scholarly journals Browder’s Convergence Theorem for Multivalued Mappings in Banach Spaces without the Endpoint Condition

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Thanomsak Laokul
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convergence Theorem ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Multivalued Mappings ◽  
Uniformly Convex ◽  
Uniformly Gâteaux Differentiable Norm ◽  
Differentiable Norm

We prove Browder’s convergence theorem for multivalued mappings in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm by using the notion of diametrically regular mappings. Our results are significant improvement on results of Jung (2007) and Panyanak and Suantai (2020).

scholarly journals Weak convergence theorem for Passty type asymptotically nonexpansive mappings

1999 ◽  
Vol 22 (1) ◽  
pp. 217-220
Author(s):  
B. K. Sharma ◽  
B. S. Thakur ◽  
Y. J. Cho
Keyword(s):  
Banach Space ◽  
Convergence Theorem ◽  
Nonexpansive Mappings ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive ◽  
Differentiable Norm ◽  
Weak Convergence Theorem ◽  
Fréchet Differentiable

In this paper, we prove a convergence theorem for Passty type asymptotically nonexpansive mappings in a uniformly convex Banach space with Fréchet-differentiable norm.

scholarly journals Weak Convergence of the Projection Type Ishikawa Iteration Scheme for Two Asymptotically Nonexpansive Nonself-Mappings

2011 ◽  
Vol 2011 ◽  
pp. 1-19
Author(s):  
Tanakit Thianwan
Keyword(s):  
Banach Space ◽  
Weak Convergence ◽  
Iteration Scheme ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Ishikawa Iteration ◽  
Asymptotically Nonexpansive ◽  
Differentiable Norm ◽  
Fréchet Differentiable

We study weak convergence of the projection type Ishikawa iteration scheme for two asymptotically nonexpansive nonself-mappings in a real uniformly convex Banach spaceEwhich has a Fréchet differentiable norm or its dualE*has the Kadec-Klee property. Moreover, weak convergence of projection type Ishikawa iterates of two asymptotically nonexpansive nonself-mappings without any condition on the rate of convergence associated with the two maps in a uniformly convex Banach space is established. Weak convergence theorem without making use of any of the Opial's condition, Kadec-Klee property, or Fréchet differentiable norm is proved. Some results have been obtained which generalize and unify many important known results in recent literature.

scholarly journals COMMON FIXED POINTS FOR SEMIGROUPS OF POINTWISE LIPSCHITZIAN MAPPINGS IN BANACH SPACES

2011 ◽  
Vol 84 (3) ◽  
pp. 353-361 ◽  
Author(s):  
W. M. KOZLOWSKI
Keyword(s):  
Banach Space ◽  
Fixed Points ◽  
Banach Spaces ◽  
Convex Subset ◽  
Common Fixed Points ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Lipschitzian Mappings ◽  
Nonlinear Mappings

AbstractLet C be a bounded, closed, convex subset of a uniformly convex Banach space X. We investigate the existence of common fixed points for pointwise Lipschitzian semigroups of nonlinear mappings Tt:C→C, where each Tt is pointwise Lipschitzian. The latter means that there exists a family of functions αt:C→[0,∞) such that $\|T_t(x)-T_t(y)\| \leq \alpha _{t}(x)\|x-y\|$ for x,y∈C. We also demonstrate how the asymptotic aspect of the pointwise Lipschitzian semigroups can be expressed in terms of the respective Fréchet derivatives.

scholarly journals Convergence results of a faster iterative scheme including multi-valued mappings in Banach spaces

Filomat ◽  
2021 ◽  
Vol 35 (4) ◽  
pp. 1359-1368
Author(s):  
Kifayat Ullah ◽  
Junaid Ahmad ◽  
Muhammad Khan ◽  
Naseer Muhammad
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Current Literature ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Numerical Efficiency ◽  
Convergence Results

In this paper, we study M-iterative scheme in the new context of multi-valued generalized ?-nonexpansive mappings. A uniformly convex Banach space is used as underlying setting for our approach. We also provide a new example of generalized ?-nonexpasive mappings. We connect M iterative scheme and other well known schemes with this example, to show the numerical efficiency of our results. Our results improve and extend many existing results in the current literature.

scholarly journals Ishikawa iteration process with errors for nonexpansive mappings in uniformly convex Banach spaces

2000 ◽  
Vol 24 (1) ◽  
pp. 49-53 ◽  
Author(s):  
Deng Lei ◽  
Li Shenghong
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Ishikawa Iteration ◽  
Uniformly Convex Banach Spaces ◽  
Ishikawa Iteration Process

We shall consider the behaviour of Ishikawa iteration with errors in a uniformly convex Banach space. Then we generalize the two theorems of Tan and Xu without the restrictions thatCis bounded andlimsupnsn<1.

scholarly journals Strong convergence theorems of three-step iterations for multi-valued mappings in Banach spaces

Filomat ◽  
2011 ◽  
Vol 25 (1) ◽  
pp. 173-184
Author(s):  
S. Homaeipour ◽  
A. Razani
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Convergence Theorems

In this paper, two classes of three-step iteration schemes for multi-valued mappings in a uniformly convex Banach space are presented. Moreover, their strong convergence are proved.

scholarly journals Composite Iterative Algorithms for Variational Inequality and Fixed Point Problems in Real Smooth and Uniformly Convex Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-21 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Ching-Feng Wen
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Iterative Algorithm ◽  
Nonexpansive Mappings ◽  
Iterative Algorithms ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Point Problem ◽  
Uniformly Convex ◽  
Differentiable Norm

We introduce composite implicit and explicit iterative algorithms for solving a general system of variational inequalities and a common fixed point problem of an infinite family of nonexpansive mappings in a real smooth and uniformly convex Banach space. These composite iterative algorithms are based on Korpelevich's extragradient method and viscosity approximation method. We first consider and analyze a composite implicit iterative algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space and then another composite explicit iterative algorithm in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literatures.

scholarly journals Approximating Fixed Points of Operators Satisfying (RCSC) Condition in Banach Spaces

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Thabet Abdeljawad ◽  
Kifayat Ullah ◽  
Junaid Ahmad ◽  
Manuel de la Sen ◽  
Junaid Khan
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Fixed Points ◽  
Banach Spaces ◽  
Nonempty Subset ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Weak And Strong Convergence ◽  
Numerical Efficiency

Let K be a nonempty subset of a Banach space E. A mapping T:K→K is said to satisfy (RCSC) condition if each a,b∈K, 1/2a−Fa≤a−b⇒Fa−Fb≤1/3a−b+a−Fb+b−Fa. In this paper, we study, under some appropriate conditions, weak and strong convergence for this class of maps through M iterates in uniformly convex Banach space. We also present a new example of mappings with condition (RCSC). We connect M iteration and other well-known processes with this example to show the numerical efficiency of our results. The presented results improve and extend the corresponding results of the literature.

scholarly journals p-Uniform Convexity andq-Uniform Smoothness of Absolute Normalized Norms onℂ2

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Tomonari Suzuki
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Banach Space ◽  
Uniform Convexity ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Uniform Smoothness

We first prove characterizations ofp-uniform convexity andq-uniform smoothness. We next give a formulation on absolute normalized norms onℂ2. Using these, we present some examples of Banach spaces. One of them is a uniformly convex Banach space which is notp-uniformly convex.

scholarly journals A note on best approximation and invertibility of operators on uniformly convex Banach spaces

1991 ◽  
Vol 14 (3) ◽  
pp. 611-614 ◽  
Author(s):  
James R. Holub
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Best Approximation ◽  
Bounded Linear Operator ◽  
Convex Banach Space ◽  
Sufficient Condition ◽  
Uniformly Convex ◽  
Bounded Linear ◽  
Uniformly Convex Banach Spaces

It is shown that ifXis a uniformly convex Banach space andSa bounded linear operator onXfor which‖I−S‖=1, thenSis invertible if and only if‖I−12S‖<1. From this it follows that ifSis invertible onXthen either (i)dist(I,[S])<1, or (ii)0is the unique best approximation toIfrom[S], a natural (partial) converse to the well-known sufficient condition for invertibility thatdist(I,[S])<1.

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