Strong convergence by a hybrid algorithm for solving generalized mixed equilibrium problems and fixed point problems of a Lipschitz pseudo-contraction in Hilbert spaces
© Ungchittrakool and Jarernsuk; licensee Springer 2012
Received: 18 February 2012
Accepted: 29 August 2012
Published: 12 September 2012
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.
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 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  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.
However, the following examples show that the converse is not true.
Example 1.1 (Chidume and Mutangadura )
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.
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 ). 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.  introduced the hybrid iterative algorithm which just involved one sequence of closed convex set for a pseudo-contractive mapping in Hilbert spaces as follows:
Theorem 1.4 ()
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.
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.
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.
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 ;
(A4) for each , is convex and lower semi-continuous.
Lemma 2.3 (Blum and Oettli )
The proof of the following lemma appears in [, Lemma 2.8].
- (ii)is firmly nonexpansive-type mapping, i.e., for any ,
is closed and convex.
Lemma 2.5 (Zhang )
- (ii)is firmly nonexpansive-type mapping, i.e., for any ,
is closed and convex;
, , .
Remark 2.6 In the framework of a Hilbert space, it is well known that and then is firmly nonexpansive.
Lemma 2.7 ()
is a closed convex subset of C.
is demiclosed at zero, i.e., if is a sequence in C such that and , then .
Lemma 2.8 ()
where f is continuous and concave functional. Then the set K is closed and convex.
Thus K is convex. □
The following lemma provides some useful properties of a firmly nonexpansive mapping on a Hilbert space.
Lemma 2.10 ([, Lemma 2.5])
T is firmly nonexpansive if and only ifis firmly nonexpansive.
3 Main result
for all ,
for all ,
for all with .
Thenconverges strongly to.
Therefore, . By mathematical induction, we have for all .
This implies that is bounded and then , and are bounded too.
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.
Assume the sequence, andare as in Theorem 3.1. Thenconverges strongly to.
Corollary 3.4 (Yao et al. [, 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. □
B is monotone ⟺ is pseudo-contractive.
B is inverse strongly monotone ⟺ is strictly pseudo-contractive.
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. □
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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