398 research outputs found
Extragradient algorithms for equilibrium problems and symmetric generalized hybrid mappings
In this paper, we propose new algorithms for finding a common point of the
solution set of a pseudomonotone equilibrium problem and the set of fixed
points of a symmetric generalized hybrid mapping in a real Hilbert space. The
convergence of the iterates generated by each method is obtained under
assumptions that the fixed point mapping is quasi-nonexpansive and demiclosed
at , and the bifunction associated with the equilibrium problem is weakly
continuous. The bifunction is assumed to be satisfying a Lipschitz-type
condition when the basic iteration comes from the extragradient method. It
becomes unnecessary when an Armijo back tracking linesearch is incorporated in
the extragradient method.Comment: 12 page
Extragradient algorithms for split equilibrium problem and nonexpansive mapping
In this paper, we propose new extragradient algorithms for solving a split
equilibrium and nonexpansive mapping SEPNM( where
are nonempty closed convex subsets in real Hilbert spaces respectively, is a
bounded linear operator, is a pseudomonotone bifunction on and is a
monotone bifunction on , are nonexpansive mappings on and
respectively. By using extragradient method combining with cutting techniques,
we obtain algorithms for the problem. Under certain conditions on parameters,
the iteration sequences generated by the algorithms are proved to be weakly and
strongly convergent to a solution of this problem.Comment: 13 pages, Some typos were corrected
An explicit iterative method to solve generalized mixed equilibrium problem, variational inequality problem and hierarchical fixed point problem for a nearly nonexpansive mapping
In this paper, we introduce a new iterative method to find a common solution
of a generalized mixed equilibrium problem, a variational inequality problem
and a hierarchical fixed point problem for a demicontinuous nearly nonexpansive
mapping. We prove that the proposed method converges strongly to a common
solution of above problems under the suitable conditions. It is also noted that
the main theorem is proved without usual demiclosedness condition. Also, under
the appropriate assumptions on the control sequences and operators, our
iterative method can be reduced to recent methods. So, the results here improve
and extend some recent corresponding results given by many other authors.Comment: arXiv admin note: text overlap with arXiv:1403.360
Weak convergence theorems for a symmetric generalized hybrid mapping and an equilibrium problem
In this paper, we introduce three new iterative methods for finding a common
point of the set of fixed points of a symmetric generalized hybrid mapping and
the set of solutions of an equilibrium problem in a real Hilbert space. Each
method can be considered as an combination of Ishikawa's process with the
proximal point algorithm, the extragradient algorithm with or without
linesearch. Under certain conditions on parameters, the iteration sequences
generated by the proposed methods are proved to be weakly convergent to a
solution of the problem. These results extend the previous results given in the
literature. A numerical example is also provided to illustrate the proposed
algorithms.Comment: arXiv admin note: text overlap with arXiv:1508.0390
Comments on relaxed -cocoercive mappings
We show that the variational inequality has a unique solution for a
relaxed -cocoercive, -Lipschitzian mapping with
, where is a nonempty closed convex subset of a Hilbert
space . From this result, it can be derived that, for example, the recent
algorithms given in the references of this paper, despite their becoming more
complicated, are not general as they should be
Convergence results for a common solution of a finite family of equilibrium problems and quasi-Bregman nonexpansive mappings in Banach space
We introduce an iterative process for finding common fixed point of finite
family of quasi-Bregman nonexpansive mappings which is a unique solution of
some equilibrium problem.Comment: 17 pages. arXiv admin note: text overlap with arXiv:1512.00243 by
other author
Extragradient and linesearch algorithms for solving equilibrium problems and fixed point problems in Banach spaces
In this paper, using sunny generalized nonexpansive retraction, we propose
new extragradient and linesearch algorithms for finding a common element of the
set of solutions of an equilibrium problem and the set of fixed points of a
relatively nonexpansive mapping in Banach spaces. To prove strong convergence
of iterates in the extragradient method, we introduce a -Lipschitz-type
condition and assume that the equilibrium bifunction satisfies in this
condition. This condition is unnecessary when the linesearch method is used
instead of the extragradient method. A numerical example is given to illustrate
the usability of our results. Our results generalize, extend and enrich some
existing results in the literature.Comment: 25 pages, 2 figures, 2 tabel. arXiv admin note: text overlap with
arXiv:1509.0201
An iterative algorithm for a common fixed point of Bregman Relatively Nonexpansive Mappings
We introduce and investigate an iterative scheme for approximating common
fixed point of a family of Bregman relatively-nonexpansive mappings in real
reflexive Banach spaces. We prove strong convergence theorem of the sequence
generated by our scheme under some appropriate conditions. Furthermore, our
scheme and results unify some known results obtained in this direction
Moduli of regularity and rates of convergence for Fej\'er monotone sequences
In this paper we introduce the concept of modulus of regularity as a tool to
analyze the speed of convergence, including the finite termination, for classes
of Fej\'er monotone sequences which appear in fixed point theory, monotone
operator theory, and convex optimization. This concept allows for a unified
approach to several notions such as weak sharp minima, error bounds, metric
subregularity, H\"older regularity, etc., as well as to obtain rates of
convergence for Picard iterates, the Mann algorithm, the proximal point
algorithm and the cyclic algorithm. As a byproduct we obtain a quantitative
version of the well-known fact that for a convex lower semi-continuous function
the set of minimizers coincides with the set of zeros of its subdifferential
and the set of fixed points of its resolvent
The Split Common Null Point Problem
We introduce and study the Split Common Null Point Problem (SCNPP) for
set-valued maximal monotone mappings in Hilbert spaces. This problem
generalizes our Split Variational Inequality Problem (SVIP) [Y. Censor, A.
Gibali and S. Reich, Algorithms for the split variational inequality problem,
Numerical Algorithms 59 (2012), 301--323]. The SCNPP with only two set-valued
mappings entails finding a zero of a maximal monotone mapping in one space, the
image of which under a given bounded linear transformation is a zero of another
maximal monotone mapping. We present four iterative algorithms that solve such
problems in Hilbert spaces, and establish weak convergence for one and strong
convergence for the other three.Comment: Journal of Nonlinear and Convex Analysis, accepted for publicatio
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