scholarly journals Some Results on Iterative Proximal Convergence and Chebyshev Center

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Laishram Shanjit ◽  
Yumnam Rohen ◽  
Sumit Chandok ◽  
M. Bina Devi
Keyword(s):  
Banach Space ◽  
Reflexive Banach Space ◽  
Best Proximity Point ◽  
Normal Structure ◽  
Sufficient Condition ◽  
Chebyshev Center ◽  
Opial’S Condition ◽  
Relatively Nonexpansive

In this paper, we prove a sufficient condition that every nonempty closed convex bounded pair M , N in a reflexive Banach space B satisfying Opial’s condition has proximal normal structure. We analyze the relatively nonexpansive self-mapping T on M ∪ N satisfying T M ⊆ M and T N ⊆ N , to show that Ishikawa’s and Halpern’s iteration converges to the best proximity point. Also, we prove that under relatively isometry self-mapping T on M ∪ N satisfying T N ⊆ N and T M ⊆ M , Ishikawa’s iteration converges to the best proximity point in the collection of all Chebyshev centers of N relative to M . Some illustrative examples are provided to support our results.

scholarly journals On the modulus ofu-convexity of Ji Gao

2003 ◽  
Vol 2003 (1) ◽  
pp. 49-54 ◽  
Author(s):  
Eva María Mazcuñán-Navarro
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Normal Structure ◽  
Fixed Point Property ◽  
Sufficient Condition ◽  
Point Property

We consider the modulus ofu-convexity of a Banach space introduced by Ji Gao (1996) and we improve a sufficient condition for the fixed-point property (FPP) given by this author. We also give a sufficient condition for normal structure in terms of the modulus ofu-convexity.

scholarly journals The approximation problem for compact operators

1969 ◽  
Vol 1 (3) ◽  
pp. 397-401 ◽  
Author(s):  
S.R. Caradus
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Finite Rank ◽  
Reflexive Banach Space ◽  
Approximation Problem ◽  
Compact Operators ◽  
Sufficient Condition

The following sufficient condition is obtained for the uniform approximability of compact operators on a reflexive Banach space by operators of finite rank: if S is the unit ball of X and ø: X* → C(S) is the imbedding ø(x*)x = x*(x) then we require ø(X*) to be complemented in C(S).

A Note on Asymptotic Normal Structure and Close-to-Normal Structure

1982 ◽  
Vol 25 (3) ◽  
pp. 339-343 ◽  
Author(s):  
Kok-Keong Tan
Keyword(s):  
Banach Space ◽  
Convex Subset ◽  
Convex Set ◽  
Reflexive Banach Space ◽  
Compact Convex ◽  
Normal Structure ◽  
Weakly Compact ◽  
Compact Convex Set ◽  
Asymptotic Normal ◽  
Open Question

AbstractA closed convex subset X of a Banach space E is said to have (i) asymptotic normal structure if for each bounded closed convex subset C of X containing more than one point and for each sequence in C satisfying ‖xn − xn + 1‖ → 0 as n → ∞, there is a point x ∈ C such that ; (ii) close-to-normal structure if for each bounded closed convex subset C of X containing more than one point, there is a point x ∈ C such that ‖x − y‖ < diam‖ ‖(C) for all y ∈ C While asymptotic normal structure and close-to-normal structure are both implied by normal structure, they are not related. The example that a reflexive Banach space which has asymptotic normal structure but not close-to normal structure provides us a non-empty weakly compact convex set which does not have close-to-normal structure. This answers an open question posed by Wong in [9] and hence also provides us a Kannan map defined on a weakly compact convex set which does not have a fixed point.

scholarly journals BLURRED MAXIMAL CYCLICALLY MONOTONE SETS AND BIPOTENTIALS

2010 ◽  
Vol 08 (04) ◽  
pp. 323-336 ◽  
Author(s):  
MARIUS BULIGA ◽  
GÉRY DE SAXCÉ ◽  
CLAUDE VALLÉE
Keyword(s):  
Banach Space ◽  
Differential Inclusion ◽  
Reflexive Banach Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

Let X be a reflexive Banach space and Y its dual. In this paper, we find necessary and sufficient conditions for the existence of a bipotential for a blurred maximal cyclically monotone set. Equivalently, we find a necessary and sufficient condition on ϕ ∈ Γ0(X) so that the differential inclusion [Formula: see text] can be put in the form y ∈ ∂b(·, y)(x), with b a bipotential.

scholarly journals Weak proximal normal structure and coincidence quasi-best proximity points

2020 ◽  
Vol 21 (2) ◽  
pp. 331
Author(s):  
Farhad Fouladi ◽  
Ali Abkar ◽  
Erdal Karapinar
Keyword(s):  
Point Theorem ◽  
Best Proximity Point ◽  
Normal Structure ◽  
Point Problem ◽  
Best Proximity Points ◽  
Relatively Nonexpansive

We introduce the notion of pointwise cyclic-noncyclic relatively nonexpansive pairs involving orbits. We study the best proximity point problem for this class of mappings. We also study the same problem for the class of pointwise noncyclic-noncyclic relatively nonexpansive pairs involving orbits. Finally, under the assumption of weak proximal normal structure, we prove a coincidence quasi-best proximity point theorem for pointwise cyclic-noncyclic relatively nonexpansive pairs involving orbits. Examples are provided to illustrate the observed results.

scholarly journals Strong Convergence of an Inertial Iterative Algorithm for Generalized Mixed Variational-like Inequality Problem and Bregman Relatively Nonexpansive Mapping in Reflexive Banach Space

2021 ◽  
Vol 2021 ◽  
pp. 1-25
Author(s):  
Saud Fahad Aldosary ◽  
Watcharaporn Cholamjiak ◽  
Rehan Ali ◽  
Mohammad Farid
Keyword(s):  
Banach Space ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Real Banach Space ◽  
Reflexive Banach Space ◽  
Existence Of A Solution ◽  
Relatively Nonexpansive Mapping ◽  
Inequality Problem ◽  
Relatively Nonexpansive ◽  
Hybrid Iterative Method

In this paper, we consider a generalized mixed variational-like inequality problem and prove a Minty-type lemma for its related auxiliary problems in a real Banach space. We prove the existence of a solution of these auxiliary problems and also prove some properties for the solution set of generalized mixed variational-like inequality problem. Furthermore, we introduce and study an inertial hybrid iterative method for solving the generalized mixed variational-like inequality problem involving Bregman relatively nonexpansive mapping in Banach space. We study the strong convergence for the proposed algorithm. Finally, we list some consequences and computational examples to emphasize the efficiency and relevancy of the main result.

scholarly journals Some aspects of generalized Zbăganu and James constant in Banach spaces

2021 ◽  
Vol 54 (1) ◽  
pp. 299-310
Author(s):  
Qi Liu ◽  
Muhammad Sarfraz ◽  
Yongjin Li
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Banach Spaces ◽  
Normal Structure ◽  
Inner Product ◽  
Sufficient Condition ◽  
Point Property ◽  
James Constant ◽  
Geometric Constant

Abstract We shall introduce a new geometric constant C Z ( λ , μ , X ) {C}_{Z}\left(\lambda ,\mu ,X) based on a generalization of the parallelogram law, which was proposed by Moslehian and Rassias. First, it is shown that, for a Banach space, C Z ( λ , μ , X ) {C}_{Z}\left(\lambda ,\mu ,X) is equal to 1 if and only if the norm is induced by an inner product. Next, a characterization of uniformly non-square is given, that is, X X has the fixed point property. Also, a sufficient condition which implies weak normal structure is presented. Moreover, a generalized James constant J ( λ , X ) J\left(\lambda ,X) is also introduced. Finally, some basic properties of this new coefficient are presented.

scholarly journals Modulus of Convexity, the CoeffcientR(1,X), and Normal Structure in Banach Spaces

2008 ◽  
Vol 2008 ◽  
pp. 1-5 ◽  
Author(s):  
Hongwei Jiao ◽  
Yunrui Guo ◽  
Fenghui Wang
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Normal Structure ◽  
Geometric Parameters ◽  
Sufficient Condition ◽  
Modulus Of Convexity

LetδX(ϵ)andR(1,X)be the modulus of convexity and the Domínguez-Benavides coefficient, respectively. According to these two geometric parameters, we obtain a sufficient condition for normal structure, that is, a Banach spaceXhas normal structure if2δX(1+ϵ)>max{(R(1,x)-1)ϵ,1-(1-ϵ/R(1,X)-1)}for someϵ∈[0,1]which generalizes the known result by Gao and Prus.

scholarly journals Geometric Constants in Banach Spaces Related to the Inscribed Quadrilateral of Unit Balls

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1294
Author(s):  
Asif Ahmad ◽  
Qi Liu ◽  
Yongjin Li
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Normal Structure ◽  
Inner Product ◽  
Sufficient Condition ◽  
Basic Properties ◽  
Geometric Constant ◽  
Geometric Constants

We introduce a new geometric constant Jin(X) based on a generalization of the parallelogram law, which is symmetric and related to the length of the inscribed quadrilateral side of the unit ball. We first investigate some basic properties of this new coefficient. Next, it is shown that, for a Banach space, Jin(X) becomes 16 if and only if the norm is induced by an inner product. Moreover, its properties and some relations between other well-known geometric constants are studied. Finally, a sufficient condition which implies normal structure is presented.

Projections on Tree-Like banach Spaces

1985 ◽  
Vol 37 (5) ◽  
pp. 908-920
Author(s):  
A. D. Andrew
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Bounded Linear Operator ◽  
Unconditional Basis ◽  
Reflexive Banach Space ◽  
Separable Space ◽  
Bounded Linear ◽  
Tree Space

1. In this paper, we investigate the ranges of projections on certain Banach spaces of functions defined on a diadic tree. The notion of a “tree-like” Banach space is due to James 4], who used it to construct the separable space JT which has nonseparable dual and yet does not contain l1. This idea has proved useful. In [3], Hagler constructed a hereditarily c0 tree space, HT, and Schechtman [6] constructed, for each 1 ≦ p ≦ ∞, a reflexive Banach space, STp with a 1-unconditional basis which does not contain lp yet is uniformly isomorphic to for each n.In [1] we showed that if U is a bounded linear operator on JT, then there exists a subspace W ⊂ JT, isomorphic to JT such that either U or (1 — U) acts as an isomorphism on W and UW or (1 — U)W is complemented in JT. In this paper, we establish this result for the Hagler and Schechtman tree spaces.

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