Research Article | Open Access
Strong Convergence Theorems for Equilibrium Problems and -Strict Pseudocontractions in Hilbert Spaces
We introduce a new iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed point of a finite family of -strictly pseudo-contractive nonself-mappings. Strong convergence theorems are established in a real Hilbert space under some suitable conditions. Our theorems presented in this paper improve and extend the corresponding results announced by many others.
Let be a real Hilbert space with inner product and norm , respectively. Let be a nonempty closed convex subset of . Let be a bifunction from into , where denotes the set of real numbers. We consider the following problem: Find such that which is called equilibrium problem. We use to denote the set of solution of the problem (1.1). 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 (1.1). Some methods have been proposed to solve the equilibrium problem (see, e.g., [1–3]).
Recall that a nonself-mapping is called a -strict pseudocontraction if there exists a constant such that We use to denote the fixed point set of the mapping , that is, . As , is said to be nonexpansive, that is, . is said to be pseudocontractive if and is also said to be strongly pseudocontractive if there exists a positive constant such that is pseudocontractive. Clearly, the class of -strict pseudocontractions falls into the one between classes of nonexpansive mappings and pseudocontractions. We remark also that the class of strongly pseudocontractive mappings is independent of the class of -strict pseudocontractions (see, e.g., [4, 5]).
Iterative methods for equilibrium problem and nonexpansive mappings have been extensively investigated; see, for example, [1–18] and the references therein. However, iterative methods for strict pseudocontractions are far less developed than those for nonexpansive mappings though Browder and Petryshyn  initiated their work in 1967; the reason is probably that the second term appearing in the right-hand side of (1.2) impedes the convergence analysis for iterative algorithms used to find a fixed point of the strict pseudocontraction . On the other hand, strict pseudocontractions have more powerful applications than nonexpansive mappings do in solving inverse problems (see, e.g., ). Therefore, it is interesting to develop the theory of iterative methods for equilibrium problem and strict pseudocontractions.
In 2007, Acedo and Xu  proposed the following parallel algorithm for a finite family of -strict pseudocontractions in Hilbert space : where , and is a finite sequence of positive numbers such that . They proved that the sequence defined by (1.3) converges weakly to a common fixed point of under some appropriate conditions. Moreover, by applying additional projections, they further proved that algorithm can be modified to have strong convergence.
Recently, S. Takahashi and W. Takahashi  studied the equilibrium problem and fixed point of nonexpansive self-mappings in Hilbert spaces by a viscosity approximation methods for finding an element of . Very recently, by using the general approximation method, Qin et al.  obtained a strong convergence theorem for finding an element of . On the other hand, Ceng et al.  proposed an iterative scheme for finding an element of and then obtained some weak and strong convergence theorems.
In this paper, inspired and motivated by research going in this area, we introduce a modified parallel iteration, which is defined in the following way: where is a given point, is a finite family of -strictly pseudocontractive nonself-mappings, is a finite sequences of positive numbers, , , and are some sequences in (0,1).
Our purpose is not only to modify the parallel algorithm (1.3) to the case of equilibrium problems and common fixed point for a finite family of -strictly pseudocontractive nonself-mappings, but also to establish strong convergence theorems in a real Hilbert space under some different conditions. Our theorems presented in this paper improve and extend the main results of [9, 12–14, 16].
Let be a nonempty closed and convex subset of a Hilbert space . We use to denote the metric or nearest point projection of onto ; that is, for , is the only point in such that . we write and indicate that the sequence convergence weakly and strongly to , respectively.
It is well known that Hilbert space satisfies Opial's condition , that is, for any sequence with and every with , we have
To study the equilibrium problem (1.1), we may assume that the bifunction of into 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.
In order to prove our main results, we need the following Lemmas and Propositions.
Lemma 2.1 (see [1, 3]). Let be a bifunction from into satisfying (A1)–(A4). Then, for any and , there exists such that Further, if , then the following holds. (1) is single-valued. (2) is firmly nonexpansive, that is, , .(3). (4) is closed and convex.
Lemma 2.2 (see ). Let be an inner product space. Then, for all and with , we have
Lemma 2.3 (see ). Let and be bounded sequence in Banach space , and let be a sequence in such that . Suppose and Then .
Lemma 2.5 (see ). If is a -strict pseudocontraction, then the fixed point set is closed convex so that the projection is well defined.
Lemma 2.6 (see ). Let be a nonempty bounded closed convex subset of . Given and , then if and only if there holds the relation:
Lemma 2.7 (see ). Assume is a sequence of nonnegative real numbers such that where is a sequence in (0,1) and is a real sequence such that (i),(ii) or . Then .
Proposition 2.8 (see, e.g., Acedo and Xu ). Let be a nonempty closed convex subset of Hilbert space . Given an integer , assume that is a finite family of -strict pseudocontractions. Suppose that is a positive sequence such that . Then is a -strict pseudocontraction with .
3. Main Results
Theorem 3.1. Let be a nonempty closed convex subset of Hilbert space , and let be a bifunction from into satisfying (A1)–(A4). Let be a finite family of -strict pseudocontractions such that and . Assume is a finite sequences of positive numbers such that for all . Given and , , and are some sequences in (0,1); the following control conditions are satisfied. (i) for all and ,(ii) and ,(iii),(iv). Then the sequence generated by (1.4) converges strongly to , where .
Proof. From Lemma 2.1, we see that , and note that can be rewritten as . Putting , we have is a -strict pseudocontraction and by Propositions 2.8 and 2.9, where .
From (1.4), condition (i), and Lemma 2.2, taking a point , we have Furthermore, we have It follows from (1.4) and (3.2) that Consequently, sequence is bounded and so are and .
Define a mapping for each . Then is nonexpansive. Indeed, by using (1.1), condition (i), and Lemma 2.2, we have for all that which shows that is nonexpansive.
Next we show that . Setting , we have It follows that From (1.4), we have and By Lemma 2.1, and , we have Putting in (3.8) and in (3.9), we obtain So, from (A2) and , we have and hence which implies that Combining (3.6), (3.7), and (3.13), we have This together with (i), (ii) and (iv) imply that Hence, by Lemma 2.3, we obtain Consequently,
On the other hand, by (1.4) and (iii), we have which implies that Combining (ii), (3.17), and (3.19), we have Note that which together with (3.17) and (3.20) implies Moreover, for , we have and hence From Lemma 2.2, (3.2) and (3.24), we have and hence By (ii) and (3.17), we obtain It follows from (3.22) and (3.27) that Define by . Then, is a nonexpansive with by Lemma 2.4. Note that by condition (i) and which combines with condition (i) and (3.28) yielding that
We now show that , where . To see this, we choose a subsequence of such that Since is bounded, there exists a subsequence of converging weakly to . Without loss of generality, we assume that as . Form (3.27), we obtain as . Since is closed and convex, is weakly closed. So, we have and . Otherwise, from and Opial's condition, we obtain This is a contradiction. Hence, we get . Moreover, by , we have It follows from (A2) that Replacing by , we have Since and , it follows from (A4) that for all . Put for all and . Then, we have , and hence, . By (A1) and (A4), we have which implies . From (A3), we have for all , and hence, . Therefore, . From Lemma 2.6, we know that It follows from (3.31) and (3.37) that
Finally, we prove that as . From (1.4) again, we have which implies that It follows from (3.38), (3.40), and Lemma 2.7 that . This completes the proof.
As , that is, and in Theorem 3.1, we have the following results immediately.
Theorem 3.2. Let be a nonempty closed convex subset of Hilbert space , and let be a bifunction from into satisfying (A1)–(A4). Let be a -strict pseudocontractions such that . Let be a sequence generated in the following manner: where and , , and are some sequences in (0,1). If the following control conditions are satisfied: (i) for all and ,(ii) and ,(iii),then converges strongly to , where .
This work was supported by the National Science Foundation of China (11001287), the Natural Science Foundation Project of Chongqing (CSTC 2010BB9254), and the Science and Technology Research Project of Chongqing Municipal Education Commission (KJ 110701).
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