scholarly journals On Generalized Nonexpansive Maps in Banach Spaces

Computation ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 61 ◽  
Author(s):  
Kifayat Ullah ◽  
Junaid Ahmad ◽  
Manuel de la Sen
Keyword(s):  
Banach Spaces ◽  
General Class ◽  
Iterative Algorithms ◽  
Uniformly Convex ◽  
Demiclosed Principle ◽  
New Class ◽  
Nonexpansive Maps ◽  
Convergence Results ◽  
Uniformly Convex Banach Spaces ◽  
Expansive Maps

We introduce a very general class of generalized non-expansive maps. This new class of maps properly includes the class of Suzuki non-expansive maps, Reich–Suzuki type non-expansive maps, and generalized α -non-expansive maps. We establish some basic properties and demiclosed principle for this class of maps. After this, we establish existence and convergence results for this class of maps in the context of uniformly convex Banach spaces and compare several well known iterative algorithms.

scholarly journals A one-step-two-mappings iterative scheme for multi-valued maps in W-hyperbolic spaces

Filomat ◽  
2018 ◽  
Vol 32 (4) ◽  
pp. 1403-1411 ◽  
Author(s):  
Birol Gunduz ◽  
Sezgin Akbulut
Keyword(s):  
Banach Spaces ◽  
Hyperbolic Space ◽  
Iterative Scheme ◽  
Hyperbolic Spaces ◽  
Uniformly Convex ◽  
Convergence Theorems ◽  
Nonexpansive Maps ◽  
Uniformly Convex Banach Spaces ◽  
One Step ◽  
Geodesic Spaces

In this paper, we study a one-step iterative scheme for two multi-valued nonexpansive maps in W-hyperbolic spaces. We establish strong and ?-convergence theorems for the proposed algorithm in a uniformly convex W-hyperbolic space which improve and extend the corresponding known results in uniformly convex Banach spaces as well as CAT(0) spaces. Our new results are also valid in geodesic spaces.

scholarly journals Fixed point approximation of Presic nonexpansive mappings in product of CAT (0) spaces

2016 ◽  
Vol 32 (3) ◽  
pp. 315-322
Author(s):  
HAFIZ FUKHAR-UD-DIN ◽  
◽  
VASILE BERINDE ◽  
ABDUL RAHIM KHAN ◽  
◽  
...  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fixed Points ◽  
Banach Spaces ◽  
Point Theorem ◽  
Nonexpansive Mappings ◽  
Iterative Algorithms ◽  
Control Parameters ◽  
Uniformly Convex ◽  
Uniformly Convex Banach Spaces

We obtain a fixed point theorem for Presiˇ c nonexpansive mappings on the product of ´ CAT (0) spaces and approximate this fixed points through Ishikawa type iterative algorithms under relaxed conditions on the control parameters. Our results are new in the literature and are valid in uniformly convex Banach spaces.

scholarly journals Some Convergence Results for a Class of Generalized Nonexpansive Mappings in Banach Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Thabet Abdeljawad ◽  
Kifayat Ullah ◽  
Junaid Ahmad ◽  
Manuel de la Sen ◽  
Muhammad Naveed Khan
Keyword(s):  
Weak Convergence ◽  
Iterative Method ◽  
Fixed Points ◽  
Banach Spaces ◽  
Current Literature ◽  
Nonexpansive Mappings ◽  
Uniformly Convex ◽  
Strong And Weak Convergence ◽  
Convergence Results ◽  
Uniformly Convex Banach Spaces

This paper investigates fixed points of Reich-Suzuki-type nonexpansive mappings in the context of uniformly convex Banach spaces through an M ∗ iterative method. Under some appropriate situations, some strong and weak convergence theorems are established. To support our results, a new example of Reich-Suzuki-type nonexpansive mappings is presented which exceeds the class of Suzuki-type nonexpansive mappings. The presented results extend some recently announced results of current literature.

scholarly journals Fixed Point Approximation of Nonexpansive Mappings on a Nonlinear Domain

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Safeer Hussain Khan
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Hyperbolic Space ◽  
Iterative Process ◽  
Nonexpansive Mappings ◽  
Hyperbolic Spaces ◽  
Uniformly Convex ◽  
Convergence Results ◽  
Uniformly Convex Banach Spaces ◽  
Uniformly Convex Hyperbolic Space

We use a three-step iterative process to prove some strong andΔ-convergence results for nonexpansive mappings in a uniformly convex hyperbolic space, a nonlinear domain. Three-step iterative processes have numerous applications and hyperbolic spaces contain Banach spaces (linear domains) as well as CAT(0) spaces. Thus our results can be viewed as extension and generalization of several known results in uniformly convex Banach spaces as well as CAT(0) spaces.

scholarly journals Approximation of Fixed Points for Suzuki’s Generalized Non-Expansive Mappings

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 522 ◽  
Author(s):  
Javid Ali ◽  
Faeem Ali ◽  
Puneet Kumar
Keyword(s):  
Strong Convergence ◽  
Fixed Points ◽  
Banach Spaces ◽  
Iterative Scheme ◽  
Uniformly Convex ◽  
Weak And Strong Convergence ◽  
Numerical Example ◽  
Matlab Program ◽  
Convergence Results ◽  
Uniformly Convex Banach Spaces

In this paper, we study a three step iterative scheme to approximate fixed points of Suzuki’s generalized non-expansive mappings. We establish some weak and strong convergence results for such mappings in uniformly convex Banach spaces. Further, we show numerically that the considered iterative scheme converges faster than some other known iterations for Suzuki’s generalized non-expansive mappings. To support our claim, we give an illustrative numerical example and approximate fixed points of such mappings using Matlab program. Our results are new and generalize several relevant results in the literature.

scholarly journals Approximation of Fixed Points for Suzuki's Generalized Non-Expansive Mappings

2019 ◽  
Author(s):  
Javid Ali ◽  
Faeem Ali ◽  
Puneet Kumar
Keyword(s):  
Strong Convergence ◽  
Fixed Points ◽  
Banach Spaces ◽  
Iterative Scheme ◽  
Uniformly Convex ◽  
Weak And Strong Convergence ◽  
Matlab Program ◽  
Convergence Results ◽  
Uniformly Convex Banach Spaces

In this paper, we study a three step iterative scheme to approximate fixed points of Suzuki's generalized non-expansive mappings. We establish some weak and strong convergence results for such mappings in uniformly convex Banach spaces. Further, we show numerically that iterative scheme (1.8) converges faster than some other known iterations for Suzuki's generalized non-expansive mappings. To support our claim, we give an illustrative example and approximate fixed points of such mappings using Matlab program. Our results are new and generalize several relevant results in the literature.

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.

scholarly journals On $$\alpha $$-Firmly Nonexpansive Operators in r-Uniformly Convex Spaces

2021 ◽  
Vol 76 (4) ◽  
Author(s):  
Arian Bërdëllima ◽  
Gabriele Steidl
Keyword(s):  
Banach Spaces ◽  
Semigroup Theory ◽  
General Setting ◽  
Weak Limit ◽  
Uniformly Convex ◽  
Fixed Point Set ◽  
Uniformly Convex Banach Spaces ◽  
Differentiable Norm ◽  
Firmly Nonexpansive ◽  
Nonexpansive Operators

AbstractWe introduce the class of $$\alpha $$ α -firmly nonexpansive and quasi $$\alpha $$ α -firmly nonexpansive operators on r-uniformly convex Banach spaces. This extends the existing notion from Hilbert spaces, where $$\alpha $$ α -firmly nonexpansive operators coincide with so-called $$\alpha $$ α -averaged operators. For our more general setting, we show that $$\alpha $$ α -averaged operators form a subset of $$\alpha $$ α -firmly nonexpansive operators. We develop some basic calculus rules for (quasi) $$\alpha $$ α -firmly nonexpansive operators. In particular, we show that their compositions and convex combinations are again (quasi) $$\alpha $$ α -firmly nonexpansive. Moreover, we will see that quasi $$\alpha $$ α -firmly nonexpansive operators enjoy the asymptotic regularity property. Then, based on Browder’s demiclosedness principle, we prove for r-uniformly convex Banach spaces that the weak cluster points of the iterates $$x_{n+1}:=Tx_{n}$$ x n + 1 : = T x n belong to the fixed point set $${{\,\mathrm{Fix}\,}}T$$ Fix T whenever the operator T is nonexpansive and quasi $$\alpha $$ α -firmly. If additionally the space has a Fréchet differentiable norm or satisfies Opial’s property, then these iterates converge weakly to some element in $${{\,\mathrm{Fix}\,}}T$$ Fix T . Further, the projections $$P_{{{\,\mathrm{Fix}\,}}T}x_n$$ P Fix T x n converge strongly to this weak limit point. Finally, we give three illustrative examples, where our theory can be applied, namely from infinite dimensional neural networks, semigroup theory, and contractive projections in $$L_p$$ L p , $$p \in (1,\infty ) \backslash \{2\}$$ p ∈ ( 1 , ∞ ) \ { 2 } spaces on probability measure spaces.

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