Best approximation mappings in Hilbert spaces

Mathematical Programming - The notion of best approximation mapping (BAM) with respect to a closed affine subspace in finite-dimensional spaces was introduced by Behling, Bello Cruz and Santos to...     

Best proximity points of non-self mappings

Let A,B be nonempty subsets of a Banach space X and let T:A→B be a non-self mapping. Under appropriate conditions, we study the existence of solutions for the minimization problem min _x∈A ∥x?Tx∥.     

Best proximity point theorems for contractive mappings

This article is concerned with some new best proximity point theorems for principal cyclic contractive mappings, proximal cyclic contractive mappings, and proximal contractive mappings. As a consequence, an interesting fixed point theorem, due to Edelstein, for a contractive mapping is obtained from all those best proximity point theorems.     

Best proximity points for asymptotic cyclic contraction mappings

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.     

Best proximity point theorems for cyclic strongly quasi-contraction mappings

In this paper, we introduce a new class of maps, called cyclic strongly quasi-contractions, which contains the cyclic contractions as a subclass. Then we give some convergence and existence results of best proximity point theorems for cyclic strongly quasi-contraction maps. An example is given to support our main results.     

Best proximity point theorems for single- and set-valued non-self mappings

We study the existence of best proximity points for single-valued non-self mappings. Also, we prove a best proximity point theorem for set-valued non-self mappings in metric spaces with an appropriate geometric property. Examples are given to support the usability of our results.     

Best proximity points: global minimization of multivalued non-self 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.     

Best approximation theorems for nonexpansive and condensing mappings in hyperconvex spaces

In hyperconvex metric spaces we consider best approximation, invariant approximation and best proximity pair problems for multivalued mappings that are condensing or nonexpansive.     

Best simultaneous approximations, asymptotically nonexpansive mappings and variational inequalities in Banach spaces

The existence of common fixed points is established for three mappings where T is either generalized (f,g)-nonexpansive or asymptotically (f,g)-nonexpansive on a nonempty subset of a Banach space. As applications, the invariant best simultaneous approximation results are proved and the existence of solution of variational inequalities is obtained. Our results unify and substantially improve several recent results existing in the current literature.     

Best uniform approximation of a continuous compact-valued mapping by sets of continuous single-valued mappings

We prove existence theorems and establish necessary and sufficient conditions and criteria for an extremal element for the problem of the best uniform approximation of a continuous compact-valued mapping by sets of continuous single-valued mappings. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 12, pp. 1601–1618, December, 2005.     

Best uniform approximation in the metric space of continuous mappings with compact convex images

For the problem of the best uniform approximation of a continuous mapping with compact convex images by sets of other continuous mappings with compact convex images, we establish necessary and sufficient conditions and a criterion for an element to be extremal; the criterion obtained is a generalization of the classic Kolmogorov criterion for a polynomial of the best approximation.     

Harmonic mappings and quasiconformal mappings

Work done with support from NSF grant MCS 83-00248.     

Generating continuous mappings with Lipschitz mappings

If is a metric space, then and denote the semigroups of continuous and Lipschitz mappings, respectively, from to itself. The relative rank of modulo is the least cardinality of any set where generates . For a large class of separable metric spaces we prove that the relative rank of modulo is uncountable. When is the Baire space , this rank is . A large part of the paper emerged from discussions about the necessity of the assumptions imposed on the class of spaces from the aforementioned results.

     

Extremal polygonal quasiconformal mappings and biLipschitz mappings

We show that the extremal polygonal quasiconformal mappings are biLipschitz with respect to the hyperbolic metric in the unit disk.     

Contractive mappings, Kannan mappings and metric completeness

In this paper, we first study the relationship between weakly contractive mappings and weakly Kannan mappings. Further, we discuss characterizations of metric completeness which are connected with the existence of fixed points for mappings. Especially, we show that a metric space is complete if it has the fixed point property for Kannan mappings.