#### Abstract

We prove a new strong convergence theorem for an element in the intersection of the set of common fixed points of a countable family of nonexpansive mappings, the set of solutions of some variational inequality problems, and the set of solutions of some equilibrium problems using a new iterative scheme. Our theorem generalizes and improves some recent results.

#### 1. Introduction

Let be a real Hilbert space; a mapping is said to be monotone if for all

For some the mapping is said to be -inverse strongly monotone if

A -inverse strongly monotone map is some time called -cocoercive. A map is said to be relaxed -cocoercive if there exists a constant such that

is said to be relaxed -cocoercive, if there exist such that

A map is said to be -Lipschitzian if there exists a real number such that

is a contraction map, if in the above inequality and nonexpansive if

Let be a nonempty, closed, and convex subset of a real Hilbert space A variational inequality problem is searched for such that

where is some nonlinear mapping of into Inequality (1.6) is called the variational inequality.

Recall that for each there exists a unique nearest point in to denoted by That is, for all . is called a metric projection of onto The mapping is nonexpansive in this setting, that is, for all It is also known that satisfies the following inequality

The solution set of the problem (1.6) is denoted by It is well known (see ) that if and only if

A monotone map is said to be maximal if the graph of is not properly contained in the graph of any other monotone map, where for a multivalued map It is also known that is maximal monotone if and only if for for every implies Let be a monotone mapping defined from into and let be a normal cone to at that is, Define a map by

Then, is maximal monotone and see, for example, .

Let be a bifunction on a closed convex nonempty subset of a real Hilbert space An equilibrium problem is searched for such that

The set of solutions of the equilibrium problem above is denoted by

Several physical problems (such as the theories of lubrications, filterations and flows, moving boundary problems, see, e.g., [1, 3]) can be reduced to variational inequality or equilibrium problems. Consequently, these problems have solutions as the solutions of these resultant variational inequality or equilibrium problems.

Maingé  introduced a Halpern-type scheme and proved a strong convergence theorem for family of nonexpansive mappings in Hilbert space.

Recently, S. Takahashi and W. Takahashi  introduced an iterative scheme which they used to study the problem of approximating a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping.

More recently, Kumam and Katchang , W. Kumam and P. Kumam , Li and Su  and many others (see, e.g.  and the references contained in them) studied the problem of fixed point approximations and solutions of some equilibrium and, or solutions of some variational inequalities problems.

In this paper we introduce a new iterative scheme for approximation of a common element in the intersection of the set of fixed points of some countable family of nonexpansive mappings, the set of solutions of some equilibrium problem, and the set of solutions of some variational inequality problem and prove a new theorem. Our theorem generalizes and improves some recent results.

#### 2. Preliminaries

For a sequence the notation and means that the sequence converges strongly and weakly to , repectively. A Banach space is said to satisfy an Opial's condition (or in other words is an Opial's space) if for a sequence in with then

It is well known that Hilbert spaces are Opial’s spaces (see ).

In the sequel we shall make use of the following results.

Lemma 2.1 (see ). Let be a nonempty closed convex subset of and let be a bifunction of into satisfying; is monotone, i.e. ; is convex and lower semicontinuous.Let and Then there exists such that

Lemma 2.2 (see ). Let be a nonempty closed convex subset of and let be a bifunction of into satisfying For and define a map by Then, the following holds:(1) is single-valued;(2) is firmly nonexpansive, that is, for any (3)(4) is closed and convex.

Lemma 2.3 (see ). Let be a real inner product space. Then, the following inequality holds:

Lemma 2.4 (see ). Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose for all integers and . Then,

Lemma 2.5 (see ). Let be a sequence of nonnegative real numbers satisfying the following relation: Where (i), ; (ii); (iii) Then, as

#### 3. Main Results

In the sequel we assume that the sequences satisfy

Theorem 3.1. Let be a nonempty, closed, and convex subset of a real Hilbert space Let be a bifunction from to which satisfies conditions Let be family of nonexpansive mappings of into and let be a -Lipschitzian, relaxed -cocoercive map of into such that For an arbitrary but fixed let and be sequences generated by where and are sequences in and and are sequences in satisfying for some satisfying , , Then, both and converge strongly to

Proof. First, we show that is nonexpansive, actually, using the property of we have for , and thus is nonexpansive.
Let since , and the fact that is firmly nonexpansive (and hence nonexpansive) we have the following: We claim that satisfies We prove this by induction. Cearly the result is true for Assume that the result holds for for some Then, for we have Hence the result, and so is bounded. Furthermore, , and are each bounded.
We now show that Note that , so that Using (3.6) and we have which implies that from which we get If, without loss of generality , are real numbers such that for all and we then have Now, define two sequences and by and Then, Observe that is bounded and that Using (3.10) we have that This implies and by Lemma 2.4, Hence, Using (3.10) we have We now have so that But, Putting (3.18) in (3.19), we have and hence, Observe also that if then Also, which implies so that
We go further to prove that for each Consider the following estimates: Using (3.1), we have which implies Using this and (3.27), we get Since , are each bounded and using (3.21), we have that Using this and (3.21), we also have Next we show that where Let be a subsequence of such that
Since is bounded, there exists a subsequence of such that . Then, otherwise, for we have which is a contradiction, so
Next we show that Since we have It follows from that and so Since , and using we have for all For a real number , and let Clearly so that using and we have This implies and using this and we have that for all and hence
Next we show that
Let then, we have the following: so that We then have so that and as , and as we get Using this and (3.21) we also have As is a relaxed -cocoercive and using condition we have for and so is monotone. If then is maximal monotone. Let denote the graph of
Let Since and we have by definition of Also, as (using property of the projection ), we have and hence Using this, we obtain the following estimates: which implies (letting ).
Since is maximal monotone, we obtained that and hence
We mention here that since we proved that and then So, clearly
Since , we have Hence,
From the recursion formula (3.1) and Lemma 2.3, we have Using Lemma 2.5, we have that and consequently converge to and the proof is complete.

The following corollaries follow from Theorem 3.1.

Corollary 3.2. Let and be as in Theorem 3.1 and let be finite family of nonexpansive mappings of into Let . For an arbitrary but fixed let and be sequences generated by where and are sequences in and and are sequences in satisfying , for some satisfying , , Then, both and converge strongly to

Corollary 3.3. Let and be as in Theorem 3.1. Let be a nonexpansive map of into Let For an arbitrary but fixed let and be sequences generated by where is a sequence in and is a sequences in satisfying Then, both and converge strongly to

Remark 3.4. Prototypes of the sequences and in our theorem are the following: