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Consider a self-mapping T defined on a union of two subsets A and B of a Banach space such that T(A) ⊆ B and T(B) ⊆ A. In this work we survey the existence of an optimal approximate solution, known as a best proximity point for a class of cyclic mappings, called cyclic Kannan nonexpansive mappings, in Banach spaces under appropriate sufficient conditions. In this order, the notion of T-uniformly semi-normal structure is introduced and used to investigate the existence of best proximity points. As an application of the existence theorem, we conclude an old fixed point problem in Banach spaces which are not reflexive necessarily. Examples are given to support the usability of our main conclusions.Keywords: Best proximity point, fixed point, cyclic Kannan nonexpansive mapping, T-uniformly semi-normal structure, uniformly convex Banach space

In this article, we prove that every nonempty and convex pair of subsets of uniformly convex in every direction Banach spaces has the proximal normal structure and then we present a best proximity point theorem for cyclic relatively nonexpansive mappings in such spaces. We also study the structure of minimal sets of cyclic relatively nonexpansive mappings and obtain the existence results of best proximity points for cyclic mappings using some new geometric notions on minimal sets. Finally, we prove a best proximity point theorem for a new class of cyclic contraction-type mappings in the setting of uniformly convex Banach spaces and so, we improve the main conclusions of Eldred and Veeramani.

In this article, we formulate a best proximity pair theorem for noncyclic relatively nonexpansive mappings in convex metrc spaces by using a geometric notion of semi-normal structure. In this way, we generalize a corresponding result in [W. Takahashi, A convexity in metric space and nonexpansive mappings, Kodai Math. Sem. Rep. 22 (1970) 142-149]. We also establish a best proximity pair theorem for pointwise noncyclic contractions in the setting of convex metric spaces. Our result generalizes a result due to Sankara Raju Kosuru and Veeramani [G. Sankara Raju Kosuru and P. Veeramani, A note on existence and convergence of best proximity points for pointwise cyclic contractions, Numer. Funct. Anal. Optim., 82 (2011) 821-830].

We introduce the concept of cyclic Kannan orbital C-nonexpansive mappings and obtain the existence of a best proximity point on a pair of bounded, closed and convex subsets of a strictly convex metric space by using the geometric notion of seminormal structure. We also study the structure of minimal sets for cyclic Kannan C-nonexpansive mappings and show that results similar to the celebrated Goebel– Karlovitz lemma for nonexpansive self-mappings can be obtained for cyclic Kannan C-nonexpansive mappings.

In this article, we establish some new existence theorems for best proximity point and fixed point problems for certain mappings in Banach spaces. The main results of this article improve and extend the results presented by Wong [25]. Examples are given to support the usability of our main conclusions.

We prove a best proximity point theorem for proximal generalized contractive type mappings in metric spaces, which is a generalization of recent best proximity point theorems and some famous fixed point theorems due to Berinde and Suzuki. We also introduce a new class of proximal non-self mappings and obtain sufficient conditions, which ensure the existence of a best proximity point. Moreover, we define algorithms and prove that they find a best proximity point for these classes of non-self mappings in the setting of metric and Banach spaces.

Let us consider the mappings $${S : A \to A}$$ and $${T : A \to B}$$ , where A and B are two nonempty subsets of a metric space (X, d). The purpose of this paper is to provide sufficient conditions to ensure the existence and uniqueness of an element $${p \in A}$$ such that $${d(Sp, Tp) = {\rm dist}(A, B) := {\rm inf}\{d(x, y) : (x, y) \in A \times B\}}$$ . In particular, the existence and uniqueness of a best proximity point for non-self-mappings are obtained. Examples are given to show usability of our results.

In this article, we study the structure of minimal sets of relatively non-expansive mappings. We consider the cyclic and the noncyclic cases and show that results alike to the celebrated Goebel-Karlovitz lemma for non-expansive self-mappings can be obtained for relatively non-expansive mappings.

In this article, we give a best proximity point theorem for generalized contractions in metric spaces with appropriate geometric property. We also, give an example which ensures that our result cannot be obtained from a similar result due to Amini-Harandi (Best proximity points for proximal generalized contractions in metric spaces. Optim Lett, 2012). Moreover, we prove a best proximity point theorem for multivalued non-self mappings which generalizes the Mizoguchi and Takahashi’s fixed point theorem for multivalued mappings.

The aim of this paper is to prove some best proximity point theorems for new classes of cyclic mappings, called pointwise cyclic orbital contractions and asymptotic pointwise cyclic orbital contractions. We also prove a convergence theorem of best proximity point for relatively nonexpansive mappings in uniformly convex Banach spaces.

In this work, we consider two classes of non-self-mappings, which are called proximal weakly contractive and proximal nonexpansive mappings, and study the existence of solutions of a minimization problem. Existence results of best proximity points for these two classes of non-self-mappings in metric and Banach spaces are also obtained.

Let A, B be nonempty subsets of a metric space (X, d) and T : A → 2B be a multivalued non-self-mapping. The purpose of this paper is to establish some theorems on the existence of a point \({x^*\in A}\) , called best proximity point, which satisfies \({{\rm inf}\{d(x^*,y):y\in Tx^*\}=dist(A,B).}\) This will be done for contraction multivalued non-self-mappings in metric spaces, as well as for nonexpansive multivalued non-self-mappings in Banach spaces having appropriate geometric property.

In this paper, we prove the existence and convergence of best proximity points for asymptotic cyclic contractions in metric spaces with the property UC, as well as for asymptotic proximal pointwise contractions in uniformly convex Banach spaces. Moreover, we consider a generalized cyclic contraction mapping and prove the existence of best proximity points for such a mapping in Banach spaces which do not necessarily satisfy the geometric property UC.

Abkar and Gabeleh in (J. Optim. Theory. Appl. doi: 10.1007/s10957-011-9818-2) proved some theorems which ensure the existence and convergence of fixed points, as well as best proximity points for cyclic mappings in ordered metric spaces. In this paper we extend these results to generalized cyclic contractions and obtain some new results on the existence and convergence of fixed points for weakly contractive mappings, as well as on best proximity points for cyclic φ-contraction mappings in partially ordered metric spaces.

In this paper we consider a cyclic mapping on a partially ordered complete metric space. We prove some fixed point theorems, as well as some theorems on the existence and convergence of best proximity points.

[EN] In this article, we prove a fixed point theorem for cyclic relatively nonexpansive mappings in the setting of generalized semimetric spaces by using a geometric notion of seminormal structure and then we conclude some results in uniformly convex Banach spaces. We also discuss on the stability of seminormal structure in generalized semimetric spaces., This research was in part supported by a grant from IPM (No. 93470047).

LetAandBbe two nonempty subsets of a Banach spaceX. A mappingT:A∪B→A∪Bis said to be cyclic relatively nonexpansive ifT(A)⊆BandT(B)⊆AandTx-Ty≤x-yfor all (x,y)∈A×B. In this paper, we introduce a geometric notion of seminormal structure on a nonempty, bounded, closed, and convex pair of subsets of a Banach spaceX. It is shown that if (A,B) is a nonempty, weakly compact, and convex pair and (A,B) has seminormal structure, then a cyclic relatively nonexpansive mappingT:A∪B→A∪Bhas a fixed point. We also discuss stability of fixed points by using the geometric notion of seminormal structure. In the last section, we discuss sufficient conditions which ensure the existence of best proximity points for cyclic contractive type mappings.Erratum to “Seminormal Structure and Fixed Points of Cyclic Relatively Nonexpansive Mappings”

It has been remarked in [1] that the pair (A, A) has seminormal structure if and only if A has normal structure in the sense of Brodskĭi and Mil’man [2]. We revise this remark as follows. If the pair (A, A) has seminormal structure, then A has normal structure in the sense of Brodskĭi and Mil’man. Indeed, if the set A has normal structure, then (A, A) may not have seminormal structure. We illustrate this with the following example.

We first consider a cyclic -contraction map on a reflexive Banach space and provide a positive answer to a question raised by Al-Thagafi and Shahzad on the existence of best proximity points for cyclic -contraction maps in reflexive Banach spaces in one of their works (2009). In the second part of the paper, we will discuss the existence of best proximity points in the framework of more general metric spaces. We obtain some new results on the existence of best proximity points in hyperconvex metric spaces as well as in ultrametric spaces.

In this paper, we introduce a new class of multivalued contractions with respect to b-generalized pseudodistances and prove a best proximity point theorem for this class of non-self mappings. In this way, we improve and extend several existing results in the literature. Examples are given to support our main results. This work is a continuation of studies on the use of a new type of distances in the fixed point theory. The pioneering effort in direction of defining distance is inter alia paper of O. Kada, T. Suzuki and W. Takahashi.

In this article, we prove a best proximity point theorem for generalized cyclic contractions in convex metric spaces. Then we investigate the structure of minimal sets of cyclic relatively nonexpansive mappings in the setting of convex metric spaces. In this way, we obtain an extension of the Goebel-Karlovitz lemma, which is a key lemma in fixed point theory.