Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand
The purpose of this paper is to introduce a new hybrid projection method for finding a common
element of the set of common fixed points of two relatively quasi-nonexpansive mappings, the set of
the variational inequality for an -inverse-strongly monotone, and the set of solutions of the generalized
equilibrium problem in the framework of a real Banach space. We obtain a strong convergence theorem
for the sequences generated by this process in a 2-uniformly convex and uniformly smooth Banach space.
Base on this result, we also get some new and interesting results. The results in this paper generalize,
extend, and unify some well-known strong convergence results in the literature.
1. Introduction
Let be a real Banach space, the dual space of . A Banach space is said to be strictly convex if for all with and . Let be the unit sphere of . Then a Banach space is said to be smooth if the limit
exists for each It is also said to be uniformly smooth if the limit is attained uniformly for . Let be a Banach space. The modulus of convexity of is the function defined by
A Banach space is uniformly convex if and only if for all . Let be a fixed real number with . A Banach space is said to be -uniformly convex if there exists a constant such that for all ; see [1, 2] for more details. Observe that every -uniform convex is uniformly convex. One should note that no Banach space is -uniform convex for . It is well known that a Hilbert space is 2-uniformly convex, uniformly smooth. For each , the generalized duality mapping is defined by
for all . In particular, is called the normalized duality mapping. If is a Hilbert space, then , where is the identity mapping. It is also known that if is uniformly smooth, then is uniformly norm-to-norm continuous on each bounded subset of .
Let be a real Banach space with norm and denotes the dual space of . Consider the functional defined by
Observe that, in a Hilbert space , (1.4) reduces to , The generalized projection is a map that assigns to an arbitrary point the minimum point of the functional that is, where is the solution to the minimization problem
existence and uniqueness of the mapping follow from the properties of the functional and strict monotonicity of the mapping (see, e.g., [3–7]). In Hilbert spaces, It is obvious from the definition of function that
Remark 1.1. If is a reflexive, strictly convex, and smooth Banach space, then for , if and only if . It is sufficient to show that if then . From (2.13), we have . This implies that From the definition of one has . Therefore, we have see [5, 7] for more details.
Next, we give some examples which are closed relatively quasi-nonexpansive (see [8]).
Example 1.2. Let be the generalized projection from a smooth, strictly convex and reflexive Banach space onto a nonempty closed and convex subset of . Then, is a closed relatively quasi-nonexpansive mapping from onto with .
Let be a real Banach space and let be a nonempty closed and convex subset of and be a mapping. The classical variational inequality problem for a mapping A is to find such that
The set of solutions of (1.4) is denoted by . Recall that A is called
(i)monotone if
(ii)an -inverse-strongly monotone if there exists a constant such that
Such a problem is connected with the convex minimization problem, the complementary problem, and the problem of finding a point satisfying .
Let be a bifunction from to , where denotes the set of real numbers. The equilibrium problem (for short, EP) is to find such that
The set of solutions of (1.10) is denoted by . Given a mapping let for all Then if and only if for all that is, is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.10). Some methods have been proposed to solve the equilibrium problem; see, for instance, [9–11].
Let be a closed convex subset of ; a mapping is said to be nonexpansive if
A point is a fixed point of provided that . Denote by the set of fixed points of ; that is, . Recall that a point in is said to be an asymptotic fixed point of [12] if contains a sequence which converges weakly to such that . The set of asymptotic fixed points of will be denoted by . A mapping from into itself is said to be relatively nonexpansive [13–15] if and for all and . The asymptotic behavior of a relatively nonexpansive mapping was studied in [16–18]. is said to be -nonexpansive, if for . is said to be relatively quasi-nonexpansive if and for and . A mapping in a Banach space is closed if and then .
Remark 1.3. The class of relatively quasi-nonexpansive mappings is more general than the class of relatively nonexpansive mappings [16–19] which requires the strong restriction .
In Hilbert spaces , Iiduka et al. [20] proved that the sequence defined by: and
where is the metric projection of onto , and is a sequence of positive real numbers, and converges weakly to some element of .
It is well know that if is a nonempty closed and convex subset of a Hilbert space and is the metric projection of onto , then is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. In this connection, Alber [4] recently introduced a generalized projection mapping in a Banach space which is an analogue of the metric projection in Hilbert spaces.
In 2008, Iiduka and Takahashi [21] introduced the following iterative scheme for finding a solution of the variational inequality problem for inverse-strongly monotone in a -uniformly convex and uniformly smooth Banach space : and
for every where is the generalized metric projection from onto , is the duality mapping from into , and is a sequence of positive real numbers. They proved that the sequence generated by (1.13) converges weakly to some element of .
Matsushita and Takahashi [22] introduced the following iteration: a sequence defined by
where the initial guess element is arbitrary, is a real sequence in , is a relatively nonexpansive mapping, and denotes the generalized projection from onto a closed convex subset of . They proved that the sequence converges weakly to a fixed point of .
In 2005, Matsushita and Takahashi [19] proposed the following hybrid iteration method (it is also called the CQ method) with generalized projection for relatively nonexpansive mapping in a Banach space :
They proved that converges strongly to , where is the generalized projection from onto .
Recently, Takahashi and Zembayashi [23] proposed the following modification of iteration (1.15) for a relatively nonexpansive mapping:
where is the duality mapping on . Then, converges strongly to where is the generalized projection of onto Also, Takahashi and Zembayashi [24] proved the following iteration for a relatively nonexpansive mapping:
where is the duality mapping on . Then, converges strongly to where is the generalized projection of onto Qin and Su [25] proved the following iteration for relatively nonexpansive mappings in a Banach space :
the sequence generated by (1.18) converges strongly to
In 2009, Wei et al. [26] proved the following iteration for two relatively nonexpansive mappings in a Banach space :
if and are sequences in such that and for some then generated by (1.19) converges strongly to a point where the mapping of onto is the generalized projection. Very recently, Cholamjiak [27] proved the following iteration:
where is the duality mapping on . Assume that , , and are sequences in . Then converges strongly to , where
Motivated and inspired by Iiduka and Takahashi [21], Takahashi and Zembayashi [23, 24], Wei et al. [26], Cholamjiak [27], and Kumam and Wattanawitoon [28], we introduce a new hybrid projection iterative scheme which is difference from the algorithm (1.20) of Cholamjiak in [27, Theorem 3.1] for two relatively quasi-nonexpansive mappings in a Banach space. For an initial point with and , define a sequence as follows:
where is the duality mapping on . Then, we prove that under certain appropriate conditions on the parameters, the sequences and generated by (1.21) converge strongly to
The results presented in this paper improve and extend the corresponding results announced by Iiduka and Takahashi [21], Wei et al. [26], Kumam and Wattanawitoon [28], and many other authors in the literature.
2. Preliminaries
We also need the following lemmas for the proof of our main results.
Lemma 2.1 (Beauzamy [29] and Xu [30]). If is a 2-uniformly convex Banach space, then, for all we have
where is the normalized duality mapping of and .
The best constant in the Lemma is called the -uniformly convex constant of .
Lemma 2.2 (Beauzamy [29] and Zǎlinescu [31]). If is a p-uniformly convex Banach space and is a given real number with , then for all and
where is the generalized duality mapping of and is the p-uniformly convexity constant of .
Lemma 2.3 (Kamimura and Takahashi [6]). Let be a uniformly convex and smooth Banach space and let and be two sequences of . If and either or is bounded, then
Lemma 2.4 (Alber [4]). Let be a nonempty closed and convex subset of a smooth Banach space and . Then, if and only if
Lemma 2.5 (Alber [4]). Let be a reflexive, strictly convex, and smooth Banach space, let be a nonempty closed and convex subset of and let Then
Lemma 2.6 (Qin et al. [8]). Let be a uniformly convex and smooth Banach space, let be a closed and convex subset of , and let be a closed relatively quasi-nonexpansive mapping from into itself. Then is a closed and convex subset of .
For solving the equilibrium problem for a bifunction , let us assume that satisfies the following conditions:
(A1) for all ; (A2) is monotone, that is, for all ; (A3)for each ,
(A4)for each , is convex and lower semi-continuous. Lemma 2.7 (Blum and Oettli [9]). Let be a closed and convex subset of a smooth, strictly convex and reflexive Banach space , let be a bifunction from to satisfying (A1)–(A4), and let and . Then, there exists such that
Lemma 2.8 (Combettes and Hirstoaga [10]). Let be a closed and convex subset of a uniformly smooth, strictly convex and reflexive Banach space and let be a bifunction from to satisfying (A1)–(A4). For and , define a mapping as follows:
for all . Then the following holds: (1) is single-valued; (2) is a firmly nonexpansive-type mapping, for all ,
(3)(4) is closed and convex.
Lemma 2.9 (Takahashi and Zembayashi [24]). Let be a closed and convex subset of a smooth, strictly convex, and reflexive Banach space , let be a bifunction from to satisfying (A1)–(A4), and let . Then, for and
Let be a reflexive, strictly convex, and smooth Banach space and the duality mapping from into . Then is also single value, one-to-one, surjective, and it is the duality mapping from into . We make use of the following mapping studied in Alber [4]:
for all and , that is, .
Lemma 2.10 (Alber [4]). Let be a reflexive, strictly convex, and smooth Banach space and let be as in (2.10). Then
for all and .
Let be an inverse-strongly monotone of into which is said to be hemicontinuous if for all , the mapping of into , defined by , is continuous with respect to the weak topology of . We define by the normal cone for at a point ; that is,
Theorem 2.11 (Rockafellar [32]). Let be a nonempty, closed and convex subset of a Banach space , and a monotone, hemicontinuous mapping of into . Let be a mapping defined as follows:
Then is maximal monotone and
3. Main Results
In this section, we establish a new hybrid iterative scheme for finding a common element of the set of solutions of an equilibrium problems, the common fixed point set of two relatively quasi-nonexpansive mappings, and the solution set of variational inequalities for -inverse strongly monotone in a 2-uniformly convex and uniformly smooth Banach space.
Theorem 3.1. Let be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space . Let be a bifunction from to satisfying (A1)–(A4) and let be an -inverse-strongly monotone mapping of into satisfying and Let be closed relatively quasi-nonexpansive mappings such that For an initial point with and , we define the sequence as follows:
where is the duality mapping on , and are sequences in such that and , for some , and for some with , where is the 2-uniformly convexity constant of . Then converges strongly to , where .
Proof. We have several steps to prove this theorem as follows:Step 1. We show that is closed and convex.
Clearly is closed and convex. Suppose that is closed and convex for each . Since for any , we know that
is equivalent to
So, is closed and convex. Then, by induction, is closed and convex for all .Step 2. We show that is well defined.
Put for all . On the other hand, from Lemma 2.8 one has is relatively quasi-nonexpansive mappings and Supposing for by the convexity of for each , we have
and so
For all , we know from Lemma 2.10, that
Since and from being an -inverse-strongly monotone, we get
From Lemma 2.1 and being an -inverse-strongly monotone, we obtain
Substituting (3.7) and (3.8) into (3.6), we have
Replacing (3.9) into (3.5), we get
Substituting (3.10) into (3.4), we also have
This shows that and hence, . Hence, for all . This implies that the sequence is well defined.Step 3. We show that exists and is bounded.
From and we have
and from Lemma 2.5, we have
From (3.12) and (3.13), then are nondecreasing and bounded. So, we obtain that exists. In particular, by (1.6), the sequence is bounded. This implies that is also bounded.Step 4. We show that is a Cauchy sequence in .
Since , for , by Lemma 2.5, we have
Taking , we have We have . From Lemma 2.3, we get . Thus is a Cauchy sequence.Step 5. We cliam that , as .
By the completeness of , the closedness of and is a Cauchy sequence (from Step 4); we can assume that there exists such that as .
By definition of , we have
Since exists, we get
It follow form Lemma 2.3, that
Since and from the definition of , we have
and so
Hence
By using the triangle inequality, we obtain
By (3.17), (3.20), we get
Since is uniformly norm-to-norm continuous on bounded subsets of , we have
Step 6. Show that .
Applying (3.4) and (3.11), we get From Lemma 2.9 and , we observe that
From (3.22), (3.23) and Lemma 2.3, we get
Since is uniformly norm-to-norm continuous, we obtain
From , we have as and
By (A2), that
and , we get for all . For , define . Then which implies that From (A1), we obtain that
Thus From (A3), we have for all . Hence .Step 7. We show that .
From definition of , we have
Since , we have
It follows from (3.16) that
again from Lemma 2.3, we get
By using the triangle inequality, we get
Again by (3.17) and (3.33), we also have
Since is uniformly norm-to-norm continuous, we obtain
Since
from (3.22), (3.25), and (3.35), we have
Since is uniformly norm-to-norm continuous, we also have
From (3.1), we get
it follows that
and hence
Since for some , (3.36), and (3.39), one has Since is uniformly norm-to-norm continuous, we get
Since
from (3.35) and (3.43), we obtain
Since is closed and , we have .
On the other hand, we note that
It follows from and , that
Furthermore, from (3.4) and (3.5),
and hence
From (3.47) and (3.49), we have
From Lemma 2.5, Lemma 2.10, and (3.8), we compute
Applying Lemmas 2.3 and (3.50), we obtain that
Again since is uniformly norm-to-norm continuous on bounded set, we have
Since
by (3.35) and (3.52), we have
and hence
From (3.1) we obtain that
and hence
so
By (3.53), (3.56) andcondition for some we obtain
Since is uniformly norm-to-norm continuous on bounded set, we obtain
Since , then Thus by the closedness of and we get . Hence .Step 8. We show that .
Define by Theorem 2.11; is maximal monotone and . Let . Since , we get .
From , we have
On the other hand, since , then by Lemma 2.4, we have
and hence
It follows from (3.62) and (3.64), that
Where . Taking the limit as and (3.53), we obtain . By the maximality of , we have ; that is, .Step 9. We show that .
From , we have , Since , we also have
By taking limit , we obtain that
By Lemma 2.4, we can conclude that and as . This completes the proof.
Setting in Theorem 3.1., so, we obtain the following corollary.
Corollary 3.2. Let be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space . Let be a bifunction from to satisfying (A1)–(A4) and let be an -inverse-strongly monotone mapping of into satisfying and . Let be closed relatively quasi-nonexpansive mappings such that . For an initial point with and , define a sequence as follows:
where is the duality mapping on . Assume that and are sequences in such that and , for some , , and for some with , where is the 2-uniformly convexity constant of . Then converges strongly to , where .
If in Theorem 3.1, then we obtain the following corollary.
Corollary 3.3. Let be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space . Let be a bifunction from to satisfying (A1)–(A4). Let is closed relatively quasi-nonexpansive mappings such that . For an initial point with and , define a sequence as follows:
where is the duality mapping on . Assume that and are sequences in such that and , for some and . Then converges strongly to , where .
4. Application
4.1. Complementarity Problem
Let be a nonempty, closed and convex cone , A a mapping of into . We define its polar in to be the set
Then the element is called a solution of the complementarity problem if
The set of solutions of the complementarity problem is denoted by
Theorem 4.1. Let be a nonempty and closed convex subset of a 2-uniformly convex and uniformly smooth Banach space . Let be a bifunction from to satisfying (A1)–(A4) and let be an -inverse-strongly monotone of into satisfying and Let be closed relatively quasi-nonexpansive mappings and For an initial point with and , we define the sequence as follows:
where is the duality mapping on , and are sequences in such that and , for some , and for some with , where is the 2-uniformly convexity constant of . Then converges strongly to , where .
Proof. As in the proof of Takahashi in [7, Lemma ], we get that . So, we obtain the result.
4.2. Approximation of a Zero of a Maximal Monotone Operator
Let be a multivalued mapping from to with domain and range A mapping is said to be a monotone operator if for each and A monotone operator is said to be maximal if its graph is not property contained in the graph of any other monotone operator. We know that if is a maximal monotone operator, then is closed and convex. Let be a reflexive, strictly convex, and smooth Banach space, and let be a monotone operator from to , we know that is maximal if and only if for all Let be defined by and such a is called the resolvent of . We know that is a relatively nonexpansive (closed relatively quasi-nonexpansive for example; see [8]), and for all (see [7, 33–35] for more details).
Theorem 4.2. Let be a nonempty and closed convex subset of a 2-uniformly convex and uniformly smooth Banach space . Let be a bifunction from to satisfying (A1)–(A4) and let be -inverse-strongly monotone of into satisfying and Let be a maximal monotone operator of into and let be a resolvent of and a closed mapping such that For an initial point with and , we define the sequence as follows:
where is the duality mapping on , and are sequences in such that and , for some , and for some with , where is the 2-uniformly convexity constant of . Then converges strongly to , where .
Proof. Since is a closed relatively nonexpansive mapping and . So, we obtain the result.
Corollary 4.3. Let be a nonempty and closed convex subset of a 2-uniformly convex and uniformly smooth Banach space . Let be a bifunction from to satisfying (A1)–(A4) and let be -inverse-strongly monotone of into satisfying and Let be a maximal monotone operator of into and let be a resolvent of and closed such that For an initial point with and , we define the sequence as follows:
where is the duality mapping on , and are sequences in such that and , for some , and for some with , where is the 2-uniformly convexity constant of . Then converges strongly to , where .
Acknowledgments
The authors would like to thank the referee for the valuable suggestions on the manuscript. Siwaporn Saewan would like to thank the Office of the Higher Education Commission, Thailand for supporting by grant fund under the program Strategic Scholarships for Frontier Research Network for the Join Ph.D. Program Thai Doctoral degree for this research. Moreover, Poom Kumam was supported by the Thailand Research Fund and the Commission on Higher Education (MRG5180034).