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Strong convergence by a hybrid algorithm for solving generalized mixed equilibrium problems and fixed point problems of a Lipschitz pseudo-contraction in Hilbert spaces
Fixed Point Theory and Applications volume 2012, Article number: 147 (2012)
Abstract
In this paper, we construct a sequence by using some appropriated closed convex sets based on the hybrid shrinking projection methods to find a common solution of fixed point problems of a Lipschitz pseudo-contraction and generalized mixed equilibrium problems in Hilbert spaces. The strong convergence theorems are proved under some mild conditions on scalars. The results not only cover the research work of Yao et al. (Nonlinear Anal. 71:4997-5002, 2009) but can also be applied for finding the common element of the set of zeroes of a Lipschitz monotone mapping and the set of generalized mixed equilibrium problems in Hilbert spaces.
MSC:47H05, 47H09, 47H10, 47J25.
1 Introduction
The equilibrium problem theory provides a novel and unified treatment of a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity and optimization, and it has been extended and generalized in many directions; see [1, 2]. In particular, equilibrium problems are related to the problem of finding fixed points problems of some nonlinear mappings. Therefore, it is natural to construct a unified approach to these problems. In this direction, several authors have introduced some iterative schemes for finding a common element of the set of the solutions of equilibrium problems and the set of fixed points (see also [3–7] and the references therein). In this paper, we suggest and analyze a hybrid algorithm for solving generalized mixed equilibrium problems and fixed point problems of a Lipschitz pseudo-contraction in the framework of Hilbert spaces.
Let E be a real Banach space, and the dual space of E. Let C be a nonempty closed convex subset of E. Let be a bifunction, be a real-valued function, and be a nonlinear mapping. The generalized mixed equilibrium problem is to find such that
The solution set of (1.1) is denoted by , i.e.,
If , the problem (1.1) reduces to the mixed equilibrium problem for Θ, denoted by , which is to find such that
If , the problem (1.1) reduces to the mixed variational inequality of Browder type, denoted by , which is to find such that
If and , the problem (1.1) reduces to the equilibrium problem for Θ (for short, EP), denoted by , which is to find such that
Let for all . Then if and only if for all , i.e., p is a solution of the variational inequality; there are several other problems, for example, the complementarity problem, fixed point problem and optimization problem, which can also be written in the form of an EP. In other words, the EP is a unifying model for several problems arising in physics, engineering, science, optimization, economics, etc. Many papers on the existence of solutions of EP have appeared in the literature (see, for example, [1, 8–10] and references therein). Motivated by the work [3, 11, 12], Takahashi and Takahashi [4] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of the EP (1.2) and the set of fixed points of a nonexpansive mapping in the setting of a Hilbert space. They also studied the strong convergence of the sequences generated by their algorithm for a solution of the EP which is also a fixed point of a nonexpansive mapping defined on a closed convex subset of a Hilbert space.
Recall, a mapping T with domain and range in H is called firmly nonexpansive if
nonexpansive if
Throughout this paper, I stands for an identity mapping. The mapping T is said to be a strict pseudo-contraction if there exists a constant such that
In this case, T may be called a κ-strict pseudo-contraction mapping. In the even that , T is said to be a pseudo-contraction, i.e.,
It is easy to see that (1.3) is equivalent to
By definition, it is clear that
However, the following examples show that the converse is not true.
Example 1.1 (Chidume and Mutangadura [13])
Take , , , . If , we define to be . Define by
Then, T is Lipschitz and a pseudo-contraction but not a strict pseudo-contraction.
Example 1.2 Take and define by . Then, T is a strict pseudo-contraction but not a nonexpansive mapping.
Indeed, it is clear that T is not nonexpansive. On the other hand, let us consider
for all . Thus T is a strict pseudo-contraction.
Example 1.3 Take and let , it is not hard to verify that T is nonexpansive but not firmly nonexpansive.
From a practical point of view, strict pseudo-contractions have more powerful applications than nonexpansive mappings do in solving inverse problems (see [14]). Therefore, it is important to develop a theory of iterative methods for strict pseudo-contractions.
Takahashi and Zembayashi [5, 6] proposed some hybrid methods to find the solution of a fixed point problem and an equilibrium problem in Banach spaces. Subsequently, many authors (see, e.g.[15–19] and references therein) have used the hybrid methods to solve fixed point problems and equilibrium problems.
Recently, Yao et al. [20] introduced the hybrid iterative algorithm which just involved one sequence of closed convex set for a pseudo-contractive mapping in Hilbert spaces as follows:
Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a pseudo-contraction. Let be a sequence in . Let . For and , define a sequence of C as follows:
Theorem 1.4 ([20])
Let C be a nonempty closed convex subset of a real Hilbert space H. Letbe an L-Lipschitz pseudo-contraction such that. Assume the sequencefor some. Then the sequencegenerated by (1.4) converges strongly to.
Very recently, Tang et al. [21] generalized the hybrid algorithm (1.4) in the case of the Ishikawa iterative process as follows:
Under some appropriate conditions of and , they proved that (1.5) converges strongly to .
Motivated and inspired by the above research work, in this paper, by employing (1.4) and (1.5), we construct a sequence by using some appropriated closed convex sets based on the hybrid shrinking projection methods to find a common solution of fixed point problems of a Lipschitz pseudo-contraction and generalized mixed equilibrium problems in Hilbert spaces. More precisely, we also provide some applications of the main theorem for finding the common element of the set of zeroes of a Lipschitz monotone mapping and the set of generalized mixed equilibrium problems in Hilbert spaces.
2 Preliminaries
Let H be a real Hilbert space with inner product and norm and let C be a closed convex subset of H. For every point , there exists a unique nearest point in C, denoted by , such that
where is called the metric projection of H onto C. We know that is a nonexpansive mapping. It is also known that H satisfies Opial’s condition, i.e., for any sequence with , the inequality
holds for every with .
For a given sequence , let denote the weak ω-limit set of .
Now we recall some lemmas which will be used in the proof of the main result in the next section. We note that Lemmas 2.1 and 2.2 are well known.
Lemma 2.1 Let H be a real Hilbert space. There holds the following identity
-
(i)
.
Lemma 2.2 Let C be a closed convex subset of a real Hilbert space H. Givenand. Thenif and only if there holds the relation
For solving the equilibrium problem for a bifunction , let us assume that Θ satisfies the following condition:
(A1) for all ;
(A2) Θ is monotone, i.e., for all ;
(A3) for each ,
(A4) for each , is convex and lower semi-continuous.
For a real Banach space E with norm , duality product and dual space , the normalized duality mapping is defined by
Lemma 2.3 (Blum and Oettli [1])
Let C be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space E, and let Θ be a bifunction of C × C intosatisfying (A 1)-(A 4). Letand. Then, there existssuch that
The proof of the following lemma appears in [[5], Lemma 2.8].
Lemma 2.4 Let C be a closed convex subset of a uniformly smooth, strictly convex and reflexive Banach space E, and let Θ be a bifunction fromtosatisfying (A 1)-(A 4). Forand, define a mappingas follows:
for all. Then, the following hold:
-
(i)
is single-valued;
-
(ii)
is firmly nonexpansive-type mapping, i.e., for any ,
-
(iii)
;
-
(iv)
is closed and convex.
Lemma 2.5 (Zhang [22])
Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space E. Letbe a continuous and monotone mapping, be a lower semi-continuous and convex function, and Θ be a bifunction oftosatisfying (A 1)-(A 4). Forand. Then, there existssuch that
Define a mappingas follows:
for all. Then, the following conclusions hold:
-
(i)
is single-valued;
-
(ii)
is firmly nonexpansive-type mapping, i.e., for any ,
-
(iii)
;
-
(iv)
is closed and convex;
-
(v)
, , .
Remark 2.6 In the framework of a Hilbert space, it is well known that and then is firmly nonexpansive.
Lemma 2.7 ([23])
Let H be a real Hilbert space, C a closed convex subset of H anda continuous pseudo-contractive mapping, then
-
(i)
is a closed convex subset of C.
-
(ii)
is demiclosed at zero, i.e., if is a sequence in C such that and , then .
Lemma 2.8 ([24])
Let C be a closed convex subset of H. Letbe a sequence in H and. Let. Ifis such thatand satisfies the condition
Then.
Lemma 2.9 Letbe a closed convex set, and
where f is continuous and concave functional. Then the set K is closed and convex.
Proof It is easy to see that the continuity of f yields the closeness of K. Notice that for all and , we have , , , and then the concavity of f allows
Thus K is convex. □
The following lemma provides some useful properties of a firmly nonexpansive mapping on a Hilbert space.
Lemma 2.10 ([[7], Lemma 2.5])
T is firmly nonexpansive if and only ifis firmly nonexpansive.
3 Main result
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H, be an L-Lipschitz pseudo-contraction. Let Θ be a bifunction fromintosatisfying (A 1)-(A 4), be a lower semicontinuous and convex function, be a continuous and monotone mapping such that. Let. Forand, define a sequenceof C as follows:
Assume the sequence, andare such that
-
(1)
for all ,
-
(2)
for all ,
-
(3)
for all with .
Thenconverges strongly to.
Proof By Lemma 2.7(i) and Lemma 2.5(iv), we see that and are closed and convex respectively, then Ω is also. Hence is well defined. Next, we will prove by induction that for all . Note that . Assume that holds for some . Let , thus . We observe that
Consider the last term of (3.2), we obtain
By connecting (3.2) and (3.3), and then by the assumption (1) on , we obtain
Notice that and by Lemma 2.10, we observe that
So, we have
Joining (3.4) and (3.5), we obtain
Notice that
By (3.6) and (3.7), we have
Combining (3.8) and (3.5), we obtain
Therefore, . By mathematical induction, we have for all .
Let , it is not hard to see that the linearity of and together with the continuity and concavity of allow to be continuous and concave. By Lemma 2.9, is closed and convex for all . Therefore, is well defined. From , we have for all . Using , we also have for all . So, for , we have
Hence,
This implies that is bounded and then , and are bounded too.
From and , we have
Hence,
and therefore
which implies that exists. From Lemma 2.1 and (3.10), we obtain
Since , we have
Therefore, we obtain
We note that
that is,
Next, we will show that
Since is bounded, the reflexivity of H guarantees that . Let , then there exists a subsequence of such that and by Lemma 2.7(ii), we have . On the other hand, since and , we have . Define by for all . It is not hard to verify that G satisfies conditions (A1)-(A4). It follows from and (A2) that
Replacing n by , we have
By using (A4) and the assumption (3) on , we obtain for all . For and , let . So, from (A1) and (A4) we have
Dividing by t, we have
From (A3) we have for all , and hence . So, and then we have (3.11). Therefore, by inequality (3.9) and Lemma 2.8, we obtain converges strongly to . This completes the proof. □
Remark 3.2 It is interesting that the assumption on a sequence of scalars is a very mild condition. This is a direct result of the firmly nonexpansiveness of together with the structure and the definition of the set . If for all n, then and the sequence and are independent. However, the properties of still force to produce the sequence to cause a convergence to the common solution .
If and , then we have the following corollary.
Corollary 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H, be an L-Lipschitz pseudo-contraction. Let Θ be a bifunction fromintosatisfying (A 1)-(A 4), such that. Let. Forand, define a sequenceof C as follows:
Assume the sequence, andare as in Theorem 3.1. Thenconverges strongly to.
Corollary 3.4 (Yao et al. [[20], Theorem 3.1])
Let C be a nonempty closed convex subset of a real Hilbert space H. Letbe an L-Lipschitz pseudo-contraction such that. Assume thatis a sequence such thatfor all n. Then the sequencegenerated by (1.4) converges strongly to.
Proof Put , , and for all in Theorem 3.1. Then, for all . So, for all (note that ). Since for all , so we have and then for all . Thus for all . For this reason, (1.4) is a special case of (3.1). Applying Theorem 3.1, we have the desired result. □
Recall that a mapping B is said to be monotone, if for all and inverse strongly monotone if there exists a real number such that for all . For the second case, B is said to be γ-inverse strongly monotone. It follows immediately that if B is γ-inverse strongly monotone, then B is monotone and Lipschitz continuous, that is, . The pseudo-contractive mapping and strictly pseudo-contractive mapping are strongly related to the monotone mapping and inverse strongly monotone mapping, respectively. It is well known that
-
(i)
B is monotone ⟺ is pseudo-contractive.
-
(ii)
B is inverse strongly monotone ⟺ is strictly pseudo-contractive.
Indeed, for (ii), we notice that the following equality always holds in a real Hilbert space:
without loss of generality, we can assume that , and then it yields
Corollary 3.5 Let C, H, Θ, A and φ be as in Theorem 3.1 and letbe an L-Lipschitz monotone mapping such that. Let. Forand, define a sequenceof C as follows:
Assumefor all, andare as in Theorem 3.1. Thenconverges strongly to.
Proof Let . Then T is pseudo-contractive and -Lipschitz. Hence, it follows from Theorem 3.1, we have the desired result. □
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Acknowledgements
The authors would like to thank the Centre of Excellence in Mathematics under the Commission on Higher Education, Ministry of Education, Thailand. They also thank the Editor and two anonymous referees for reading this paper carefully and providing valuable comments to improve the original version of this paper. The project was supported by Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok, 10400, Thailand.
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Ungchittrakool, K., Jarernsuk, A. Strong convergence by a hybrid algorithm for solving generalized mixed equilibrium problems and fixed point problems of a Lipschitz pseudo-contraction in Hilbert spaces. Fixed Point Theory Appl 2012, 147 (2012). https://doi.org/10.1186/1687-1812-2012-147
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DOI: https://doi.org/10.1186/1687-1812-2012-147