Effective rates of convergence for Lipschitzian pseudocontractive mappings in general Banach spaces

This paper gives an explicit and effective rate of convergence for an asymptotic regularity result ‖Tx_n−x_n‖→0 due to Chidume and Zegeye in 2004 [14] where (x_n) is a certain perturbed Krasnoselski-Mann iteration schema for Lipschitz pseudocontractive self-mappings T of closed and convex subsets of a real Banach space. We also give a qualitative strengthening of the theorem by Chidume and Zegeye, by weakening the assumption of the existence of a fixed point. For the bounded case, our bound is polynomial in the data involved.     

A quadratic rate of asymptotic regularity for CAT(0)-spaces

In this paper we obtain a quadratic bound on the rate of asymptotic regularity for the Krasnoselski-Mann iterations of nonexpansive mappings in CAT(0)-spaces, whereas previous results guarantee only exponential bounds. The method we use is to extend to the more general setting of uniformly convex hyperbolic spaces a quantitative version of a strengthening of Groetsch's theorem obtained by Kohlenbach using methods from mathematical logic (so-called “proof mining”).     

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.     

Noncompact simplexes in Banach spaces with the Radon-Nikodým property

It is well known that a compact convex subset C of a locally convex topological vector space is a simplex if and only if each point x of C admits a unique probability measure on the extreme points of C with barycenter x. An exact analog of this result is proved for a closed and bounded separable convex subset of a Banach space with the Radon-Nikodým Property, and a weaker analog is proved in the nonseparable case.     

Fixed point iteration processes for asymptotic pointwise nonexpansive mappings in Banach spaces

Let X be a uniformly convex Banach space with the Opial property. Let T:C→C be an asymptotic pointwise nonexpansive mapping, where C is bounded, closed and convex subset of X. In this paper, we prove that the generalized Mann and Ishikawa processes converge weakly to a fixed point of T. In addition, we prove that for compact asymptotic pointwise nonexpansive mappings acting in uniformly convex Banach spaces, both processes converge strongly to a fixed point.     

Approximation of the limit distance function in Banach spaces

In this paper we study the behavior of the limit distance function d(x)=limdist(x,Cn) defined by a nested sequence (Cn) of subsets of a real Banach space X. We first present some new criteria for the non-emptiness of the intersection of a nested sequence of sets and of their ε-neighborhoods from which we derive, among other results, Dilworth's characterization [S.J. Dilworth, Intersections of centred sets in normed spaces, Far East J. Math. Sci. (Part II) (1988) 129-136 (special volume)] of Banach spaces not containing c₀ and Marino's result [G. Marino, A remark on intersection of convex sets, J. Math. Anal. Appl. 284 (2003) 775-778]. Passing then to the approximation of the limit distance function, we show three types of results: (i) that the limit distance function defined by a nested sequence of non-empty bounded closed convex sets coincides with the distance function to the intersection of the weak^∗-closures in the bidual; this extends and improves the results in [J.M.F. Castillo, P.L. Papini, Distance types in Banach spaces, Set-Valued Anal. 7 (1999) 101-115]; (ii) that the convexity condition is necessary; and (iii) that in spaces with separable dual, the distance function to a weak^∗-compact convex set is approximable by a (non-necessarily nested) sequence of bounded closed convex sets of the space.     

Generalized Dini theorems for nets of functions on arbitrary sets

We characterize the uniform convergence of pointwise monotonic nets of bounded real functions defined on arbitrary sets, without any particular structure. The resulting condition trivially holds for the classical Dini theorem. Our vector-valued Dini-type theorem characterizes the uniform convergence of pointwise monotonic nets of functions with relatively compact range in Hausdorff topological ordered vector spaces. As a consequence, for such nets of continuous functions on a compact space, we get the equivalence between the pointwise and the uniform convergence. When the codomain is locally convex, we also get the equivalence between the uniform convergence and the weak-pointwise convergence; this also merges the Dini-Weston theorem on the convergence of monotonic nets from Hausdorff locally convex ordered spaces. Most of our results are free of any structural requirements on the common domain and put compactness in the right place: the range of the functions.     

Endpoints of set-valued dynamical systems of asymptotic contractions of Meir–Keeler type and strict contractions in uniform spaces

In this paper, we introduce the concepts of the set-valued dynamical systems of asymptotic contractions of Meir–Keeler type and set-valued dynamical systems of strict contractions in uniform spaces and we present a method which is useful for establishing conditions guaranteeing the existence and uniqueness of endpoints of these contractions and the convergence to these endpoints of all generalized sequences of iterations of these contractions. The result, concerning the investigations of problems of the set-valued asymptotic fixed point theory, include some well-known results of Meir and Keeler, Kirk and Suzuki concerning the asymptotic fixed point theory of single-valued maps in metric spaces. The result, concerning set-valued strict contractions (in which the contractive coefficient is not constant), is different from the result of Yuan concerning the existence of endpoints of Tarafdar–Vyborny generalized contractions (in which the contractive coefficient is constant) in bounded metric spaces and provides some examples of Tarafdar–Yuan topological contractions in compact uniform spaces. Definitions and results presented here are new for set-valued dynamical systems in uniform, locally convex and metric spaces and even for single-valued maps. Examples show a fundamental difference between our results and the well-known ones.     

Uniformly normal structure and uniformly Lipschitzian semigroups

Assume that X is a real Banach space with uniformly normal structure and C is a nonempty closed convex subset of X. We show that a κ-uniformly Lipschitzian semigroup of nonlinear self-mappings of C admits a common fixed point if the semigroup has a bounded orbit and if κ is appropriately greater than one. This result applies, in particular, to the framework of uniformly convex Banach spaces.     

A fixed point result in strictly convex Banach spaces

We define an alternate convexically nonexpansive map T on a bounded, closed, convex subset C of a Banach space X and prove that if X is a strictly convex Banach space and C is a nonempty weakly compact convex subset of X, then every alternate convexically nonexpansive map T : C → C has a fixed point. As its application, we give an existence result for the solution of an integral equation.     

Smooth approximation of convex functions in Banach spaces

This paper shows that every extended-real-valued lower semi-continuous proper (respectively Lipschitzian) convex function defined on an Asplund space can be represented as the point-wise limit (respectively uniform limit on every bounded set) of a sequence of Lipschitzian convex functions which are locally affine (hence, C^∞) at all points of a dense open subset; and shows an analogous for w^∗-lower semi-continuous proper (respectively Lipschitzian) convex functions defined on dual spaces whose pre-duals have the Radon-Nikodym property.     

Moscow spaces and selection theory

A well-known result on Moscow spaces states that every Gδ-dense subset of a Moscow space X is C-embedded in X. We present here the selection version of this result and also (by means of two different approaches) we use selection theory to characterize the open bounded subsets of a uniform space (X,U) in the case when its completion is a Moscow space.     

A QUANTITATIVE VERSION OF A THEOREM DUE TO BORWEIN-REICH-SHAFRIR

We give a quantitative analysis of a result due to Borwein, Reich and Shafrir on the asymptotic behaviour of the general Krasnoselski-Mann iteration for nonexpansive self-mappings of convex sets in arbitrary normed spaces. Besides providing explicit bounds we also get new qualitative results concerning the independence of the rate of asymptotic regularity of that iteration from various input data. In the special case of bounded convex sets, where by well-known results of Ishikawa, Edelstein/O'Brien and Goebel/Kirk the norm of the iteration converges to zero, we obtain uniform bounds which do not depend on the starting point of the iteration and the nonexpansive function, but only depend on the error ε, an upper bound on the diameter of C and some very general information on the sequence of scalars λ _k used in the iteration. Only in the special situation, where λ _k := λ is constant, uniform bounds were known in that bounded case. For the unbounded case, no quantitative information was known before. Our results were obtained in a case study of analysing non-effective proofs in analysis by certain logical methods. General logical meta-theorems of the author guarantee (at least under some additional restrictions) the extractability of such bounds from proofs of a certain kind and provide an algorithm to extract them. Our results in the present paper (which we present here without any reference to that logical background) were found by applying that method to the original proof of the Borwein/Reich/Shafrir theorem. The general logical method which led to these results will be discussed (with further examples) in [22] Kohlenbach, U. On the computational content of the Krasnoselski and Ishikawa fixed point theorems. Proceedings of the Fourth Workshop on Computability and Complexity in Analysis. San Sebastian. Edited by: Blanck, J., Brattka, V. and Hertling, P. Springer LNCS 2064.  [Google Scholar].

     

The eigenvalue distribution of products of Toeplitz matrices - Clustering and attraction

We use a recent result concerning the eigenvalues of a generic (non-Hermitian) complex perturbation of a bounded Hermitian sequence of matrices to prove that the asymptotic spectrum of the product of Toeplitz sequences, whose symbols have a real-valued essentially bounded product h, is described by the function h in the “Szegö way”. Then, using Mergelyan’s theorem, we extend the result to the more general case where h belongs to the Tilli class. The same technique gives us the analogous result for sequences belonging to the algebra generated by Toeplitz sequences, if the symbols associated with the sequences are bounded and the global symbol h belongs to the Tilli class. A generalization to the case of multilevel matrix-valued symbols and a study of the case of Laurent polynomials not necessarily belonging to the Tilli class are also given.     

Effective results on compositions of nonexpansive mappings

This paper provides uniform bounds on the asymptotic regularity for iterations associated to a finite family of nonexpansive mappings. We obtain our quantitative results in the setting of (r,δ)$(r,\delta )$-convex spaces, a class of geodesic spaces which generalizes metric spaces with a convex geodesic bicombing.     

Long-time asymptotic solutions of convex Hamilton-Jacobi equations with Neumann type boundary conditions

We study the long-time asymptotic behavior of solutions u of the Hamilton-Jacobi equation u _t (x, t)?+?H(x, Du(x, t))?=?0 in ?? × (0, ??), where ?? is a bounded open subset of ${\mathbb{R}^n}$ , with Hamiltonian H?=?H(x, p) being convex and coercive in p, and establish the uniform convergence of u to an asymptotic solution as t ?? ??.     

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.     

Convergence theorems for λ-strict pseudo-contractions in q-uniformly smooth Banach spaces

In this paper, we continue to discuss the properties of iterates generated by a strict pseudo- contraction or a finite family of strict pseudo-contractions in a real q-uniformly smooth Banach space. The results presented in this paper are interesting extensions and improvements upon those known ones of Marino and Xu [Marino, G., Xu, H. K.： Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl., 324, 336-349 （2007）]. In order to get a strong convergence theorem, we modify the normal Mann＇s iterative algorithm by using a suitable convex combination of a fixed vector and a sequence in C. This result extends a recent result of Kim and Xu [Kim, T. H., Xu, H. K.： Strong convergence of modified Mann iterations. Nonl. Anal., 61, 51- 60 （2005）] both from nonexpansive mappings to λ-strict pseudo-contractions and from Hilbert spaces to q-uniformly smooth Banach spaces.     

Boundary value problems for the Laplacian in convex and semiconvex domains

We study the fully inhomogeneous Dirichlet problem for the Laplacian in bounded convex domains in Rn, when the size/smoothness of both the data and the solution are measured on scales of Besov and Triebel-Lizorkin spaces. As a preamble, we deal with the Dirichlet and Regularity problems for harmonic functions in convex domains, with optimal nontangential maximal function estimates. As a corollary, sharp estimates for the Green potential are obtained in a variety of contexts, including local Hardy spaces. A substantial part of this analysis applies to bounded semiconvex domains (i.e., Lipschitz domains satisfying a uniform exterior ball condition).