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

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.

scholarly journals On Best Proximity Results for a Generalized Modified Ishikawa’s Iterative Scheme Driven by Perturbed 2-Cyclic Like-Contractive Self-Maps in Uniformly Convex Banach Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-15 ◽  
Author(s):  
M. De la Sen ◽  
Mujahid Abbas
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Iterative Process ◽  
Iterative Scheme ◽  
Convex Banach Space ◽  
The Self ◽  
Uniformly Convex ◽  
Cyclic Structure ◽  
Best Proximity Points ◽  
Uniformly Convex Banach Spaces

This paper proposes a generalized modified iterative scheme where the composed self-mapping driving can have distinct step-dependent composition order in both the auxiliary iterative equation and the main one integrated in Ishikawa’s scheme. The self-mapping which drives the iterative scheme is a perturbed 2-cyclic one on the union of two sequences of nonempty closed subsets Ann=0∞ and Bnn=0∞ of a uniformly convex Banach space. As a consequence of the perturbation, such a driving self-mapping can lose its cyclic contractive nature along the transients of the iterative process. These sequences can be, in general, distinct of the initial subsets due to either computational or unmodeled perturbations associated with the self-mapping calculations through the iterative process. It is assumed that the set-theoretic limits below of the sequences of sets Ann=0∞ and Bnn=0∞ exist. The existence of fixed best proximity points in the set-theoretic limits of the sequences to which the iterated sequences converge is investigated in the case that the cyclic disposal exists under the asymptotic removal of the perturbations or under its convergence of the driving self-mapping to a limit contractive cyclic structure.

scholarly journals Common fixed point iterations for a class of multi-valued mappings in $$\mathbf {CAT}(0)$$ spaces

2020 ◽  
Vol 9 (3) ◽  
pp. 681-690
Author(s):  
Khairul Saleh ◽  
Hafiz Fukhar-ud-din
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Finite Family ◽  
Uniformly Convex ◽  
Uniformly Convex Banach Spaces ◽  
Fixed Point Iterations

Abstract In this work, we propose an iterative scheme to approach common fixed point(s) of a finite family of generalized multi-valued nonexpansive mappings in a CAT(0) space. We establish and prove convergence theorems for the algorithm. The results are new and interesting in the theory of $$CAT\left( 0\right) $$ C A T 0 spaces and are the analogues of corresponding ones in uniformly convex Banach spaces and Hilbert spaces.

BEST PROXIMITY POINT THEOREMS FOR CYCLIC QUASI-CONTRACTION MAPS IN UNIFORMLY CONVEX BANACH SPACES

2016 ◽  
Vol 95 (1) ◽  
pp. 149-156 ◽  
Author(s):  
NGUYEN VAN DUNG ◽  
VO THI LE HANG
Keyword(s):  
Banach Spaces ◽  
Point Theorem ◽  
Best Proximity Point ◽  
Negative Answer ◽  
Uniformly Convex ◽  
Contraction Mappings ◽  
Uniformly Convex Banach Spaces ◽  
Contraction Maps

In this paper we first give a negative answer to a question of Amini-Harandi [‘Best proximity point theorems for cyclic strongly quasi-contraction mappings’, J. Global Optim.56 (2013), 1667–1674] on a best proximity point theorem for cyclic quasi-contraction maps. Then we prove some new results on best proximity point theorems that show that results of Amini-Harandi for cyclic strongly quasi-contractions are true under weaker assumptions.

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