# Top PDF Smooth Banach spaces and approximations ### Smooth Banach spaces and approximations

rrhe fol1owing two problems were posed by Bonic and Frampton for non- cP smooth B-spaces and they can also be acked for nonCp,q smooth B-spaces: suppose that E is non-Cp,q smooth, that F[r] ### Iterative solutions of K positive definite operator equations in real uniformly smooth Banach spaces

 S. S. Chang, Y. J. Cho, B. S. Lee, J. S. Jung, and S. M. Kang, Iterative approximations of ﬁxed points and solutions for strongly accretive and strongly pseudo-contractive mappings in Banach spaces, J. Math. Anal. Appl. 224 (1998), no. 1, 149–165. MR 99g:47146. Zbl 933.47040. ### Fixed point solutions for variational inequalities in image restoration over q uniformly smooth Banach spaces

A Banach space X is said to be strictly convex if x+y  <  for all x, y ∈ X with x = y =  and x = y. A Banach space X is said to be uniformly convex if, for each > , there exists δ >  such that for x, y ∈ X with x, y ≤  and x – y ≥ , x+y  ≤  – δ holds. Let S(X) = {x ∈ X : x = }. The norm of X is said to be Gâteaux diﬀerentiable (or X is said to be smooth) if the limit ### Best Approximations Theorem for a Couple in Cone Banach Space

Banach, valued metric space was considered by Rzepecki 1, Lin 2, and lately by Huang and Zhang 3. Basically, for nonempty set X, the definition of metric d : X ×X → R 0, ∞ is replaced by a new metric, namely, by an ordered Banach space E: d : X × X → E. Such metric spaces are called cone metric spaces in short CMSs. In 1980, by using this idea Rzepecki 1 generalized the fixed point theorems of Maia type. Seven years later, Lin 2 extends some results of Khan and Imdad 4 by considering this new metric space construction. In 2007, Huang and Zhang 3 discussed some properties of convergence of sequences and proved the fixed point theorems of contractive mapping for cone metric spaces: any mapping T of a complete cone metric space X into itself that satisfies, for some 0 ≤ k < 1, the inequality ### General convergence analysis of projection methods for a system of variational inequalities in q uniformly smooth Banach spaces

This alternative formula enables us to suggest and analyze a two-step explicit projection method for solving system (.), and this is the main motivation of our next result. Theorem . Let C be a nonempty closed convex subset of a q-uniformly smooth Banach space E. Let T i : C × C → E be relaxed (γ i , r i )-cocoercive and μ i -Lipschitz continuous in the ### The convergence theorems of Ishikawa iterative process with errors for Φ-hemi-contractive mappings in uniformly smooth Banach spaces

Keywords: generalized Lipschitz mapping; -hemi-contractive mapping; Ishikawa iterative sequence with errors; uniformly smooth real Banach space.. 1 Introduction and preliminary.[r] ### A New System of Nonlinear Fuzzy Variational Inclusions Involving Accretive Mappings in Uniformly Smooth Banach Spaces

Recently, Lan and Verma 54, by using the concept of A, η-accretive mappings, the resolvent operator technique associated with A, η-accretive mappings, introduced and studied a new class of nonlinear fuzzy variational inclusion systems with A, η-accretive mappings in Banach spaces and construct some new iterative algorithms to approximate the solutions of the nonlinear fuzzy variational inclusion systems. ### A STABLE METHOD FOR LINEAR EQUATION IN BANACH SPACES WITH SMOOTH NORMS

Abstract: A stable method for numerical solution of a linear operator equation in reflexive Banach spaces is proposed. The operator and the right-hand side of the equation are assumed to be known approximately. The corresponding error levels may remain unknown. Approximate operators and their conjugate ones must possess the property of strong pointwise convergence. The exact normal solution is assumed to be sourcewise representable and some upper estimate for the norm of its source element must be known. The norm in the Banach space of solutions is supposed to satisfy the following smoothness-type condition: some function of the norm must be differentiable. Under these conditions a stability of the method with respect to nonuniform perturbations in operator is shown and the strong convergence to the normal solution is proved. A boundary control problem for the one-dimensional wave equation is considered as an example of possible application. The results of the model numerical experiments are presented. ### Fixed Points of Non-Smooth Functions on Finite Dimensional Ordered Banach Spaces via Clarke Generalized Jacobian

Ordered Banach spaces are very significant class of vector spaces which are studied widely in theory and applications of mathematics. This class of vector spaces is considered in nonlinear integral equations , nonlinear boundary value problems , optimal control theory , operator equations  and etc. On the other hand, an important theory in mathematical analysis is fixed point theory. This theory and its applications in orederd Banach spaces have been considered by many researchers. (see [2, 3, 5, 6, 11, 14,16, 17] and the references therein.) ### A new general iterative algorithm with Meir-Keeler contractions for variational inequality problems in q-uniformly smooth Banach spaces

Recall that if C and D are nonempty subsets of a Banach space X such that C is nonempty closed convex and D ⊂ C, then a mapping P : C → D is sunny [] provided P(x + t(x – P(x))) = P(x) for all x ∈ C and t ≥ , whenever x + t(x – P(x)) ∈ C. A mapping P : C → D is called a retraction if Px = x for all x ∈ D. Furthermore, P is a sunny nonexpansive retraction from C onto D if P is a retraction from C onto D which is also sunny and nonexpansive. A subset D of C is called a sunny nonexpansive retraction of C if there exists a sunny nonexpansive retraction from C onto D. ### Almost stability of the Mann type iteration method with error term involving strictly hemicontractive mappings in smooth Banach spaces

Let K be a nonempty subset of an arbitrary Banach space X and X ∗ be its dual space. The symbols D(T), R(T ) and F(T) stand for the domain, the range and the set of ﬁxed points of T : X → X respectively (x is called a ﬁxed point of T iﬀ T(x) = x). We denote by J the normalized duality mapping from X to  X ∗ deﬁned by ### Approximation of common solutions for variational inequalities and fixed point of strict pseudo-contractions in q-uniformly smooth Banach spaces

11. Aoyama, K, Kimura, Y, Takahashi, W, Toyoda, M: Approximation of common ﬁxed points of a countable family of nonexpansive mapping in a Banach space. Nonlinear Anal., Theory Methods Appl. 67, 2350-2360 (2007) 12. Reich, S: Asymptotic behavior of contractions in Banach spaces. J. Math. Anal. Appl. 44, 57-70 (1973) 13. Song, Y, Luchuan, C: A general iteration scheme for variational inequality problem and common ﬁxed point ### Asymptotically Pseudocontractions, Banach Operator Pairs and Best Simultaneous Approximations

29 M. A. Khamsi and W. M. Kozlowski, “On asymptotic pointwise contractions in modular function spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 9, pp. 2957–2967, 2010. 30 K. Deimling, “Zeros of accretive operators,” Manuscripta Mathematica, vol. 13, pp. 365–374, 1974. 31 L. Habiniak, “Fixed point theorems and invariant approximations,” Journal of Approximation Theory, ### Implicit and explicit iterative algorithms for hierarchical variational inequality in uniformly smooth Banach spaces

Let X be a real Banach space with its topological dual X ∗ , and C be a nonempty closed convex subset of X. Let T : C → X be a nonlinear mapping on C. We denote by Fix(T) the set of ﬁxed points of T and by R the set of all real numbers. A mapping T : C → X is called L-Lipschitz continuous if there exists a constant L ≥  such that ### A Generalized Nonlinear Random Equations with Random Fuzzy Mappings in Uniformly Smooth Banach Spaces

We introduce and study the general nonlinear random H, η-accretive equations with random fuzzy mappings. By using the resolvent technique for the H, η-accretive operators, we prove the existence theorems and convergence theorems of the generalized random iterative algorithm for this nonlinear random equations with random fuzzy mappings in q-uniformly smooth Banach spaces. Our result in this paper improves and generalizes some known corresponding results in the literature. ### Approximation of fixed points of strongly pseudocontractive mappings in uniformly smooth Banach spaces

the unique fixed point of T . It is our purpose in this note to solve the above question by proving the following much more general result: E is a real uniformly smooth Banach space, and K is a nonempty closed convex subset of E . Assume that T : K → K is a con- tinuous and strong pseudocontraction, and T neither is Lipschizian nor has the bounded range, then the Ishikawa iteration sequence converges strongly to the unique fixed point of T . ### Fixed point theorems and iterative approximations for monotone nonexpansive mappings in ordered Banach spaces

the proof is the same as ones of Theorem . and so we omit it. Theorem . Let E be a uniformly convex Banach space with the partial order ‘≤’ with re- spect to closed convex cone P and T : P → P be a monotone nonexpansive mapping. Assume that the sequence {T n } ∞ n= is bounded. Then F(T ) = ∅. ### Mangasarian-Fromovitz and Zangwill Conditions For Non-Smooth Infinite Optimization problems in Banach Spaces

Abstract. In this paper we study optimization problems with infinity many inequality constraints on a Banach space where the objective func- tion and the binding constraints are Lipschitz near the optimal solution. Necessary optimality conditions and constraint qualifications in terms of Michel-Penot subdifferential are given. ### The viscosity iterative algorithms for the implicit midpoint rule of nonexpansive mappings in uniformly smooth Banach spaces

Throughout this paper, we assume that E and E ∗ is a real Banach space and the dual space of E, respectively. Let T be a mapping from C into itself, where C is a subset of E. We denoted by F(T) the set of ﬁxed points of T. It is well known that the duality mapping J : E →  E ∗ is deﬁned by ### Multismoothness in Banach Spaces

In this paper, motivated by the results published by R. Khalil and A. Saleh in 2005, we study the notion of k-smooth points and the notion of k-smoothness, which are dual to the notion of k-rotundity. Generalizing these notions and combining smoothness with the recently introduced notion of unitary, we study classes of Banach spaces for which the vector space, spanned by the state space corresponding to a unit vector, is a closed set. Copyright © 2007 B.-L. Lin and T. S. S. R. K. Rao. This is an open access article dis- tributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is prop- erly cited.