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 real-valued function, and B : C → E* be a mapping. The generalized mixed equilibrium problem, is to find x ∈ C such that

(1.1)

The set of solutions to (1.1) is denoted by GMEP(f, B, φ), i.e.,

(1.2)

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

(1.3)

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

(1.4)

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

(1.5)

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

(1.6)

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

(1.7)

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

(1.8)

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 p-uniformly convex if there exists a constant c > 0 such that δ (ε) ≥ cεp for all ε ∈ [0, 2]; see [21, 22] for more details. Observe that every p-uniformly convex is uniformly convex. One should note that no Banach space is p-uniformly convex for 1 < p < 2. It is well known that a Hilbert space is 2-uniformly convex, uniformly smooth. For each p > 1, the generalized duality mapping J p : E → 2E*is defined by

for all x ∈ E. In particular, J = J2 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

  1. (i)

    monotone if

  2. (ii)

    α-inverse-strongly monotone if there exists a constant α > 0 such that

The class of inverse-strongly 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 quasi-nonexpansive 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 quasi-nonexpansive if F(T) ≠ ∅ and there exists a sequence {k n } ⊂ [1, ∞) with k n → 1 as n → ∞ such that

T is called uniformly L-Lipschitzian 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

(1.9)

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:

(1.10)

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. (1)

    (||y|| - ||x||)2 ≤ ϕ(y, x) ≤ (||y|| + ||x||)2 for all x, y ∈ E.

  2. (2)

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

  3. (3)

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

  4. (4)

    If E is a reflexive, strictly convex and smooth Banach space, then, for all x, y ∈ E,

By the Hahn-Banach 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 norm-to-norm 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 → 2E*is single-valued, one-to-one 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 limn →∞||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 limn→∞||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 hemi-relatively 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 hemi-relatively nonexpansive. Moreover, the class of relatively asymptotically nonexpansive is more general than the class of relatively nonexpansive mappings.

We note that hemi-relatively nonexpansive mappings are sometimes called quasi-ϕ-nonexpansive mappings.

We recall the following :

  1. (i)

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

  2. (ii)

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

  3. (iii)

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

  4. (iv)

    T : C → C is said to be asymptotically quasi-ϕ-nonexpansive [43] if F(T) ≠ ∅ and there exists a sequence {k n } ⊂ [0, ∞) with k n → 1 as n → ∞ such that ϕ(p, Tnx) ≤ k n ϕ (p, x) for all x ∈ C, p ∈ F(T) and n ≥ 1.

Remark 1.3. (i) The class of (asymptotically) quasi-ϕ-nonexpansive mappings is more general than the class of relatively (asymptotically) nonexpansive mappings, which requires the strong restriction .

  1. (ii)

    In real Hilbert spaces, the class of (asymptotically) quasi-ϕ-nonexpansive mappings is reduced to the class of (asymptotically) quasi-nonexpansive 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 limn→∞x n = x0 and limn→∞Tx n = y0, then Tx0 = y0.

We give some examples which are closed and asymptotically quasi-ϕ-nonexpansive.

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-10 is nonempty. Then, J r = (J + rU)-1J is a closed and asymptotically quasi-ϕ-nonexpansive mapping from E onto D(U) and F(J r ) = U-10.

(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 quasi-ϕ-nonexpansive 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 supn→∞< 1. Suppose that {x n } is given by

(1.11)

where J is the duality mapping on E. If F(T) is nonempty, then {x n } converges strongly to P F (T) x , where PF(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 inverse-strongly monotone operator A in a 2-uniformly convex and uniformly smooth Banach space E : x1 = x ∈ C and

(1.12)

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 quasi-ϕ-nonexpansive 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 quasi-ϕ-nonexpansive mapping with the sequence such that as n → ∞ and S : C → C a closed and asymptotically quasi-ϕ-nonexpansive 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:

(1.13)

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 :

  1. (a)

    β n + γ n + δ n = 1, ∀n ≥ 1;

  2. (b)

    lim infn→∞ γ n δ n , limn→∞ β n = 0;

  3. (c)

    0 ≤ α n < 1 and lim supn→∞ 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 α-inverse-strongly 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

(1.14)

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

  1. (a)

    lim infn →∞ α n (1 - α n ) > 0;

  2. (b)

    lim infn →∞ β n (1 -β n ) > 0;

then, {x n } converges strongly to ΠΩx0, where ΠΩ 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 quasi-ϕ-nonexpansive mappings in a uniformly smooth and strictly convex Banach space which has the Kadec-Klee 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 quasi-ϕ-nonexpansive 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 Kadec-Klee 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 2-uniformly 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 p-uniformly convex constant of E.

Lemma 2.3. ([62]). If E be a p-uniformly 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 p-uniformly 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 quasi-ϕ-nonexpansive 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 limn→∞ϕ(x n , y n ) = 0, then limn→∞||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, x0 = 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, one-to-one, surjective, and it is the duality mapping from E* into E. We make use of the following mapping V studied in Alber [31]

(2.1)

for all x ∈ E and x* ∈ E*; that is, V (x, x*) = ϕ(x, J-1x*).

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 α-inverse 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 supt↓0f (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. (1)

    T r is single-valued;

  2. (2)

    T r is a firmly nonexpansive-type 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 semi-continuous 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:

(2.3)

for all x ∈ C. Then, the following conclusions holds:

  1. (a)

    K r is single-valued ;

  2. (b)

    K r is a firmly nonexpansive-type mapping, i.e.;

  3. (c)

    ;

  4. (d)

    GMEP(f, B, φ) is a closed convex,

  5. (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 quasi-ϕ-nonexpansive.

Let C be a nonempty closed convex subset of a Banach space E and let A be an inverse-strongly 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,

(2.4)

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:

(2.5)

Then, U is maximal monotone and U-10 = 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 α-inverse strongly monotone mapping and the set of common fixed points for a pair of asymptotically quasi-ϕ-nonexpansive mappings in Banach spaces.

Theorem 3.1. Let E be a uniformly smooth and 2-uniformly convex Banach space, C be a nonempty closed convex subset of E. Let A be an α-inverse-strongly 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 semi-continuous and convex function. Let T : C → C be a closed and asymptotically quasi-ϕ-nonexpansive mapping with the sequencesuch thatas n → ∞ and S : C → C be a closed and asymptotically quasi-ϕ-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 x0 ∈ E and

(3.1)

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 < c2α/2, where is the 2-uniformly convexity constant of E and {r n } ⊂ [d, ∞) for some d > 0. Suppose that the following conditions are satisfied: lim infn→∞(1 -α n ) > 0 and lim infn→∞(1 -β n ) > 0. Then, the sequence {x n } converges strongly to ΠΩx0, where ΠΩ is generalized projection of E onto Ω.

Proof. We have several steps to prove this theorem as follows:

Step 1. We first show that Cn+1is closed and convex for each n ≥ 1. Indeed, it is obvious that C1 = C is closed and convex. Suppose that C i is closed and convex for each i ∈ ℕ. Next, we prove that Ci+1is closed and convex. For any z ∈ Ci+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, Ci+1is 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-10 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, ΠΩ 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 ⊂ C1 = 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

(3.2)

Thus, q ∈ VI(A, C) and A is α-inverse-strongly monotone, we have

(3.3)

From Lemma 2.2 and ||Ay|| ≤ ||Ay - Au|| for all y ∈ C and q ∈ Ω, we obtain

(3.4)

Substituting (3.3) and (3.4) into (3.2), we have

(3.5)

As Ti is asymptotically quasi-ϕ-nonexpansive mapping, we also have

(3.6)

It follows that

(3.7)

This shows that q ∈ Ci+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

(3.8)

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 , x0) is bounded, and so are {x n }, {w n }, {y n }, {z n }, {Tnw n } and {Snx n }.

Step 4. We show that {x n } is a Cauchy sequence.

Since and , we have

This implies that {ϕ(x n , x0)} is nondecreasing, and limn →∞ϕ(x n , x0) exists.

For m > n and from Lemma 2.1, we have

(3.9)

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

(3.10)

Since , from definition of Cn+1, we have

(3.11)

and from (3.5) and (3.6), we also have

(3.12)

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

(3.13)

and by using triangle inequality, we have

(3.14)

Since J is uniformly norm-to-norm continuous, we also have

(3.15)

and

(3.16)

Since , and from (3.7), we have

(3.17)

Since ||x n - u n || → 0 and J is uniformly continuous, we have

(3.18)

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

(3.19)

Since J is uniformly norm-to-norm continuous, we also have

(3.20)

Again from (3.1) and (3.16), we have

(3.21)

This implies that ||JTnw n - Jx n || → 0. Again since J-1 is uniformly norm-to-norm continuous, we also have

(3.22)

For p ∈ Ω, we note that

(3.23)

It follows from (3.22) and x n → p as n → ∞, that

(3.24)

On other hand, we have

Since T is uniformly asymptotically regular and from (3.24), we obtain that

(3.25)

Thai is, TTnwn → 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

(3.26)

From (3.18) and lim infn→∞(1 -β n ) > 0, obtain that

(3.27)

From Lemma 2.1, Lemma 2.8 and (3.4), we compute

Applying Lemma 2.6 and (3.27) that

(3.28)

Since J is uniformly norm-to-norm continuous on bounded sets, by (3.28), we have

(3.29)

From(3.1), (3.20) and (ii), we have

(3.30)

Since J-1 is uniformly norm-to-norm continuous on bounded sets

(3.31)

We observe that

(3.32)

It follows from (3.31) and x n → p as n → ∞, we obtain

(3.33)

On other hand, we have

Since S is uniformly asymptotically regular and (3.33), we obtain that

(3.34)

that is, SSnz 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

(3.35)

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 set-valued U : E ⇉ E* by Lemma 2.14, U is maximal monotone and U-10 = 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

(3.36)

On the other hand, since . Then from Lemma 2.7, we have

and thus

(3.37)

It follows from (3.36) and (3.37) that

where M = supn≥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-10 and hence p ∈ VI(A, C). Hence, by (a), (b) and (c), we obtain p ∈ Ω.

Step 5. Finally, we prove that p = ΠΩx0. Taking the limit as n → ∞ in (3.8), we obtain that

and hence, p = ΠΩx0 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 2-uniformly convex Banach space, and C be a nonempty closed convex subset of E. Let A be an α-inverse-strongly 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 semi-continuous and convex function. Let T : C → C be a closed and asymptotically quasi-ϕ-nonexpansive mapping with the sequencesuch thatas n → ∞ and S : C → C be a closed and asymptotically quasi-ϕ-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) ∩ MEP(f, φ) ≠ ∅. Let {x n } be the sequence defined by x0 ∈ E and

(3.38)

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 < c2α/2, where is the 2-uniformly convexity constant of E and {r n } ⊂ [d, ∞) for some d > 0. Suppose that the following conditions are satisfied:

  1. (i)

    lim infn→∞(1 -α n ) > 0,

  2. (ii)

    lim infn→∞(1 -β n ) > 0.

Then, the sequence {x n } converges strongly to ΠΩx0, where ΠΩ 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 2-uniformly convex Banach space, C be a nonempty closed convex subset of E. Let A be an α-inverse-strongly 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 semi-continuous and convex function. Let T : C → C be a closed and asymptotically quasi-ϕ-nonexpansive mapping with the sequencesuch thatas n → ∞ and S : C → C be a closed and asymptotically quasi-ϕ-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) ∩ MVI(B, C) ≠ ∅. Let {x n } be the sequence defined by x0 ∈ E and

(3.39)

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 < c2α/2, where is the 2-uniformly convexity constant of E and {r n } ⊂ [d, ∞) for some d > 0. Suppose that the following conditions are satisfied:

  1. (i)

    lim infn→∞(1 -α n ) > 0;

  2. (ii)

    lim infn→∞(1 -β n ) > 0.

Then, the sequence {x n } converges strongly to ΠΩx0, where ΠΩ 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 quasi-ϕ-nonexpansive, we obtain the following corollary.

Corollary 3.4. Let E be a uniformly smooth and 2-uniformly convex Banach space, C be a nonempty closed convex subset of E. Let A be an α-inverse-strongly 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 semi-continuous 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 x0 ∈ E and

(3.40)

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 < c2α/ 2, whereis the 2-uniformly convexity constant of E and {r n } ⊂ [d, ∞) for some d > 0. Suppose that the following conditions are satisfied:

  1. (i)

    lim infn→∞(1 - α n ) > 0;

  2. (ii)

    lim infn→∞(1 - β n ) > 0.

Then, the sequence {x n } converges strongly to ΠΩx0, where ΠΩis generalized projection of E onto Ω.

Since every closed relatively nonexpansive mapping is asymptotically quasi-ϕ-nonexpansive, we obtain the following corollary.

Corollary 3.5. Let E be a uniformly smooth and 2-uniformly convex Banach space, C be a nonempty closed convex subset of E. Let A be an α-inverse-strongly 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 semi-continuous 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 x0 ∈ E and

(3.41)

where {α n } and {β n } are sequences in [0, 1], {λ n } ⊂ [a, b] for some a, b with 0 < a < b < c2α/2, whereis the 2-uniformly convexity constant of E and {r n } ⊂ [d, ∞) for some d > 0. Suppose that the following conditions are satisfied:

  1. (i)

    lim infn→∞(1 - α n ) > 0,

  2. (ii)

    lim infn→∞(1 - β n ) > 0.

Then, the sequence {x n } converges strongly to ΠΩx0, where ΠΩis generalized projection of E onto Ω.

Corollary 3.6. Let E be a uniformly smooth and 2-uniformly convex Banach space, C be a nonempty closed convex subset of E. Let A be an α-inverse-strongly 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 semi-continuous and convex function. Let T, S : C → C be a closed quasi-ϕ-nonexpansive mappings Ω := F(T) ∩ F(S) ∩ VI(A, C) ∩ GMEP(f, B,φ) ≠ ∅. Let {x n } be the sequence defined by x0 ∈ E and

(3.42)

where {α n } and {β n } are sequences in [0, 1], {λ n } ⊂ [a, b] for some a, b with 0 < a < b < c2α/2, where is the 2-uniformly convexity constant of E and {r n } ⊂ [d, ∞) for some d > 0. Suppose that the following conditions are satisfied:

  1. (i)

    lim infn→∞(1 - α n ) > 0;

  2. (ii)

    lim infn→∞(1 - β n ) > 0.

Then, the sequence {x n } converges strongly to ΠΩx0, where ΠΩ is generalized projection of E onto Ω

Proof. Since every closed quasi-ϕ-nonexpansive mapping is asymptotically quasi-ϕ-nonexpansive, the result is implied by Theorem 3.1.

Corollary 3.7. Let E be a uniformly smooth and 2-uniformly convex Banach space, C be a nonempty closed convex subset of E. Let A be an α-inverse-strongly 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 semi-continuous 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 x0 ∈ 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 < c2α/2, whereis the 2-uniformly convexity constant of E and {r n } ⊂ [d, ∞) for some d > 0. Suppose that the following conditions are satisfied:

  1. (i)

    lim infn→∞(1 - α n ) > 0;

  2. (ii)

    lim infn→∞(1 - β n ) > 0.

Then, the sequence {x n } converges strongly to ΠΩx0, where ΠΩis generalized projection of E onto Ω.

Proof. Since every closed relatively nonexpansive mapping is quasi-ϕ-nonexpansive, 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 quasi-nonexpansive mappings to be the more general than the closed and asymptotically quasi-ϕ-nonexpansive mappings and adjusting a problem from the classical equilibrium problem to be the generalized equilibrium problem.