scholarly journals On a Class of Generalized Nonexpansive Mappings

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1085 ◽  
Author(s):  
Simeon Reich ◽  
Alexander J. Zaslavski
Keyword(s):  
Hilbert Space ◽  
Nonexpansive Mapping ◽  
Fixed Points ◽  
Convex Set ◽  
Nonexpansive Mappings ◽  
General Function ◽  
New Class ◽  
Generalized Nonexpansive Mapping ◽  
Closed And Convex Set ◽  
Definition Of

In our recent work we have introduced and studied a notion of a generalized nonexpansive mapping. In the definition of this notion the norm has been replaced by a general function satisfying certain conditions. For this new class of mappings, we have established the existence of unique fixed points and the convergence of iterates. In the present paper we construct an example of a generalized nonexpansive self-mapping of a bounded, closed and convex set in a Hilbert space, which is not nonexpansive in the classical sense.

scholarly journals ON STRONG CONVERGENCE OF HALPERN’S METHOD FOR QUASI-NONEXPANSIVE MAPPINGS IN HILBERT SPACES

2016 ◽  
Vol 21 (1) ◽  
pp. 63-82 ◽  
Author(s):  
Jesus Garcia Falset ◽  
Enrique Llorens-Fuster ◽  
Giuseppe Marino ◽  
Angela Rugiano
Keyword(s):  
Hilbert Space ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Fixed Points ◽  
Hilbert Spaces ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Type Method ◽  
Numerical Example

In this paper, we introduce a Halpern’s type method to approximate common fixed points of a nonexpansive mapping T and a strongly quasi-nonexpansive mappings S, defined in a Hilbert space, such that I − S is demiclosed at 0. The result shows as the same algorithm converges to different points, depending on the assumptions of the coefficients. Moreover, a numerical example of our iterative scheme is given.

scholarly journals Fixed Point Theorems for an Elastic Nonlinear Mapping in Banach Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Hiroko Manaka
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Nonexpansive Mapping ◽  
Fixed Points ◽  
Fixed Point Theorems ◽  
Duality Mapping ◽  
Nonlinear Mapping ◽  
Normalized Duality Mapping ◽  
Strongly Nonexpansive Mapping ◽  
Weak Convergence Theorem

LetEbe a smooth Banach space with a norm·. LetV(x,y)=x2+y2-2 x,Jyfor anyx,y∈E, where·,·stands for the duality pair andJis the normalized duality mapping. We define aV-strongly nonexpansive mapping byV(·,·). This nonlinear mapping is nonexpansive in a Hilbert space. However, we show that there exists aV-strongly nonexpansive mapping with fixed points which is not nonexpansive in a Banach space. In this paper, we show a weak convergence theorem and strong convergence theorems for fixed points of this elastic nonlinear mapping and give the existence theorem.

scholarly journals Approximating fixed points of enriched nonexpansive mappings in Banach spaces by using a retraction-displacement condition

2020 ◽  
Vol 36 (1) ◽  
pp. 27-34 ◽  
Author(s):  
VASILE BERINDE
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Iterative Algorithm ◽  
Hilbert Spaces ◽  
Nonexpansive Mappings ◽  
Convergence Theorems ◽  
Displacement Condition ◽  
Nonlinear Mappings

In this paper, we prove convergence theorems for a fixed point iterative algorithm of Krasnoselskij-Mann typeassociated to the class of enriched nonexpansive mappings in Banach spaces. The results are direct generaliza-tions of the corresponding ones in [Berinde, V.,Approximating fixed points of enriched nonexpansive mappings byKrasnoselskij iteration in Hilbert spaces, Carpathian J. Math., 35 (2019), No. 3, 293–304.], from the setting of Hilbertspaces to Banach spaces, and also of some results in [Senter, H. F. and Dotson, Jr., W. G.,Approximating fixed pointsof nonexpansive mappings, Proc. Amer. Math. Soc.,44(1974), No. 2, 375–380.], [Browder, F. E., Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197–228.], byconsidering enriched nonexpansive mappings instead of nonexpansive mappings. Many other related resultsin literature can be obtained as particular instances of our results.

scholarly journals New Family of Multivalent Analytic Functions Defined on Complex Hilbert Space

2020 ◽  
pp. 155-165 ◽  
Author(s):  
Abbas Kareem Wanas ◽  
S. R. Swamy
Keyword(s):  
Hilbert Space ◽  
Analytic Functions ◽  
Convex Set ◽  
Extreme Points ◽  
Complex Hilbert Space ◽  
Coefficient Estimates ◽  
Geometric Properties ◽  
New Class ◽  
New Family

In this article, we define a certain new class of multivalent analytic functions with negative coefficients on complex Hilbert space. We derive a number of important geometric properties, such as, coefficient estimates, radii of starlikeness and convexity, extreme points and convex set.

scholarly journals Common fixed points of generalized almost nonexpansive mappings

Filomat ◽  
2010 ◽  
Vol 24 (3) ◽  
pp. 11-18 ◽  
Author(s):  
Mujahid Abbas ◽  
Dejan Ilic
Keyword(s):  
Fixed Points ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Approximation Result ◽  
New Class ◽  
Invariant Approximation

The concept of a generalized almost nonexpansive mappings is introduced and the existence of common fixed points for this new class of mappings is proved. As an application, an invariant approximation result is obtained.

Adjoint Abelian Operators on Banach Space

1969 ◽  
Vol 21 ◽  
pp. 505-512 ◽  
Author(s):  
J. G. Stampfli
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Spectral Type ◽  
New Class ◽  
Definition Of ◽  
Adjoint Operators

I. In the first part of this paper we introduce a new class of operators, mentioned in the title. It is easy to say that these are a generalization of self-adjoint operators for Hilbert space. This is deceptive since it implies that the definition of self-adjointness is forced into the unnatural setting of a Banach space. We feel that the definition of adjoint abelian preserves the obvious distinction between a space and its dual. Certain attractive properties of self-adjoint operators have already been singled out and carried over to Banach space. Specifically, we mention the notion of hermitian (see 17; 11), and spectral type operators (see 4). There is some comparison of these concepts later.

scholarly journals Strong Convergence Theorems for Equilibrium Problems and Fixed Point Problems in Hilbert Spaces

2010 ◽  
Vol 2010 ◽  
pp. 1-13
Author(s):  
Jian-Wen Peng ◽  
Yan Wang
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Fixed Points ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Common Element ◽  
Convergence Theorems

We introduce an Ishikawa iterative scheme by the viscosity approximate method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in Hilbert space. Then, we prove some strong convergence theorems which extend and generalize S. Takahashi and W. Takahashi's results (2007).

scholarly journals Characterizations of fixed points of nonexpansive mappings

2005 ◽  
Vol 2005 (11) ◽  
pp. 1723-1735 ◽  
Author(s):  
Tomonari Suzuki
Keyword(s):  
Banach Space ◽  
Nonexpansive Mapping ◽  
Fixed Points ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Strict Convexity ◽  
Banach Limit ◽  
Banach Limits

Using the notion of Banach limits, we discuss the characterization of fixed points of nonexpansive mappings in Banach spaces. Indeed, we prove that the two sets of fixed points of a nonexpansive mapping and some mapping generated by a Banach limit coincide. In our discussion, we may not assume the strict convexity of the Banach space.

scholarly journals Some Results on the Approximation of Solutions of Variational Inequalities for Multivalued Maps on Banach Spaces

2021 ◽  
Vol 18 (4) ◽  
Author(s):  
Luigi Muglia ◽  
Giuseppe Marino
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Nonexpansive Mapping ◽  
Fixed Points ◽  
Banach Spaces ◽  
Unique Solution ◽  
Nonexpansive Mappings ◽  
Multivalued Maps ◽  
Demiclosedness Principle

AbstractMultivalued $$*$$ ∗ -nonexpansive mappings are studied in Banach spaces. The demiclosedness principle is established. Here we focus on the problem of solving a variational inequality which is defined on the set of fixed points of a multivalued $$*$$ ∗ -nonexpansive mapping. For this purpose, we introduce two algorithms approximating the unique solution of the variational inequality.

scholarly journals On weak and strong convergence of an explicit iteration process for a total asymptotically quasi-I-nonexpansive mapping in Banach space

Filomat ◽  
2014 ◽  
Vol 28 (8) ◽  
pp. 1699-1710
Author(s):  
Hukmi Kiziltunc ◽  
Yunus Purtas
Keyword(s):  
Banach Space ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Iterative Process ◽  
Iteration Process ◽  
Nonempty Closed Convex Subset ◽  
Weak And Strong Convergence ◽  
New Class ◽  
Convergence Results ◽  
Definition Of

In this paper, we introduce a new class of Lipschitzian maps and prove some weak and strong convergence results for explicit iterative process using a more satisfactory definition of self mappings. Our results approximate common fixed point of a total asymptotically quasi-I-nonexpansive mapping T and a total asymptotically quasi-nonexpansive mapping I, defined on a nonempty closed convex subset of a Banach space.

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