Advances in Pure Mathematics
Vol.4 No.6(2014), Article
ID:46625,7
pages
DOI:10.4236/apm.2014.46033
Convergence Theorem of Hybrid Iterative Algorithm for Equilibrium Problems and Fixed Point Problems of Finite Families of Uniformly Asymptotically Nonexpansive Semigroups
Hongbo Liu, Yi Li*
School of Science, Southwest University of Science and Technology, Mianyang, China
Email: *liyi@swust.edu.cn
Copyright © 2014 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Received 11 April 2014; revised 11 May 2014; accepted 18 May 2014
ABSTRACT
Throughout this paper, we introduce a new hybrid iterative algorithm for finding a common element of the set of common fixed points of a finite family of uniformly asymptotically nonexpansive semigroups and the set of solutions of an equilibrium problem in the framework of Hilbert spaces. We then prove the strong convergence theorem with respect to the proposed iterative algorithm. Our results in this paper extend and improve some recent known results.
Keywords:Hybrid Iterative Algorithm, Uniformly Asymptotically Nonexpansive Semigroups, Equilibrium Problem, Common Fixed Point
1. Introduction
Recall the following equilibrium problem. Let be a closed convex subset of a real Hilbert space
with inner produce
and norm
. Let
be a bifunction, where
is the set of real numbers. The equilibrium problem for
is to to find
such that
the set of solutions is denoted by.
A mapping of a normed space
into itself is said to be nonexpansive if
for each
. We denote by
the set of fixed point of
. Given a mapping
, let
for all
.Then
if and only if
for all
, i.e.,
is a solution of the variational inequality, there are several other problems, for example, the complementarity problem, minimax problems, the Nash equilibrium problem in noncooperative games, fixed point problem and optimization problem, which can also be written in the form of an EP. In other words, the EP is an unifying model for several problems arising in physics, engineering, science, optimization, economics, etc. In the last two decades, many papers have appeared in the literature on the existence of solutions of EP; see, for example ([1] -[3] ) and references therein.
Iterative methods for finding fixed points of nonexpansivemappings are an important topic in the theory of nonexpansive mappings and have wide applications in a number of applied areas, such as the convex feasibility problem (see [4] -[7] ), the split feasibility problem (see [8] -[10] ) and image recovery and signal processing (see [6] ).
In 1953, Mann [11] introduced the following iterative process to approximate a fixed point of a nonexpansive single valued mapping in a Hilbert space
:
where the initial point is taken in
arbitrarily and
is a sequence in
. However, we note that Mann’s iteration process has only weak convergence. To obtain strong converges for Mann iteration, Nakajo and Takahashi [12] and Takahashi et al. [13] introduce some hybrid iterative process. Motivated by Suzuki’s result [14] and Nakajo-Takahashi’s results [12] .
On the other hand, Tada and Takahashi [15] introduce a new iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping T in a Hilbert space H.
A family of mappings on a closed convex subset
of a Hibert space
is called a nonexpansive semigroup if it satisfies the following conditions:
1) for all
;
2) for all
;
3) for all
and
4) for all
,
is continuous.
Takahashi and Chen [16] proved a strong convergence theorem for nonexpansive semigroups in Hilbert spaces by hybrid method in themathematical programming. Recently Saejung [17] improved the result in [16] .
Takahashi’s result gives us new idea that a finite family of uniformly asymptotically nonexpansive semigroups is introduced.
Definition 1.1 A family of mappings on a closed convex subset
of a Hibert space
is called an uniformly asymptotically nonexpansive semigroup with sequence
(
and
) if it satisfies the following conditions:
1) for all
;
2) for all
;
3) for all
,
,
4) for all,
is continuous.
In this paper, we introduce a new hybrid iterative process for finding a common element of the set of common fixed points of a finite family of uniformly asymptotically nonexpansive semigroups and the set of solutions of an equilibrium problem in the framework of Hilbert spaces. Then we prove some strong convergence theorems of the proposed iterative process. Our results generalize results of Tada and Takahashi [15] , Takahashi et al. [13] , He and Chen [16] and Saejung [17] .
2. Preliminaries
Throughout the paper, we denote weak convergence of by
, and strong convergence by
. Let
be a closed convex subset of
, we use
to denote the common fixed points set of the semigroup
. i.e.,
.
Next, We present an example of an uniformly asymptotically nonexpansive semigroup.
Example 2.1 As an example, we consider the nonempty closed convex subset of a Hilbert space
. define
. Observe that
is an uniformly asymptotically nonexpansive semigroup.
For every point, there exists a unique nearest point in
, denoted by
such that
that is,.
is called the metric projection of
onto
. It is well known that
is a nonexpansive mapping. It is also known that H satisfies Opial’s condition, i.e., for any sequence
with
, following the inequality holds:
To prove our result, we recall the following Lemma.
Lemma 2.1 (see [18] ). Let be a closed convex subset of
. Given
and a point
. Then
if and only if
for all
.
Lemma 2.2 (see [12] ). Let be a closed convex subset of
. Then for all
and
we have
.
Lemma 2.3 (see [18] ). Let be a real Hilbert space, there hold the following identities:
1), for all
and
.
2), for all
.
Lemma 2.4 (see [19] ) Let be a real Hilbert space. For
,
with
.
For solving the equilibrium problem, let us assume the following conditions for a bifunction (see [1] ):
1), for all
.
2), for all
.
3) For each,
4) is convex and lower semicontinuous for each
.
Lemma 2.5 (see [1] ) Let be a nonempty closed convex subset of
and let
be a bifunction of
into
satisfying (A1)-(A4). Let
and
. Then, there exists
such that
Lemma 2.6 Let satisfies (A1)-(A4). For
and
, define a mapping
as follows:
Then, the following holds:
1) is single valued;
2) is firmly nonexpansive, i.e., for any
,
;
3);
4) is closed and convex.
In 2013, Mohammad, E. introduce a new hybrid iterative process for finding a common element of the set of common fixed points of a finite family of nonexpansive semigroups and the set of solutions of an equilibrium problem in the framework of Hilbert spaces. He then prove strong convergence of the proposed iterative process. In this paper, we improve Mohammad’s result, and obtain follwing main results.
Mohammad’s Theorem 3.1 (see [20] ) about nonexpansive semigroups is the special case of our results. Our results improve chang’s result in [21] .
3. Main Results
First, we show the following theorem to our main results.
Theorem 3.1 Let be nonempty closed convex subset of
.
be an uniformly asymptotically nonexpansive semigroups with nonnegative real sequences
with
and
(as
), then
is a closed and convex subset of
.
Proof. Let be a sequence in
, such that
. Since
be an uniformly asymptotically nonexpansive semigroups, we have
for and for all
. Therefore,
We obtain. Hence,
. So, we have
. This implies
is closed.
Let and
, and put
. Next we prove that
. Indeed, in view of Lemma 2.3 2), let
, we have
(1)
Since
(2)
Substituting (1) into (2) and simplifying it we have
Hence, we have. This implies that
. Since
is closed, we have
, i.e.,
. This completes the proof of theorem 3.1.
Theorem 3.2 Let be a nonempty closed convex subset of a real Hilbert space
and
be a bifunction of
into
satisfying (A1)-(A4). Let
be a finite family of uniformly asymptotically semigroups with sequence
(
and
). Assume that
. For an initial piont
, let
and
be sequences generated by
(3)
where is the metric projection of
onto
. If
,
,
and
satisfying the following conditions:
1);
2) (for
) and
;
3) and
;
4),
then, the sequences
and
converge strongly to
.
Proof. 1) First, we prove.
Indeed, is obvious. Suppose that
, then for
and
, by Lemma 2.6 we have
(4)
Since
be a finite family of uniformly asymptotically semigroups,we have
which implies that.Therefore we have
for all
. Note
is closed and convex.this implies that
is well defined. From Lemma 2.5, sequence
is also well defined.
2) Next, we prove that exists.
Since is closed and convex subset of
, there exists a unique
such that
. From
, we have
Since, we get that
It follows that the sequence is bounded and non decreasing, this implies that
exists
3) Now we show that,
.
Infact, from Lemma 2.2 we have
witch implies that we get is Cauchy. Hence there exists
such that
. Since
, thus
. By Lemma 2.4, we have
(5)
from condition (C1), so we have
this implies for all
. We know that
, hence we have
that is,
Using we get that
that is,
which implies. Hence for all
we get that
Without loss of generality, as in Saejung’s article [17] , let. For
and
,
where denotes the maximal integer that is not larger than
. Since for
mapping
for a fixed
and
, then
.
4) Now we prove that.
First, since and
, by (A2) we get that
and hence
Since,
and A(4), we get that
If and
, let
, then
. So, from (A1)-(A4) we have
which gives for all
. Hence by (A3) we have
which is.
For, we have
Since, then
i.e.,
for all
and thus
.
5) Now we prove that.
Since and
, we get that
Since, we have
which implies. The proof is completed.
From Theorem 3.1, taking and
, we obtain Corollary 3.1 Let
be a nonempty closed convex subset of a real Hilbert space
and
be a bifunction of
into
satisfying (A1)-(A4). Let
be a finite family of uniformly asymptotically semigroups with sequence
(
and
). Assume that
. For an initial piont
, let
and
be sequences generated by
(6)
where is the metric projection of
onto
. If
,
,
and
satisfying the following conditions:
1);
2) (for
) and
;
3),
then, the sequences
converge strongly to
.
Competing Interests
The authors declare that they have no competing interests.
Acknowledgements
The authors are very grateful to reviewers for carefully reading this paper and their comments. This work is supported by the Doctoral Program Research Foundation of Southwest University of Science and Technology (No. 11zx7129) and Applied Basic Research Project of Sichuan Province (No. 2013JY0096).
References
- Blum, E. and Oettli, W. (1994) From Optimization and Variational Inequalities to Equilibrium Problems. Mathematics Students, 63, 123-145.
- Flam, S.D. and Antipin, A.S. (1997) Equilibrium Programming Using Proximal-Link Algolithms. Mathematical Programming, 78, 29-41. http://dx.doi.org/10.1007/BF02614504
- Moudafi, A. and Thera, M. (1999) Proximal and Dynamical Approaches to Equilibrium Problems. Lecture Note in Economics and Mathematical Systems, 477, 187-201.
- Bauschke, H.H. and Borwein, J.M. (1996) On Projection Algorithms for Solving Convex Feasibility Problems. SIAM Review, 38, 367-426. http://dx.doi.org/10.1137/S0036144593251710
- Butnariu, D., Censor, Y., Gurfil, P. and Hadar, E. (2008) On the Behavior of Subgradient Projections Methods for Convex Feasibility Problems in Euclidean Spaces. SIAM Journal on Optimization, 19, 786-807. http://dx.doi.org/10.1137/070689127
- Hale, E.T., Yin, W. and Zhang, Y. (2010) Fixed-Point Continuation Applied to Compressed Sensing: Implementation and Numerical Experiments. Journal of Computational Mathematics, 28, 170-194.
- Maruster, S. and Popirlan, C. (2008) On the Mann-Type Iteration and the Convex Feasibility Problem. Journal of Computational and Applied Mathematics, 212, 390-396. http://dx.doi.org/10.1016/j.cam.2006.12.012
- Byrne, C. (2004) A Unified Treatment of Some Iterative Algorithms in Signal Processing and Image Reconstruction. Inverse Problems, 20, 103-120. http://dx.doi.org/10.1088/0266-5611/20/1/006
- Censor, Y., Elfving, T., Kopf, N. and Bortfeld, T. (2005) The Multiple-Sets Split Feasibility Problem and Its Applications for Inverse Problems. Inverse Problems, 21, 2071-2084. http://dx.doi.org/10.1088/0266-5611/21/6/017
- Xu, H.K. (2006) A variable Krasnoselskii-Mann Algorithm and Themultiple-Set Split Feasibility Problem. Inverse Problems, 22, 2021-2034. http://dx.doi.org/10.1088/0266-5611/22/6/007
- Mann, W.R. (1953) Mean Value Methods in Iteration. Proceedings of the American Mathematical Society, 4, 506-510. http://dx.doi.org/10.1090/S0002-9939-1953-0054846-3
- Nakajo, K. and Takahashi, W. (2003) Strong Convergence Theorems for Nonexpansive Mappings and Nonexpansive Semigroups. Journal of Mathematical Analysis and Applications, 279, 372-379. http://dx.doi.org/10.1016/S0022-247X(02)00458-4
- Takahashi, W., Takeuchi, Y. and Kubota, R. (2008) Strong Convergence Theorems by Hybrid Methods for Families of Nonexpansive Mappings in Hilbert Spaces. Journal of Mathematical Analysis and Applications, 341, 276-286. http://dx.doi.org/10.1016/j.jmaa.2007.09.062
- Suzuki, T. (2003) On Strong Convergence to Common Fixed Points of Nonexpansive Semigroups in Hilbert Spaces. Proceedings of the American Mathematical Society, 131, 2133-2136. http://dx.doi.org/10.1090/S0002-9939-02-06844-2
- Tada, A. and Takahashi, W. (2007) Weak and Strong Convergence Theorems for a Nonexpansive Mapping and an Equilibrium Problem. Journal of Optimization Theory and Applications, 133, 359-370. http://dx.doi.org/10.1007/s10957-007-9187-z
- He, H. and Chen, R. (2007) Strong Convergence Theorems of the CQ Method for Nonexpansive Semigroups. Fixed Point Theory and Applications, 2007, Article ID 59735.
- Saejung, S. (2008) strong Convergence Theorems for Nonexpansive Semigroups without Bochner Integrals. Fixed Point Theory and Applications, 2008, Article ID 745010.
- Marino, G. and Xu, H.K. (2007) Weak and Strong Convergence Theorems for Strict Pseudo-Contractions in Hilbert Space. Journal of Mathematical Analysis and Applications, 329, 336-346. http://dx.doi.org/10.1016/j.jmaa.2006.06.055
- Cholamjiak, W. and Suantai, S. (2010) Ahybrid Method for a Countable Family of Multivalued Maps, Equilibrium Problems, and Variational Inequality Problems. Discrete Dynamics in Nature and Society, 2010, Article ID: 349158.
- Mohammad, E. (2013) Hybid Method for Equilibrium Problems and Fixed Piont Problems of Finite of Nonexpansive Semigroups. Revista Serie A Matemáticas, 107, 299-307.
- Chang, S.S., Wang, L., Tang, Y.K., Wang, B. and Qin, L.J. (2012) Strong Convergence Theorems for a Countable Family of Quasi-ψ-Asymptotically Nonexpansive Nonself Mappings. Applied Mathematics and Computation, 218, 7864-7870. http://dx.doi.org/10.1016/j.amc.2012.02.002
NOTES
*Corresponding author.