Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems

The purpose of this paper is to investigate the problem of finding a common element of the set of fixed points F(S) of a nonexpansive mapping S and the set of solutions Ω _A of the variational inequality for a monotone, Lipschitz continuous mapping A. We introduce a hybrid extragradient-like approximation method which is based on the well-known extragradient method and a hybrid (or outer approximation) method. The method produces three sequences which are shown to converge strongly to the same common element of \({F(S)\cap\Omega_{A}}\). As applications, the method provides an algorithm for finding the common fixed point of a nonexpansive mapping and a pseudocontractive mapping, or a common zero of a monotone Lipschitz continuous mapping and a maximal monotone mapping.     

On the quantitative asymptotic behavior of strongly nonexpansive mappings in banach and geodesic spaces

We give explicit rates of asymptotic regularity for iterations of strongly nonexpansive mappings T in general Banach spaces as well as rates of metastability (in the sense of Tao) in the context of uniformly convex Banach spaces when T is odd. This, in particular, applies to linear norm-one projections as well as to sunny nonexpansive retractions. The asymptotic regularity results even hold for strongly quasi-nonexpansive mappings (in the sense of Bruck), the addition of error terms and very general metric settings. In particular, we get the first quantitative results on iterations (with errors) of compositions of metric projections in CAT(?)-spaces (? > 0). Under an additional compactness assumption we obtain, moreover, a rate of metastability for the strong convergence of such iterations.     

On proximinality of convex sets in superspaces

In this paper, we show that a closed convex subset C of a Banach space is strongly proximinal (proximinal, resp.) in every Banach space isometrically containing it if and only if C is locally (weakly, resp.) compact. As a consequence, it is proved that local compactness of C is also equivalent to that for every Banach space Y isometrically containing it, the metric projection from Y to C is nonempty set-valued and upper semi-continuous.     

A monotone path-connected set with outer radially lower continuous metric projection is a strict sun

A monotone path-connected set is known to be a sun in a finite-dimensional Banach space. We show that a B-sun (a set whose intersection with each closed ball is a sun or empty) is a sun. We prove that in this event a B-sun with ORL-continuous (outer radially lower continuous) metric projection is a strict sun. This partially converses one well-known result of Brosowski and Deutsch. We also show that a B-solar LG-set (a global minimizer) is a B-connected strict sun.     

Some refined bounds for the perturbation of the orthogonal projection and the generalized inverse

In this paper, we consider the perturbation of the orthogonal projection and the generalized inverse for an n × n matrix A and present some perturbation bounds for the orthogonal projections on the rang spaces of A and A^?, respectively. A combined bound for the orthogonal projection on the rang spaces of A and A^? is also given. The proposed bounds are sharper than the existing ones. From the combined bounds of the orthogonal projection on the rang spaces of A and A^?, we derived new perturbation bounds for the generalized inverse, which always improve the existing ones. The combined perturbation bound for the orthogonal projection and the generalized inverse is also given. Some numerical examples are given to show the advantage of the new bounds.     

An Extension of a Simply Inequality Between von Neumann–Jordan and James Constants in Banach Spaces

For a non-trivial Banach space X, let J(X), CNJ(X), C_(NJ)~(p)(X) respectively stand for the James constant, the von Neumann–Jordan constant and the generalized von Neumann–Jordan constant recently inroduced by Cui et al. In this paper, we discuss the relation between the James and the generalized von Neumann–Jordan constants, and establish an inequality between them: C_(NJ)~(p)(X) ≤J(X) with p ≥ 2, which covers the well-known inequality CNJ(X) ≤ J(X). We also introduce a new constant, from which we establish another inequality that extends a result of Alonso et al.     

Approximating fixed points of mappings satisfying condition (E) in Busemann space

It is well-known that in a Banach space, using the Ishikawa iterative process, one can find fixed points of nonexpansive mappings via asymptotic center’s method. In this paper, we obtain the fixed points of mappings satisfying so-called condition (E) in a uniformly convex Busemann space. Many known results in CAT (0) spaces are improved and extended by our results.     

Numerical ranges for pairs of operators,duality mappings with gauge function,and spectra of nonlinear operators

We define and study numerical ranges for pairs of nonlinear operators F and J which act between some Banach space X and its dual X*, with respect to some increasing gauge function φ. Connections with spectra for certain classes of nonlinear operators introduced recently in the literature are also established. As a sample example, we consider the case when F is the duality map of the Lebesgue space L ^p (Ω), J is the duality map of the corresponding Sobolev space W ₀ ^1,p (Ω), and φ(t)=t ^p?1 (1<p<∞). This leads to existence, uniqueness, and perturbation results for a homogeneous eigenvalue problem involving the p-Laplace operator.     

G-matrices,J-orthogonal matrices,and their sign patterns

A real matrix A is a G-matrix if A is nonsingular and there exist nonsingular diagonal matrices D₁ and D₂ such that A^?T = D₁AD₂, where A^?T denotes the transpose of the inverse of A. Denote by J = diag(±1) a diagonal (signature) matrix, each of whose diagonal entries is +1 or ?1. A nonsingular real matrix Q is called J-orthogonal if Q^TJQ = J. Many connections are established between these matrices. In particular, a matrix A is a G-matrix if and only if A is diagonally (with positive diagonals) equivalent to a column permutation of a J-orthogonal matrix. An investigation into the sign patterns of the J-orthogonal matrices is initiated. It is observed that the sign patterns of the G-matrices are exactly the column permutations of the sign patterns of the J-orthogonal matrices. Some interesting constructions of certain J-orthogonal matrices are exhibited. It is shown that every symmetric staircase sign pattern matrix allows a J-orthogonal matrix. Sign potentially J-orthogonal conditions are also considered. Some examples and open questions are provided.     

On the extension of Hölder maps with values in spaces of continuous functions

We study the isometric extension problem for Hölder maps from subsets of any Banach space intoc ₀ or into a space of continuous functions. For a Banach spaceX, we prove that anyα-Hölder map, with 0<α ≤1, from a subset ofX intoc ₀ can be isometrically extended toX if and only ifX is finite dimensional. For a finite dimensional normed spaceX and for a compact metric spaceK, we prove that the set ofα’s for which allα-Hölder maps from a subset ofX intoC(K) can be extended isometrically is either (0, 1] or (0, 1) and we give examples of both occurrences. We also prove that for any metric spaceX, the above described set ofα’s does not depend onK, but only on finiteness ofK.     

An iterative algorithm for solving split feasibility problems and fixed point problems in Banach spaces

The purpose of this paper is to study split feasibility problems and fixed point problems concerning left Bregman strongly relatively nonexpansive mappings in p-uniformly convex and uniformly smooth Banach spaces. We suggest an iterative scheme for the problem and prove strong convergence theorem of the sequences generated by our scheme under some appropriate conditions in real p-uniformly convex and uniformly smooth Banach spaces. Finally, we give numerical examples of our result to study its efficiency and implementation. Our result complements many recent and important results in this direction.     

Birational Darboux Coordinates on (Co)Adjoint Orbits of GL(N, ℂ)

The set of all linear transformations with a fixed Jordan structure J is a symplectic manifold isomorphic to the coadjoint orbit O(J) of the general linear group GL(N, C). Any linear transformation can be projected along its eigenspace onto a coordinate subspace of complementary dimension. The Jordan structure \(\tilde J\) of the image under the projection is determined by the Jordan structure J of the preimage; consequently, the projection \(O\left( J \right) \to O\left( {\tilde J} \right)\) is a mapping of symplectic manifolds.     

Perturbation Analysis of Moore–Penrose Quasi-linear Projection Generalized Inverse of Closed Linear Operators in Banach Spaces

In this paper, we investigate the perturbation problem for the Moore–Penrose bounded quasi-linear projection generalized inverses of a closed linear operaters in Banach space. By the method of the perturbation analysis of bounded quasi-linear operators, we obtain an explicit perturbation theorem and error estimates for the Moore–Penrose bounded quasi-linear generalized inverse of closed linear operator under the T-bounded perturbation, which not only extend some known results on the perturbation of the oblique projection generalized inverse of closed linear operators, but also extend some known results on the perturbation of the Moore–Penrose metric generalized inverse of bounded linear operators in Banach spaces.     

The mapping taking three points of a Banach space to their Steiner point

A mapping St taking any three points a, b, c of a Banach space X into a set St(a,b,c) of their Steiner points and the corresponding operator P_D of metric projection of a space X × X × X onto its diagonal subspace D = {(x,x,x): x ∈ X}, P_D(a,b,c) = {(s,s,s): s ∈ St(a,b,c)}, are considered. The linearity coefficient of an arbitrary selection from P_D is estimated depending on properties of the space X. Estimates for the Lipschitz constant of an arbitrary selection from the mapping St are obtained as a corollary.     

Finite-dimensional subspaces of <Emphasis Type="Italic">L</Emphasis><Subscript>p</Subscript> with Lipschitz metric projection

We prove that the metric projection onto a finite-dimensional subspace Y ? L _p, p ∈ (1, 2) ∪ (2, ∞), satisfies the Lipschitz condition if and only if every function in Y is supported on finitely many atoms. We estimate the Lipschitz constant of such a projection for the case in which the subspace is one-dimensional.     

A Midpoint Method for Generalized Equations Under Mild Differentiability Condition

The aim of this study is the approximation of a solution x ^? of the generalized equation 0∈f(x)+F(x) in Banach spaces, where f is a single function whose second order Fréchet derivative ?² f verifies an Hölder condition, and F stands for a set-valued map with closed graph. Using a fixed point theorem and proceeding by induction under the pseudo-Lipschitz property of F, we obtain a sequence defined by a midpoint formula whose convergence to x ^? is superquadratic. Taking a weaker condition, we present the result obtained when ?² f satisfies a center-Hölder conditioning.     

Detecting fixed points of nonexpansive maps by illuminating the unit ball

We give necessary and sufficient conditions for a nonexpansive map on a finite-dimensional normed space to have a nonempty, bounded set of fixed points. Among other results we show that if f: V → V is a nonexpansive map on a finite-dimensional normed space V, then the fixed point set of f is nonempty and bounded if and only if there exist w₁,..., w _m in V such that {f(w _i ) ? w _i : i = 1,..., m} illuminates the unit ball. This yields a numerical procedure for detecting fixed points of nonexpansive maps on finite-dimensional spaces. We also discuss applications of this procedure to certain nonlinear eigenvalue problems arising in game theory and mathematical biology.     

Parallel iterative methods for Bregman strongly nonexpansive operators in reflexive Banach spaces

The purpose of this paper is to introduce two parallel iterative methods which are based on the hybrid projection and shrinking projection methods to find a common fixed point of a finite family of Bregman strongly nonexpansive operators in a reflexive Banach space X. And we also give some applications of main results for some related problems.     

Approximate Solutions of Variational Inequalities on Sets of Common Fixed Points of a One-Parameter Semigroup of Nonexpansive Mappings

Let \(\mathcal{T}\) be a one-parameter semigroup of nonexpansive mappings on a nonempty closed convex subset C of a strictly convex and reflexive Banach space X. Suppose additionally that X has a uniformly Gâteaux differentiable norm, C has normal structure, and \(\mathcal{T}\) has a common fixed point. Then it is proved that, under appropriate conditions on nonexpansive semigroups and iterative parameters, the approximate solutions obtained by the implicit and explicit viscosity iterative processes converge strongly to the same common fixed point of \(\mathcal{T}\), which is a solution of a certain variational inequality.     

Generalized Limits and Statistical Convergence

Consider the Banach space m of real bounded sequences, x, with \({\Vert x\Vert =\sup_{k}|x_{k}|}\). A positive linear functional L on m is called an S-limit if \({L(\chi _{K})=0}\) for every characteristic sequence \({\chi _{K} }\) of sets, K, of natural density zero. We provide regular sublinear functionals that both generate as well as dominate S-limits. The paper also shows that the set of S-limits and the collection of Banach limits are distinct but their intersection is not empty. Furthermore, we show that the generalized limits generated by translative regular methods is equal to the set of Banach limits. Some applications are also provided.