We proposed in this paper a new iterative scheme for finding common elements of the set of fixed points of a finite family of quasinonexpansive mappings, the set of solutions of variational inclusion, and the set of solutions of generalized equilibrium problems. Some strong convergence results were derived by using the concept of mappings for a finite family of quasinonexpansive mappings. Strong convergence results are derived under suitable conditions in Hilbert spaces.
1. Introduction
Let be a real Hilbert space with inner product and inducted norm , and let be a nonempty closed and convex subset of . Then, a mapping is said to be
(1)nonexpansive if , for all ;
(2)quasinonexpansive if , for all and ;
(3)Lipschitzian if there exists a constant such that , for all . We denoted by the set of fixed points of .
In 1953, Mann [1] introduced the following iterative procedure to approximate a fixed point of a nonexpansive 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 (1.1) has only weak convergence, in general; for instance, see [2, 3].
Many authors attempt to modify the process (1.1) so that strong convergence is guaranteed that has recently been made. Nakajo and Takahashi [4] proposed the following modification which is the socalled CQ method and proved the following strong convergence theorem for a nonexpansive mapping in a Hilbert space .
Theorem 1.1 (see [4]).
Let be a nonempty closed convex subset of a Hilbert space and let be a nonexpansive mapping of into itself such that . Suppose that and is given by
where . Then, converges strongly to .
Let be a function and let be a bifunction from to such that , where is the set of real numbers and . The generalized equilibrium problem is to find such that
The set of solutions of (1.3) is denoted by ; see also [5–7].
If is replaced by a realvalued function , problem (1.3) reduces to the following mixed equilibrium problem introduced by Ceng and Yao [8]: find such that
Let , for all . Here denotes the indicator function of the set ; that is, if and otherwise. Then problem (1.3) reduces to the following equilibrium problem: find such that
The set of solutions (1.5) is denoted by . Problem (1.5) includes, as special cases, the optimization problem, the variational inequality problem, the fixed point problem, the nonlinear complementarity problem, the Nash equilibrium problem in noncooperative games, and the vector optimization problem; see [9–12] and the reference cited therein.
Recently, Tada and Takahashi [13] proposed a new iteration for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space and then obtain the following theorem.
Theorem 1.2 (see [13]).
Let be a real Hilbert space, let be a closed convex subset of , let be a bifunction, and let be a nonexpansive mapping such that . For an initial point , let a sequence be generated by
where and . Then, converges strongly to .
Let be a singlevalued nonlinear mapping and let be a setvalued mapping. The variational inclusion is to find such that
where is the zero vector in . The set of solutions of problem (1.7) is denoted by . Recall that a mapping is called inverse strongly monotone if there exists a constant such that
A setvalued mapping is called monotone if for all , and imply . A monotone mapping is maximal if its graph of is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if for , for all imply . We define the resolvent operator associated with and as follows:
It is known that the resolvent operator is singlevalued, nonexpansive, and 1inverse strongly monotone; see [14], and that a solution of problem (1.7) is a fixed point of the operator for all ; see also [15]. If , it is easy to see that is a nonexpansive mapping; consequently, is closed and convex.
The equilibrium problems, generalized equilibrium problems, variational inequality problems, and variational inclusions have been intensively studied by many authors; for instance, see [8, 16–43].
Motivated by Tada and Takahashi [13] and Peng et al. [7], we introduce a new approximation scheme for finding a common element of the set of fixed points of a finite family of quasinonexpansive and Lipschitz mappings, the set of solutions of a generalized equilibrium problem, and the set of solutions of a variational inclusion with setvalued maximal monotone and inverse strongly monotone mappings in the framework of Hilbert spaces.
2. Preliminaries and Lemmas
Let be a closed convex subset of a real Hilbert space with norm and inner product . For each , there exists a unique nearest point in , denoted by , such that . is called the metric projection of on to . It is also known that for and , is equivalent to for all . Furthermore
for all , ; see also [4, 44]. In a real Hilbert space, we also know that
for all and .
Lemma 2.1 (see [45]).
Let be a nonempty closed convex subset of a Hilbert space . Then for points and a real number , the set
For solving the generalized equilibrium problem, let us give the following assumptions for , and the set :
(A1) for all ;
(A2) is monotone, that is, for all ;
(A3) for each is weakly upper semicontinuous;
(A4) for each is convex;
(A5) for each , is lower semicontinuous;
(B1) for each and , there exists a bounded subset and such that for any ,
(B2) is a bounded set.
Lemma 2.2 (see [7]).
Let be a nonempty closed convex subset of a real Hilbert . Let be a bifunction from to satisfying (A1)–(A5) and let be a proper lower semicontinuous and convex function such that . For and , define a mapping as follows:
Assume that either (B1) or (B2) holds. Then, the following conclusions hold:
(1)for each , ;
(2) is singlevalued;
(3) is firmly nonexpansive, that is, for any ,
(4);
(5) is closed and convex.
Lemma 2.3 (see [14]).
Let be a maximal monotone mapping and let be a Lipshitz continuous mapping. Then the mapping is a maximal monotone mapping.
Lemma 2.4.
Let be a nonempty closed convex subset of a real Hilbert space . Let be a quasinonexpansive and Lipschitz mapping of into itself. Then, is closed and convex.
Proof.
Since is Lipschitz, it is easy to show that is closed.
Let and where . From (2.2), we have
which implies ; consequently, is convex. This completes the proof.
Lemma 2.5 (see [46]).
In a strictly convex Banach space , if
for all and , then .
In 1999, Atsushiba and Takahashi [47] introduced the concept of the mapping as follows:
where is a finite mapping of into itself and for all with .
Such a mapping is called the mapping generated by and ; see also [48–50]. Throughout this paper, we denote .
Next, we prove some useful lemmas concerning the mapping.
Lemma 2.6.
Let be a nonempty closed convex subset of a strictly convex Banach space . Let be a finite family of quasinonexpansive and Lipschitz mappings of into itself such that and let be real numbers such that for all , , and . Let be the mapping generated by and . Then, the followings hold:
(i) is quasinonexpansive and Lipschitz;
(ii).
Proof.
(i) For each and , we observe that
Let , then
Hence,
This shows that is a quasinonexpansive mapping.
Next, we claim that is a Lipschitz mapping. Note that is Lipschitz for all . For each , we observe
Let , then
Hence,
Since for all , we get that is a Lipschitz mapping.
(ii) Since is trivial, it suffices to show that . To end this, let and . Then, we have
This shows that
and hence
Again by (2.16), we see that . Hence
Applying Lemma 2.5 to (2.19), we get that and hence .
Again by (2.16), we have
and hence
From (2.16), we know that . Since , we have
Applying Lemma 2.5 to (2.22), we get that and hence .
By proving in the same manner, we can conclude that and for all . Finally, we also have
which yields that since . Hence .
Lemma 2.7.
Let be a nonempty closed convex subset of a Banach space . Let be a finite family of quasinonexpansive and Lipschitz mappings of into itself and sequences in such that as . Moreover, for every , let and be the mappings generated by and and and , respectively. Then
Proof.
Let and and be generated by and and and , respectively. Then
Let and . Then
It follows that
Since as , we obtain the result.
3. Strong Convergence Theorems
In this section, we prove a strong convergence theorem which solves the problem of finding a common element of the set of solutions of a generalized equilibrium problem and the set of solutions of a variational inclusion and the set of common fixed points of a finite family of quasinonexpansive and Lipschitz mappings.
Theorem 3.1.
Let be a nonempty closed convex subset of a real Hilbert space , let be a bifunction satisfying (A1)–(A5), let be a proper lower semicontinuous and convex function, let be an inverse strongly monotone mapping, let be a maximal monotone mapping, and let be a finite family of quasinonexpansive and Lipschitz mappings of into itself. Assume that and either (B1) or (B2) holds. Let be the mapping generated by and . For an initial point with and , let , , , and be sequences generated by
where for some , for some and for some .
Then, , , , and converge strongly to .
Proof.
Since for all , we get that is nonexpansive for all . Hence, is closed and convex. By Lemma 2.2(5), we know that is closed and convex. By Lemma 2.4, we also know that is closed and convex. Hence, is a nonempty closed convex set; consequently, is well defined for every .
Next, we divide the proof into seven steps.
Step 1.
Show that for all .
By Lemma 2.1, we see that is closed and convex for all . Hence is well defined for every , . Let . From and for all , we have
It follows that , and hence for all .
Step 2.
Show that exists.
Since is a nonempty closed convex subset of , there exists a unique element . From , we obtain
Hence is bounded; so are , , and .
Since , we also have
From (3.3) and (3.4), we get that exists.
Step 3.
Show that is a Cauchy sequence.
By the construction of the set , we know that for . From (2.1), it follows that
as . Hence is a Cauchy sequence. By the completeness of and the closeness of , we can assume that .
Step 4.
Show that .
From (3.5), we get
as . Since , we have
as . Hence, as . By the nonexpansiveness of and the inverse strongly monotonicity of , we obtain that
This implies that
It follows from (3.7) that
Since is inverse strongly monotone, we have
This implies that
It follows that
From (3.7) and (3.10) we get
It follows from (3.7) and (3.14) that
as . Since is firmly nonexpansive and , we have
which implies that
It follows from (3.17) that
which yields that
Hence, from (3.7) and (3.14), we also have
It follows from (3.15) and (3.20) that
By Lemma 2.7, we also get that . From Lemma 2.6(i), we know that is Lipschitz. Since as , it is easy to verify that . Moreover, by Lemma 2.6(ii), we can conclude that .
Step 5.
Show that .
Since , we have
From (A2), we have
It follows from (A5) and the weakly lower semicontinuity of , , and that
Put for all and . Since and , we obtain , and hence . So by (A1), (A4), and the convexity of , we have
Hence,
Letting , it follows from (A3) and the weakly semicontinuity of that
for all . Observe that if , then holds. Hence .
Step 6.
Show that .
First observe that is an Lipschitz monotone mapping and . From Lemma 2.3, we know that is maximal monotone. Let , that is, . Since , we get , that is,
By the maximal monotonicity of , we have
and so
It follows from , and that
By the maximal monotonicity of , we have ; consequently, .
Step 7.
Show that .
Since and , we obtain
By taking the limit in (3.32), we obtain
This shows that .
From Steps 1–7, we can conclude that , , , and converge strongly to . This completes the proof.
4. Applications
As a direct consequence of Theorem 3.1, we obtain some new and interesting results in a Hilbert space as the following theorems. Recall that is the solution set of the classical variational inequality
Theorem 4.1.
Let be a nonempty closed convex subset of a real Hilbert space , let be a bifunction satisfying (A1)–(A5), let be a proper lower semicontinuous and convex function, let be an inverse strongly monotone mapping, and let be a finite family of quasinonexpansive and Lipschitz mappings of into itself. Assume that and either (B1) or (B2) holds. Let be the mapping generated by and . For an initial point with and , let , , , and be sequences generated by
where for some , for some and for some .
Then, , , , and converge strongly to .
Proof.
In Theorem 3.1, take , where is the indicator function of . It is well known that the subdifferential is a maximal monotone operator. Then, problem (1.7) is equivalent to problem (4.1) and the resolvent operator for all . This completes the proof.
Next, we give a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem, the set of solutions of a variational inclusion and the set of common fixed points of a finite family of quasinonexpansive and Lipschitz mappings. In order to do this, let us assume that
(B3) for each and , there exists a bounded subset and such that for any ,
Theorem 4.2.
Let be a nonempty closed convex subset of a real Hilbert space , let be a bifunction satisfying (A1)–(A5), let be an inverse strongly monotone mapping, let be a maximal monotone mapping, and let be a finite family of quasinonexpansive and Lipschitz mappings of into itself. Assume that and either (B1) or (B3) holds. Let be the mapping generated by and . For an initial point with and , let , , , and be sequences generated by
where for some , for some , and for some .
Then, , , , and converge strongly to .
Proof.
In Theorem 3.1, take , for all . Then problem (1.3) reduces to the equilibrium problem (1.5).
Remark 4.3.
Theorem 3.1 improves and extends the main results in [4, 13] and the corresponding results.
Acknowledgments
The authors would like to thank the referee for the valuable suggestions on the manuscript. The authors were supported by the Commission on Higher Education, the Thailand Research Fund, and the Graduate School of Chiang Mai University.
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