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Strong convergence of a Halpern-type algorithm for common solutions of fixed point and equilibrium problems
Journal of Inequalities and Applications volume 2014, Article number: 313 (2014)
Abstract
In this article, fixed points of nonexpansive mappings and equilibrium problems based on a Halpern-type algorithm are investigated. Strong convergence theorems for common solutions of the two problems are obtained in the framework of real Hilbert spaces.
1 Introduction
The study of equilibrium problems is an important branch of optimization theory and nonlinear functional analysis. Numerous problems in physics, optimization, transportation, signal processing, and economics are reduced to find a solution to equilibrium problems, which cover fixed point problems, variational inequalities, saddle problems, inclusion problems, and so on. A closely related subject of current interest is the problem of finding common elements in the fixed point set of nonlinear operators and in the solution set of monotone variational inequalities; see [1–15] and the references therein. The motivation for this subject is mainly due to its possible applications to mathematical modeling of concrete complex problems. The aim of this paper is to investigate a common element problem based on a Halpern-type algorithm. Strong convergence of the algorithm is obtained in the framework of real Hilbert spaces. The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, a Halpern-type algorithm is proposed and analyzed. Strong convergence theorems for common solutions of two problems are established in the framework of Hilbert spaces. In Section 4, applications of the main results are provided.
2 Preliminaries
Let H be a real Hilbert space with inner product and norm . Let C be a nonempty, closed, and convex subset of H and let be the metric projection from H onto C.
Let be a mapping. In this paper, we use to denote the fixed point set of T. Recall that T is said to be contractive iff there exists a constant such that
For such a case, T is also said to be α-contractive. Recall that T is said to be nonexpansive iff
It is well known that the fixed point set of nonexpansive mappings is nonempty provided that the subset C is bounded, convex, and closed.
Let be a mapping. Recall that A is said to be monotone iff
Recall that A is said to be inverse-strongly monotone iff there exists a constant such that
For such a case, A is also said to be α-inverse-strongly monotone.
Recall that the classical variational inequality is to find an such that
In this paper, we use to denote the solution set of (2.1). It is well known that is a solution of the variational inequality (2.1) iff x is a solution of the fixed point equation , where is a constant.
Recall that a set-valued mapping is said to be monotone iff, for all , and imply . M is maximal iff the graph of R is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mapping M is maximal if and only if, for any , , for all implies .
For a maximal monotone operator M on H, and , we may define the single-valued resolvent , where denotes the domain of M. It is known that is firmly nonexpansive, and , where , and .
Let be a monotone mapping, and let F be a bifunction of into ℝ, where ℝ denotes the set of real numbers. We consider the following generalized equilibrium problem:
In this paper, we use to denote the solution set of the generalized equilibrium problem (2.2).
Next, we give some special cases of the generalized equilibrium problem (2.2).
-
(I)
If , then problem (2.2) is reduced to the classical variational inequality (2.1).
-
(II)
If , the zero mapping, then problem (2.2) is reduced to the following equilibrium problem:
(2.3)
In this paper, we use to denote the solution set of the equilibrium problem (2.3).
To study the equilibrium problems, we may assume that F satisfies the following conditions:
-
(A1)
for all ;
-
(A2)
F is monotone, i.e., for all ;
-
(A3)
for each ,
-
(A4)
for each , is convex and weakly lower semi-continuous.
Recently, many authors have studied fixed point problems of nonexpansive mappings and solution problems of the equilibrium problems (2.2) and (2.3); for more details, see [16–25] and the references therein. In this paper, motivated and inspired by the research going on in this direction, we consider common element problems based on a mean iterative process. Strong convergence of the iterative process is obtained in the framework of real Hilbert spaces. The results presented in this paper improve and extend the corresponding results in Hao [1], Qin et al. [24], Chang et al. [25].
In order to prove our main results, we need the following lemmas.
Lemma 2.1 [26]
Assume that is a sequence of nonnegative real numbers such that
where is a sequence in and is a sequence such that
-
(1)
;
-
(2)
or .
Then .
Lemma 2.2 [27]
Let be a bifunction satisfying (A1)-(A4). Then, for any and , there exists such that
Define a mapping as follows:
then the following conclusions hold.
-
(1)
is single-valued;
-
(2)
is firmly nonexpansive, i.e., for any ,
-
(3)
;
-
(4)
is closed and convex.
Let be a family of infinitely nonexpansive mappings and be a nonnegative real sequence with , . For define a mapping as follows:
Such a mapping is nonexpansive from C to C and it is called a W-mapping generated by and ; see [28] and the references therein.
Lemma 2.3 [28]
Let be a family of infinitely nonexpansive mappings with a nonempty common fixed point set and let be a real sequence such that , where l is some real number, . Then
-
(1)
is nonexpansive and , for each ;
-
(2)
for each and for each positive integer k, the limit exists;
-
(3)
the mapping defined by
(2.5)
is a nonexpansive mapping satisfying and it is called the W-mapping generated by and .
Lemma 2.4 [29]
Let be a mapping and let be a maximal monotone operator. Then , where r is some positive real number.
Lemma 2.5 [25]
Let be a family of infinitely nonexpansive mappings with a nonempty common fixed point set and let be a real sequence such that , . If K is any bounded subset of C, then
Throughout this paper, we always assume that , .
Lemma 2.6 [30]
Let a Lipschitz monotone mapping and let be the normal cone to C at ; that is, . Define
Then W is maximal monotone and if and only if .
Lemma 2.7 [31]
Let and be bounded sequences in H and let be a sequence in with . Suppose that for all and
Then .
3 Main results
Theorem 3.1 Let C be a nonempty closed convex subset of a Hilbert space H and let F be a bifunction from to ℝ which satisfies (A1)-(A4). Let be an α-inverse-strongly monotone mapping and let be a β-inverse-strongly monotone mapping. Let be a maximal monotone operator. Let be a family of infinitely nonexpansive mappings. Assume that . Let be a κ-contraction. Let be a sequence generated by the process: and
where is the sequence generated in (2.4), , , and are sequences in such that and and are positive number sequences. Assume that the above control sequences satisfy the following restrictions:
-
(a)
, ;
-
(b)
and ;
-
(c)
;
-
(d)
.
Then the sequence converges strongly to .
Proof First, we show that is nonexpansive. For , we have
Using restriction (a), we have is nonexpansive, so is . Fix . It follows that . Putting , one finds that . It follows that
Hence, we have . This yields the result that is bounded. Therefore, both and are also bounded. Next, without loss of generality, we assume that there exists a bounded set such that . Notice that , , and , . It follows that
Hence, we have
This yields the result that
where is an appropriate constant such that
Since is firmly nonexpansive, one sees that
Substituting (3.1) into (3.2), one finds that
where is an appropriate constant such that . On the other hand, one has
where K is the bounded subset of C defined above. Combining (3.3) with (3.4), one finds
Letting we see that
Substituting (3.5) into (3.6), we see that
It follows from restrictions (a), (c), and (d) that . Using Lemma 2.7, we find that . It follows that
For any , we see that
Since
we find from (3.8) that
It also follows from (3.8) that
Using (3.7), one arrives at
Since is firmly nonexpansive, we find that
which implies that . Hence
Using (3.7) and (3.10), one has
Similarly, one also has
Since
we find from (3.11) and (3.12) that
Now, we are in a position to show , where . To prove this, we choose a subsequence of such that
Since is bounded, without loss of generality, we may assume that . Using (3.11) and (3.12), we have . Therefore, we see that . Now, we are in a position to prove . Notice that . Let . Since M is monotone, we find that . This implies that . This implies that , that is, . Next, we show that . Since , we find from (A2) that
Putting for any and , we see that . Using (3.15), we find that
Since B is monotone, we obtain from (A4) that . Using (A1) and (A4), we find that
Hence, , . It follows from (A3) that . Next, we prove that . Suppose to the contrary, , i.e., . Since and the space satisfies Opial’s condition, one has
Since , we find from Lemma 2.5 that . It follows that . This leads to a contradiction. Thus, we have . This proves that . Therefore, one has
Finally, we show that , as . Note that
which implies that
Using Lemma 2.1, we find that . This completes the proof. □
4 Applications
Recall that a mapping is said to be a k-strict pseudo-contraction if there exists a constant such that
for all . Note that the class of k-strict pseudo-contractions strictly includes the class of nonexpansive mappings. Put , where is a k-strict pseudo-contraction. Then A is -inverse-strongly monotone. Now, we are in a position to state a results on fixed points of strict pseudo-contractions.
Theorem 4.1 Let C be a nonempty closed convex subset of a Hilbert space H and let F be a bifunction from to ℝ which satisfies (A1)-(A4). Let be a k-strict pseudo-contraction, be a β-inverse-strongly monotone mapping, and be a family of infinitely nonexpansive mappings. Assume that . Let be a κ-contraction. Let be a sequence generated by and
where is the sequence generated in (2.4), , , and are sequences in such that and , and are positive number sequences. Assume that the above control sequences satisfy the following restrictions:
-
(a)
, ;
-
(b)
and ;
-
(c)
;
-
(d)
.
Then the sequence converges strongly to .
Proof Taking , wee see that is a α-strict pseudo-contraction with and . Using Theorem 3.1, we find the desired conclusion immediately. □
Let be a proper convex lower semi-continuous function. Then the subdifferential ∂g of g is defined as follows:
From Rockafellar [30], we know that ∂g is maximal monotone. It is not hard to verify that if and only if .
Let be the indicator function of C, i.e.,
Since is a proper lower semi-continuous convex function on H, we see that the subdifferential of is a maximal monotone operator. It is clear that , . Notice that . Now, we are in a position to state the result on variational inequalities.
Theorem 4.2 Let C be a nonempty closed convex subset of a Hilbert space H and let F be a bifunction from to ℝ which satisfies (A1)-(A4). Let be an α-inverse-strongly monotone mapping, be a β-inverse-strongly monotone mapping, and be a family of infinitely nonexpansive mappings. Assume that . Let be a κ-contraction. Let be a sequence generated by and
where is the sequence generated in (2.4), , , and are sequences in such that and , and are positive number sequences. Assume that the above control sequences satisfy the following restrictions:
-
(a)
, ;
-
(b)
and ;
-
(c)
;
-
(d)
.
Then the sequence converges strongly to .
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The author is grateful to Prof. Tian Cheng and the reviewers’ suggestions which improved the contents of the article.
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Zhang, Q. Strong convergence of a Halpern-type algorithm for common solutions of fixed point and equilibrium problems. J Inequal Appl 2014, 313 (2014). https://doi.org/10.1186/1029-242X-2014-313
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DOI: https://doi.org/10.1186/1029-242X-2014-313