On farthest and simultaneous farthest points in Kothe spaces

Let X be a real Banach space and G be a closed bounded subset of X. For x e X, we set ρ (x, G) = sup {kx − yk : y e G} . The set G is called remotal if for any x e X, there exists g e G such that ρ (x, G) = kx − gk . In this paper we show that for a separable remotal set G ⊂ X, the set E(G) is remotal in E(X), where E(X) is the Kothe Bochner function space. Some other results are presented.

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On Simultaneous Farthest Points in

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2011/890598 ◽

2011 ◽

Vol 2011 ◽

pp. 1-10

Author(s):

Sh. Al-Sharif ◽

M. Rawashdeh

Keyword(s):

Banach Space ◽

Bounded Subset ◽

Farthest Points ◽

Integrable Functions

Let be a Banach space and let be a closed bounded subset of . For , we set . The set is called simultaneously remotal if, for any , there exists such that . In this paper, we show that if is separable simultaneously remotal in , then the set of -Bochner integrable functions, , is simultaneously remotal in . Some other results are presented.

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Iterative solution of nonlinear equations of the monotone type in Banach spaces

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700028112 ◽

1990 ◽

Vol 42 (1) ◽

pp. 21-31 ◽

Cited By ~ 18

Author(s):

C.E. Chidume

Keyword(s):

Banach Space ◽

Nonlinear Equations ◽

Real Banach Space ◽

A farthest-point characterisation of the relative Chebyshev centre

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700015045 ◽

1996 ◽

Vol 54 (1) ◽

pp. 27-33 ◽

Cited By ~ 1

Author(s):

R. Huotari ◽

M.P. Prophet ◽

J. Shi

Keyword(s):

Banach Space ◽

Hilbert Space ◽

Convex Hull ◽

Compact Subset ◽

Real Banach Space ◽

Metric Projection ◽

Closed Convex Hull ◽

Farthest Points ◽

Farthest Point ◽

Gateaux Derivative

We characterise the relative Chebyshev centre of a compact subsetFof a real Banach space in terms of the Gateaux derivative of the distance to farthest points. We present a relative-Chebyshev-centre characterisation of Hilbert space. In Hilbert space we show that the relative Chebyshev centre is in the closed convex hull of the metric projection ofF, and we estimate the relative Chebyshev radius ofF.

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The local spectral radius of a nonnegative orbit of compact linear operators

Mathematica Slovaca ◽

10.1515/ms-2015-0172 ◽

2016 ◽

Vol 66 (3) ◽

Author(s):

Mihály Pituk

Keyword(s):

Banach Space ◽

Spectral Radius ◽

Additional Assumption ◽

Real Banach Space ◽

Linear Operators ◽

Partial Ordering ◽

Positive Eigenvector ◽

Local Spectral Radius

AbstractWe consider orbits of compact linear operators in a real Banach space which are nonnegative with respect to the partial ordering induced by a given cone. The main result shows that under a mild additional assumption the local spectral radius of a nonnegative orbit is an eigenvalue of the operator with a positive eigenvector.

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A characterisation of Hilbert spaces via orthogonality and proximinality

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700038053 ◽

2005 ◽

Vol 71 (1) ◽

pp. 107-111

Author(s):

Fathi B. Saidi

Keyword(s):

Banach Space ◽

Hilbert Space ◽

Banach Spaces ◽

Hilbert Spaces ◽

Real Banach Space ◽

Nonzero Element ◽

Dimensional Subspace ◽

Two Dimensional ◽

Orthogonal Direction ◽

Open Problems

In this paper we adopt the notion of orthogonality in Banach spaces introduced by the author in [6]. There, the author showed that in any two-dimensional subspace F of E, every nonzero element admits at most one orthogonal direction. The problem of existence of such orthogonal direction was not addressed before. Our main purpose in this paper is the investigation of this problem in the case where E is a real Banach space. As a result we obtain a characterisation of Hilbert spaces stating that, if in every two-dimensional subspace F of E every nonzero element admits an orthogonal direction, then E is isometric to a Hilbert space. We conclude by presenting some open problems.

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Perturbation of the fixed point problem for continuous mappings

Russian Universities Reports. Mathematics ◽

10.20310/2686-9667-2021-26-135-241-249 ◽

2020 ◽

pp. 241-249

Author(s):

Zukhra T. Zhukovskaya ◽

Sergey E. Zhukovskiy

Keyword(s):

Banach Space ◽

Fixed Point ◽

Fixed Points ◽

Continuous Mapping ◽

Fixed Point Problem ◽

Point Problem ◽

Bounded Subset ◽

Completely Continuous ◽

Continuous Map ◽

Continuous Mappings

We consider the problem of a double fixed point of pairs of continuous mappings defined on a convex closed bounded subset of a Banach space. It is shown that if one of the mappings is completely continuous and the other is continuous, then the property of the existence of fixed points is stable under contracting perturbations of the mappings. We obtain estimates for the distance from a given pair of points to double fixed points of perturbed mappings. We consider the problem of a fixed point of a completely continuous mapping on a convex closed bounded subset of a Banach space. It is shown that the property of the existence of a fixed point of a completely continuous map is stable under contracting perturbations. Estimates of the distance from a given point to a fixed point are obtained. As an application of the obtained results, the solvability of a difference equation of a special type is proved.

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Strong Convergence Theorems for Common Fixed Points of an Infinite Family of Asymptotically Nonexpansive Mappings

Abstract and Applied Analysis ◽

10.1155/2014/809528 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6

Author(s):

Yuanheng Wang

Keyword(s):

Banach Space ◽

Fixed Point ◽

Real Banach Space ◽

Nonexpansive Mappings ◽

Infinite Family ◽

Common Fixed Points ◽

Uniformly Gâteaux Differentiable Norm ◽

Asymptotically Nonexpansive Mappings ◽

Asymptotically Nonexpansive ◽

Differentiable Norm

In the framework of a real Banach space with uniformly Gateaux differentiable norm, some new viscosity iterative sequences{xn}are introduced for an infinite family of asymptotically nonexpansive mappingsTii=1∞in this paper. Under some appropriate conditions, we prove that the iterative sequences{xn}converge strongly to a common fixed point of the mappingsTii=1∞, which is also a solution of a variational inequality. Our results extend and improve some recent results of other authors.

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The Linearity of Riemann Integral on Functions from ℝ into Real Banach Space

Formalized Mathematics ◽

10.2478/forma-2013-0020 ◽

2013 ◽

Vol 21 (3) ◽

pp. 185-191

Author(s):

Keiko Narita ◽

Noboru Endou ◽

Yasunari Shidama

Keyword(s):

Banach Space ◽

Integral Operator ◽

Real Banach Space ◽

Closed Interval ◽

Continuous Functions ◽

Real Numbers ◽

Riemann Integral ◽

Basic Properties ◽

Bounded Functions ◽

Definition Of

Summary In this article, we described basic properties of Riemann integral on functions from R into Real Banach Space. We proved mainly the linearity of integral operator about the integral of continuous functions on closed interval of the set of real numbers. These theorems were based on the article [10] and we referred to the former articles about Riemann integral. We applied definitions and theorems introduced in the article [9] and the article [11] to the proof. Using the definition of the article [10], we also proved some theorems on bounded functions.

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