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 Split Common Fixed Point Problem for a Class of Total Asymptotic Pseudocontractions

2016 ◽  
Vol 2016 ◽  
pp. 1-10
Author(s):  
E. E. Chima ◽  
M. O. Osilike
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Fixed Point Set ◽  
Pseudocontractive Mappings ◽  
Common Fixed Point Problem ◽  
Asymptotically Pseudocontractive Mappings ◽  
Point Set ◽  
Split Common Fixed Point

We study the split common fixed point problem (SCFP) for a class of total asymptotically pseudocontractive mappings. We obtain some important properties of our class of mappings including the demiclosedness property and the closedness and convexity of the fixed point set. We then propose an algorithm and prove weak and strong convergence theorems for the approximation of solutions of the SCFP for certain class of these mappings.

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 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.

scholarly journals A Strong Convergence Algorithm for the Two-Operator Split Common Fixed Point Problem in Hilbert Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Chung-Chien Hong ◽  
Young-Ye Huang
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Nonexpansive Mappings ◽  
Intensity Modulated Radiation Therapy ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Numerical Result ◽  
Common Fixed Point Problem ◽  
Operator Split ◽  
Split Common Fixed Point

The two-operator split common fixed point problem (two-operator SCFP) with firmly nonexpansive mappings is investigated in this paper. This problem covers the problems of split feasibility, convex feasibility, and equilibrium and can especially be used to model significant image recovery problems such as the intensity-modulated radiation therapy, computed tomography, and the sensor network. An iterative scheme is presented to approximate the minimum norm solution of the two-operator SCFP problem. The performance of the presented algorithm is compared with that of the last algorithm for the two-operator SCFP and the advantage of the presented algorithm is shown through the numerical result.

scholarly journals Strong Convergence Theorems for Split Common Fixed Point Problem of Bregman Generalized Asymptotically Nonexpansive Mappings in Banach Spaces

2019 ◽  
pp. 35-50
Author(s):  
Yusuf Ibrahim
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Common Fixed Point Problem ◽  
Asymptotically Nonexpansive ◽  
Split Common Fixed Point

In this paper, a new iterative scheme is introduced and also strong convergence theorems for solving split common fixed point problem for uniformly continuous Bregman generalized asymptotically nonexpansive mappings in uniformly convex and uniformly smooth Banach spaces are presented. The results are proved without the assumption of semicompactness property and or Opial condition

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