scholarly journals Common Fixed Points Iteration Processes for a Finite Family of Asymptotically Nonexpansive Mappings

2004 ◽  
Vol 11 (1) ◽  
pp. 83-92
Author(s):  
Jui-Chi Huang
Keyword(s):  
Fixed Points ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Nonempty Closed Convex Subset ◽  
Convex Banach Space ◽  
Finite Family ◽  
Uniformly Convex ◽  
Asymptotically Nonexpansive ◽  
Iteration Processes ◽  
Suitable Sequence

Abstract Let 𝐸 be a uniformly convex Banach space which satisfies Opial's condition or its dual 𝐸* has the Kadec–Klee property, 𝐶 a nonempty closed convex subset of 𝐸, and 𝑇𝑗 : 𝐶 → 𝐶 an asymptotically nonexpansive mapping for each 𝑗 = 1, 2, . . . , 𝑟. Suppose {𝑥𝑛} is generated iteratively by where 𝑈𝑛(0) = 𝐼, 𝐼 is the identity map and {α 𝑛(𝑗)} is a suitable sequence in [0, 1]. If the set of common fixed points of is nonempty, then weak convergence of {𝑥𝑛} to some is obtained.

scholarly journals Convergence theorems of the sequence of iterates for a finite family asymptotically nonexpansive mappings

2001 ◽  
Vol 27 (11) ◽  
pp. 653-662 ◽  
Author(s):  
Jui-Chi Huang
Keyword(s):  
Banach Space ◽  
Nonexpansive Mappings ◽  
Iteration Scheme ◽  
Nonempty Closed Convex Subset ◽  
Convex Banach Space ◽  
Finite Family ◽  
Uniformly Convex ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive ◽  
Bounded Sequences

LetEbe a uniformly convex Banach space,Ca nonempty closed convex subset ofE. In this paper, we introduce an iteration scheme with errors in the sense of Xu (1998) generated by{Tj:C→C}j=1ras follows:Un(j)=an(j)I+bn(j)TjnUn(j−1)+cn(j)un(j),j=1,2,…,r,x1∈C,xn+1=an(r)xn+bn(r)TrnUn(r−1)xn+cn(r)un(r),n≥1, whereUn(0):=I,Ithe identity map; and{un(j)}are bounded sequences inC; and{an(j)},{bn(j)}, and{cn(j)}are suitable sequences in[0,1]. We first consider the behaviour of iteration scheme above for a finite family of asymptotically nonexpansive mappings. Then we generalize theorems of Schu and Rhoades.

scholarly journals Approximating common fixed points of two nonexpansive mappings in Banach spaces

1998 ◽  
Vol 57 (1) ◽  
pp. 117-127 ◽  
Author(s):  
Sachiko Atsushiba ◽  
Wataru Takahashi
Keyword(s):  
Banach Space ◽  
Fixed Points ◽  
Real Banach Space ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Nonempty Closed Convex Subset ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Fréchet Differentiable ◽  
Image Position

Let C be a nonempty closed convex subset of a real Banach space E and let S, T be nonexpansive mappings of C into itself. In this paper, we consider the following iteration procedure of Mann's type for approximating common fixed points of two mappings S and T:where {αn is a sequence in [0,1]. Using some ideas in the nonlinear ergodic theory, we prove that the iterates converge weakly to a common fixed point of the nonexpansive mappings T and S in a uniformly convex Banach space which satisfies Opial's condition or whose norm is Fréchet differentiable.

scholarly journals Approximating common fixed points of families of quasi-nonexpansive mappings

1995 ◽  
Vol 18 (2) ◽  
pp. 287-292 ◽  
Author(s):  
M. K. Ghosh ◽  
Lokenath Debnath
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fixed Points ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Convex Banach Space ◽  
Finite Family ◽  
Uniformly Convex ◽  
Special Cases

This paper deals with a family of quasi-nonexpansive mappings in a uniformly convex Banach space, and the convergence of iterates generated by this family. A fixed point theorem for two quasi-nonexpansive mappings is then proved. This theorem is then extended for a finite family of quasinonexpansive mappings. It is shown that Ishikawa's [1] result follows as special cases of results proved in this paper.

scholarly journals On Unification of the Strong Convergence Theorems for a Finite Family of Total Asymptotically Nonexpansive Mappings in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Farrukh Mukhamedov ◽  
Mansoor Saburov
Keyword(s):  
Strong Convergence ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Nonempty Closed Convex Subset ◽  
Finite Family ◽  
Convergence Theorems ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive ◽  
Total Asymptotically Nonexpansive Mappings

We unify all known iterative methods by introducing a new explicit iterative scheme for approximation of common fixed points of finite families of total asymptoticallyI-nonexpansive mappings. Note that such a scheme contains a particular case of the method introduced by (C. E. Chidume and E. U. Ofoedu, 2009). We construct examples of total asymptotically nonexpansive mappings which are not asymptotically nonexpansive. Note that no such kind of examples were known in the literature. We prove the strong convergence theorems for such iterative process to a common fixed point of the finite family of total asymptoticallyI-nonexpansive and total asymptotically nonexpansive mappings, defined on a nonempty closed-convex subset of uniformly convex Banach spaces. Moreover, our results extend and unify all known results.

Two-step iterative approximation of common fixed points of two nonself asymptotically quasi-nonexpansive mappings

2015 ◽  
Vol 08 (03) ◽  
pp. 1550060
Author(s):  
Amit Singh ◽  
R. C. Dimri ◽  
Darshana J. Prajapati
Keyword(s):  
Banach Space ◽  
Weak Convergence ◽  
Fixed Points ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Iterative Approximation ◽  
Convergence Theorems ◽  
Strong And Weak Convergence

In this paper, we study an iterative approximation of common fixed points of two nonself asymptotically quasi-nonexpansive mappings and we prove some strong and weak convergence theorems for such mappings in a uniformly convex Banach space.

scholarly journals Common Fixed Points of Two Multivalued Asymptotically Nonexpansive Mappings

2019 ◽  
Vol 12 (2) ◽  
pp. 348-357
Author(s):  
Safeer Hussain Khan ◽  
Hira Iqbal ◽  
Mujahid Abbas
Keyword(s):  
Fixed Points ◽  
Iterative Process ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Hyperbolic Spaces ◽  
Uniformly Convex ◽  
Convergence Theorems ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive ◽  
Ishikawa Iterative Process

In this paper, we construct a modified Ishikawa iterative process to approximate common fixed points for two multivalued asymptotically nonexpansive mappings and prove some convergence theorems in uniformly convex hyperbolic spaces.

scholarly journals Approximating common fixed points of finite family of asymptotically nonexpansive non-self mappings

Filomat ◽  
2008 ◽  
Vol 22 (2) ◽  
pp. 23-42
Author(s):  
G.S. Saluja
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Common Fixed Points ◽  
Convex Banach Space ◽  
Finite Family ◽  
Uniformly Convex ◽  
Asymptotically Nonexpansive ◽  
Nonexpansive Retract ◽  
Bounded Sequences ◽  
Condition B

Let K be a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T1 , T2 , ... , TN : K ? E be N asymptotically nonexpansive nonself mappings with sequences {rin} such that ??(n=1) rin < ?, for all 1 ? i ? N and n n=1 n F = ?N(i-1) F (Ti) ? ?. Let {?in}, {?in} and {?in} are sequences in [0, 1] with i=1 ?in + ?in + ?in = 1 for all i = 1, 2, ... , N . From arbitrary x1 ? K , define the sequence {xn} iteratively by (6), where {uin} are bounded sequences in K with ??(n=1) uin < ?. (i) If the dual E*of E has the Kadec-Klee property, then {xn} converges weakly to a common fixed point x*? F ; (ii) if {T1 , T2 , ... , TN} satisfies condition (B), then {xn} converges strongly to a common fixed point x*? F. .

scholarly journals Strong Convergence Theorems for Solutions of Equilibrium Problems and Common Fixed Points of a Finite Family of Asymptotically Nonextensive Nonself Mappings

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Lijuan Zhang ◽  
Hui Tong ◽  
Ying Liu
Keyword(s):  
Strong Convergence ◽  
Fixed Points ◽  
Equilibrium Problems ◽  
Common Fixed Points ◽  
Convex Banach Space ◽  
Finite Family ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Uniformly Convex ◽  
Uniformly Smooth

An iterative algorithm for finding a common element of the set of common fixed points of a finite family of asymptotically nonextensive nonself mappings and the set of solutions for equilibrium problems is discussed. A strong convergence theorem of common element is established in a uniformly smooth and uniformly convex Banach space.

scholarly journals Iterative Approximation of Common Fixed Points of Two Nonself Asymptotically Nonexpansive Mappings

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Esref Turkmen ◽  
Safeer Hussain Khan ◽  
Murat Ozdemir
Keyword(s):  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Nonempty Closed Convex Subset ◽  
Uniformly Convex ◽  
Iterative Approximation ◽  
Sunny Nonexpansive Retraction ◽  
Strong And Weak Convergence ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive ◽  
Weakly Inward

Suppose thatKis nonempty closed convex subset of a uniformly convex and smooth Banach spaceEwithPas a sunny nonexpansive retraction andF:=F(T1)∩F(T2)={x∈K:T1x=T2x=x}≠∅. LetT1,T2:K→Ebe two weakly inward nonself asymptotically nonexpansive mappings with respect toPwith two sequences{kn(i)}⊂[1,∞)satisfying∑n=1∞(kn(i)-1)<∞(i=1,2), respectively. For any givenx1∈K, suppose that{xn}is a sequence generated iteratively byxn+1=(1-αn)(PT1)nyn+αn(PT2)nyn,yn=(1-βn)xn+βn(PT1)nxn,n∈N, where{αn}and{βn}are sequences in[a,1-a]for somea∈(0,1). Under some suitable conditions, the strong and weak convergence theorems of{xn}to a common fixed point ofT1andT2are obtained.

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