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
(C1) is -inverse-strongly monotone,
(C3) for all and .
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.
The set of solutions of (1.5) is denoted by .
For solving the equilibrium problem, let us assume that a bifunction satisfies the following conditions:
(A1) for all ;
(A2) is monotone, that is, for all ;
(A4)for all is convex and lower semicontinuous.
Recently, Takahashi and Zembayashi , introduced the following iterative scheme which is called the shrinking projection method:
where is the duality mapping on and is the generalized projection from onto . They proved that the sequence converges strongly to under appropriate conditions.
Under suitable conditions over , and , they obtain that the sequence generated by (1.8) converges strongly to .
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.
where is the generalized duality mapping of and is the -uniformly convexity constant of .
for all . In a Hilbert space , we have for all .
Recall that a mapping is called nonexpansive if for all and relatively nonexpansive if satisfies the following conditions:
(1) , where is the set of fixed points of ;
(2) for all and ;
(3) , where is the set of all asymptotic fixed points of ;
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 .
Let be a uniformly smooth and strictly convex Banach space and be a maximal monotone mapping such that its zero set . Then, is a closed relatively quasi-nonexpansive mapping from onto and .
Let be the generalized projection from a smooth, strictly convex, and reflexive Banach space onto a nonempty closed convex subset of . Then, is a closed relatively quasi-nonexpansive mapping with .
Lemma 2.4 (Kamimura and Takahashi ).
Let be a uniformly convex and smooth Banach space and let be two sequences of . If and either or is bounded, then as .
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 ).
Let be a nonempty closed convex subset of a smooth Banach space and . Then if and only if for all .
Lemma 2.6 (Alber ).
Lemma 2.7 (Qin et al. ).
Let be a uniformly convex, smooth Banach space, let be a closed convex subset of , let be a closed and relatively quasi-nonexpansive mapping from into itself. Then is a closed convex subset of .
Lemma 2.8 (Cho et al. ).
for all , and with .
Lemma 2.9 (Blum and Oettli ).
Lemma 2.10 (Qin et al. ).
Then, the following hold:
(1) is single-valued;
(4) is closed and convex.
Lemma 2.11 (Takahashi and Zembayashi ).
for all and , that is, .
Lemma 2.12 (Alber ).
for all and .
Theorem 2.13 (Rockafellar ).
Then is maximal monotone and .
3. Strong Convergence Theorems
where is the duality mapping on . Assume that , and are sequences in satisfying the restrictions:
(B2) , ;
(B3) for some ;
(B4) for some with , where is the -uniformly convexity constant of .
Then, and converge strongly to .
We divide the proof into eight steps.
Show that and are well defined.
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.
So, is closed and convex. By induction, is closed and convex for all . This shows that is well-defined.
Show that for all .
This shows that ; consequently, . Hence for all .
Show that exists.
Combining (3.8) and (3.9), we obtain that exists.
Show that is a Cauchy sequence in .
Show that .
From (3.19), (3.26) and by the closedness of and , we get .
Show that .
From (A4) and , we get for all . For and . Define , then , which implies that . From (A1), we obtain that . Thus, . From (A3), we have for all . Hence .
Show that .
where . By taking the limit as and from (3.24) and (3.25), we obtain . By the maximality of , we have and hence .
Show that .
By Lemma 2.5, we can conclude that . Furthermore, it is easy to see that as . This completes the proof.
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 .
Putting and in Theorem 3.1, we obtain the result.
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 .
Next, we consider the problem of finding a zero point of an inverse-strongly monotone operator of into . Assume that satisfies the conditions:
(D1) is -inverse-strongly monotone,
where is the duality mapping on . Assume that , and are sequences in satisfying the conditions (B1)–(B4) of Theorem 3.1.
Then, and converge strongly to .
Putting in Theorem 3.1, we have . We also have and then the condition (C3) of Theorem 3.1 holds for all and . So, we obtain the result.
The set of solutions of the complementarity problem is denoted by .
Assume that is an operator satisfying the conditions:
(E1) is -inverse-strongly monotone,
(E3) for all and .
where is the duality mapping on . Assume that and are sequences in satisfying the conditions (B1)–(B4) of Theorem 3.1.
Then, and converge strongly to .
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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