scholarly journals A Real-Time Iterative Projection Scheme for Solving the Common Fixed Point Problem and Its Applications

2018 ◽  
Vol 64 (4) ◽  
pp. 616-636
Author(s):  
A Gibali ◽  
D Teller
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Real Time ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Fixed Point Problem ◽  
Feasibility Problem ◽  
Point Problem ◽  
Common Fixed Point Problem ◽  
The Common

In this paper, we are concerned with the Common Fixed Point Problem (CFPP) with demicontractive operators and its special instance, the Convex Feasibility Problem (CFP) in real Hilbert spaces. Motivated by the recent result of Ordon˜ ez et al. [35] and in general, the field of online/real-time algorithms, e.g., [20, 21, 30], in which the entire input is not available from the beginning and given piece-by-piece, we propose an online/real-time iterative scheme for solving CFPPs and CFPs in which the involved operators/sets emerge along time. This scheme is capable of operating on any block, for any finite number of iterations, before moving, in a serial way, to the next block. The scheme is based on the recent novel result of Reich and Zalas [37] known as the Modular String Averaging (MSA) procedure. The convergence of the scheme follows [37] and other classical results in the fields of fixed point theory and variational inequalities, such as [34]. Numerical experiments for linear and non-linear feasibility problems with applications to image recovery are presented and demonstrate the validity and potential applicability of our scheme, e.g., to online/real-time scenarios.

scholarly journals A New Approach for the Approximations of Solutions to a Common Fixed Point Problem in Metric Fixed Point Theory

2016 ◽  
Vol 2016 ◽  
pp. 1-5 ◽  
Author(s):  
Ishak Altun ◽  
Nassir Al Arifi ◽  
Mohamed Jleli ◽  
Aref Lashin ◽  
Bessem Samet
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Fixed Point Theory ◽  
Sufficient Conditions ◽  
Point Theory ◽  
Fixed Point Problem ◽  
Point Problem ◽  
New Approach ◽  
Common Fixed Point Problem ◽  
Unique Common Fixed Point

We provide sufficient conditions for the existence of a unique common fixed point for a pair of mappings T,S:X→X, where X is a nonempty set endowed with a certain metric. Moreover, a numerical algorithm is presented in order to approximate such solution. Our approach is different to the usual used methods in the literature.

scholarly journals An alternating iteration algorithm for solving the split equality fixed point problem with L-Lipschitz and quasi-pseudo-contractive mappings

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Meixia Li ◽  
Xueling Zhou ◽  
Haitao Che
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Split Feasibility Problem ◽  
Fixed Point Problem ◽  
Feasibility Problem ◽  
Iteration Algorithm ◽  
Point Problem ◽  
Contractive Mappings ◽  
Common Fixed Point Problem ◽  
Significant Generalization

Abstract In this paper, we are concerned with the split equality common fixed point problem. It is a significant generalization of the split feasibility problem, which can be used in various disciplines, such as medicine, military and biology, etc. We propose an alternating iteration algorithm for solving the split equality common fixed point problem with L-Lipschitz and quasi-pseudo-contractive mappings and prove that the sequence generated by the algorithm converges weakly to the solution of this problem. Finally, some numerical results are shown to confirm the feasibility and efficiency of the proposed algorithm.

scholarly journals Iterative Algorithms for the Split Problem and Its Convergence Analysis

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Zhangsong Yao ◽  
Arif Rafiq ◽  
Shin Min Kang ◽  
Li-Jun Zhu
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Iterative Algorithms ◽  
Split Feasibility Problem ◽  
Fixed Point Problem ◽  
Convergence Result ◽  
Feasibility Problem ◽  
Point Problem ◽  
Common Fixed Point Problem ◽  
Split Common Fixed Point

Now, it is known that the split common fixed point problem is a generalization of the split feasibility problem and of the convex feasibility problem. In this paper, the split common fixed point problem associated with the pseudocontractions is studied. An iterative algorithm has been presented for solving the split common fixed point problem. Strong convergence result is obtained.

scholarly journals Common Fixed-Point Problem for a Family Multivalued Mapping in Banach Space

2011 ◽  
Vol 2011 ◽  
pp. 1-9
Author(s):  
Zhanfei Zuo
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Nonexpansive Mappings ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Fixed Point Set ◽  
Common Fixed Point Problem ◽  
The Family ◽  
Point Set ◽  
The Common

It is our purpose in this paper to prove two convergents of viscosity approximation scheme to a common fixed point of a family of multivalued nonexpansive mappings in Banach spaces. Moreover, it is the unique solution in to a certain variational inequality, where stands for the common fixed-point set of the family of multivalued nonexpansive mapping .

scholarly journals Strong Convergence Theorems for the Generalized Split Common Fixed Point Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Cuijie Zhang
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Convergence Theorems ◽  
Common Fixed Point Problem ◽  
Nonexpansive Operators ◽  
The Common ◽  
Split Common Fixed Point

We introduce the generalized split common fixed point problem (GSCFPP) and show that the GSCFPP for nonexpansive operators is equivalent to the common fixed point problem. Moreover, we introduce a new iterative algorithm for finding a solution of the GSCFPP and obtain some strong convergence theorems under suitable assumptions.

scholarly journals String-averaging methods for best approximation to common fixed point sets of operators: the finite and infinite cases

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yair Censor ◽  
Ariel Nisenbaum
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Best Approximation ◽  
Fixed Point Problem ◽  
Feasibility Problem ◽  
Point Problem ◽  
Fixed Point Set ◽  
Averaging Methods ◽  
Point Set ◽  
The Common

AbstractString-averaging is an algorithmic structure used when handling a family of operators in situations where the algorithm in hand requires to employ the operators in a specific order. Sequential orderings are well known, and a simultaneous order means that all operators are used simultaneously (in parallel). String-averaging allows to use strings of indices, constructed by subsets of the index set of all operators, to apply the operators along these strings, and then to combine their end-points in some agreed manner to yield the next iterate of the algorithm. String-averaging methods were discussed and used for solving the common fixed point problem or its important special case of the convex feasibility problem. In this paper we propose and investigate string-averaging methods for the problem of best approximation to the common fixed point set of a family of operators. This problem involves finding a point in the common fixed point set of a family of operators that is closest to a given point, called an anchor point, in contrast with the common fixed point problem that seeks any point in the common fixed point set.We construct string-averaging methods for solving the best approximation problem to the common fixed points set of either finite or infinite families of firmly nonexpansive operators in a real Hilbert space. We show that the simultaneous Halpern–Lions–Wittman–Bauschke algorithm, the Halpern–Wittman algorithm, and the Combettes algorithm, which were not labeled as string-averaging methods, are actually special cases of these methods. Some of our string-averaging methods are labeled as “static” because they use a fixed pre-determined set of strings. Others are labeled as “quasi-dynamic” because they allow the choices of strings to vary, between iterations, in a specific manner and belong to a finite fixed pre-determined set of applicable strings. For the problem of best approximation to the common fixed point set of a family of operators, the full dynamic case that would allow strings to unconditionally vary between iterations remains unsolved, although it exists and is validated in the literature for the convex feasibility problem where it is called “dynamic string-averaging”.

scholarly journals A CYCLIC ALGORITHM FOR THE SPLIT COMMON FIXED POINT PROBLEM OF DEMICONTRACTIVE MAPPINGS IN HILBERT SPACES

2012 ◽  
Vol 17 (4) ◽  
pp. 457-466 ◽  
Author(s):  
Yu-Chao Tang ◽  
Ji-Gen Peng ◽  
Li-Wei Liu
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Split Feasibility Problem ◽  
Fixed Point Problem ◽  
Feasibility Problem ◽  
Point Problem ◽  
Common Fixed Point Problem ◽  
Cyclic Algorithm ◽  
Demicontractive Mappings ◽  
Split Common Fixed Point

The split common fixed point problem has been investigated recently, which is a generalization of the split feasibility problem and of the convex feasibility problem. We construct a cyclic algorithm to approximate a solution of the split common fixed point problem for the demicontractive mappings in a Hilbert space. Our results improve and extend previously discussed related problems and algorithms.

scholarly journals Common fixed point theorem for modified Kannan enriched contraction pair in Banach spaces and its applications

Filomat ◽  
2021 ◽  
Vol 35 (8) ◽  
pp. 2485-2495
Author(s):  
Rizwan Anjum ◽  
Mujahid Abbas
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Nonexpansive Mappings ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Existence Of A Solution ◽  
Common Fixed Point Problem ◽  
The Common ◽  
Well Posed ◽  
Common Fixed Point Theorem

The purpose of this paper is to introduce the class of (a,b,c)-modified enriched Kannan pair of mappings (T,S) in the setting of Banach space that includes enriched Kannan mappings, contraction and nonexpansive mappings and some other mappings. Some examples are presented to support the concepts introduced herein. We establish the existence of common fixed point of the such pair. We also show that the common fixed point problem studied herein is well posed. A convergence theorem for the Krasnoselskij iteration is used to approximate fixed points of the (a,b,c)-modified enriched Kannan pair. As an application of the results proved in this paper, the existence of a solution of integral equations is established. The presented results improve, unify and generalize many known results in the literature.

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