Abstract
The strong convergence of a hybrid algorithm to a common element of the fixed point sets of multivalued strictly pseudocontractive-type mappings and the set of solutions of an equilibrium problem in Hilbert spaces is obtained using a strict fixed point set condition. The obtained results improve, complement, and extend the results on multivalued and single-valued mappings in the contemporary literature.
1. Introduction
Let be a nonempty set and let be a map. A point is called a fixed point of if . If is a multivalued map then is a fixed point of if . If then is called a strict fixed point of . The set (resp., ) is called the fixed point set of multivalued (resp., single-valued) map , while the set is called the strict fixed point set of .
Let be a normed space. A subset of is called proximinal if for each there exists such that It is known that every closed convex subset of a uniformly convex Banach space is proximinal. We will denote the family of all nonempty closed and bounded subsets of by , the family of all nonempty subsets of by , and the family of all proximinal subsets of by , for a nonempty set .
Let denote the Hausdorff metric induced by the metric on ; that is, for every , Let be a normed space. Let be a multivalued mapping on . A multivalued mapping is called -Lipschitzian if there exists such that for all In (3) if is said to be a contraction while is nonexpansive if . is called quasi-nonexpansive if and for all , Clearly every nonexpansive mapping with nonempty fixed point set is quasi-nonexpansive. is said to be -strictly pseudocontractive-type of Isiogugu [1] if there exists such that, given any pair and , there exists satisfying and If in (5), is said to be pseudocontractive-type, while is nonexpansive-type if . Every multivalued nonexpansive mapping is nonexpansive-type. In a real Hilbert space , is said to be -strictly pseudocontractive of Chidume et al. [2] if there exists such that for all If , is said to be pseudocontractive. It is easy to see that every -strictly pseudocontractive mapping is -strictly pseudocontractive-type.
Let be a real Hilbert space with inner product and norm , respectively, and let be a nonempty closed convex subset of . Given an operator and a closed convex set , the variational inequality problem is the problem of finding such that , for all . This variational inequality problem is usually denoted as .
Let be a bifunction, where is the set of real numbers. The equilibrium problem for is to find such that
The set of solutions of (7) is denoted by . Several algorithms were introduced by authors for approximating solutions of equilibrium problems for a bifunction (or finite family of bifunctions) (see, e.g., [3] and references therein). 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 are reduced to the problem of finding the solutions of (7) (see, e.g., [4–6] and the references therein).
The purpose of this work is to first establish closed and convexity property for a strict fixed point set of a multivalued strictly pseudocontractive-type mappings. Second, establish with a strict fixed point set condition a strong convergence of a hybrid algorithm to a common element of the fixed point sets of two multivalued strictly pseudocontractive-type mappings and the set of solutions of an equilibrium problem in Hilbert spaces. The obtained results extend, complement, and improve the results on equilibrium problems as well as multivalued and single-valued mappings in the contemporary literature.
2. Preliminaries
In the sequel, we will need the following definitions and lemmas.
Definition 1. Let be a multivalued mapping; for each , is defined by For solving the equilibrium problems for a bifunction , let us assume that satisfies the following conditions:(A1) for all .(A2) is monotone; that is, , for all (A3)For each , .(A4)For each , is convex and lower semicontinuous.
Lemma 2 (see [4]). Let be a nonempty closed convex subset of a real Hilbert space and a bifunction satisfying (A1)–(A4). Let and . Then, there exists such that
Lemma 3 (see [6]). Let be a nonempty closed convex subset of a real Hilbert space . Assume that satisfying (A1)–(A4). Let and . Define by Then the following hold:(1) is single valued.(2) is firmly nonexpansive; that is, for any , .(3).(4) is closed and convex.
Lemma 4 (see [7]). Let be a nonempty closed convex subset of a real Hilbert space and a bifunction satisfying (A1)–(A4). Let and . Then for all and
Lemma 5. Let be a real Hilbert space, and let be a nonempty closed convex subset of . Let be the convex projection onto . Then, convex projection is characterized by the following relations:(i), for all (ii).(iii).
3. Main Results
Proposition 6. Let be a nonempty subset of a real Hilbert space . And let be a -strictly pseudocontractive-type mapping such that is nonempty. Then is closed and convex.
Proof. Let such that converges to . We show that . Let be arbitrary: Taking limits as , we have that . Hence, . Since was arbitrary, we have that .
We now prove that is convex. Let and and then and :Now, -strictly pseudocontractive-type condition on and a strict fixed point condition on and imply that, for all , and and and . It then follows that In particular, for each , Hence, . Since is proximinal, there exists such that ; consequently, . Also, if , then which shows that . Thus, .
We now prove a strong convergence of multivalued version of the hybrid algorithm considered in [8] to a common element of the set of fixed points of two -strictly pseudocontractive-type mappings and the set of solutions of an equilibrium problem in Hilbert spaces. As a corollary, we obtain a hybrid algorithm for finding common elements of the set of fixed points of two multivalued strictly pseudocontractive mappings of [2] and the set of solutions of an equilibrium problem, with a strict fixed point set condition.
Theorem 7. Let be a nonempty closed convex subset of a real Hilbert space , let be a bifunction satisfying (A1)–(A4), and let be two strictly pseudocontractive-type mappings with contractive coefficients and , respectively, such that . Let be a sequence generated from an arbitrary as follows:where and . and are sequences in satisfying (i),(ii) and ,(iii) for some .Then converges strongly to .
Proof. Observe that is closed and convex for all ; therefore is well defined and note that . Next we show that , for all . is obvious. Suppose , set , and then using Lemma 3, for all , we have Also, Using (19) we obtain from (18) that Also, Using (21) we obtain from (20) that This shows that . It then follows that for all . From we have from Lemma 5(i) that Since for all , we have Using Lemma 5(ii) we obtain for each and for all . Consequently the sequence is bounded, and so are and . Furthermore, since and then from definition of we have for all . Therefore the sequence is nondecreasing. It then follows that exists. From the construction of we have that and for any integer . It also follows from Lemma 5(iii) that Letting in (26), we have . Hence is a Cauchy sequence. Since is Hilbert and is closed and convex we can assume that as ; that is, . We now show that . In particular when in (26) we obtain Also, since , we obtain It then follows from (27) that Combining (27) and (29) we obtain It follows from and (30) that Setting in (22) we have Observe that It then follows from (30) that Using and we obtain from (32) that and . Hence . It remains to show that is in . Now from (32) Also, using , Lemma 4, and (35) we have It then follows from (34) and (36) that Consequently, we obtain from (31) and (37) that From the assumption that , Since implies we have from (A2) that By taking limit as of the above inequality and from (A4), (31), and (38) we have , for all . Let and for all , since , we have that . Hence . It follows from (A1) that that is, . Letting , from (A3) we obtain for all so that . Hence .
Finally we show that . By taking the limits as in (23) we have It then follows from Lemma 5(i) that . This completes the proof.
Corollary 8. Let be a nonempty closed convex subset of a real Hilbert space , let be a bifunction satisfying (A1)–(A4), and let be two strictly pseudocontractive mappings with contractive coefficients and , respectively, such that . Let be a sequence generated from an arbitrary as follows: where and . and are sequences in satisfying (i),(ii) and ,(iii) for some .Then converges strongly to .
Proof. The proof follows easily from Theorem 7.
Competing Interests
The author declares that there are no competing interests.