A New Hybrid Iterative Scheme for Countable Families of Relatively Quasi-Nonexpansive Mappings and System of Equilibrium Problems

We construct a new iterative scheme by hybrid methods and prove strong convergence theorem for approximation of a common fixed point of two countable families of closed relatively quasinonexpansive mappings which is also a solution to a system of equilibrium problems in a uniformly smooth and strictly convex real Banach space with Kadec-Klee property using the properties of generalized f -projection operator. Using this result, we discuss strong convergence theorem concerning variational inequality and convex minimization problems in Banach spaces. Our results extend many known recent results in the literature.


Introduction
Let E be a real Banach space with dual E * and C a nonempty, closed, and convex subset of E. A mapping T : The set of fixed points of T is denoted by F T : {x ∈ C : Tx x}.We denote by J the normalized duality mapping from E to 2 E * defined by

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The following properties of J are well known the reader can consult 1-3 for more details .
1 If E is uniformly smooth, then J is norm-to-norm uniformly continuous on each bounded subset of E.
3 If E is reflexive, then J is a mapping from E onto E * .
4 If E is smooth, then J is single valued.
Throughout this paper, we denote by φ the functional on E × E defined by φ x, y x 2 − 2 x, J y y 2 , ∀x, y ∈ E. 1.3 It is obvious from 1.3 that x − y 2 ≤ φ x, y ≤ x y 2 , ∀x, y ∈ E.

1.4
Definition 1.1.Let C be a nonempty subset of E, and let T be a mapping from C into E.A point p ∈ C is said to be an asymptotic fixed point of T if C contains a sequence {x n } ∞ n 0 which converges weakly to p and lim n → ∞ x n − Tx n 0. The set of asymptotic fixed points of T is denoted by F T .We say that a mapping T is relatively nonexpansive see, e.g., 4-9 if the following conditions are satisfied: R1 F T / ∅, R2 φ p, Tx ≤ φ p, x , for all x ∈ C, p ∈ F T , R3 F T F T .
If T satisfies R1 and R2 , then T is said to be relatively quasi-nonexpansive.It is easy to see that the class of relatively quasi-nonexpansive mappings contains the class of relatively nonexpansive mappings.Many authors have studied the methods of approximating the fixed points of relatively quasi-nonexpansive mappings see, e.g., 10-12 and the references cited therein .Clearly, in Hilbert space H, relatively quasi-nonexpansive mappings and quasinonexpansive mappings are the same, for φ x, y x−y 2 , for all x, y ∈ H, and this implies that φ p, Tx ≤ φ p, x ⇐⇒ Tx − p ≤ x − p , ∀x ∈ C, p ∈ F T .1.5 The examples of relatively quasi-nonexpansive mappings are given in 11 .
Let F be a bifunction of C × C into Ê.The equilibrium problem see, e.g., 13-25 is to find x * ∈ C such that F x * , y ≥ 0, 1.6 for all y ∈ C. We will denote the solutions set of 1.6 by EP F .Numerous problems in physics, optimization, and economics reduce to find a solution of problem 1.6 .The equilibrium problems include fixed point problems, optimization problems, and variational inequality problems as special cases see, e.g., 26 .
In 7 , Matsushita and Takahashi introduced a hybrid iterative scheme for approximation of fixed points of relatively nonexpansive mapping in a uniformly convex real Banach space which is also uniformly smooth: H n w ∈ C : φ w, y n ≤ φ w, x n , W n {w ∈ C : x n − w, Jx 0 − Jx n ≥ 0}, x n 1 Π H n ∩W n x 0 , n ≥ 0.

1.7
They proved that {x n } ∞ n 0 converges strongly to Π F T x 0 , where F T / ∅.In 27 , Plubtieng and Ungchittrakool introduced the following hybrid projection algorithm for a pair of relatively nonexpansive mappings: where {α n }, {β 1 and T and S are relatively nonexpansive mappings and J is the single-valued duality mapping on E. They proved under the appropriate conditions on the parameters that the sequence {x n } generated by 1.8 converges strongly to a common fixed point of T and S.
In 9 , Takahashi and Zembayashi introduced the following hybrid iterative scheme for approximation of fixed point of relatively nonexpansive mapping which is also a solution to an equilibrium problem in a uniformly convex real Banach space which is also uniformly smooth: where J is the duality mapping on E.Then, they proved that {x n } ∞ n 0 converges strongly to Π F x 0 , where F EP F ∩ F T / ∅.

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Furthermore, in 28 , Qin et al. introduced the following hybrid iterative algorithm for approximation of common fixed point of two countable families of closed relatively quasinonexpansive mappings in a uniformly convex and uniform smooth real Banach space:

1.10
They proved that the sequence {x n } converges strongly to a common fixed point of the countable families {T i } and {S i } of closed relatively quasi-nonexpansive mappings in a uniformly convex and uniformly smooth Banach space under some appropriate conditions on {β n,i }, and {α n,i }.Recently, Li et al. 29 introduced the following hybrid iterative scheme for approximation of fixed points of a relatively nonexpansive mapping using the properties of generalized f-projection operator in a uniformly smooth real Banach space which is also uniformly convex:

1.11
They proved a strong convergence theorem for finding an element in the fixed points set of T. We remark here that the results of Li et al. 29 extended and improved on the results of Matsushita and Takahashi 7 .
Quite recently, motivated by the results of Takahashi and Zembayashi 9 , Cholamjiak and Suantai 30 proved the following strong convergence theorem by hybrid iterative scheme for approximation of common fixed point of a countable family of closed relatively quasi-nonexpansive mappings which is also a solution to a system of equilibrium problems in uniformly convex and uniformly smooth Banach space.Theorem 1.2.Let E be a uniformly convex real Banach space which is also uniformly smooth, and let C be a nonempty, closed, and convex subset of E. For each k 1, 2, . . ., m, let F k be a bifunction from

an infinitely countable family of closed and relatively quasi-nonexpansive mappings of C into itself such that
x n 1 Π C n 1 x 0 , n ≥ 0.

1.12
Assume that {α n } ∞ n 1 and {r k,n } ∞ n 1 k 1, 2, . . ., m are sequences which satisfy the following conditions: Motivated by the above-mentioned results and the on-going research, it is our purpose in this paper to prove a strong convergence theorem for two countable families of closed relatively quasi-nonexpansive mappings which is also a solution to a system of equilibrium problems in a uniformly smooth and strictly convex real Banach space with Kadec-Klee property using the properties of generalized f-projection operator.Our results extend the results of Matsushita and Takahashi 7 , Takahashi and Zembayashi 9 , Qin et al. 28

Preliminaries
Let E be a real Banach space.The modulus of smoothness of E is the function ρ E : 0, ∞ → 0, ∞ defined by and lower semicontinuous.From the definitions of G and f, it is easy to see the following properties: i G ξ, ϕ is convex and continuous with respect to ϕ when ξ is fixed, ii G ξ, ϕ is convex and lower semicontinuous with respect to ξ when ϕ is fixed.
Definition 2.1 see Wu and Huang 35 .Let E be a real Banach space with its dual E * .Let C be a nonempty, closed, and convex subset of E. We say that Π For the generalized f-projection operator, Wu and Huang 35 proved the following theorem basic properties.

Lemma 2.2 see Wu and Huang 35 .
Let E be a real reflexive Banach space with its dual E * .Let C be a nonempty, closed, and convex subset of E.Then, the following statements hold: Recall that J is a single-valued mapping when E is a smooth Banach space.There exists a unique element ϕ ∈ E * such that ϕ Jx for each x ∈ E. This substitution in 2.5 gives Now, we consider the second generalized f-projection operator in a Banach space.
Definition 2.4.Let E be a real Banach space and C a nonempty, closed, and convex subset of E. We say that Obviously, the definition of T : C → C is a relatively quasi-nonexpansive mapping and is equivalent to Lemma 2.5 see Li et al. 29 .Let E be a Banach space, and let f : E → Ê ∪ { ∞} be a lower semicontinuous convex functional.Then, there exists x * ∈ E * and α ∈ Ê such that

2.10
We know that the following lemmas hold for operator Lemma 2.6 see Li et al. 29 .Let C be a nonempty, closed, and convex subset of a smooth and reflexive Banach space E.Then, the following statements hold: Also, this following lemma will be used in the sequel.Lemma 2.9 see Cho et al. 38 .Let E be a uniformly convex real Banach space.For arbitrary r > 0, let B r 0 : {x ∈ E : x ≤ r} and λ, μ, γ ∈ 0, 1 such that λ μ γ 1.Then, there exists a continuous strictly increasing convex function g : 0, 2r −→ Ê, g 0 0, 2.13 such that, for every x, y, z ∈ B r 0 , the following inequality holds: For solving the equilibrium problem for a bifunction F : C × C → Ê, let us assume that F satisfies the following conditions: for all z ∈ E.Then, the following hold: r is firmly nonexpansive-type mapping, that is, for any x, y ∈ E, Then, for each x ∈ E and q ∈ F T F r , φ q, T F r x φ T F r x, x ≤ φ q, x .

2.18
For the rest of this paper, the sequence {x n } ∞ n 0 converges strongly to p and will be denoted by x n → p as n → ∞, {x n } ∞ n 0 converges weakly to p and will be denoted by x n p and we will assume that β 1, for all n ≥ 0.
We recall that a Banach space E has Kadec-Klee property if, for any sequence {x n } ∞ n 0 ⊂ E and x ∈ E with x n x and x n → x , x n → x as n → ∞.We note that every uniformly convex Banach space has the Kadec-Klee property.For more details on Kadec-Klee property, the reader is referred to 2, 33 .
Lemma 2.13 see Li et al. 29 .Let E be a Banach space and y ∈ E. Let f : E → Ê ∪ { ∞} be a proper, convex, and lower semicontinuous mapping with convex domain

Main Results
Theorem 3.1.Let E be a uniformly smooth and strictly convex real Banach space which also has Kadec-Klee property.Let C be a nonempty, closed, and convex subset of E. For each k 1, 2, . . ., m, let F k be a bifunction from C × C satisfying (A1)-(A4).Suppose {T i } ∞ i 1 and {S i } ∞ i 1 are two countable families of closed relatively quasi-nonexpansive mappings of C into itself such that Ω :

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with the conditions Proof.We first show that C n , for all n ≥ 1 is closed and convex.It is obvious that C 1,i C is closed and convex.Suppose C k,i is closed and convex for some By the construction of the set C k 1,i , we see that C k 1,i is closed and convex.Therefore, C k 1 So, x * ∈ C n .This implies that Ω ⊂ C n , for all n ≥ 1.Therefore, {x n } is well defined.

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We now show that lim n → ∞ G x n , Jx 0 exists.Since f : E → Ê is convex and lower semicontinuous, applying Lemma 2.5, we see that there exists u * ∈ E * and α ∈ Ê such that f y ≥ y, u * α, ∀y ∈ E.

3.4
It follows that

3.5
Since for each x * ∈ F. This implies that {x n } ∞ n 0 is bounded and so is {G x n , Jx 0 } ∞ n 0 .By the construction of C n , we have that It is obvious that and so {G x n , Jx 0 } ∞ n 0 is nondecreasing.It follows that the limit of {G x n , Jx 0 } ∞ n 0 exists.Now since {x n } ∞ n 0 is bounded in C and E is reflexive, we may assume that x n p, and since C n is closed and convex for each n ≥ 1, it is easy to see that p ∈ C n for each n ≥ 1. Again since

3.10
International Journal of Mathematics and Mathematical Sciences then we obtain This implies that lim n → ∞ G x n , Jx 0 G p, Jx 0 .By Lemma 2.13, we obtain lim n → ∞ x n p .In view of Kadec-Klee property of E, we have that lim n → ∞ x n p.
We next show that p 3.12 Now, 3.7 implies that Taking the limit as n → ∞ in 3. Ju n,i Jp , ∀i ≥ 1.

3.17
This implies that { Ju n,i } ∞ n 0 , i ≥ 1 is bounded in E * .Since E is reflexive, and so E * is reflexive, we can then assume that Ju n,i f 0 ∈ E * , for all i ≥ 1.In view of reflexivity of E, we see that J E E * .Hence, there exists x ∈ E such that Jx f 0 .Since

3.18
International Journal of Mathematics and Mathematical Sciences 13 taking the limit inferior of both sides of 3.18 and in view of weak lower semicontinuity of Jp , for all i ≥ 1 and Kadec-Klee property of E * that Ju n,i → Jp, for all i ≥ 1.Note that J −1 : E * → E is hemicontinuous; it yields that u n,i p, for all i ≥ 1.It then follows from lim n → ∞ u n,i p , for all i ≥ 1 and Kadec-Klee property of E that lim n → ∞ u n,i p, for all i ≥ 1.Hence, Since J is uniformly norm-to-norm continuous on bounded sets and lim n → ∞ x n − u n,i 0, for all i ≥ 1, we obtain

3.21
Since {x n } is bounded, so are {z n,i }, {JT i x n }, and {JS i x n }.Also, since E is uniformly smooth, E * is uniformly convex.Then, from Lemma 2.9, we have International Journal of Mathematics and Mathematical Sciences

3.22
It then follows that

3.23
But

3.24
From lim n → ∞ x n − u n,i 0 and lim n → ∞ Jx n − Ju n,i 0, we obtain By property of g, we have lim n → ∞ Jx n − JT i x n 0, for all i ≥ 1.Since J −1 is also uniformly norm-to-norm continuous on bounded sets, we have

3.27
Similarly, we can show that lim n → ∞ x n − S i x n 0, ∀i ≥ 1.

3.28
Since x n → p and T i , S i are closed, we have p , by Lemma 2.12, we obtain

3.29
It then yields that Jy n,i Jp , i ≥ 1.

3.31
This implies that { Jy n,i } ∞ n 0 is bounded in E * .Since E is reflexive, and so E * is reflexive, we can then assume that Jy n,i f 1 ∈ E * .In view of reflexivity of E, we see that J E E * .Hence, there exists z ∈ E such that Jz f 1 .Since Jy n,i 2 , 3.32 taking the limit inferior of both sides of 3.32 and in view of weak lower semicontinuity of • , we have that is, p z.This implies that f 1 Jp and so Jy n,i Jp.It follows from lim n → ∞ Jy n,i Jp and Kadec-Klee property of E * that Jy n,i → Jp.Note that J −1 : E * → E is hemicontinuous; it yields that y n,i p.It then follows from lim n → ∞ y n,i p and Kadec-Klee property of E that lim n → ∞ y n,i p, i ≥ 1.By the fact that θ k n , k 1, 2, . . ., m is relatively nonexpansive and using Lemma 2.12 again, we have that

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Observe that

3.35
Using 3.35 in 3.34 , we obtain

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Finally, we show that p

3.50
We know that G ξ, Jϕ is convex and lower semicontinuous with respect to ξ when ϕ is fixed.This implies that From the definition of Π f Ω x 0 and p ∈ Ω, we see that p w.This completes the proof.
Take f x 0 for all x ∈ E in Theorem 3.1, then G ξ, Jx φ ξ, x and Π f C x 0 Π C x 0 .Then we obtain the following corollary.Corollary 3.2.Let E be a uniformly smooth and strictly convex real Banach space which also has Kadec-Klee property.Let C be a nonempty, closed, and convex subset of E. For each k 1, 2, . . ., m, let F k be a bifunction from C × C satisfying (A1)-(A4).Suppose {T i } ∞ i 1 and {S i } ∞ i 1 are two countable families of closed relatively quasi-nonexpansive mappings of C into itself such that Ω :

3.52
with the conditions Corollary 3.4 see Takahashi and Zembayashi 9 .Let E be a uniformly convex real Banach space which is also uniformly smooth.Let C be a nonempty, closed, and convex subset of E. Let F be a bifunction from C × C satisfying (A1)-(A4).Suppose T is a relatively nonexpansive mapping of C into itself such that Ω : EP F ∩ F T / ∅.Let {x n } ∞ n 0 be iteratively generated by

3.54
where J is the duality mapping on E. Suppose converges strongly to Π Ω x 0 .

Applications
Let A be a monotone operator from C into E * , the classical variational inequality is to find The set of solutions of 4.1 is denoted by VI C, A .
Let ϕ : C → Ê be a real-valued function.The convex minimization problem is to find The set of solutions of 4.2 is denoted by CMP ϕ .

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The following lemmas are special cases of Lemmas 2.8 and Lemma 2.9 of 39 .
Lemma 4.1.Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E. Assume that A : C → E * is a continuous and monotone operator.For r > 0 and x ∈ E, define a mapping T A r : E → C as follows: Then, the following hold: Then, the following hold: 3 CMP ϕ is closed and convex, 4 φ q, T ϕ r x φ T ϕ r x, x ≤ φ q, x , for all q ∈ F T ϕ r .
Then we obtain the following theorems from Theorem 3.1.
Theorem 4.3.Let E be a uniformly smooth and strictly convex real Banach space which also has Kadec-Klee property.Let C be a nonempty, closed, and convex subset of E. For each k 1, 2, . . ., m, let A k be a continuous and monotone operator from C into E * .Suppose {T i } ∞ i 1 and {S i } ∞ i 1 are two countable families of closed relatively quasi-nonexpansive mappings of C into itself such that Ω : / ∅.Let f : E → Ê be a convex and lower semicontinuous mapping with C ⊂ int D f , and suppose {x n } ∞ n 0 is iteratively generated by

4 EP
F is closed and convex.Lemma 2.12 see Takahashi and Zembayashi 39 .Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E. Assume that F : C × C → Ê satisfies (A1)-(A4), and let r > 0.
closed and convex set, from Lemma 2.6, we know thatΠ f F x 0 is single valued and denote w Π f Ω x 0 .Since x n Π f C n x 0 and w ∈ Ω ⊂ C n , we have G x n , Jx 0 ≤ G w, Jx 0 , ∀n ≥ 1.

Lemma 4 . 2 .
Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E. Assume that ϕ : C → Ê is lower semicontinuous and convex.For r > 0 and x ∈ E, define a mapping T ϕ r : E → C as follows: T ϕ r x z ∈ C : ϕ y 1 r y − z, Jz − Jx ≥ ϕ z , ∀y ∈ C .4.4
, Cholamjiak and Suantai 30 , Li et al. 29 , and many other recent known results in the literature.
Fan et al. 36 showed that the condition f is positive homogeneous which appeared in Lemma 2.2 can be removed.Lemma 2.3 see Fan et al. 36 .Let E be a real reflexive Banach space with its dual E * and C a nonempty, closed, and convex subset of 37ternational Journal of Mathematics and Mathematical SciencesThe fixed points set F T of a relatively quasi-nonexpansive mapping is closed and convex as given in the following lemma.Lemma 2.8 see Chang et al.37.Let C be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex real Banach space E which also has Kadec-Klee property.Let T be a closed relatively quasi-nonexpansive mapping of C into itself.Then, F T is closed and convex.
is a single-valued mapping.Lemma 2.7 see Li et al. 29 .Let C be a nonempty, closed, and convex subset of a smooth and reflexive Banach space E. Let x ∈ E and x ∈ Π f C x.Then, φ y, x G x, Jx ≤ G y, Jx , ∀y ∈ C. 2.12 29rollary 3.3 see Li et al.29.Let E be a uniformly convex real Banach space which is also uniformly smooth.Let C be a nonempty, closed, and convex subset of E. Suppose T is a relatively nonexpansive mapping of C into itself such that Ω : F T / ∅.Let f : E → Ê be a convex and lower semicontinuous mapping with C ⊂ int D f , and suppose {x n } ∞ n 0 is iteratively generated by x 2, . . ., m .Then, {x n } ∞ n 0 converges strongly to Π Ω x 0 .
Let f : E → Ê be a convex and lower International Journal of Mathematics and Mathematical Sciences 21 semicontinuous mapping with C ⊂ int D f , and suppose {x n } ∞ n 0 is iteratively generated by x0 ∈ C, C 1,i C, C 1 ∩ ∞ i 1 C 1,i , x 1 Π −1 α n,i Jx n 1 − α n,i Jz n,i , u n,i T A m r m ,n T A m−1 r m−1 ,n • • • T A 2 r 2 ,n T A 1 r 1 ,n y n,i , C n 1,i {z ∈ C n,i : G z, Ju n,i ≤ G z, Jx n }, 2, . .., m satisfying lim inf n → ∞ r k,n > 0 k 1, 2, . .., m .Then, {x n } ∞n 0 converges strongly to Π Ê be lower semicontinuous and convex.Suppose {T i } ∞ i 1 and {S i } ∞ i 1 are two countable families of closed relatively quasi-nonexpansive mappings of C into itself such that Ω : f Ω x 0 .Theorem 4.4.Let E be a uniformly smooth and strictly convex real Banach space which also has Kadec-Klee property.Let C be a nonempty, closed, and convex subset of E. For each k 1, 2, . . ., m, let ϕ k : C →