Common best proximity pairs in strictly convex Banach spaces

2017 ◽  
Vol 24 (3) ◽  
pp. 363-372 ◽  
Author(s):  
Moosa Gabeleh
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Banach Space ◽  
Existence Results ◽  
Current Paper ◽  
Strictly Convex ◽  
Space Setting ◽  
Strictly Convex Banach Space ◽  
Strictly Convex Banach Spaces ◽  
Best Proximity Pair

AbstractA mapping {T\colon A\cup B\to A\cup B} such that {T(A)\subseteq A} and {T(B)\subseteq B} is called a noncyclic mapping, where A and B are two nonempty subsets of a Banach space X. A best proximity pair {(p,q)\in A\times B} for such a mapping T is a point such that {p=Tp,q=Tq} and {\|p-q\|=\operatorname{dist}(A,B)}. In the current paper, we establish some existence results of best proximity pairs in strictly convex Banach spaces. The presented theorems improve and extend some recent results in the literature. We also obtain a generalized version of Markov–Kakutani’s theorem for best proximity pairs in a strictly convex Banach space setting.

NONEXPANSIVE BIJECTIONS TO THE UNIT BALL OF THE -SUM OF STRICTLY CONVEX BANACH SPACES

2018 ◽  
Vol 97 (2) ◽  
pp. 285-292 ◽  
Author(s):  
V. KADETS ◽  
O. ZAVARZINA
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Convex Banach Space ◽  
Strictly Convex ◽  
Strictly Convex Banach Space ◽  
Strictly Convex Banach Spaces

Extending recent results by Cascales et al. [‘Plasticity of the unit ball of a strictly convex Banach space’, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat.110(2) (2016), 723–727], we demonstrate that for every Banach space $X$ and every collection $Z_{i},i\in I$, of strictly convex Banach spaces, every nonexpansive bijection from the unit ball of $X$ to the unit ball of the sum of $Z_{i}$ by $\ell _{1}$ is an isometry.

scholarly journals On geometric properties of ⊥BJCϵ symmetric in some types of Banach spaces

2020 ◽  
Vol 1664 (1) ◽  
pp. 012038
Author(s):  
Saied A. Jhonny ◽  
Buthainah A. A. Ahmed
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Unit Sphere ◽  
Real Banach Space ◽  
Convex Banach Space ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Geometric Properties ◽  
Strictly Convex ◽  
Strictly Convex Banach Space

Abstract In this paper, we ⊥ B J C ϵ -orthogonality and explore ⊥ B J C ϵ -symmetricity such as a ⊥ B J C ϵ -left-symmetric ( ⊥ B J C ϵ -right-symmetric) of a vector x in a real Banach space (𝕏, ‖·‖𝕩) and study the relation between a ⊥ B J C ϵ -right-symmetric ( ⊥ B J C ϵ -left-symmetric) in ℐ(x). New results and proofs are include the notion of norm attainment set of a continuous linear functionals on a reflexive and strictly convex Banach space and using these results to characterize a smoothness of a vector in a unit sphere.

scholarly journals Range-Kernel orthogonality and elementary operators on certain Banach spaces

2021 ◽  
Vol 54 (1) ◽  
pp. 272-279
Author(s):  
Ahmed Bachir ◽  
Abdelkader Segres ◽  
Nawal Ali Sayyaf ◽  
Khalid Ouarghi
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Banach Space ◽  
Strictly Convex ◽  
Strictly Convex Banach Space ◽  
Arbitrary Operator ◽  
Elementary Operators

Abstract The characterization of the points in C p : 1 ≤ p < ∞ ( ℋ ) {C}_{p{:}_{1\le p\lt \infty }}\left({\mathcal{ {\mathcal H} }}) , the Von Neuman-Schatten p-classes, that are orthogonal to the range of elementary operators has been done for certain kinds of elementary operators. In this paper, we shall study this problem of characterization on an abstract reflexive, smooth and strictly convex Banach space for arbitrary operator. As an application, we consider other kinds of elementary operators defined on the spaces C p : 1 ≤ p < ∞ ( ℋ ) {C}_{p{:}_{1\le p\lt \infty }}\left({\mathcal{ {\mathcal H} }}) , and finally, we give a counterexample to Mecheri’s result given in this context.

Projections in the Convex Hull of Surjective Isometries

2010 ◽  
Vol 53 (3) ◽  
pp. 398-403 ◽  
Author(s):  
Fernanda Botelho ◽  
James Jamison
Keyword(s):  
Banach Space ◽  
Convex Hull ◽  
Banach Spaces ◽  
Convex Combination ◽  
Continuous Functions ◽  
Convex Banach Space ◽  
Strictly Convex ◽  
Strictly Convex Banach Space ◽  
Spaces Of Continuous Functions

AbstractWe characterize those linear projections represented as a convex combination of two surjective isometries on standard Banach spaces of continuous functions with values in a strictly convex Banach space.

scholarly journals Strong and weak convergence of Ishikawa iterations for best proximity pairs

2020 ◽  
Vol 18 (1) ◽  
pp. 10-21
Author(s):  
Moosa Gabeleh ◽  
S. I. Ezhil Manna ◽  
A. Anthony Eldred ◽  
Olivier Olela Otafudu
Keyword(s):  
Weak Convergence ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Relatively Nonexpansive Mapping ◽  
Strictly Convex ◽  
Strong And Weak Convergence ◽  
Uniformly Convex Banach Spaces ◽  
Relatively Nonexpansive ◽  
Strictly Convex Banach Spaces ◽  
Best Proximity Pair

Abstract Let A and B be nonempty subsets of a normed linear space X. A mapping T : A ∪ B → A ∪ B is said to be a noncyclic relatively nonexpansive mapping if T(A) ⊆ A, T(B) ⊆ B and ∥Tx − Ty∥ ≤ ∥x − y∥ for all (x, y) ∈ A × B. A best proximity pair for such a mapping T is a point (p, q) ∈ A × B such that p = Tp, q = Tq and d(p, q) = dist(A, B). In this work, we introduce a geometric notion of proximal Opiaľs condition on a nonempty, closed and convex pair of subsets of strictly convex Banach spaces. By using this geometric notion, we study the strong and weak convergence of the Ishikawa iterative scheme for noncyclic relatively nonexpansive mappings in uniformly convex Banach spaces. We also establish a best proximity pair theorem for noncyclic contraction type mappings in the setting of strictly convex Banach spaces.

scholarly journals The Denjoy–Wolff Theorem in the Open Unit Ball of a Strictly Convex Banach Space

1999 ◽  
Vol 143 (1) ◽  
pp. 111-123 ◽  
Author(s):  
Jaroslaw Kapeluszny ◽  
Tadeusz Kuczumow ◽  
Simeon Reich
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Convex Banach Space ◽  
Open Unit ◽  
Open Unit Ball ◽  
Strictly Convex ◽  
Strictly Convex Banach Space

scholarly journals Strong convergence of a modified Halpern-type iteration for asymptotically quasi-∅-nonexpansive mappings

2013 ◽  
Vol 21 (1) ◽  
pp. 261-276
Author(s):  
Chang-Qun Wu ◽  
Yan Hao
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Iterative Algorithms ◽  
Convex Banach Space ◽  
Halpern Iteration ◽  
Uniformly Smooth ◽  
Strictly Convex Banach Space

Abstract In this paper, the problem of modifying Halpern iteration for approximating a common fixed point of a family of asymptotically quasi- ∅-nonexpansive mappings is considered. Strong convergence theorems are established in a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. The results presented in this paper mainly improve the corresponding results announced in [Y.J. Cho, X. Qin, S.M. Kang, Strong convergence of the modified Halpern- type iterative algorithms in Banach spaces, An. Stiint. Univ. Ovidius Constanta Ser. Mat. 17 (2009) 51-68].

scholarly journals Isometries between Spaces of Vector-Valued Differentiable Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Alireza Ranjbar-Motlagh
Keyword(s):  
Banach Space ◽  
Real Line ◽  
Convex Banach Space ◽  
Compact Interval ◽  
The Real ◽  
Strictly Convex ◽  
Differentiable Functions ◽  
Strictly Convex Banach Space ◽  
Vector Valued

This article characterizes the isometries between spaces of all differentiable functions from a compact interval of the real line into a strictly convex Banach space.

scholarly journals Common Fixed Points of a Subfamily of a Nonexpansive Periodic Evolution Family on a Strictly Convex Banach Space

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Gul Rahmat ◽  
Tariq Shah ◽  
Muhammad Sarwar ◽  
Hassen Aydi ◽  
Habes Alsamir
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Common Fixed Points ◽  
Convex Banach Space ◽  
Evolution Family ◽  
Linear Operators ◽  
Bounded Subset ◽  
Bounded Linear ◽  
Strictly Convex ◽  
Strictly Convex Banach Space

In this study, we establish some results for strong convergence of a sequence to a common fixed point of a subfamily of a nonexpansive and periodic evolution family of bounded linear operators acting on a closed and bounded subset J of a strictly convex Banach space X . In fact, we generalized the results from semigroups of the operator to an evolution family of operators.

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