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The shrinking projection method for solving generalized equilibrium problems and common fixed points for asymptotically quasiϕnonexpansive mappings
Fixed Point Theory and Applications volume 2011, Article number: 9 (2011)
Abstract
In this article, we introduce a new hybrid projection iterative scheme based on the shrinking projection method for finding a common element of the set of solutions of the generalized mixed equilibrium problems and the set of common fixed points for a pair of asymptotically quasiϕnonexpansive mappings in Banach spaces and set of variational inequalities for an αinverse strongly monotone mapping. The results obtained in this article improve and extend the recent ones announced by Matsushita and Takahashi (Fixed Point Theory Appl. 2004(1):3747, 2004), Qin et al. (Appl. Math. Comput. 215:38743883, 2010), Chang et al. (Nonlinear Anal. 73:22602270, 2010), Kamraksa and Wangkeeree (J. Nonlinear Anal. Optim.: Theory Appl. 1(1):5569, 2010) and many others.
AMS Subject Classification: 47H05, 47H09, 47J25, 65J15.
1. Introduction
Let E be a Banach space with norm ·, C be a nonempty closed convex subset of E, and let E* denote the dual of E. Let f : C × C → ℝ be a bifunction, φ: C → ℝ be a realvalued function, and B : C → E* be a mapping. The generalized mixed equilibrium problem, is to find x ∈ C such that
The set of solutions to (1.1) is denoted by GMEP(f, B, φ), i.e.,
If B ≡ 0, then the problem (1.1) reduces into the mixed equilibrium problem for f, denoted by MEP(f, φ), is to find x ∈ C such that
If φ ≡ 0, then the problem (1.1) reduces into the generalized equilibrium problem, denoted by GEP(f, B), is to find x ∈ C such that
If f ≡ 0, then the problem (1.1) reduces into the mixed variational inequality of Browder type, denoted by MVI(B, C), is to find x ∈ C such that
If φ ≡ 0, then the problem (1.5) reduces into the classical variational inequality, denoted by VI(B, C), which is to find x ∈ C such that
If B ≡ 0 and φ ≡ 0, then the problem (1.1) reduces into the equilibrium problem for f, denoted by EP(f), which is to find x ∈ C such that
If f ≡ 0, then the problem (1.3) reduces into the minimize problem, denoted by Argmin (φ), which is to find x ∈ C such that
The above formulation (1.6) was shown in [1] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, variational inequality problems, vector equilibrium problems, and Nash equilibria in noncooperative games. In addition, there are several other problems, for example, the complementarity problem, fixed point problem and optimization problem, which can also be written in the form of an EP(f). In other words, the EP(f) is an unifying model for several problems arising in physics, engineering, science, optimization, economics, etc. In the last two decades, many articles have appeared in the literature on the existence of solutions of EP(f); see, for example [1–4] and references therein. Some solution methods have been proposed to solve the EP(f) in Hilbert spaces and Banach spaces; see, for example [5–20] and references therein.
A Banach space E is said to be strictly convex if < 1 for all x, y ∈ E with x = y = 1 and x ≠ y. Let U = {x ∈ E : x = 1} be the unit sphere of E. Then, a Banach space E is said to be smooth if the limit exists for each x, y ∈ U. It is also said to be uniformly smooth if the limit exists uniformly in x, y ∈ U. Let E be a Banach space. The modulus of convexity of E is the function δ : [0, 2] → [0, 1] defined by
A Banach space E is uniformly convex if and only if δ (ε) > 0 for all ε ∈ (0, 2]. Let p be a fixed real number with p ≥ 2. A Banach space E is said to be puniformly convex if there exists a constant c > 0 such that δ (ε) ≥ cε^{p} for all ε ∈ [0, 2]; see [21, 22] for more details. Observe that every puniformly convex is uniformly convex. One should note that no Banach space is puniformly convex for 1 < p < 2. It is well known that a Hilbert space is 2uniformly convex, uniformly smooth. For each p > 1, the generalized duality mapping J_{ p } : E → 2^{E*}is defined by
for all x ∈ E. In particular, J = J_{2} is called the normalized duality mapping. If E is a Hilbert space, then J = I, where I is the identity mapping.
A set valued mapping U : E ⇉ E* with graph G(U) = {(x, x*) : x* ∈ Ux}, domain D(U) = {x ∈ E : Ux ≠ ∅}, and rang R(U) = ∪{Ux : x ∈ D(U)}. U is said to be monotone if 〈x  y, x*  y*〉 ≥ 0 whenever x* ∈ Ux, y* ∈ Uy. A monotone operator U is said to be maximal monotone if its graph is not properly contained in the graph of any other monotone operator. We know that if U is maximal monotone, then the solution set U^{1} 0 = {x ∈ D(U) : 0 ∈ Ux} is closed and convex. It is knows that U is a maximal monotone if and only if R(J + rU) = E* for all r > 0 when E is a reflexive, strictly convex and smooth Banach space (see [23]).
Recall that let A : C → E* be a mapping. Then, A is called

(i)
monotone if

(ii)
αinversestrongly monotone if there exists a constant α > 0 such that
The class of inversestrongly monotone mappings has been studied by many researchers to approximating a common fixed point; see [24–29] for more details.
Recall that a mappings T : C → C is said to be nonexpansive if
T is said to be quasinonexpansive if F(T) ≠ ∅, and
T is said to be asymptotically nonexpansive if there exists a sequence {k_{ n } } ⊂ [1, ∞) with k_{ n } → 1 as n → ∞ such that
T is said to be asymptotically quasinonexpansive if F(T) ≠ ∅ and there exists a sequence {k_{ n } } ⊂ [1, ∞) with k_{ n } → 1 as n → ∞ such that
T is called uniformly LLipschitzian continuous if there exists L > 0 such that
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [30] in 1972. Since 1972, a host of authors have studied the weak and strong convergence of iterative processes for such a class of mappings.
If C is a nonempty closed convex subset of a Hilbert space H and P_{ C } : H → C is the metric projection of H onto C, then P_{ C } is a nonexpansive mapping. This fact actually characterizes Hilbert spaces and, consequently, it is not available in more general Banach spaces. In this connection, Alber [31] recently introduced a generalized projection operator C in Banach space E which is an analogue of the metric projection in Hilbert spaces.
Let E be a smooth, strictly convex and reflexive Banach spaces and C be a nonempty, closed convex subset of E. We consider the Lyapunov functional ϕ : E × E → ℝ^{+} defined by
for all x, y ∈ E, where J is the normalized duality mapping from E to E*.
Observe that, in a Hilbert space H, (1.9) reduces to ϕ(y, x) = x  y^{2} for all x, y ∈ H. The generalized projection Π _{ C } : E → C is a mapping that assigns to an arbitrary point x ∈ E the minimum point of the functional ϕ(y, x); that is, Π _{ C }x = x*, where x* is the solution to the minimization problem:
The existence and uniqueness of the operator Π _{ C } follows from the properties of the functional ϕ(y, x) and strict monotonicity of the mapping J (see, for example, [9, 32–34]). In Hilbert spaces, Π _{ C } = P_{ C } .^{.} It is obvious from the definition of the function ϕ that

(1)
(y  x)^{2} ≤ ϕ(y, x) ≤ (y + x)^{2} for all x, y ∈ E.

(2)
ϕ(x, y) = ϕ (x, z) + ϕ (z, y) + 2 〈x  z, Jz  Jy〉 for all x, y, z ∈ E.

(3)
ϕ(x, y) = 〈x, Jx  Jy〉 + 〈y  x, Jy〉 ≤ x Jx  Jy + y  x y for all x, y ∈ E.

(4)
If E is a reflexive, strictly convex and smooth Banach space, then, for all x, y ∈ E,
By the HahnBanach theorem, J(x) ≠ ∅ for each x ∈ E, for more details see [35, 36].
Remark 1.1. It is also known that if E is uniformly smooth, then J is uniformly normtonorm continuous on each bounded subset of E. Also, it is well known that if E is a smooth, strictly convex and reflexive Banach space, then the normalized duality mapping J : E → 2^{E*}is singlevalued, onetoone and onto (see [35]).
Let C be a closed convex subset of E, and let T be a mapping from C into itself. We denote by F(T) the set of fixed point of T. A point p in C is said to be an asymptotic fixed point of T[37] if C contains a sequence {x_{ n }} which converges weakly to p such that lim_{n →∞}x_{ n } Tx_{ n } = 0. The set of asymptotic fixed points of T will be denoted by .
A point p in C is said to be a strong asymptotic fixed point of T[37] if C contains a sequence {x_{ n } } which converges strong to p such that lim_{n→∞}x_{ n } Tx_{ n } = 0. The set of strong asymptotic fixed points of S will be denoted by .
A mapping T is called relatively nonexpansive[38–40] if and
The asymptotic behavior of relatively nonexpansive mappings were studied in [38, 39].
A mapping T : C → C is said to be weak relatively nonexpansive if and
A mapping T is called hemirelatively nonexpansive if F(T) ≠ ∅ and
A mapping T is said to be relatively asymptotically nonexpansive[32, 41] if and there exists a sequence {k_{ n } } ⊂ [0, ∞) with k_{ n } → 1 as n → ∞ such that
Remark 1.2. Obviously, relatively nonexpansive implies weak relatively nonexpansive and both also imply hemirelatively nonexpansive. Moreover, the class of relatively asymptotically nonexpansive is more general than the class of relatively nonexpansive mappings.
We note that hemirelatively nonexpansive mappings are sometimes called quasiϕnonexpansive mappings.
We recall the following :

(i)
T : C → C is said to be ϕnonexpansive [42, 43] if ϕ (Tx, Ty) ≤ ϕ (x, y) for all x, y ∈ C.

(ii)
T : C → C is said to be quasiϕnonexpansive [42, 43] if F(T) ≠ ∅ and ϕ(p, Tx) ≤ ϕ(p, x) for all x ∈ C and p ∈ F(T).

(iii)
T : C → C is said to be asymptotically ϕnonexpansive [43] if there exists a sequence {k_{ n } } ⊂ [0, ∞) with k_{ n } → 1 as n → ∞ such that ϕ (T^{n}x, T^{n}y) ≤ k_{ n } ϕ(x, y) for all x, y ∈ C.

(iv)
T : C → C is said to be asymptotically quasiϕnonexpansive [43] if F(T) ≠ ∅ and there exists a sequence {k_{ n } } ⊂ [0, ∞) with k_{ n } → 1 as n → ∞ such that ϕ(p, T^{n}x) ≤ k_{ n } ϕ (p, x) for all x ∈ C, p ∈ F(T) and n ≥ 1.
Remark 1.3. (i) The class of (asymptotically) quasiϕnonexpansive mappings is more general than the class of relatively (asymptotically) nonexpansive mappings, which requires the strong restriction .

(ii)
In real Hilbert spaces, the class of (asymptotically) quasiϕnonexpansive mappings is reduced to the class of (asymptotically) quasinonexpansive mappings.
Let T be a nonlinear mapping, T is said to be uniformly asymptotically regular on C if
T : C → C is said to be closed if for any sequence {x_{ n } } ⊂ C such that lim_{n→∞}x_{ n }= x_{0} and lim_{n→∞}Tx_{ n }= y_{0}, then Tx_{0} = y_{0}.
We give some examples which are closed and asymptotically quasiϕnonexpansive.
Example 1.4. (1). Let E be a uniformly smooth and strictly convex Banach space and U ⊂ E × E* be a maximal monotone mapping such that its zero set U^{1}0 is nonempty. Then, J_{ r }= (J + rU)^{1}J is a closed and asymptotically quasiϕnonexpansive mapping from E onto D(U) and F(J_{ r }) = U^{1}0.
(2). Let Π _{ C } be the generalized projection from a smooth, strictly convex and reflexive Banach space E onto a nonempty closed and convex subset C of E. Then Π _{ C } is a closed and asymptotically quasiϕnonexpansive mapping from E onto C with F(Π_{ C }) = C.
Recently, Matsushita and Takahashi [44] obtained the following results in a Banach space.
Theorem MT. Let E be a uniformly convex and uniformly smooth Banach space, let C be a nonempty closed convex subset of E, let T be a relatively nonexpansive mapping from C into itself, and let {α_{ n } } be a sequence of real numbers such that 0 ≤ α_{ n } < 1 and lim sup_{n→∞}< 1. Suppose that {x_{ n }} is given by
where J is the duality mapping on E. If F(T) is nonempty, then {x_{ n } } converges strongly to P_{ F } (T) ^{x} , where P_{F(T)}is the generalized projection from C onto F(T). In 2008, Iiduka and Takahashi [45] introduced the following iterative scheme for finding a solution of the variational inequality problem for an inversestrongly monotone operator A in a 2uniformly convex and uniformly smooth Banach space E : x_{1} = x ∈ C and
for every n = 1, 2, 3,..., where Π _{ C } is the generalized metric projection from E onto C, J is the duality mapping from E into E* and {λ_{ n } } is a sequence of positive real numbers. They proved that the sequence {x_{ n } } generated by (1.12) converges weakly to some element of VI(A, C).
A popular method is the shrinking projection method which introduced by Takahashi et al. [46] in year 2008. Many authors developed the shrinking projection method for solving (mixed) equilibrium problems and fixed point problems in Hilbert and Banch spaces; see, [12, 15, 16, 47–57] and references therein.
Recently, Qin et al. [58] further extended Theorem MT by considering a pair of asymptotically quasiϕnonexpansive mappings. To be more precise, they proved the following results.
Theorem QCK. Let E be a uniformly smooth and uniformly convex Banach space and C a nonempty closed and convex subset of E. Let T : C → C be a closed and asymptotically quasiϕnonexpansive mapping with the sequence such that as n → ∞ and S : C → C a closed and asymptotically quasiϕnonexpansive mapping with the sequence such that as n → ∞. Let {α_{ n } }, {β_{ n } }, {γ_{ n } } and {δ_{ n } } be real number sequences in [0, 1].
Assume that T and S are uniformly asymptotically regular on C and Ω = F(T) ∩ F(S) is nonempty and bounded. Let {x_{ n } } be a sequence generated in the following manner:
where for each n ≥ 1, J is the duality mapping on E, and M_{ n } = sup{ϕ(z, x_{ n } ) : z ∈ Ω } for each n ≥ 1. Assume that the control sequences {α_{ n } }, {β_{ n } }, {γ_{ n } } and {δ_{ n } } satisfy the following restrictions :

(a)
β_{ n } + γ_{ n } + δ_{ n } = 1, ∀n ≥ 1;

(b)
lim inf_{n→∞} γ _{ n } δ _{ n }, lim_{n→∞} β _{ n }= 0;

(c)
0 ≤ α_{ n } < 1 and lim sup_{n→∞} α _{ n }< 1.
On the other hand, Chang, Lee and Chan [59] proved a strong convergence theorem for finding a common element of the set of solutions for a generalized equilibrium problem (1.4) and the set of common fixed points for a pair of relatively nonexpansive mappings in Banach spaces. They proved the following results.
Theorem CLC. Let E be a uniformly smooth and uniformly convex Banach space, C be a nonempty closed convex subset of E. Let A : C → E* be a αinversestrongly monotone mapping and f : C × C → ℝ be a bifunction satisfying the conditions (A 1)  (A 4). Let S, T : C → C be two relatively nonexpansive mappings such that Ω := F(T) ∩ F(S) ∩ GEP(f, A). Let {x_{ n } } be the sequence generated by
where {α_{ n } } and {β_{ n } } are sequences in [0, 1] and {γ_{ n } } ⊂ [a, 1) for some a > 0. If the following conditions are satisfied

(a)
lim inf_{n →∞} α _{ n }(1  α _{ n }) > 0;

(b)
lim inf_{n →∞} β _{ n }(1 β _{ n }) > 0;
then, {x_{ n }} converges strongly to Π_{Ω}x_{0}, where Π_{Ω} is the generalized projection of E onto Ω.
Very recently, Kim [60], considered the shrinking projection methods which were introduced by Takahashi et al. [46] for asymptotically quasiϕnonexpansive mappings in a uniformly smooth and strictly convex Banach space which has the KadecKlee property.
In this article, motivated and inspired by the study of Matsushita and Takahashi [44], Qin et al. [58], Kim [60], and Chang et al. [59], we introduce a new hybrid projection iterative scheme based on the shrinking projection method for finding a common element of the set of solutions of the generalized mixed equilibrium problems, the set of the variational inequality and the set of common fixed points for a pair of asymptotically quasiϕnonexpansive mappings in Banach spaces. The results obtained in this article improve and extend the recent ones announced by Matsushita and Takahashi [44], Qin et al. [58], Chang et al. [59] and many others.
2. Preliminaries
For the sake of convenience, we first recall some definitions and conclusions which will be needed in proving our main results.
In the sequel, we denote the strong convergence, weak convergence and weak* convergence of a sequence {x_{ n } } by x_{ n } → x, x_{ n } ⇀* × and x_{ n } ⇀* x, respectively.
It is well known that a uniformly convex Banach space has the KadecKlee property, i.e. if x_{ n } ⇀ x and x_{ n }  → x, then x_{ n } → x.
Lemma 2.1. ([31, 61]) Let E be a smooth, strictly convex and reflexive Banach space and C be anonempty closed convex subset. Then, the following conclusion hold:
Lemma 2.2. ([34]). If E be a 2uniformly convex Banach space and 0 < c ≤ 1. Then, for all x, y ∈ E we have
where J is the normalized duality mapping of E.
The best constant in Lemma is called the puniformly convex constant of E.
Lemma 2.3. ([62]). If E be a puniformly convex Banach space and p be a given real number with p ≥ 2, then for all x, y ∈ E, j_{ x } ∈ J_{ p }x and j_{ y } ∈ J_{ p }y
where J_{ p } is the generalized duality mapping of E andis the puniformly convexity constant of E.
Lemma 2.4. ([63]) Let E be a uniformly convex Banach space and B_{ r } (0) a closed ball of E. Then, there exists a continuous strictly increasing convex function g : [0, ∞) → [0, ∞) with g(0) = 0 such that
for all x, y ∈ B_{ r } (0) and α ∈ [0, 1].
Lemma 2.5. ([58]) Let E be a uniformly convex and smooth Banach space, C a nonempty closed convex subset of E and T : C → C a closed asymptotically quasiϕnonexpansive mapping. Then, F(T) is a closed convex subset of C.
Lemma 2.6. ([61]) Let E be a smooth and uniformly convex Banach space. Let x_{ n }and y_{ n }be sequences in E such that either {x_{ n }} or {y_{ n }} is bounded. If lim_{n→∞}ϕ(x_{ n }, y_{ n }) = 0, then lim_{n→∞}x_{ n } y_{ n } = 0.
Lemma 2.7. (Alber[31]). Let C be a nonempty closed convex subset of a smooth Banach space E and x ∈ E. Then, x_{0} = Π_{ C }x if and only if
Let E be a reflexive, strictly convex, smooth Banach space and J the duality mapping from E into E*. Then, J^{1} is also single valued, onetoone, surjective, and it is the duality mapping from E* into E. We make use of the following mapping V studied in Alber [31]
for all x ∈ E and x* ∈ E*; that is, V (x, x*) = ϕ(x, J^{1}x*).
Lemma 2.8. (Kohsaka and Takahashi [[64], Lemma 3.2]). Let E be a reflexive, strictly convex smooth Banach space and let V be as in (2.1). Then,
for all × ∈ E and x*, y* ∈ E*.
Proof. Let x ∈ E. Define g(x*) = V (x, x*) and f(x*) = x*^{2} for all x* ∈ E*. Since J^{1} is the duality mapping from E* to E, we have
Hence, we get
that is,
for all x*, y* ∈ E*.
For solving the generalized equilibrium problem, let us assume that the nonlinear mapping A : C → E* is αinverse strongly monotone and the bifunction f : C × C → ℝ satisfies the following conditions:
(A1) f(x, x) = 0 ∀x ∈ C;
(A2) f is monotone, i.e., f(x, y) + f(y, x) ≤ 0, ∀x, y ∈ C;
(A3) lim sup_{t↓0}f (x + t(z  x), y) ≤ f(x, y), ∀x, y, z ∈ C;
(A4) the function y ↦ f(x, y) is convex and lower semicontinuous.
Lemma 2.9. ([1]) Let E be a smooth, strictly convex and reflexive Banach space and C be a nonempty closed convex subset of E. Let f : C × C → ℝ be a bifunction satisfying the conditions (A 1)  (A 4). Let r > 0 and × ∈ E, then there exists z ∈ C such that
Lemma 2.10. ([65]) Let C be a closed convex subset of a uniformly smooth and strictly convex Banach space E and let f be a bifunction from C × C to ℝ satisfying (A 1)  (A 4). For r > 0 and × ∈ E, define a mapping T_{ r } : E → C as follows:
for all × ∈ C. Then, the following conclusions holds:

(1)
T _{ r } is singlevalued;

(2)
T_{ r } is a firmly nonexpansivetype mapping, i.e.
(A3) F(T_{ r } ) = EP(f );
(A4) EP(f) is a closed convex.
Lemma 2.11. ([19]) Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space E, let f be a bifunction from C × C to ℝ satisfying (A 1)  (A 4) and let r > 0. Then, for × ∈ E and q ∈ F(T_{ r } ),
Lemma 2.12. ([66]) Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space E. Let B : C → E* be a continuous and monotone mapping, φ : C → ℝ be a lower semicontinuous and convex function, and f be a bifunction from C × C to ℝ satisfying (A 1)  (A 4). For r > 0 and × ∈ E, then there exists u ∈ C such that
Define a mapping K_{ r } : C → C as follows:
for all x ∈ C. Then, the following conclusions holds:

(a)
K_{ r } is singlevalued ;

(b)
K_{ r } is a firmly nonexpansivetype mapping, i.e.;

(c)
;

(d)
GMEP(f, B, φ) is a closed convex,

(e)
ϕ(q, K_{ r }z) + ϕ(K_{ r }z, z) ≤ ϕ(q, z), ∀q ∈ F (K_{ r } ), z ∈ E.
Remark 2.13. ([66]) It follows from Lemma 2.12 that the mapping K_{ r } : C → C defined by (2.3) is a relatively nonexpansive mapping. Thus, it is quasiϕnonexpansive.
Let C be a nonempty closed convex subset of a Banach space E and let A be an inversestrongly monotone mapping of C into E* which is said to be hemicontinuous if for all x, y ∈ C, the mapping F of [0, 1] into E*, defined by F(t) = A(tx + (1  t)y), is continuous with respect to the weak* topology of E*. We define by N_{ C } (v) the normal cone for C at a point v ∈ C, that is,
Lemma 2.14. (Rockafellar[23]). Let C be a nonempty, closed convex subset of a Banach space E, and A a monotone, hemicontinuous operator of C into E*. Let U : E ⇉ E* be an operator defined as follows:
Then, U is maximal monotone and U^{1}0 = VI(A, C).
3. Main results
In this section, we shall prove a strong convergence theorem for finding a common element of the set of solutions for a generalized mixed equilibrium problem (1.2), set of variational inequalities for an αinverse strongly monotone mapping and the set of common fixed points for a pair of asymptotically quasiϕnonexpansive mappings in Banach spaces.
Theorem 3.1. Let E be a uniformly smooth and 2uniformly convex Banach space, C be a nonempty closed convex subset of E. Let A be an αinversestrongly monotone mapping of C into E* satisfying Ay ≤ Ay  Au, ∀y ∈ C and u ∈ VI(A, C) ≠ ∅. Let B : C → E* be a continuous and monotone mapping and f : C × C → ℝ be a bifunction satisfying the conditions (A 1)  (A 4), and φ : C → ℝ be a lower semicontinuous and convex function. Let T : C → C be a closed and asymptotically quasiϕnonexpansive mapping with the sequencesuch thatas n → ∞ and S : C → C be a closed and asymptotically quasiϕnonexpansive mapping with the sequencesuch thatas n → ∞. Assume that T and S are uniformly asymptotically regular on C and Ω := F(T) ∩ F(S) ∩ VI(A, C) ∩ GMEP(f, B, φ.) ≠ ∅.
Let {x_{ n }} be the sequence defined by x_{0} ∈ E and
where as n → ∞, for each n ≥ 1, M_{ n } = sup{ϕ(z, x_{ n } ) : z ∈ Ω } for each n ≥ 1, {α_{ n } } and {β_{ n } } are sequences in [0, 1], {λ_{ n } } ⊂ [a, b] for some a, b with 0 < a < b < c^{2}α/2, where is the 2uniformly convexity constant of E and {r_{ n } } ⊂ [d, ∞) for some d > 0. Suppose that the following conditions are satisfied: lim inf_{n→∞}(1 α_{ n }) > 0 and lim inf_{n→∞}(1 β_{ n } ) > 0. Then, the sequence {x_{ n }} converges strongly to Π_{Ω}x_{0}, where Π_{Ω} is generalized projection of E onto Ω.
Proof. We have several steps to prove this theorem as follows:
Step 1. We first show that C_{n+1}is closed and convex for each n ≥ 1. Indeed, it is obvious that C_{1} = C is closed and convex. Suppose that C_{ i }is closed and convex for each i ∈ ℕ. Next, we prove that C_{i+1}is closed and convex. For any z ∈ C_{i+1}, we know that ϕ(z, u_{ i }) ≤ ϕ (z, x_{ i }) + θ_{ i }is equivalent to
where and M_{ i } = sup{ϕ(z, x_{ i } ) : z ∈ Ω} for each i ≥ 1. Hence, C_{i+1}is closed and convex. Then, for each n ≥ 1, we see that C_{ n } is closed and convex. Hence, is well defined.
By the same argument as in the proof of [[43], Lemma 2.4], one can show that F(T) ∩ F(S) is closed and convex. We also know that VI(A, C) = U^{1}0 is closed and convex, and hence from Lemma 2.12(d), Ω := F(S) ∩ F(T) ∩ VI(A, C) ∩ GMEP(f, B, φ) is a nonempty, closed and convex subset of C. Consequently, Π_{Ω} is well defined.
Step 2. We show that the sequence {x_{ n } } is well defined. Next, we prove that Ω ⊂ C_{ n } for each n ≥ 1. If n = 1, Ω ⊂ C_{1} = C is obvious. Suppose that Ω ⊂ C_{ i } for some positive integer i. For every q ∈ Ω, we obtain from the assumption that q ∈ C_{ i } . It follows, from Lemma 2.1 and Lemma 2.8, that
Thus, q ∈ VI(A, C) and A is αinversestrongly monotone, we have
From Lemma 2.2 and Ay ≤ Ay  Au for all y ∈ C and q ∈ Ω, we obtain
Substituting (3.3) and (3.4) into (3.2), we have
As T^{i} is asymptotically quasiϕnonexpansive mapping, we also have
It follows that
This shows that q ∈ C_{i+1}. This implies that Ω ⊂ C_{ n } for each n ≥ 1.
From , we see that
Since Ω ⊂ C_{ n } for each n ≥ 1, we arrive at
Hence, the sequence {x_{ n } } is well defined.
Step 3. Now, we prove that {x_{ n } } is bounded.
In view of Lemma 2.1, we see that
for each q ∈ C_{ n } . Therefore, we obtain that the sequence ϕ(x_{ n } , x_{0}) is bounded, and so are {x_{ n } }, {w_{ n } }, {y_{ n } }, {z_{ n } }, {T^{n}w_{ n } } and {S^{n}x_{ n } }.
Step 4. We show that {x_{ n } } is a Cauchy sequence.
Since and , we have
This implies that {ϕ(x_{ n } , x_{0})} is nondecreasing, and lim_{n →∞}ϕ(x_{ n } , x_{0}) exists.
For m > n and from Lemma 2.1, we have
Letting m, n → ∞ in (3.9), we see that ϕ(x_{ m } , x_{ n } ) → 0. It follows from Lemma 2.6 that x_{ m }  x_{ n }  → 0 as m, n → ∞. Hence, {x_{ n } } is a Cauchy sequence. Since E is a Banach space and C is closed and convex, we can assume that p ∈ C such that x_{ n } → p as n → ∞.
Step 5. We will show that p ∈ Ω:= F(T) ∩ F(S) ∩ VI(A, C) ∩ GMEP(f, B, φ).
(a) First, we show that p ∈ F(T) ∩ F(S).
By taking m = n + 1 in (3.9), we obtain that
Since , from definition of C_{n+1}, we have
and from (3.5) and (3.6), we also have
Since E is uniformly smooth and uniformly convex, from (3.10)(3.12), θ_{ n } → 0 as n → ∞ and
Lemma 2.6, it follows that
and by using triangle inequality, we have
Since J is uniformly normtonorm continuous, we also have
and
Since , and from (3.7), we have
Since x_{ n }  u_{ n } → 0 and J is uniformly continuous, we have
Since {x_{ n } } and {u_{ n } } are bounded, it follows from (3.14) and (3.15) that ϕ(y_{ n } , u_{ n } ) → 0 as n → ∞. Since E is smooth and uniformly convex, from Lemma 2.6, we have
Since J is uniformly normtonorm continuous, we also have
Again from (3.1) and (3.16), we have
This implies that JT^{n}w_{ n }  Jx_{ n }  → 0. Again since J^{1} is uniformly normtonorm continuous, we also have
For p ∈ Ω, we note that
It follows from (3.22) and x_{ n } → p as n → ∞, that
On other hand, we have
Since T is uniformly asymptotically regular and from (3.24), we obtain that
Thai is, TT^{nw}n → p as n → ∞. From the closedness of T, we see that p ∈ F(T). Furthermore, For q ∈ Ω, from (3.7) and (3.18) that
and hence
From (3.18) and lim inf_{n→∞}(1 β_{ n }) > 0, obtain that
From Lemma 2.1, Lemma 2.8 and (3.4), we compute
Applying Lemma 2.6 and (3.27) that
Since J is uniformly normtonorm continuous on bounded sets, by (3.28), we have
From(3.1), (3.20) and (ii), we have
Since J^{1} is uniformly normtonorm continuous on bounded sets
We observe that
It follows from (3.31) and x_{ n } → p as n → ∞, we obtain
On other hand, we have
Since S is uniformly asymptotically regular and (3.33), we obtain that
that is, SS^{n}z_{ n } → p as n → ∞. From the closedness of S, we see that p ∈ F(S). Hence, p ∈ F(T) ∩ F(S).
(b) We show that p ∈ GMEP(f, B, φ). From (A2), we have
and hence
For t with 0 < t ≤ 1 and y ∈ C, let y_{ t } = t_{ y } + (1  t)p. Then, we get y_{ t } ∈ C. From (3.35), it follows that
we know that y_{ n } , u_{ n } → p as n → ∞, and as n → ∞. Since B is monotone, we know that 〈By_{ t }  Bu_{ n } , y_{ t }  u_{ n } 〉 ≥ 0. Thus, it follows from (A4) that
Based on the conditions (A1), (A4) and convexity of φ, we have
and hence
From (A3) and the weakly lower semicontinuity of φ, and letting t → 0, we also have
This implies that p ∈ GMEP(f, B, φ).
(c) We show that p ∈ VI(A, C). Indeed, define a setvalued U : E ⇉ E* by Lemma 2.14, U is maximal monotone and U^{1}0 = VI(A, C). Let (v, w) ∈ G(U). Since w ∈ Uv = Av + N_{ C } (v), we get w  Av ∈ N_{ C } (v).
From w_{ n } ∈ C, we have
On the other hand, since . Then from Lemma 2.7, we have
and thus
It follows from (3.36) and (3.37) that
where M = sup_{n≥1}v  w_{ n }. Takeing the limit as n → ∞, (3.28) and (3.29), we obtain 〈v  p, w〉 ≥ 0. Based on the maximality of U, we have p ∈ U^{1}0 and hence p ∈ VI(A, C). Hence, by (a), (b) and (c), we obtain p ∈ Ω.
Step 5. Finally, we prove that p = Π_{Ω}x_{0}. Taking the limit as n → ∞ in (3.8), we obtain that
and hence, p = Π_{Ω}x_{0} by Lemma 2.1. This completes the proof.
The following Theorems can readily be derived from Theorem 3.1.
Corollary 3.2. Let E be a uniformly smooth and 2uniformly convex Banach space, and C be a nonempty closed convex subset of E. Let A be an αinversestrongly monotone mapping of C into E* satisfying Ay ≤ Ay  Au, ∀y ∈ C and u ∈ VI(A, C) ≠ Ø,. Let f : C × C → ℝ be a bifunction satisfying the conditions (A 1)  (A 4), and φ : C → ℝ be a lower semicontinuous and convex function. Let T : C → C be a closed and asymptotically quasiϕnonexpansive mapping with the sequencesuch thatas n → ∞ and S : C → C be a closed and asymptotically quasiϕnonexpansive mapping with the sequencesuch thatas n → ∞. Assume that T and S are uniformly asymptotically regular on C and Ω:= F(T) ∩ F(S) ∩ VI(A, C) ∩ MEP(f, φ) ≠ ∅. Let {x_{ n } } be the sequence defined by x_{0} ∈ E and
where as n → ∞, for each n ≥ 1, M_{ n } = sup{ϕ(z, x_{ n } ) : z ∈ Ω} for each n ≥ 1, {α_{ n } } and {β_{ n } } are sequences in [0, 1], {λ_{ n } } ⊂ [a, b] for some a, b with 0 < a < b < c^{2}α/2, where is the 2uniformly convexity constant of E and {r_{ n } } ⊂ [d, ∞) for some d > 0. Suppose that the following conditions are satisfied:

(i)
lim inf_{n→∞}(1 α_{ n } ) > 0,

(ii)
lim inf_{n→∞}(1 β_{ n } ) > 0.
Then, the sequence {x_{ n } } converges strongly to Π_{Ω}x_{0}, where Π_{Ω} is generalized projection of E onto Ω.
Proof. Putting B ≡ 0 in Theorem 3.1, the conclusion of Theorem 3.2 can be obtained.
Corollary 3.3. Let E be a uniformly smooth and 2uniformly convex Banach space, C be a nonempty closed convex subset of E. Let A be an αinversestrongly monotone mapping of C into E* satisfying Ay ≤ Ay  Au, ∀y ∈ C and u ∈ VI(A, C) ≠ Ø. Let B : C → E* be a continuous and monotone mapping and φ : C → ℝ be a lower semicontinuous and convex function. Let T : C → C be a closed and asymptotically quasiϕnonexpansive mapping with the sequencesuch thatas n → ∞ and S : C → C be a closed and asymptotically quasiϕnonexpansive mapping with the sequencesuch thatas n → ∞. Assume that T and S are uniformly asymptotically regular on C and Ω := F(T) ∩ F(S) ∩ VI(A, C) ∩ MVI(B, C) ≠ ∅. Let {x_{ n } } be the sequence defined by x_{0} ∈ E and
where as n → ∞, for each n ≥ 1, M_{ n } = sup{ϕ(z, x_{ n } ) : z ∈ Ω} for each n ≥ 1, {α_{ n } } and {β_{ n } } are sequences in [0, 1], {λ_{ n } } ⊂ [a, b] for some a, b with 0 < a < b < c^{2}α/2, where is the 2uniformly convexity constant of E and {r_{ n } } ⊂ [d, ∞) for some d > 0. Suppose that the following conditions are satisfied:

(i)
lim inf_{n→∞}(1 α_{ n } ) > 0;

(ii)
lim inf_{n→∞}(1 β_{ n } ) > 0.
Then, the sequence {x_{ n } } converges strongly to Π_{Ω}x_{0}, where Π_{Ω} is generalized projection of E onto Ω.
Proof. Putting f ≡ 0 in Theorem 3.1, the conclusion of Theorem 3.2 can be obtained.
Since every closed relatively asymptotically nonexpansive mapping is asymptotically quasiϕnonexpansive, we obtain the following corollary.
Corollary 3.4. Let E be a uniformly smooth and 2uniformly convex Banach space, C be a nonempty closed convex subset of E. Let A be an αinversestrongly monotone mapping of C into E* satisfying Ay ≤ Ay  Au, ∀y ∈ C and u ∈ VI(A, C) ≠ Ø. Let B : C → E* be a continuous and monotone mapping and f : C × C → ℝ be a bifunction satisfying the conditions (A 1)  (A 4), and φ. C → ℝ be a lower semicontinuous and convex function. Let T. C → C be a closed and relatively asymptotically nonexpansive mapping with the sequencesuch thatas n → ∞ and S. C →C be a closed and relatively asymptotically nonexpansive mapping with the sequencesuch thatas n → ∞. Assume that T and S are uniformly asymptotically regular on C and Ω := F(T) ∩ F(S) ∩ VI(A, C) ∩ GMEP(f, B,φ) ≠ ∅. Let {x_{ n } } be the sequence defined by x_{0} ∈ E and
whereas n → ∞, for each n ≥ 1, M_{ n } = sup{ϕ (z, x_{ n }). z ∈ Ω} for each n ≥ 1, {α_{ n } } and {β_{ n } } are sequences in [0, 1], {λ_{ n } } ⊂ [a, b] for some a, b with 0 < a < b < c^{2}α/ 2, whereis the 2uniformly convexity constant of E and {r_{ n } } ⊂ [d, ∞) for some d > 0. Suppose that the following conditions are satisfied:

(i)
lim inf_{n→∞}(1  α_{ n } ) > 0;

(ii)
lim inf_{n→∞}(1  β_{ n } ) > 0.
Then, the sequence {x_{ n } } converges strongly to Π_{Ω}x_{0}, where Π_{Ω}is generalized projection of E onto Ω.
Since every closed relatively nonexpansive mapping is asymptotically quasiϕnonexpansive, we obtain the following corollary.
Corollary 3.5. Let E be a uniformly smooth and 2uniformly convex Banach space, C be a nonempty closed convex subset of E. Let A be an αinversestrongly monotone mapping of C into E* satisfying Ay ≤ Ay Au, ∀y ∈ C and u ∈ VI(A, C) ≠ Ø. Let B : C → E* be a continuous and monotone mapping and f : C × C → ℝ be a bifunction satisfying the conditions (A 1)  (A 4), and φ : C → ℝ be a lower semicontinuous and convex function. Let T, S : C → C be closed relatively nonexpansive mappings such that Ω := F(T) ∩ F(S) ∩ VI(A, C) ∩ GMEP(f, B,φ) ≠ ∅. Let {x_{ n } } be the sequence defined by x_{0} ∈ E and
where {α_{ n } } and {β_{ n } } are sequences in [0, 1], {λ_{ n } } ⊂ [a, b] for some a, b with 0 < a < b < c^{2}α/2, whereis the 2uniformly convexity constant of E and {r_{ n } } ⊂ [d, ∞) for some d > 0. Suppose that the following conditions are satisfied:

(i)
lim inf_{n→∞}(1  α_{ n } ) > 0,

(ii)
lim inf_{n→∞}(1  β_{ n } ) > 0.
Then, the sequence {x_{ n } } converges strongly to Π_{Ω}x_{0}, where Π_{Ω}is generalized projection of E onto Ω.
Corollary 3.6. Let E be a uniformly smooth and 2uniformly convex Banach space, C be a nonempty closed convex subset of E. Let A be an αinversestrongly monotone mapping of C into E* satisfying Ay ≤ Ay  Au, ∀y ∈ C and u ∈ VI(A, C) ≠ Ø. Let B : C → E* be a continuous and monotone mapping and f : C × C → ℝ be a bifunction satisfying the conditions (A 1)  (A 4), and φ : C → ℝ be a lower semicontinuous and convex function. Let T, S : C → C be a closed quasiϕnonexpansive mappings Ω := F(T) ∩ F(S) ∩ VI(A, C) ∩ GMEP(f, B,φ) ≠ ∅. Let {x_{ n } } be the sequence defined by x_{0} ∈ E and
where {α_{ n } } and {β_{ n } } are sequences in [0, 1], {λ_{ n } } ⊂ [a, b] for some a, b with 0 < a < b < c^{2}α/2, where is the 2uniformly convexity constant of E and {r_{ n } } ⊂ [d, ∞) for some d > 0. Suppose that the following conditions are satisfied:

(i)
lim inf_{n→∞}(1  α_{ n } ) > 0;

(ii)
lim inf_{n→∞}(1  β_{ n } ) > 0.
Then, the sequence {x_{ n } } converges strongly to Π_{Ω}x_{0}, where Π_{Ω} is generalized projection of E onto Ω
Proof. Since every closed quasiϕnonexpansive mapping is asymptotically quasiϕnonexpansive, the result is implied by Theorem 3.1.
Corollary 3.7. Let E be a uniformly smooth and 2uniformly convex Banach space, C be a nonempty closed convex subset of E. Let A be an αinversestrongly monotone mapping of C into E* satisfying Ay ≤ Ay  Au, ∀y ∈ C and u ∈ VI(A, C) ≠ Ø. Let B : C → E* be a continuous and monotone mapping and f : C × C → ℝ be a bifunction satisfying the conditions (A 1)  (A 4), and φ : C → ℝ be a lower semicontinuous and convex function. Let T, S : C → C be closed relatively nonexpansive mappings such that Ω := F(T) ∩ F(S) ∩ VI(A, C) ∩ GMEP(f, B,φ) ≠ Ø. Let {x_{ n } } be the sequence defined by x_{0} ∈ E and
where {α_{ n } } and {β_{ n } } are sequences in [0, 1], {λ_{ n } } ⊂ [a, b] for some a, b with 0 < a < b < c^{2}α/2, whereis the 2uniformly convexity constant of E and {r_{ n } } ⊂ [d, ∞) for some d > 0. Suppose that the following conditions are satisfied:

(i)
lim inf_{n→∞}(1  α_{ n } ) > 0;

(ii)
lim inf_{n→∞}(1  β_{ n } ) > 0.
Then, the sequence {x_{ n } } converges strongly to Π_{Ω}x_{0}, where Π_{Ω}is generalized projection of E onto Ω.
Proof. Since every closed relatively nonexpansive mapping is quasiϕnonexpansive, the result is implied by Theorem 3.1.
Remark 3.8. Corollaries 3.7, 3.6 and 3.7 improve and extend the corresponding results of Saewan et al. [[51], Theorem 3.1] in the sense of changing the closed relatively quasinonexpansive mappings to be the more general than the closed and asymptotically quasiϕnonexpansive mappings and adjusting a problem from the classical equilibrium problem to be the generalized equilibrium problem.
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Acknowledgements
The authors would like to thank Prof. Jong Kyu Kim and the anonymous referees for their respective helpful discussions and suggestions in preparation of this article. This research was supported by grant under the program Strategic Scholarships for Frontier Research Network for the Joint Ph.D. Program of Thai Doctoral degree from the Office of the Higher Education Commission, Thailand. Moreover, the first author was also supported by the King Mongkuts Daimond scholarship for Ph.D. program at King Mongkuts University of Technology Thonburi (KMUTT), under project NRUCSEC no.54000267, and the second author was supported by the Higher Education Commission and the Thailand Research Fund under Grant MRG5380044. Furthermore, this work was partially supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission.
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SS designed and performed all the steps of proof in this research and also wrote the paper. PK participated in the design of the study and suggest many good ideas that made this paper possible and helped to draft the first manuscript. All authors read and approved the final manuscript.
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Saewan, S., Kumam, P. The shrinking projection method for solving generalized equilibrium problems and common fixed points for asymptotically quasiϕnonexpansive mappings. Fixed Point Theory Appl 2011, 9 (2011). https://doi.org/10.1186/1687181220119
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DOI: https://doi.org/10.1186/1687181220119