Fixed point properties for semigroup of nonexpansive mappings on Fréchet spaces

2009 ◽  
Vol 70 (11) ◽  
pp. 3837-3841 ◽  
Author(s):  
Anthony To-Ming Lau ◽  
Wataru Takahashi
Keyword(s):  
Fixed Point ◽  
Nonexpansive Mappings ◽  
Fréchet Spaces ◽  
Fixed Point Properties ◽  
Semigroup Of Nonexpansive Mappings

scholarly journals Common fixed points of one-parameter nonexpansive semigroups in strictly convex Banach spaces

2006 ◽  
Vol 2006 ◽  
pp. 1-10 ◽  
Author(s):  
Tomonari Suzuki
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Convex Subset ◽  
Continuous Semigroup ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Strongly Continuous Semigroup ◽  
Nonexpansive Semigroups ◽  
Semigroup Of Nonexpansive Mappings ◽  
Strictly Convex Banach Spaces

One of our main results is the following convergence theorem for one-parameter nonexpansive semigroups: letCbe a bounded closed convex subset of a Hilbert spaceE, and let{T(t):t∈ℝ+}be a strongly continuous semigroup of nonexpansive mappings onC. Fixu∈Candt1,t2∈ℝ+witht1<t2. Define a sequence{xn}inCbyxn=(1−αn)/(t2−t1)∫t1t2T(s)xnds+αnuforn∈ℕ, where{αn}is a sequence in(0,1)converging to0. Then{xn}converges strongly to a common fixed point of{T(t):t∈ℝ+}.

scholarly journals Fixed point properties for semigroups of nonexpansive mappings on convex sets in dual Banach spaces

2018 ◽  
Vol 17 (1) ◽  
pp. 67-87
Author(s):  
Anthony To-Ming Lau ◽  
Yong Zhang
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Convex Subset ◽  
Convex Sets ◽  
Compact Convex ◽  
Nonexpansive Mappings ◽  
Point Property ◽  
Fixed Point Properties ◽  
Dual Banach Space ◽  
Standing Problem

Abstract It has been a long-standing problem posed by the first author in a conference in Marseille in 1990 to characterize semitopological semigroups which have common fixed point property when acting on a nonempty weak* compact convex subset of a dual Banach space as weak* continuous and norm nonexpansive mappings. Our investigation in the paper centers around this problem. Our main results rely on the well-known Ky Fan’s inequality for convex functions.

Amenability and Fixed Point Properties of Semitopological Semigroups in Modular Vector Spaces

2019 ◽  
Vol 63 (3) ◽  
pp. 692-704
Author(s):  
Khadime Salame
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Nonexpansive Mappings ◽  
Continuous Functions ◽  
Modular Space ◽  
Modular Spaces ◽  
Fixed Point Properties ◽  
Uniformly Continuous ◽  
Locally Convex

AbstractIn this paper, we initiate the study of fixed point properties of amenable or reversible semitopological semigroups in modular spaces. Takahashi’s fixed point theorem for amenable semigroups of nonexpansive mappings, and T. Mitchell’s fixed point theorem for reversible semigroups of nonexpansive mappings in Banach spaces are extended to the setting of modular spaces. Among other things, we also generalize another classical result due to Mitchell characterizing the left amenability property of the space of left uniformly continuous functions on semitopological semigroups by introducing the notion of a semi-modular space as a generalization of the concept of a locally convex space.

scholarly journals Common Fixed Point and Invariant Approximation Results on Non-Starshaped Domains

2005 ◽  
Vol 12 (4) ◽  
pp. 659-669
Author(s):  
Nawab Hussain ◽  
Donal O'Regan ◽  
Ravi P. Agarwal
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Fréchet Spaces ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Invariant Approximation ◽  
Commuting Maps

Abstract We extend the concept of 𝑅-subweakly commuting maps due to Shahzad [J. Math. Anal. Appl. 257: 39–45, 2001] to the case of non-starshaped domains and obtain common fixed point results for this class of maps on non-starshaped domains in the setup of Fréchet spaces. As applications, we establish Brosowski–Meinardus type approximation theorems. Our results unify and extend the results of Al-Thagafi, Dotson, Habiniak, Jungck and Sessa, Sahab, Khan and Sessa and Shahzad.

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