On Computability and Applicability of Mann-Reich-Sabach-Type Algorithms for Approximating the Solutions of Equilibrium Problems in Hilbert Spaces

and Applied Analysis 3 CC(X), and the family of all proximinal subsets ofX byP(X), for a nonempty set X. LetH denote the Hausdorff metric induced by the metric d on X; that is, for every A, B ∈ CB(X), H(A, B) = max{sup


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  :  ×  → R be a bifunction on , where R is the set of real numbers.The variational inequality problem of  in  denoted by (, ) is to find an  * ∈  such that ⟨ −  * ,  ( * )⟩ ≥ 0, ∀ ∈ , while the equilibrium problem for  is to find  * ∈  such that  ( * , ) ≥ 0, ∀ ∈ .
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][5][6][7][8][9][10] and references therein).In a real Hilbert space, many authors have studied the algorithms involving the construction of the sequences of sets {  } ∞ =1 and the metric projections {  } ∞ =1 , from an arbitrary  0 ∈ , where  +1 = { ∈   : ‖ −   ‖ 2 ≤ ‖ −   ‖ 2 },  +1 =   +1  0 , while    is the projection map and {  } ∞ =1 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-Sabachtype 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  :  ×  → R be a bifunction and  ∈ [, ∞) for some  > 0. Then from an arbitrary  0 ∈  the algorithm is generated as follows.
where V  ∈   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 {  } ∞ =1 and the sequence of sets {  } ∞ =1 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  :  ×  → R in a real Hilbert space  involving the construction of the sequences {  } ∞ =1 and {  } ∞ =1 from an arbitrary  0 ∈ , where  =  0 is a closed and convex subset of ,  +1 = { ∈   : ‖ −   ‖ 2 ≤ ‖ −   ‖ 2 }, and  +1 =   +1  0 , while    is the metric projection of  into   and {  } ∞ =1 is the sequence of the resolvents of the bifunction; there exists a selection {  } ∞ =1 of {  } ∞ =1 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  +1 =   +1  0 and the selection converges strongly to  ()  0 .Where a norm ‖.‖ on a Hilbert space  is order inclusion transitive on () if given any ,  ∈ () with  ⊆  and arbitrary  ∈ , then (, ) = inf ∈ ‖ − ‖ = ‖ − ‖ and (, ) = inf ∈ ‖ − ‖ = ‖ − ‖ imply that (, ) = inf ∈ ‖ − ‖ = ‖ − ‖ and () is the set of the solutions of the equilibrium problem for the bifunction.Also if we set   =     +(1−  )V  in Algorithm 1, where {  } ∞ =1 ⊂ [0, 1] 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 {  } ∞ =1 and {  } ∞ =1 above.

Preliminaries
Let  be a nonempty set and let  :  →  be a map.A point 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 2  , 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  : () ⊆  → 2  be a multivalued mapping on .A multivalued mapping  : () ⊆  → 2  is called  − ℎ if there exists  ≥ 0 such that for all ,  ∈ () In (7), if  ∈ [0, 1)  is said to be a contraction while  is nonexpansive if  = 1.
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.en   () 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 .en, convex projection is characterized by the following relations: Lemma 8 (see [1]).Let  be a nonempty closed convex subset of a real Hilbert space  and  :  ×  → R a bifunction satisfying (A )-(A ).Let  > 0 and  ∈ .en, there exists  ∈  such that Lemma 9 (see [2]).Let  be a nonempty closed convex subset of a real Hilbert space .Assume that  : × → R satisfies (A )-(A ).Let  > 0 and  ∈ ; define   :  → 2  by en the following conditions hold: (1)   is single-valued.

Definition
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
, then the following conditions are true: Consequently,  +1 =   +1   .(C 3 ) Since  +1 ⊆   and   is closed and convex for each n, condition  2 and order inclusion transitive property of  on () guarantee that  +1 =   +1  0 .
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  :  ×  → R be a bifunction.Let  ∈ [, ∞) for some  > 0. Then from an arbitrary  0 ∈  we generate the sequence {  } ∞ =1 as follows.
=0 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   =     0 , consequently, from Lemma 7(i) Since () ⊆   for all  ≥ 1, we have that Taking the limit as  → ∞ in (21) we have Thus, from Lemma 7(i)  =  ()  0 .This completes the proof.
Remark .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.
Remark .The above proof shows that the strong convergence of Algorithm 19 to a common solution  ∈ F 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  =  ()  0 .
Remark .It is also of a great interest to us to get the same results in normed spaces which enjoy the order inclusion transitive property.

Numerical Examples of the Computations
We shall use Algorithm 19 to recompute the example presented by Isiogugu et al. [13], when  = 1 is defined as follows.

Table 1 :
Sequences generated by Algorithms 2 and 19.