  then  has -semi-normal structure.
We present some  examples showing the useability of our main conclusions.
./files/site1/files/42/8Abstract.pdf

Keywords: Strongly relatively C-nonexpansive mapping, Best proximity pair, Uniformly convex space, T-Uniformly semi-normal structure.

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