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This paper deals with variational inclusions of the form 0∈K−f(x) where f:Rn→Rm is a semismooth function and K is a nonempty closed convex cone in Rm. We show that the previous problem can be solved by a Newton-type method using the Clarke generalized Jacobian of f. The results obtained in this paper extend those obtained by Robinson in the famous paper (Robinson in Numer. Math. 19:341-347, 1972). We provide a semilocal method with a superlinear convergence that is new in the context of semismooth functions. Finally, numerical results are also given to illustrate the convergence. [ABSTRACT FROM AUTHOR]

Abstract??We study convergence and rate of convergence of expansions of elements in a Banach spaceXinto series with regard to a given dictionary$$f\sim\sum_{j=1}^{\infty}c_{j}(f)g_{j}(f),\quad g_{j}(f)\in\mathcal{d},\ c_{j}(f)>0,\ j=1,2,\dots.$$ In building such a representation we should construct two sequences: {gj(f)}j=1?and {cj(f)}j=1?. In this paper the construction of {gj(f)}j=1?will be based on ideas used in greedy-type nonlinear approximation. This explains the use of the termgreedy expansion. We use a norming functionalof a residualfm?1obtained afterm?1 steps of an expansion procedure to select themth elementfrom the dictionary. This approach has been used in previous papers on greedy approximation. The greedy expansions in Hilbert spaces are well studied. The corresponding convergence theorems and estimates for the rate of convergence are known. Much less is known about greedy expansions in Banach spaces. The first substantial result on greedy expansions in Banach spaces has been obtained recently by Ganichev and Kalton. They proved a convergence result for theLp, 1m(f) that provides convergence of the corresponding greedy expansions in any uniformly smooth Banach space. Moreover, we obtain estimates for the rate of convergence of such greedy expansions for? the closure (inX) of the convex hull of. [ABSTRACT FROM AUTHOR]

This is an expository paper about the fundamental mathematical notion of starshapedness, emphasizing the geometric, analytical, combinatorial, and topological properties of starshaped sets and their broad applicability in many mathematical fields. The authors decided to approach the topic in a very broad way since they are not aware of any related survey-like publications dealing with this natural notion. The concept of starshapedness is very close to that of convexity, and it is needed in fields like classical convexity, convex analysis, functional analysis, discrete, combinatorial and computational geometry, differential geometry, approximation theory, PDE, and optimization; it is strongly related to notions like radial functions, section functions, visibility, (support) cones, kernels, duality, and many others. We present in a detailed way many definitions of and theorems on the basic properties of starshaped sets, followed by survey-like discussions of related results. At the end of the article, we additionally survey a broad spectrum of applications in some of the above mentioned disciplines. [ABSTRACT FROM AUTHOR]

This paper studies the problem of approximating a function f in a Banach space $$\mathcal{X}$$ from measurements $$l_j(f)$$ , $$j=1,\ldots ,m$$ , where the $$l_j$$ are linear functionals from $$\mathcal{X}^*$$ . Quantitative results for such recovery problems require additional information about the sought after function f. These additional assumptions take the form of assuming that f is in a certain model class $$K\subset \mathcal{X}$$ . Since there are generally infinitely many functions in K which share these same measurements, the best approximation is the center of the smallest ball B, called the Chebyshev ball, which contains the set $$\bar{K}$$ of all f in K with these measurements. Therefore, the problem is reduced to analytically or numerically approximating this Chebyshev ball. Most results study this problem for classical Banach spaces $$\mathcal{X}$$ such as the $$L_p$$ spaces, $$1\le p\le \infty $$ , and for K the unit ball of a smoothness space in $$\mathcal{X}$$ . Our interest in this paper is in the model classes $$K=\mathcal{K}(\varepsilon ,V)$$ , with $$\varepsilon >0$$ and V a finite dimensional subspace of $$\mathcal{X}$$ , which consists of all $$f\in \mathcal{X}$$ such that $$\mathrm{dist}(f,V)_\mathcal{X}\le \varepsilon $$ . These model classes, called approximation sets, arise naturally in application domains such as parametric partial differential equations, uncertainty quantification, and signal processing. A general theory for the recovery of approximation sets in a Banach space is given. This theory includes tight a priori bounds on optimal performance and algorithms for finding near optimal approximations. It builds on the initial analysis given in Maday et al. (Int J Numer Method Eng 102:933-965, 2015) for the case when $$\mathcal{X}$$ is a Hilbert space, and further studied in Binev et al. (SIAM UQ, 2015). It is shown how the recovery problem for approximation sets is connected with well-studied concepts in Banach space theory such as liftings and the angle between spaces. Examples are given that show how this theory can be used to recover several recent results on sampling and data assimilation. [ABSTRACT FROM AUTHOR]

We study greedy algorithms in a Banach space from the point of view of convergence and rate of convergence. There are two well-studied approximation methods: the Weak Chebyshev Greedy Algorithm (WCGA) and the Weak Relaxed Greedy Algorithm (WRGA). The WRGA is simpler than the WCGA in the sense of computational complexity. However, the WRGA has limited applicability. It converges only for elements of the closure of the convex hull of a dictionary. In this paper we study algorithms that combine good features of both algorithms, the WRGA and the WCGA. In the construction of such algorithms we use different forms of relaxation. First results on such algorithms have been obtained in a Hilbert space by A. Barron, A. Cohen, W. Dahmen, and R. DeVore. Their paper was a motivation for the research reported here. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce two implicit iteration schemes for approximating fixed points of nonexpansive mapping T and a finite family of nonexpansive mappings {Ti}i=1N, respectively, in Banach spaces and to prove weak and strong convergence theorems. The results presented in this paper improve and extend the corresponding ones of H.-K. Xu and R. Ori, 2001, Z. Opial, 1967, and others. [ABSTRACT FROM AUTHOR]

We investigate Banach spaces possessing (or not possessing) the approximation properties APs, 0 < s ≤ 1, in connection with the following known question in the geometric theory of operators: under which conditions on Banach spaces X and Y and on positive numbers r and p does the p-nuclearity of the second adjoint of a continuous operator T from X to Y imply the p-nuclearity of T? Actually, we give necessary and sufficient conditions under which this question is answered affirmatively. In addition, the corresponding counterexamples are obtained in the maximally strong form. For instance, it is shown (and this statement is a significant strengthening of the previous results of that sort) that there exists a pair of separable Banach spaces Z and W such that the spaces Z** and W have Schauder bases, while for every p, 1 ≤ p < 2, there is a non-p-nuclear operator from W to Z with a p-nuclear second adjoint. Earlier, in similar examples, the corresponding spaces did not possess even the Grothendieck approximation property. The technique developed in this paper does not allow us to treat the case p > 2. That case will be studied in a forthcoming paper of the author. Bibliography: 11 titles. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the existence and approximation problem of solutions for a class of variational inclusions with accretive mappings in Banach spaces. The results extend and improve some recent results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce iterative algorithms for finding a zero of set-valued accretive operators by the viscosity approximation method based on Meir-Keeler-type contractions in a reflexive Banach space which admits a weakly continuous duality mapping. We obtain some strong convergence theorems under suitable conditions. As applications, we apply our results for finding common fixed point of nonexpansive semigroups and for solving equilibrium problem, optimization problem, and variational inequalities. [ABSTRACT FROM AUTHOR]

Earlier the second author showed that, in the periodic case, the rate of best uniform rational approximations of a function is well described in terms of the rates of best uniform piecewise polynomial approximations of the function itself and its conjugate. In the present paper, a similar result is obtained for a closed interval. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of a Mayer-type optimal control problem for semilinear unbounded evolution inclusions in reflexive and separable Banach spaces subject to endpoint constraints described by finitely many Lipschitzian equalities and inequalities. First we construct a sequence of discrete approximations to the optimal control problem for evolution inclusions and prove that optimal solutions to discrete approximation problems uniformly converge to a given optimal solution for the original continuous-time problem. Then, based on advanced tools of variational analysis and generalized differentiation, we derive necessary optimality conditions for discrete-time problems under fairly general assumptions. Combining these results with recent achievements of variational analysis in infinite-dimensional spaces, we establish new necessary optimality conditions for continuous-time evolution inclusions by passing to the limit from discrete approximations. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of random $C^{*}$-ternary algebras and consider some properties of them. As an application we approximate a random $C^{*}$-ternary algebra homomorphism in these spaces. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new viscosity approximation method by using the shrinking projection algorithm to approximate a common fixed point of a countable family of nonlinear mappings in a Banach space. Under quite mild assumptions, we establish the strong convergence of the sequence generated by the proposed algorithm and provide an affirmative answer to an open problem posed by Maingé (Comput. Math. Appl. 59:74-79, 2010) for quasi-nonexpansive mappings. In contrast with related processes, our method does not require any demiclosedness principle condition imposed on the involved operators belonging to the wide class of quasi-nonexpansive operators. As an application, we also introduce an iterative algorithm for finding a common element of the set of common fixed points of an infinite family of quasi-nonexpansive mappings and the set of solutions of a mixed equilibrium problem in a real Banach space. We prove a strong convergence theorem by using the proposed algorithm under some suitable conditions. Our results improve and generalize many known results in the current literature. [ABSTRACT FROM AUTHOR]

This paper gives the concepts of finite dimensional irreducible operators ((FDI) operators) and infinite dimensional irreducible operators ((IDI) operators). Discusses the relationships of (FDI) operators, (IDI) operators and strongly irreducible operators ((SI) operators) and illustrates some properties of the three classes of operators. Some sufficient conditions for the finite-dimensional irreducibility of operators which have the forms of upper triangular operator matrices are given. This paper proves that every operator with a singleton spectrum is a small compact perturbation of an (FDI) operator on separable Banach spaces and shows that every bounded linear operator T can be approximated by operators in (ΣFDI)( X) with respect to the strong-operator topology and every compact operator K can be approximated by operators in (ΣFDI)( X) with respect to the norm topology on a Banach space X with a Schauder basis, where (ΣFDI)( X):= { T ∈ B( X): T = Σ ⊕ T, T ∈ (FDI), k ∈ ℕ}. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce a new iteration by the combination of the viscosity approximation with Meir-Keeler contractions and proximal point algorithm for finding common zeros of a finite family of accretive operators in a Banach space with a uniformly Gâteaux differentiable norm. The results of this paper improve and extend corresponding well-known results by many others. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new viscosity approximation method by using the shrinking projection algorithm to approximate a common fixed point of a countable family of nonlinear mappings in a Banach space. Under quite mild assumptions, we establish the strong convergence of the sequence generated by the proposed algorithm and provide an affirmative answer to an open problem posed by Maingé (Comput. Math. Appl. 59:74-79, 2010) for quasi-nonexpansive mappings. In contrast with related processes, our method does not require any demiclosedness principle condition imposed on the involved operators belonging to the wide class of quasi-nonexpansive operators. As an application, we also introduce an iterative algorithm for finding a common element of the set of common fixed points of an infinite family of quasi-nonexpansive mappings and the set of solutions of a mixed equilibrium problem in a real Banach space. We prove a strong convergence theorem by using the proposed algorithm under some suitable conditions. Our results improve and generalize many known results in the current literature. [ABSTRACT FROM AUTHOR]

In this paper, we consider impulsive integro-differential equations in Banach space and we establish the bound on the difference between two approximate solutions. We also discuss nearness and convergence of solutions of the problem under consideration. The impulsive integral inequality of Grownwall type is used to obtain results. [ABSTRACT FROM AUTHOR]

In this paper, an important inequality for $$T$$ , a contraction integral operator, is obtained. From a practical programming point of view, this inequality allows us to express our iterative algorithm with a 'for loop' rather than a 'while loop'. The main tool used in our research is the fixed point theorem in the Banach space of continuous functions, $$X:=C([a , b],\mathbb{R }^k)$$ . [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study the functional equation arising in dynamic programming of multistage decision processes f(x)=opty∈Dopt{p(x, y), q(x, y)f(a(x, y)), r(x, y)f(b(x, y)), s(x, y)f(c(x, y))} for all x ∈ S. A few iterative algorithms for solving the functional equation are suggested. The existence, uniqueness and iterative approximations of solutions for the functional equation are discussed in the Banach spaces BC(S) and B(S) and the complete metric space BB(S), respectively. The properties of solutions, nonnegative solutions, and nonpositive solutions and the convergence of iterative algorithms for the functional equation and other functional equations, which are special cases of the above functional equation, are investigated in the complete metric space BB(S), respectively. Eight nontrivial examples which dwell upon the importance of the results in this paper are also given. [ABSTRACT FROM AUTHOR]

This paper concerns common fixed points for a finite family of hemicontractions or a finite family of strict pseudocontractions on uniformly convex Banach spaces. By introducing a new iteration process with error term, we obtain sufficient and necessary conditions, as well as sufficient conditions, for the existence of a fixed point. As one will see, we derive these strong convergence theorems in uniformly convex Banach spaces and without any requirement of the compactness on the domain of the mapping. The results given in this paper extend some previous theorems. [ABSTRACT FROM AUTHOR]

In this paper, we consider the orders of approximation of functions on the whole real axis by operators of Fejér type in the Banach space with the so-called generalized Hölder metric. [ABSTRACT FROM AUTHOR]

In this paper, which is the sequel to [16], we study inverse estimates of the Bernstein type for nonlinear approximation with structured redundant dictionaries in a Banach space. The main results are for blockwise incoherent dictionaries in Hilbert spaces, which generalize the notion of joint block-diagonal mutually incoherent bases introduced by Donoho and Huo. The Bernstein inequality obtained for such dictionaries is proved to be sharp, but it has an exponent that does not match that of the corresponding Jackson inequality. [ABSTRACT FROM AUTHOR]

Let X be a real Banach space and I a nonempty interval. Let $K:I\rightsquigarrow X$ be a multi-function with the graph $\mathcal {K} $ . We give here a characterization for $\mathcal {K} $ to be approximate/near weakly invariant with respect to the differential inclusion $x^{\prime }(t)\in F(t, x(t))$ by means of an appropriate tangency concept and Lipschitz conditions on F. The tangency concept introduced in this paper extends in a natural way the quasi-tangency concept introduced by Cârjă et al. (Trans Amer Math Soc. [2009];361:343-90) (see also Cârjă et al. ([2007])). Viability, invariance and applications. Amsterdam: Elsevier Science B V) in the case when F is independent of t. As an application, we give some results concerning the set of solutions for the differential inclusion $x^{\prime }(t)\in F(t,x(t))$ . [ABSTRACT FROM AUTHOR]

In this paper, we study an iterative scheme for two different types of resolvents of a monotone operator defined on a Banach space. These resolvents are generalizations of resolvents of a monotone operator in a Hilbert space. We obtain iterative approximations of a zero point of a monotone operator generated by the shrinking projection method with errors in a Banach space. Using our result, we discuss some applications. [ABSTRACT FROM AUTHOR]

Our purpose in this paper is to prove strong convergence theorems for approximation of a fixed point of a left Bregman strongly relatively nonexpansive mapping which is also a solution to a finite system of equilibrium problems in the framework of reflexive real Banach spaces. We also discuss the approximation of a common fixed point of a family of left Bregman strongly nonexpansive mappings which is also solution to a finite system of equilibrium problems in reflexive real Banach spaces. Our results complement many known recent results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the common approximate fixed points of monotone nonexpansive semigroups of nonlinear mappings $\{T(t)\}_{t \geq0}$, i.e., a family such that $T(0)x=x$, $T(s+t)x=T(s)\circ T(t)x $, where the domain is a Banach space. In particular we prove that under suitable conditions, the common approximate fixed points are the same as the common approximate fixed points set of two mappings from the family. Then we give an algorithm of how to construct an approximate fixed point sequence of the semigroup in the case of a uniformly convex Banach space. [ABSTRACT FROM AUTHOR]

We study sparse approximate solutions to convex optimization problems. It is known that in many engineering applications researchers are interested in an approximate solution of an optimization problem as a linear combination of a few elements from a given system of elements. There is an increasing interest in building such sparse approximate solutions using different greedy-type algorithms. The problem of approximation of a given element of a Banach space by linear combinations of elements from a given system (dictionary) is well studied in nonlinear approximation theory. At first glance, the settings of approximation and optimization problems are very different. In the approximation problem, an element is given and our task is to find a sparse approximation of it. In optimization theory, an energy function is given and we should find an approximate sparse solution to the minimization problem. It turns out that the same technique can be used for solving both problems. We show how the technique developed in nonlinear approximation theory, in particular the greedy approximation technique, can be adjusted for finding a sparse solution of an optimization problem. [ABSTRACT FROM AUTHOR]

A non-Archimedean Banach space has the orthogonal finite-dimensional decomposition property (OFDDP) if it is the orthogonal direct sum of a sequence of finite-dimensional subspaces. This property has an influence in the non-Archimedean Grothendieck's approximation theory, where an open problem is the following: Let E be a non-Archimedean Banach space of countable type with the OFDDP and let D be a closed subspace of E. Does D have the OFDDP? In this paper we give a negative answer to this question; we construct a Banach space of countable type with the OFDDP having a one-codimensional subspace without the OFDDP. Next we prove that, however, for certain classes of Banach spaces of countable type, the OFDDP is preserved by taking finite-codimensional subspaces. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a sufficient condition for the convergence of positive linear operators in Banach function spaces on $${\mathbb {R}}^n$$ and derive a Korovkin type theorem for these spaces. Also, we generalized this result via statistical sense. [ABSTRACT FROM AUTHOR]

Given a Banach space X and one of its compact sets $\mathcal{F}$, we consider the problem of finding a good n-dimensional space X⊂ X which can be used to approximate the elements of $\mathcal{F}$. The best possible error we can achieve for such an approximation is given by the Kolmogorov width $d_{n}(\mathcal{F})_{X}$. However, finding the space which gives this performance is typically numerically intractable. Recently, a new greedy strategy for obtaining good spaces was given in the context of the reduced basis method for solving a parametric family of PDEs. The performance of this greedy algorithm was initially analyzed in Buffa et al. (Modél. Math. Anal. Numér. 46:595-603, ) in the case $X=\mathcal{H}$ is a Hilbert space. The results of Buffa et al. (Modél. Math. Anal. Numér. 46:595-603, ) were significantly improved upon in Binev et al. (SIAM J. Math. Anal. 43:1457-1472, ). The purpose of the present paper is to give a new analysis of the performance of such greedy algorithms. Our analysis not only gives improved results for the Hilbert space case but can also be applied to the same greedy procedure in general Banach spaces. [ABSTRACT FROM AUTHOR]

This paper deals with a functional equation and inequality arising in dynamic programming of multistage decision processes. Using several fixed-point theorems due to Krasnoselskii, Boyd-Wong and Liu, we prove the existence and/or uniqueness and iterative approximations of solutions, bounded solutions and bounded continuous solutions for the functional equation in two Banach spaces and a complete metric space, respectively. Utilizing the monotone iterative method, we establish the existence and iterative approximations of solutions and nonpositive solutions for the functional inequality in a complete metric space. Six examples which dwell upon the importance of our results are also included. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new iterative method of order six and study the semilocal convergence of the method by using the recurrence relations for solving nonlinear equations in Banach spaces. We prove an existence-uniqueness theorem and give a priori error bounds which demonstrates the R-order of the method to be six. Finally, we give some numerical applications to demonstrate our approach. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new iterative scheme and prove strong convergence theorem for approximation of a common fixed point of a countable family of relatively nonexpansive mappings, which is also a solution to an equilibrium problem in a uniformly convex and uniformly smooth real Banach space. We apply our results to approximate fixed point of a nonexpansive mapping, which is also solution to an equilibrium problem in a real Hilbert space and prove strong convergence of general H-monotone mappings in a uniformly convex and uniformly smooth real Banach space. Our results extend many known recent results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we establish a general solution and the generalized Hyers-Ulam-Rassias stability of the following general mixed additive-cubic functional equation in the quasi-Banach spaces. [ABSTRACT FROM AUTHOR]

Recently, we have developed the necessary and sufficient conditions under which a rational function F( hA) approximates the semigroup of operators exp( tA) generated by an infinitesimal operator A. The present paper extends these results to an inhomogeneous equation u′( t) = Au( t) + f( t). [ABSTRACT FROM AUTHOR]

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We introduce new general iterative approximation methods for finding a common fixed point of a countable family of nonexpansive mappings. Strong convergence theorems are established in the framework of reflexive Banach spaces which admit a weakly continuous duality mapping. Finally, we apply our results to solve the the equilibrium problems and the problem of finding a zero of an accretive operator. The results presented in this paper mainly improve on the corresponding results reported by many others. [ABSTRACT FROM AUTHOR]

In this paper, we give some equivalent conditions on a Banach frame for a Banach space by using the pseudoinverse operator. We also consider the stability of a Banach frame for a Banach space X with respect to X or an X-frame for a Banach space X under perturbation. These results generalize and improve the related works of Balan, Casazza, Christensen, Stoeva and Jian et al. [ABSTRACT FROM AUTHOR]

In 1996, Ricardo Ricardo Mañé discovered that Mather measures are in fact the minimizers of a “universal” infinite dimensional linear programming problem. This fundamental result has many applications, of which one of the most important is to the estimates of the generic number of Mather measures. Mañé obtained the first estimation of that sort by using finite dimensional approximations. Recently, we were able, with Gonzalo Contreras, to use this method of finite dimensional approximation in order to solve a conjecture of John Mather concerning the generic number of Mather measures for families of Lagrangian systems. In the present paper we obtain finer results in that direction by applying directly some classical tools of convex analysis to the infinite dimensional problem. We use a notion of countably rectifiable sets of finite codimension in Banach (and Frechet) spaces which may deserve independent interest. [ABSTRACT FROM AUTHOR]

We study the strong convergence of two kinds of viscosity iteration processes for approximating common fixed points of the pseudocontractive semigroup in uniformly convex Banach spaces with uniformly Gâteaux differential norms. As special cases, we get the strong convergence of the implicit viscosity iteration process for approximating common fixed points of the nonexpansive semigroup in Banach spaces satisfying some conditions. The results presented in this paper extend and generalize some results concerned with the nonexpansive semigroup in (Chen and He, 2007) and the pseudocontractive mapping in (Zegeye et al., 2007) to the pseudocontractive semigroup in Banach spaces under different conditions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the problem of best weighted simultaneous approximations to totally bounded sequences in Banach spaces. Characterization results from convex sets in Banach spaces are established under the assumption that the Banach space is uniformly smooth. [ABSTRACT FROM AUTHOR]

Abstract  A posteriori error estimates are provided for discretizations in time of abstract nonlinear parabolic problems u′ = F(u), by the backward Euler method in the maximal regularity framework of Banach spaces. The estimates are of conditional type, i.e., are valid under assumptions on the approximate solution, and the proofs are based on appropriate fixed point arguments. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to provide an error analysis for the multicategory support vector machine (MSVM) classificaton problems. We establish the uniform convergency approach for MSVMs and estimate the misclassification error. The main difficulty we overcome here is to bound the offset vector. As a result, we confirm that the MSVM classification algorithm with polynomial kernels is always efficient when the degree of the kernel polynomial is large enough. Finally the rate of convergence and examples are given to demonstrate the main results. [ABSTRACT FROM AUTHOR]

Let X be a uniformly convex Banach space, and let S, T be a pair of mean nonexpansive mappings. In this paper, it is proved that the sequence of Ishikawa iterations associated with S and T converges to the common fixed point of S and T. [ABSTRACT FROM AUTHOR]

In this paper we study the connection between the metric projection operator P K : B → K, where B is a reflexive Banach space with dual space B* and K is a non–empty closed convex subset of B, and the generalized projection operators ∏ K : B → K and π K : B* → K. We also present some results in non–reflexive Banach spaces. [ABSTRACT FROM AUTHOR]

The convex feasibility problem (CFP) of finding a point in the nonempty intersection ∩i=1NCi is considered, where N ≥ 1 is an integer and each Ci is assumed to be the fixed point set of a nonexpansive mapping Ti : E → E, where E is a reflexive Banach space with a weakly sequentially continuous duality mapping. By using viscosity approximation methods for a finite family of nonexpansive mappings, it is shown that for any given contractive mapping f : C → C, where C is a nonempty closed convex subset of E and for any given x0 ∈ C the iterative scheme xn+1 = P[αn+1 f (xn) + (1- α n+1)Tn+1xn] is strongly convergent to a solution of (CFP), if and only if {α n} and {xn} satisfy certain conditions, where αn∈ (0,1),Tn = Tn(mod N) and P is a sunny nonexpansive retraction of E onto C. The results presented in the paper extend and improve some recent results in Xu (2004), O'Hara et al. (2003), Song and Chen (2006), Bauschke (1996), Browder (1967), Halpern (1967), Jung (2005), Lions (1977), Moudafi (2000), Reich (1980),Wittmann (1992), Reich (1994). [ABSTRACT FROM AUTHOR]

We continue to study the efficiency of approximation and convergence of greedy-type algorithms in uniformly smooth Banach spaces. Two greedy-type approximation methods, the Weak Chebyshev Greedy Algorithm (WCGA) and the Weak Relaxed Greedy Algorithm (WRGA), have been introduced and studied in [24]. These methods (WCGA and WRGA) are very general approximation methods that work well in an arbitrary uniformly smooth Banach space $X$ for any dictionary ${\Cal D}$. It turns out that these general approximation methods are also very good for specific dictionaries. It has been observed in [7] that the WCGA and WRGA provide constructive methods in $m$-term trigonometric approximation in $L_p$, $p\in[2,\infty)$, which realize an optimal rate of $m$-term approximation for different function classes. In [25] the WCGA and WRGA have been used in constructing deterministic cubature formulas for a wide variety of function classes with error estimates similar to those for the Monte Carlo Method. The WCGA and WRGA can be considered as a constructive deterministic alternative to (or substitute for) some powerful probabilistic methods. This observation encourages us to continue a thorough study of the WCGA and WRGA. In this paper we study modifications of the WCGA and WRGA that are motivated by numerical applications. In these modifications we are able to perform steps of the WCGA (or WRGA) approximately with some controlled errors. We prove that the modified versions of the {\it WCGA and WRGA perform as well as the WCGA and WRGA}. We give two applications of greedy-type algorithms. First, we use them to provide a constructive proof of optimal estimates for best $m$-term trigonometric approximation in the uniform norm. Second, we use them to construct deterministic sets of points $\{\xi^1,\ldots,\xi^m\} \subset [0,1]^d$ with the $L_p$ discrepancy less than $Cp^{1/2}m^{-1/2}$, $C$ is an effective absolute constant. [ABSTRACT FROM AUTHOR]

Let C[-1, 1] be the Banach space of continuous complex functions f on the interval [-1, 1] equipped with the standard maximum norm ¦¦f¦¦; let ω(·) = ω(·, f) be the modulus of continuity of f; and let Rn = Rn (f) be the best uniform approximation of f by rational functions (r.f.) whose degrees do not exceed n = 1, 2, …. The space C[-1, 1] is also regarded as a pre-Hilbert space with respect to the inner product given by (f,g) (1/π) &integ;-1¹ f(x)&g(x)macr; (1 - x²)-½ dx. Let zn = [z1, z2, …, zn] be a set of points located outside the interval [-1, 1]. By F(·, f, zn) we denote an orthoprojection operator acting from the pre-Hilbert space C[-1, 1] onto its (n + 1)-dimensional subspace consisting of rational functions whose poles (with multiplicity taken into account) can only be points of the set zn. In this paper, we show that if f is not a rational function of degree ≤ n, then we can find a set of points zn = zn (f) such that ¦¦f(·) - F(·, f, zn)¦¦ ≤ 12Rnln 3 /ω-1(Rn/3). [ABSTRACT FROM AUTHOR]

We examine single step time discrete approximations to an abstract Cauchy problem considered in a pair of Banach spaces, of which one (space of solutions) is densely embedded in the other (space of initial data). The stability and error analysis of such discretizations is carried out. Our approach is applicable to the analysis of approximations to parabolic PDE problems in various pairs of function spaces arising from practical needs. In the final part of the paper, we present a possible application of our results for studying a semi-discrete version of a model initial-boundary value problem of heat conduction. [ABSTRACT FROM AUTHOR]