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 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 Common Fixed Point of a Finite-step Iteration Algorithm Under Total Asymptotically Quasi-nonexpansive Maps

2019 ◽  
Vol 16 (3) ◽  
pp. 0654
Author(s):  
Abed Et al.
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Convex Banach Space ◽  
Finite Family ◽  
Iteration Algorithm ◽  
Uniformly Convex ◽  
Nonexpansive Maps ◽  
Finite Step

      Throughout this paper, a generic iteration algorithm for a finite family of total asymptotically quasi-nonexpansive maps in uniformly convex Banach space is suggested. As well as weak / strong convergence theorems of this algorithm to a common fixed point are established. Finally, illustrative numerical example by using Matlab is presented.

scholarly journals Convergence theorems of finite-step iteration with errors for non-self asymptotically nonexpansive in the intermediate sense mappings

Filomat ◽  
2011 ◽  
Vol 25 (1) ◽  
pp. 81-103
Author(s):  
G.S. Saluja
Keyword(s):  
Banach Space ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Convergence Theorems ◽  
Asymptotically Nonexpansive ◽  
Finite Step ◽  
Nonexpansive Retract ◽  
Bounded Sequences

Let K be a nonempty closed convex nonexpansive retract of a uniformly convex Banach space E with P as a nonexpansive retraction. Let T : K ? E be non-self asymptotically nonexpansive in the intermediate sense mapping with F(T) = ?. Let {?ni}, {?ni} and {?ni} are sequences in [0, 1] with ?n(i) + ?n(i) + ?n(i) = 1 for all i = 1, 2, . . . , N. From arbitrary x1 ? K , define the sequence {xn } iteratively by (8), where {u(i) } for all i = 1, 2, . . . , N are bounded sequences in K with P? u(i) < ?. (i) If the dual E

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.

Convergence theorems for total asymptotically nonexpansive single-valued and quasi nonexpansive multi-valued mappings in hyperbolic spaces

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Preeyalak Chuadchawna ◽  
Ali Farajzadeh ◽  
Anchalee Kaewcharoen
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Hyperbolic Space ◽  
Hyperbolic Spaces ◽  
Iterative Sequence ◽  
Uniformly Convex ◽  
Convergence Theorems ◽  
Asymptotically Nonexpansive ◽  
Uniformly Convex Hyperbolic Space

Abstract In this paper, we discuss the Δ-convergence and strong convergence for the iterative sequence generated by the proposed scheme to approximate a common fixed point of a total asymptotically nonexpansive single-valued mapping and a quasi nonexpansive multi-valued mapping in a complete uniformly convex hyperbolic space. Finally, by giving an example, we illustrate our result.

scholarly journals Fixed points of mappings satisfying semicontractivity conditions

1992 ◽  
Vol 53 (1) ◽  
pp. 25-38
Author(s):  
Jürgen Schu
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Convex Subset ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Opial’S Condition ◽  
Asymptotically Nonexpansive

AbstractLet A be a subset of a Banach space E. A mapping T: A →A is called asymptoically semicontractive if there exists a mapping S: A×A→A and a sequence (kn) in [1, ∞] such that Tx=S(x, x) for all x ∈A while for each fixed x ∈A, S(., x) is asymptotically nonexpansive with sequence (kn) and S(x,.) is strongly compact. Among other things, it is proved that each asymptotically semicontractive self-mpping T of a closed bounded and convex subset A of a uniformly convex Banach space E which satisfies Opial's condition has a fixed point in A, provided s has a certain asymptoticregurity property.

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.

scholarly journals Weak and strong convergence to fixed points of asymptotically nonexpansive mappings

1991 ◽  
Vol 43 (1) ◽  
pp. 153-159 ◽  
Author(s):  
J. Schu
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Nonexpansive Mappings ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Weak And Strong Convergence ◽  
Uniformly Convex Banach Spaces ◽  
Asymptotically Nonexpansive ◽  
Fourth Kind

Let T be an asymptotically nonexpansive self-mapping of a closed bounded and convex subset of a uniformly convex Banach space which satisfies Opial's condition. It is shown that, under certain assumptions, the sequence given by xn+1 = αnTn(xn) + (1 - αn)xn converges weakly to some fixed point of T. In arbitrary uniformly convex Banach spaces similar results are obtained concerning the strong convergence of (xn) to a fixed point of T, provided T possesses a compact iterate or satisfies a Frum-Ketkov condition of the fourth kind.

scholarly journals Common fixed points of monotone ρ-nonexpansive semigroup in modular spaces

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Noureddine El Harmouchi ◽  
Karim Chaira ◽  
El Miloudi Marhrani
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Common Fixed Point ◽  
Common Fixed Points ◽  
Nonexpansive Semigroup ◽  
Iteration Algorithm ◽  
Uniformly Convex ◽  
Modular Spaces ◽  
Nonexpansive Semigroups ◽  
Convergence Results

Abstract In this paper, we consider the class of monotone ρ-nonexpansive semigroups and give existence and convergence results for common fixed points. First, we prove that the set of common fixed points is nonempty in uniformly convex modular spaces and modular spaces. Then we introduce an iteration algorithm to approximate a common fixed point for the same class of semigroups.

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