scholarly journals On the semigroup of differentiable mappings

1969 ◽  
Vol 10 (3-4) ◽  
pp. 503-510 ◽  
Author(s):  
Sadayuki Yamamuro
Keyword(s):  
Banach Space ◽  
Real Banach Space ◽  
Fréchet Differentiable ◽  
Differentiable Mappings

The purpose of this paper is to improve a result in [2] on the automorphisms of the semigroup D = D(E) of all (Fréchet)-differentiable mappings of a real Banach space E into itself.

A Non-Reflexive Smooth Space with a Smooth Dual

1974 ◽  
Vol 17 (4) ◽  
pp. 579-580
Author(s):  
J. H. M. Whitfield
Keyword(s):  
Banach Space ◽  
Real Banach Space ◽  
Gâteaux Differentiability ◽  
Fréchet Differentiability ◽  
Smooth Norm ◽  
Frechet Differentiability ◽  
Gateaux Differentiability ◽  
Fréchet Differentiable ◽  
Smooth Space ◽  
Nonreflexive Space

Let (E, ρ) and (E*ρ*) be a real Banach space and its dual. Restrepo has shown in [4] that, if p and ρ* are both Fréchet differentiable, E is reflexive. The purpose of this note is to show that Fréchet differentiability cannot be replaced by Gateaux differentiability. This answers negatively a question raised by Wulbert [5]. In particular, we will renorm a certain nonreflexive space with a smooth norm whose dual is also smooth.

scholarly journals On the semigroup of Hadamard differentiable mappings

1972 ◽  
Vol 14 (3) ◽  
pp. 329-334
Author(s):  
Sadayuki Yamamuro
Keyword(s):  
Banach Space ◽  
Real Banach Space ◽  
Differentiable Mappings

The main purpose of this paper is to prove that every automorphism of the semigroup of all Hadamard-differentiable mappings of a separable real Banach space into itself is inner. This generalizes the results of [7] which is a generalization of a result proved by Magill, Jr. [5].

scholarly journals Approximating common fixed points of two nonexpansive mappings in Banach spaces

1998 ◽  
Vol 57 (1) ◽  
pp. 117-127 ◽  
Author(s):  
Sachiko Atsushiba ◽  
Wataru Takahashi
Keyword(s):  
Banach Space ◽  
Fixed Points ◽  
Real Banach Space ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Nonempty Closed Convex Subset ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Fréchet Differentiable ◽  
Image Position

Let C be a nonempty closed convex subset of a real Banach space E and let S, T be nonexpansive mappings of C into itself. In this paper, we consider the following iteration procedure of Mann's type for approximating common fixed points of two mappings S and T:where {αn is a sequence in [0,1]. Using some ideas in the nonlinear ergodic theory, we prove that the iterates converge weakly to a common fixed point of the nonexpansive mappings T and S in a uniformly convex Banach space which satisfies Opial's condition or whose norm is Fréchet differentiable.

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

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.

scholarly journals A characterisation of Hilbert spaces via orthogonality and proximinality

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.

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

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.

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

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.

scholarly journals Strong convergence of an inertial forward-backward splitting method for accretive operators in real Banach space

2020 ◽  
Vol 21 (2) ◽  
pp. 397-412 ◽  
Author(s):  
H.A. Abass ◽  
◽  
C. Izuchukwu ◽  
O.T. Mewomo ◽  
Q.L. Dong ◽  
...  
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Real Banach Space ◽  
Splitting Method ◽  
Accretive Operators

Fixed point theorems for uniformly generalized Kannan type semigroup of self-mappings

2017 ◽  
Vol 26 (2) ◽  
pp. 231-240
Author(s):  
AHMED H. SOLIMAN ◽  
MOHAMMAD IMDAD ◽  
MD AHMADULLAH
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Common Fixed Point ◽  
Convex Subset ◽  
Fixed Point Theorems ◽  
Real Banach Space ◽  
Normal Structure ◽  
Uniform Normal Structure

In this paper, we consider a new uniformly generalized Kannan type semigroup of self-mappings defined on a closed convex subset of a real Banach space equipped with uniform normal structure and employ the same to show that such semigroup of self-mappings admits a common fixed point provided the underlying semigroup of self-mappings has a bounded orbit.

scholarly journals Iterative methods for zero points of accretive operators in Banach spaces

2012 ◽  
Vol 20 (1) ◽  
pp. 329-344
Author(s):  
Sheng Hua Wang ◽  
Sun Young Cho ◽  
Xiao Long Qin
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Convex Function ◽  
Real Banach Space ◽  
Real Hilbert Space ◽  
Lower Semicontinuous ◽  
Iterative Algorithms ◽  
Accretive Operators ◽  
Weak And Strong Convergence ◽  
Zero Points

Abstract The purpose of this paper is to consider the problem of approximating zero points of accretive operators. We introduce and analysis Mann-type iterative algorithm with errors and Halpern-type iterative algorithms with errors. Weak and strong convergence theorems are established in a real Banach space. As applications, we consider the problem of approximating a minimizer of a proper lower semicontinuous convex function in a real Hilbert space

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