A Hybrid Iterative Scheme for Equilibrium Problems, Variational Inequality Problems, and Fixed Point Problems in Banach Spaces
© Prasit Cholamjiak. 2009
Received: 5 February 2009
Accepted: 10 April 2009
Published: 5 May 2009
The purpose of this paper is to introduce a new hybrid projection algorithm for finding a common element of the set of solutions of the equilibrium problem and the set of the variational inequality for an inverse-strongly monotone operator and the set of fixed points of relatively quasi-nonexpansive mappings in a Banach space. Then we show a strong convergence theorem. Using this result, we obtain some applications in a Banach space.
The set of solutions of (1.1) is denoted by . Such a problem is connected with the convex minimization problem, the complementarity, the problem of finding a point satisfying , and so on. First, we recall that
where is the duality mapping on , and is the generalized projection from onto . Assume that for some with where is the -uniformly convexity constant of . They proved that if is weakly sequentially continuous, then the sequence converges weakly to some element in where .
The problem of finding a common element of the set of the variational inequalities for monotone mappings in the framework of Hilbert spaces and Banach spaces has been intensively studied by many authors; see, for instance, [2–4] and the references cited therein.
Recently, Takahashi and Zembayashi , introduced the following iterative scheme which is called the shrinking projection method:
The problem of finding a common element of the set of fixed points and the set of solutions of an equilibrium problem in the framework of Hilbert spaces and Banach spaces has been studied by many authors; see [5, 7–16].
Motivated by Iiduka and Takahashi , Takahashi and Zembayashi , and Qin et al. , we introduce a new general process for finding common elements of the set of the equilibrium problem and the set of the variational inequality problem for an inverse-strongly monotone operator and the set of the fixed points for relatively quasi-nonexpansive mappings.
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 . See [20, 21] for more details.
is said to be relatively quasi-nonexpansive if satisfies the conditions and . It is easy to see that the class of relatively quasi-nonexpansive mappings is more general than the class of relatively nonexpansive mappings [9, 25, 26].
We give some examples which are closed relatively quasi-nonexpansive; see .
Lemma 2.4 (Kamimura and Takahashi ).
Let be a nonempty closed convex subset of . If is reflexive, strictly convex and smooth, then there exists such that for and . The generalized projection defined by . The existence and uniqueness of the operator follows from the properties of the functional and strict monotonicity of the duality mapping ; for instance, see [20, 27–30]. In a Hilbert space, is coincident with the metric projection.
Lemma 2.5 (Alber ).
Lemma 2.6 (Alber ).
Lemma 2.7 (Qin et al. ).
Lemma 2.8 (Cho et al. ).
Lemma 2.9 (Blum and Oettli ).
Lemma 2.10 (Qin et al. ).
Then, the following hold:
Lemma 2.11 (Takahashi and Zembayashi ).
Lemma 2.12 (Alber ).
Theorem 2.13 (Rockafellar ).
3. Strong Convergence Theorems
We divide the proof into eight steps.
It is obvious that is a closed convex subset of . By Lemma 2.7, we know that is closed and convex. From Lemma 2.10 , we also have is closed and convex. Hence is a nonempty, closed, and convex subset of ; consequently, is well defined.
As a direct consequence of Theorem 3.1, we obtain the following results.
Let be a -uniformly convex and uniformly smooth Banach space, and let be a nonempty closed convex subset of . Let be a bifunction from to satisfying (A1)–(A4) and let be a closed relatively quasi-nonexpansive mapping from into itself such that . Assume that satisfies and for some . Then the sequence generated by (1.7) converges strongly to .
If in Theorem 3.1, then Theorem 3.1 reduces to Theorem of Qin et al. .
Corollary 3.2 improves Theorem of Takahashi and Zembayashi  from the class of relatively nonexpansive mappings to the class of relatively quasi-nonexpansive mappings, that is, we relax the strong restriction: . Further, the algorithm in Corollary 3.2 is also simpler to compute than the one given in .
From [20, Lemma ], we have . Hence, we obtain the result.
The author would like to thank Professor Suthep Suantai and the referee for the valuable suggestions on the manuscript. The author was supported by the Commission on Higher Education and the Thailand Research Fund.
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