#### Abstract

Let be a closed, convex, and nonempty subset of a real -uniformly smooth Banach space , which is also uniformly convex. For some , let and be family of nonexpansive maps and -inverse strongly accretive map, respectively. Let be a bifunction satisfying some conditions. Let be a nonexpansive projection of onto . For some fixed real numbers , , and arbitrary but fixed vectors , let and be sequences generated by , , , , , where is fixed, and are sequences satisfying appropriate conditions. If , under some mild conditions, we prove that the sequences and converge strongly to some element in .

#### 1. Introduction

Let be a real normed space and its dual space. For some real number , the * generalized duality mapping * is defined by
where denotes the pairing between elements of and elements of .

For , usually denoted by is called the normalised duality mapping.

Let be a real Banach space; a map is said to be * accretive* if for all , there exists such that

For some real number , is called *inverse strongly accretive* if for all , there exists such that
Observe that a -inverse strongly accretive map is -Lipschitzian.

Let be a nonempty, closed, and convex subset of , and let be an accretive mapping. A variational inequality problem is, searching for such that for some Let be a bifunction on a closed convex nonempty subset of a real Banach space ; an equilibrium problem is searching for such that

A set of solutions of the problems (4) and (5) are denoted by and , respectively.

Let be a mapping of onto . Then, is said to be * sunny* if for all and . A mapping of into is said to be a retraction if . If a mapping is a * retraction,* then for every , * range* of . A subset is said to be * sunny nonexpansive retract* of if there exists a sunny nonexpansive retraction of onto . A retraction is said to be * orthogonal* if for each , is normal to in the sense of James [1].

It is well known (see [2]) that if is a Banach space; a projection mapping is a sunny nonexpansive retraction of onto . If is uniformly smooth and there exists a nonexpansive retraction of onto , then there exists a nonexpansive projection of onto . If is a real smooth Banach space, then is an orthogonal retraction of onto if and only if and for all . It then follows that, for , we have and which implies

An accretive mapping is said to be maximal if its graph GF is not contained in the graph of any other accretive map. Equivalently, is maximal accretive if for every such that holds for all ; then, . A mapping with domain and range in is said to be * demiclosed* at if whenever is a sequence in such that and ; then, . The following proposition is known to hold; see, for example, [3].

Proposition 1. *Let be a -inverse strongly accretive map. Let be defined by
**
where , for all ; then, is maximal accretive and if and only if .*

Recently, MaingÃ© [4] studied the Halpern-type scheme for approximation of a common fixed point of * countable infinite* family of nonexpansive mappings in a real Hilbert space.

The present author [3] proved a strong convergence theorem for family of nonexpansive maps and solution of variational inequality problems. Kumam and Jaiboon [5] studied a hybrid iterative method for mixed equilibrium problem and variational inequality problem in the framework of a real Hilbert space.

Various numerous authors studied the problem of approximating solutions of equilibrium and fixed point problems in the framework of a real Hilbert space; see, for example, [5â€“18] and the references contained therein. In [19], Ceng et al. studied this problem in the framework of a uniformly smooth and uniformly convex Banach space.

Takahashi and Zembayashi [20] (see also [21â€“23]) studied the problem of approximating solutions of equilibrium problems and fixed points of some nonlinear maps in the framework of real Banach spaces. It is our purpose in this paper to introduce a new hybrid iterative method for approximating a common element in the intersection of the set of fixed points of countable infinite family of nonexpansive mappings, the set of solutions of variational inequality problem, and the set of solutions of equilibrium problem in Banach spaces. Our theorems extend and improve some recent important results, and our method of proof in this paper is of independent interest.

#### 2. Preliminaries

Let denote a unit sphere of the real Banach space . is said to have a *GÃ¢teaux differentiable * norm if the limit
exists for each ; is said to have a * uniformly GÃ¢teaux differentiable* norm if for each , the limit is attained uniformly for . Let be a normed space with . The * modulus of smoothness* of is the function defined by

The space is called * uniformly smooth* if and only if . For some constant , is called -uniformly smooth if there exists a constant such that , .

The modulus of convexity of is the function defined by
is called * uniformly convex* if and only if for all .

A Banach space is said to be strictly convex if for with and .

It is well known that if is smooth then the duality mapping is singled valued, and if has uniformly GÃ¢teaux differentiable norm then the duality mapping is norm-to-*weak** uniformly continuous on bounded subset of . Also, every -uniformly smooth space is uniformly smooth and has a uniformly *GÃ¢teaux differentiable* norm, and every uniformly convex space is strictly convex.

In the sequel, we will make use of the following results.

Lemma 2 (see Petryshyn [24]). *Let be a real normed linear space. Then, the following inequality holds:
*

Theorem 3 (see Goebel and Kirk [25]). *Let be a real uniformly convex Banach space, a closed convex subset of , and a nonexpansive mapping. Then, is demiclosed at zero, where denotes the identity map. *

Lemma 4 (see Suzuki [26]). *Let and be bounded sequences in a Banach space , and let be a sequence in with . Suppose that for all integers and . Then, .*

Lemma 5 (see Xu [27]). *Let be a sequence of nonnegative real numbers satisfying the following relation:
**
where ( i) , ; (ii) ; (iii) ; , . Then, as .*

Lemma 6 (see Xu [28]). * Let be a real q-uniformly smooth Banach space for some ; then, there exists some positive constant such that
**
for all and .*

Lemma 7 (see Kamimura and Takahashi [29]). * Let be a real smooth and uniformly convex Banach space, and let . Then, there exists a strictly increasing, continuous, and convex function such that and for all .*

The following conditions are required on the bifunction for solving equilibrium problems with respect to :(A1) for all ;(A2) is monotone; that is, for all ;(A3) for all ;(A4) for all is convex and lower semicontinuous.

Lemma 8 (see Blum and Oettli [30]). * Let be a real smooth, strictly convex, and reflexive Banach space. Let be a bifunction satisfying (A1)â€“(A4), and let , . Then, there exists such that
*

Lemma 9. *Let be a closed convex nonempty subset of a real uniformly smooth and strictly convex Banach space . Let be a bifunction satisfying (A1)â€“(A4). For and , define a map by
**Then, the following hold:*(i)* is single-valued;*(ii)*; *(iii)*if is firmly nonexpansive-type, that is, for ,
**then is closed and convex. *

*Proof. *â€‰ (i) Let , then
Adding these inequalities and using (A2), we get
which implies . Consider

(ii)

(iii) is closed and convex follows from (ii) and the fact that every firmly nonexpansive map is nonexpansive and the fixed point set of nonexpansive map is closed and convex.

Let be a real uniformly smooth Banach space, and for some , let and be the identity and inverse strongly accretive mappings, respectively. Then, for the map , we have the following estimates: If is chosen such that , we then have and so become a nonexpansive mapping of into .

For spaces, we have the following relation: if ,

Also if is a Hilbert space and we choose , then is nonexpansive.

#### 3. Path Convergence Theorems

In the sequel, we assume for each that the sequence satisfies and the sequences , satisfy .

For a countable family of nonexpansive mappings of , we denote a set ( being the identity mapping on ).

Let be a nonempty closed and convex subset of a real uniformly smooth Banach space and a nonexpansive projection of onto . For some real number , let be a inverse strongly accretive mapping. For some real numbers , , and â€‰ arbitrarily chosen but fixed and for each , define a map by , arbitrary and fixed

Then, is a strict contraction on .

For , , , we have

Thus, for each , there exists a unique such that

Lemma 10. *Let be a real uniformly smooth Banach space which is also uniformly convex. Let be a closed, convex, and nonempty subset of . For , let be a net satisfying (25), and assume that . Then, is bounded and admits at most one accumulation point in as .*

*Proof. *Let . Then, using (25), we have
which implies
Thus, is bounded.

Now, assume for the sake of contradiction that and are two distinct accumulation points of in ; then, there exists a subnet of such that as , and so we have the following estimates:
so that
and since as , we get from (29)

Applying similar argument to as an accumulation point of in , we also get

Adding these last two inequalities, we get
a contradiction, and thus . This completes the proof.

Lemma 11. *Let be a real uniformly smooth Banach space which is also uniformly convex. Let be a closed, convex, and nonempty subset of . Let be fixed and such that and for all . Let be a sequence satisfying (25), and let . Then, , for all .*

*Proof. *For and , we have the following estimates (using Lemma 7 establishing the existence of ):
Using (25), we have
which implies

Using this and (33), we get
Thus,
Since is bounded and as , we have for all . By property of ,â€‰â€‰ for all . This completes the proof.

Theorem 12. *Let be a real uniformly smooth Banach space which is also uniformly convex. Let be a closed, convex, and nonempty subset of . Let be fixed and such that and for all . Let be a sequence satisfying (25), and let . If the duality mapping of is weakly sequentially continuous, then converges strongly to an element in .*

*Proof. *Since is bounded, there exists a subsequence say of that converges weakly to some point . Using demiclosedness property of at for , and the fact that , we get that is a point in . We also observe from (33) that
which implies

Since admits weak sequential continuity, the last inequality implies that the subsequence converges strongly to , and since admits unique accumulation point in , then it converges strongly to . This completes the proof.

The following corollary follows from Theorem 12.

Corollary 13. *Let be a real space, . Let , , , and be as in Theorem 12. Then, converges strongly to an element of .*

Theorem 14. *Let be a real uniformly smooth Banach space which is also uniformly convex. Let be a closed, convex, and nonempty subset of . Let be fixed and a sequence in such that and for all . Let be a sequence satisfying (25), and let . If for at least one in , is demicompact, then converges strongly to an element of .*

*Proof. *For some fixed , let be demicompact. Since , there exists a subsequence say of that converges strongly to some point . By continuity of for all , we have that . But the sequence admits unique accumulation point in ; so, it converges strongly to .

The following corollaries follow from Theorem 14.

Corollary 15. *Let be a real space, . Let , , , and be as in Theorem 14. If for at least one , the map is demicompact, then converges strongly to an element of .*

Corollary 16. *Let be a closed, convex, and nonempty subset of a real Hilbert space . Let , , and be as in Theorem 14. Then, converges strongly to an element of .*

#### 4. Iterative Convergence Theorem

We now state and prove the following theorem.

Theorem 17. *Let be a real uniformly smooth Banach space which is also uniformly convex. Let be a closed, convex, and nonempty subset of . For some , let and be a family of nonexpansive maps and a inverse strongly accretive map, respectively. Let be a bifunction satisfying (A1)â€“(A4). Let be a nonexpansive projection of onto . For some fixed real numbers and , define a sequence iteratively by and as
**
where are sequences satisfying the following conditions:*(i)*,*(ii)*,*(iii)*.**Let . If either the duality map of is weakly sequentially continuous or for at least one , is demicompact, then converges strongly to some element in .*

*Proof. *Let then, we claim that for all . It is clear that the claim is true for . Assume that it is true for for some . Then,
Hence, the result, and so is bounded. Furthermore, , , and are each bounded.

We now show that . Note that , , so that

Define two sequences and by and . Then,
Observe that is bounded and that
for some positive real number . This implies
and by Lemma 4, . Hence,

From (42) and (46), we have

From (40), we have which implies
and thus . Let be a real sequence in satisfying the following conditions:

Let be the unique fixed point satisfying (25) for each , and let as . Using (25) and Lemma 2, we have the following estimates:

This implies
and, hence,
Moreover,
and since