scholarly journals A new iteration procedure for multivalued nonexpansive maps in uniformly convex Banach spaces

2019 ◽  
Vol 2019 (-) ◽  
Author(s):  
G.V.R. Babu ◽  
G. Satyanarayana
Keyword(s):  
Banach Spaces ◽  
Iteration Procedure ◽  
Uniformly Convex ◽  
Nonexpansive Maps ◽  
Uniformly Convex Banach Spaces

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 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 Common Fixed Points of Two G -Nonexpansive Mappings via a Faster Iteration Procedure

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Sabiya Khatoon ◽  
Izhar Uddin ◽  
Javid Ali ◽  
Reny George
Keyword(s):  
Fixed Points ◽  
Banach Spaces ◽  
Directed Graph ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Iteration Procedure ◽  
Uniformly Convex ◽  
Numerical Example ◽  
Uniformly Convex Banach Spaces

In this work, we study the convergence of a new faster iteration in which two G -nonexpansive mappings are involved in the setting of uniformly convex Banach spaces with a directed graph. Moreover, by constructing a numerical example, we show the fastness of our iteration procedure over other existing iteration procedures 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.

Uniformly convex Banach spaces are reflexive-constructively

2013 ◽  
Vol 59 (4-5) ◽  
pp. 352-356
Author(s):  
Douglas S. Bridges ◽  
Hajime Ishihara ◽  
Maarten McKubre-Jordens
Keyword(s):  
Banach Spaces ◽  
Uniformly Convex ◽  
Uniformly Convex Banach Spaces
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