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]

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[2] 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.

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

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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

Keywords: generalized Lipschitz mapping; -hemi-contractive mapping; Ishikawa iterative sequence with errors; uniformly smooth real Banach space.. 1 Introduction and preliminary.[r]

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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.

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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.

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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 [2], nonlinear boundary value problems [4], optimal control theory [8], operator equations [20] 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.)

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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.

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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

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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

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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,

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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

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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.

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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 .

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 ) = ∅.

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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.

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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

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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.

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