Abstract
We establish the existence of a strong convergent selection of a modified Mann-Reich-Sabach iteration scheme for approximating the common elements of the set of fixed points of a multivalued (or single-valued) strictly pseudocontractive-type mapping and the set of solutions of an equilibrium problem for a bifunction in a real Hilbert space . This work is a continuation of the study on the computability and applicability of algorithms for approximating the solutions of equilibrium problems for bifunctions involving the construction of a sequence of closed convex subsets of from an arbitrary and a sequence of the metric projections of into . The obtained result is a partial resolution of the controversy over the computability of such algorithms in the contemporary literature.
1. Introduction
Let be a real Hilbert space with an inner product and a norm , respectively and let be a nonempty closed convex subset of . Let be an operator on and be a bifunction on , where is the set of real numbers. The variational inequality problem of in denoted by is to find an such that while the equilibrium problem for is to find such that The set of solutions of (2) is denoted by . Suppose for all , then if and only if is a solution of (1). Many problems in optimization, economics, and physics reduce to finding a solution of (1) (see, for example, [1–3]) and the references therein. The following conditions are assumed for solving the equilibrium problems for a bifunction ,(A1) for all .(A2) is monotone, that is, , for all (A3)For each , .(A4)For each , is convex and lower semicontinuous.
Several algorithms have been introduced by authors for approximating the solutions of an equilibrium problem for a bifunction (or the common elements of the sets of solutions of equilibrium problems for a finite family of bifunctions). Many authors have also approximated the common elements of the set of fixed points of a multivalued (or single-valued) mapping and the set of solutions of an equilibrium problem for a bifunction (or the common elements of the sets of fixed points of a finite family of multivalued (or single-valued) mappings and the sets of solutions of equilibrium problems for a finite family of bifunctions) (see, for example, [4–10] and references therein). In a real Hilbert space, many authors have studied the algorithms involving the construction of the sequences of sets and the metric projections , from an arbitrary , where , , while is the projection map and is the sequence of the resolvent of the bifunctions, (see, for example, [4, 9] and references therein).
Among the iteration schemes studied are the modified Reich-Sabach-type Algorithm 1 and Mann-Reich-Sabach-type Algorithm 2 below defined for the approximation of (i) the solutions of an equilibrium problem for a bifunction; (ii) the common elements of the set of fixed points of a multivalued (or single-valued) mapping , and the set of solutions of an equilibrium problem for a bifunction respectively.
(i) Let be a real Hilbert space and a closed and convex subset of . Let be a bifunction and for some . Then from an arbitrary the algorithm is generated as follows.
Algorithm 1. (ii) Let be a real Hilbert space, a closed and convex subset of , a bifunction, and ) multivalued strictly pseudocontractive-type mapping. Let and for some . Then from an arbitrary the algorithm is generated as follows.
Algorithm 2. where for multivalued mapping .
However, despite the fact that most of these algorithms yield strong convergence theoretically, the difficulty encountered by computers with the construction of the sequence of the metric projection and the sequence of sets has made such algorithms almost impossible for real life applications. This noncomputability and nonapplicability of such algorithms has led to the introduction of other algorithms which do not involve the construction of these two sequences but require stronger conditions and many parameters in the hypothesis of their convergence theorems.
One of these important algorithms is the algorithm of Zhaoli Ma et al. [10].
The purpose of this research is to develop a computable version of Algorithms 1 and 2. In particular, it is established that given the modified Reich-Sabach-types Algorithm 1 for approximating the solutions of an equilibrium problem EP(F) for a bifunction in a real Hilbert space involving the construction of the sequences and from an arbitrary , where is a closed and convex subset of , , and , while is the metric projection of into and is the sequence of the resolvents of the bifunction; there exists a selection of which converges strongly to a solution of the equilibrium problem. Furthermore, if the norm on is order inclusion transitive on the closed convex subsets of , then and the selection converges strongly to . Where a norm on a Hilbert space is order inclusion transitive on if given any with and arbitrary , then and imply that and is the set of the solutions of the equilibrium problem for the bifunction. Also if we set in Algorithm 1, where satisfying some conditions and is a multivalued strictly pseudocontractive-type mapping a similar selection existing as well which is a selection of Algorithm 2, the numerical example of the computation is presented for the selection of Algorithm 2 which is the generalization of the selections of Algorithm 1. The results of this research are great contributions towards the resolution of the controversy over the computability and applicability of algorithms for approximating the solutions of equilibrium problems for bifunctions involving the construction of the sequences and above.
2. Preliminaries
Let be a nonempty set and let be a map. A point is called a fixed point of if . If is a multivalued map from into the family of nonempty subsets of , then is a fixed point of if . If , is called a strict fixed point of . The set (respectively, ) is called the fixed point set of multivalued (respectively, 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 shall denote the family of all nonempty closed and bounded subsets of by , the family of all nonempty subsets of by , the family of all nonempty closed and convex 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 if there exists such that for all In (7), if is said to be a contraction while is nonexpansive if .
Definitions 3 (see [11]). is said to be -strictly pseudocontractive-type of Isiogugu [11] if there exists such that given any pair and , there exists satisfying and
Definitions 4. A multivalued map is said to be of type-one (see for example [12, 13]) if given any pair , then
Lemma 5 (see [14]). Let be a real Hilbert space and let be a -strictly pseudocontractive-type mapping. Then is an -Lipschitzian.
Lemma 6 (see [7]). 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.
Lemma 7. 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).
Lemma 8 (see [1]). 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 9 (see [2]). Let be a nonempty closed convex subset of a real Hilbert space . Assume that satisfies (A1)-(A4). Let and ; define by Then the following conditions hold:(1) is single-valued.(2) is firmly nonexpansive, that is for any , .(3).(4) is closed and convex.
Lemma 10 (see [15]). Let be a nonempty closed convex subset of a real Hilbert space and a bifunction satisfying (A1)-(A4). Let and . Then for all and
Definition 11 (see [16, 17]). Let be a Banach space. Let be a multivalued mapping. is said to be at if for any sequence such that converges weakly to and a sequence with for all such that converges strongly to . Then (i.e., ).
3. Main Results
Definition 12. Let be sequence of sets. Then a sequence is called a selection of if for each .
Definition 13. A norm on a Hilbert space is order inclusion transitive on if given any with and arbitrary , then and imply that (i.e., if is the point in B closet to and is the point in closest to then is the point in closest to ).
Definition 14. A Hilbert is said to have order inclusion transitive property on if its norm is order inclusion transitive on .
It is easy to see that the set of real numbers with the usual norm has order inclusion transitive property.
Proposition 15. In the definition of the set , if we define , then the following conditions are true: is a selection of ..If has order inclusion transitive property on then,
Proof. and Therefore, ; thus, is a selection of .
Let arbitrary. Then Consequently, .
Since and is closed and convex for each n, condition and order inclusion transitive property of on guarantee that .
We now consider the following algorithm which we shall refer to as a selection of Algorithm 1.
Let be a real Hilbert space, be a nonempty closed convex subset of , and be a bifunction. Let for some . Then from an arbitrary we generate the sequence as follows.
Algorithm 16.
Theorem 17. Let , and and be as in Algorithm 16. Suppose satisfying (A1)-(A4), , then (i) converges strongly to ; (ii) if has order inclusion transitive property, then converges strongly to .
Proof. Since , given arbitrary , we have that Therefore, is monotone, nonincreasing, and bounded; hence, exists. Also, Hence which implies that . Also, so that . Consequently , , and are bounded. From we have that is a Cauchy in and hence converges strongly to . From the Opial condition of , the firmly nonexpansive and demiclosedness property of established that . (ii) If has order inclusion transitive property then , consequently, from Lemma 7(i) Since for all , we have that Taking the limit as in (21) we have Thus, from Lemma 7(i) . This completes the proof.
Remark 18. It is important to note that the strong convergence of Algorithm 16 to a does not depend on the order inclusion transitive property condition on .
Motivated by Algorithm 16 we now obtain the following algorithm which is a selection of Algorithm 2.
Let be a real Hilbert space, be a closed and convex subset of , be a bifunction, and be a multivalued strictly pseudocontractive-type mapping. Let and for some . Then from an arbitrary the algorithm is generated as follows.
Algorithm 19. where .
Theorem 20. Let , , , , , and be as in Algorithm 19. Suppose that (I-T) is weakly demiclosed at , satisfies (A1)-(A4), , and satisfies (i) ; (ii) ; (iii) . Then converges strongly to . Also, if has order inclusion transitive property, then .
Proof. Let be arbitrary. Then It then follows that exists; hence, is bounded. Also, Since from (ii), we have that . Furthermore, Consequently, which implies that is a Cauchy sequence in . Also, since is closed and convex, converges strongly to some . From the Opial condition of , weakly demiclosed of at , we have that
It remains to show that is in since It follows from and (26) that Also, Observe that It follows from (26) that Now from (28), Also, using , Lemma 10, and (31) we have Therefore, from (30) and (32), Consequently, from (27) and (33), From the assumption that , Since , we have We then deduce from (A2) that By taking limit as of the above inequality and from (A4), (27), and (34), , for all Let and for all , since , Hence Therefore, from (A1), that is, . Letting , from (A3) we obtain for all so that for all . Hence .
Finally, if has order inclusion transitive property, consequently, from Lemma 7(i) Since for all , we have that Taking the limit as in (40) we have Thus, from Lemma 7(i) . This completes the proof.
Remark 21. The above proof shows that the strong convergence of Algorithm 19 to a common solution does not depend on order inclusion transitive property condition on . However, order inclusion transitive property is only required on if we want to have that .
Remark 22. It is also of a great interest to us to get the same results in normed spaces which enjoy the order inclusion transitive property.
4. Numerical Examples of the Computations
We shall use Algorithm 19 to recompute the example presented by Isiogugu et al. [13], when is defined as follows.
Let (the reals with the usual metric and inner product) and ; we define
(i) by We have that, for any , Also, given any , , , and we can choose so that It then follows that Similarly, for any , , Furthermore, for any , Thus, is strictly pseudocontractive-type. It is easy to see that, given any pair , we have that . Therefore is not nonexpansive and . We then set
(ii)
(iii)We also let , , and to be as defined in [9] for . That is,
(iv)
Observe that , , and Now is a quadratic function of with coefficients , , and Therefore, we can compute the discriminant of as follows:Obviously, for all if it has at most one solution in . Thus and hence . Consequently,
(v) ,
(vi) .
The algorithm is computed with Microsoft word Excel 97-2003 Workbook. Table 1 shows the sequences and generated from our computation using two different values of .
Disclosure
Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the CoE-MaSS.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Acknowledgments
The first author acknowledges with thanks the bursary and financial support from Department of Science and Technology and National Research Foundation, Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DST-NRF CoE-MaSS) Postdoctoral Bursary.