Set Of Best Approximations Research Articles

The fixed point theorems and invariant approximations for random nonexpansive mappings in random normed modules

By making full use of the theory of random sequential compactness in random normed modules, in this paper we establish a noncompact Dotson fixed point theorem: if C is a σ–stable random sequentially compact L0–star–shaped subset of a random normed module, then every random nonexpansive mapping T:C→C has a fixed point. Furthermore, we obtain an existence result for best approximations in random normed modules: let E be a random normed module, T:E→E a random nonexpansive mapping with a fixed point u and C a closed, σ–stable and T–invariant subset of E such that T(C)‾ is random sequentially compact, then the set of best approximations of u in C is nonempty, which generalizes the classical result of Smoluk. In addition, we also get an existence result for invariant approximations in random normed modules. A significant distinction between the proofs of our results in random normed modules and the corresponding classical results in normed spaces is that the σ–stability of both the sets and mappings involved in the random setting plays a prominent part in the proofs of the main results of this paper.

On Stability of the Metric Projection Operator

Let $M$ be a closed linear subspace of a normed linear space $X$. For a given $f\in X$ denote by $P_M f$ the set of best approximations to $f$ from $M$. The operator $P_M$ is termed the metric projection onto $M$. In this paper we are interested in the stability of the metric projection $P_M$ relative to perturbations of the subspace $M$. We mainly consider the case where $X=L^p$, $p\in [1,\infty]$. We consider a measure of distance $d(M,N)$ between subspaces $M$ and $N$ and estimate $\|P_Mf-P_Nf\|$ in terms of $d(M,N)$ and $\|f\|$. Typically such an estimate will be of order $d(M,N)^\beta$ with some $\beta$ which, in general, depends on the geometry of the space $X$.

Common fixed points for generalized asymptotically nonexpansive mappings

A common fixed point theorem for noncommuting generalized asymptotically nonexpansive mappings has been obtained in convex metric spaces. As an application, a result on the set of best approximation is also derived for such class of mappings. The proved results unify and extend some of the known results on the subject.

Applications of Caristi's fixed point results

In the setup of locally convex spaces, applying Caristi's results we prove some fixed point results for non-self multivalued maps and common fixed point theorems for Caristi type maps utilizing two different techniques. We apply our results to obtain some common fixed points for a Banach operator pair from the set of best approximations. Consequently, we either improve or extend a number of known results in the fixed point theory.

Common Fixed Points, Invariant Approximation and Generalized Weak Contractions

Sufficient conditions for the existence of a common fixed point of generalized -weakly contractive noncommuting mappings are derived. As applications, some results on the set of best approximation for this class of mappings are obtained. The proved results generalize and extend various known results in the literature.

Some common fixed point theorems for generalized nonlinear contractive mappings

Sufficient conditions for the existence of a common fixed point for generalized nonlinear contractive mappings are derived. As applications, some results on the set of best approximation for this class of mappings are also obtained. The proved results generalize and extend various known results in the literature.

Invariant Points and ε‐Simultaneous Approximation

We generalize and extend Brosowski‐Meinardus type results on invariant points from the set of best approximation to the set of ε‐simultaneous approximation. As a consequence some results on ε‐approximation and best approximation are also deduced. The results proved in this paper generalize and extend some of the known results on the subject.

Some new results on approximation in fuzzy 2-normed spaces

In this paper, we define a fuzzy 2-normed space and study the concept of best approximation in fuzzy 2-normed spaces (FTNS). We also define a set of best approximations, proximal set and approximatively compact set and prove some interesting results in this new setup.

Common fixed points for Banach operator pairs from the set of best approximations

The existence of common fixed point results for a Banach operator pair under certain generalized $\varphi$-contractions is established. As applications, the corresponding invariant best approximation results are proved. Our results unify and generalize various known results to the more general class of noncommuting mappings.

P-Best Approximation on Probabilistic Normed Spaces

We studied the best approximation between two sets in probabilistic normed spaces. We defined the best approximation on these spaces and generalized some definitions such as set of best approximation, p-proximinal set and p-approximately compact relative to any set and proved some theorems about them.

Best approximation on probabilistic normed spaces

The main purpose of this paper is to study the best approximation in probabilistic normed spaces. We define the best approximation in these spaces and generalize some definitions such as set of best approximation, proximinal set and approximatively compact set. Then prove some theorems about them.

Coincidences of Lipschitz-Type Hybrid Maps and Invariant Approximation

The aim of this paper is to obtain new coincidence and common fixed point theorems by using Lipschitz-type conditions of hybrid maps (not necessarily continuous) on a metric space. As applications, we demonstrate the existence of common fixed points from the set of best approximations. Our work sets analogues, unifies and improves various known results existing in the literature.

Stabilité des médianes conditionnelles par rapport à des filtrations discrétisées

Let Y=( Y s ) s⩽1 be a stochastic process and X be a random vector. For the L p -minimizing criterion, the set of best approximations of X we can get when the path of Y is known until time t is the set of conditionnal L p -medians of X given ( Y s ) s⩽ t . Assume that the path of Y is only observed in a finite set of times. In some cases, we show that the multiplication of the observation times allow us to rediscover asymptotically the best approximation of X we can get. To cite this article: B. Cadre, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 545–548.

On Quasi-Chebyshev Subspaces of Banach Spaces

A quasi-Chebyshev subspace of a Banach space X has been defined as one in which the set of best approximants for every x in X is non-empty and compact. This generalizes the well known concept of pseudo-Chebyshev property. In this paper we shall give various characterizations of quasi-Chebyshev subspaces in Banach spaces. Moreover, we present a characterization of the spaces in which all closed linear subspaces are quasi-Chebyshev.

Geometry and Topology of Continuous Best and Near Best Approximations

The existence of a continuous best approximation or of near best approximations of a strictly convex space by a subset is shown to imply uniqueness of the best approximation under various assumptions on the approximating subset. For more general spaces, when continuous best or near best approximations exist, the set of best approximants to any given element is shown to satisfy connectivity and radius constraints.

ON BEST APPROXIMATIONS IN THE LEBESGUE-BOCHNER SPACE $L^1(X)$

If $\Sigma_0\subseteq \Sigma$ is a sub-$\sigma$-field and $X$ a reflexive Banach space, we show that the Lebesgue-Bochner space $L^1(\Sigma_0, X )$ is proximinal in $L^1(\Sigma, X)$. Then we examine how the set of best approximations and the distance function depend on $\Sigma_0$.

Infinite-dimensional convex programming with applications to constrained approximation

In this paper, existence and characterization of solutions and duality aspects of infinite-dimensional convex programming problems are examined. Applications of the results to constrained approximation problems are considered. Various duality properties for constrained interpolation problems over convex sets are established under general regularity conditions. The regularity conditions are shown to hold for many constrained interpolation problems. Characterizations of local proximinal sets and the set of best approximations are also given in normed linear spaces.

Absolutely proximinal subspaces of Banach spaces

Let us say that a subspace M of a Banach space X is absolutely proximinal if it is proximinal and, for each x ϵ X, ∥ x∥ can be expressed as a function of d( x, M), the distance from x to M, and d(0, P M ( x)), the distance from the origin to the best approximant set. Then this functional dependence must be given by a suitable norm on R 2. This defines a naturally occurring class of subspaces which includes all L p -summands, all M-ideals, all subspaces with the 1 1 2 -ball property, and all absolute subspaces. This paper initiates the study of this class of subspaces. amongst other things, we show that: • • The set-valued metric projection onto an absolutely proximinal subspace is Lipschitz continuous in the Hausdorff metric; • • Absolutely proximinal subspaces are, modulo renorming, the same as subspaces with the 1 1 2 -ball property; • • A subspace is absolutely proximinal if and only if its polar is absolutely proximinal in the dual space. We also obtain some numerical estimates for the inner radius of a set of best approximants.

Best approximation over the whole complex plane

Best rational approximation over the whole complex plane is investigated. While existence is elementary, there is not always uniqueness—every constant may be the best constant approximation to f( z) = z. However, under certain circumstances, the set of best approximations is, in a sense, bounded. When f has singularities of planar Lebesgue measure zero, the error corresponding to best approximation converges to zero, and the best approximations converge in measure.

Hermite—Birkhoff interpolation and approximation of a function and its derivatives

In this paper we look at some fundamental results on Hermite and Hermite-Birkhoff interpolation by elements of Haar spaces and apply these to certain problems in the approximation of a function and its derivatives. Important interpolation results have been given by Ikebe [ 61 and Haussminn 14, 5 1. We shall use the main theorem of [ 5 ] to generalize the main theorem of (6 1. As an application we obtain information about the dimension of the set of best approximations of a function and its derivatives. An example is presented.