Recently, common solution problems (i.e., to find a common element of the set of solutions of equilibrium problems and/or the set of fixed points of mappings and/or the set of solutions of variational inequalities) with their applications have been discussed. Some authors such as in references 1234567 presented various iterative schemes and showed some strong or weak convergence theorems on common solution problems in Hilbert spaces. In 2008-2009, Takahashi and Zembayashi 89 introduced several iterative sequences on finding a common solution of an equilibrium problem and a fixed-point problem for a relatively nonexpansive mapping, and established some strong or weak convergence theorems. In 2010, Chang et al. 10 discussed the common solution of a generalized equilibrium problem and a common fixed-point problem for two relatively nonexpansive mappings, and established a strong convergence theorem on the common solution problem. The frameworks of spaces in 8910 are the uniformly smooth and uniformly convex Banach spaces. Chang et al. 11 established a strong convergence theorem on solving the common fixed-point problem for a family of uniformly quasi-ϕ-asymptotically nonexpansive and uniformly Lipschitz continuous mappings in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. Some other problems such as optimization problems (e.g. see 146) and common zero-point problems (e.g. see 10) are closely related to common solution problems.

Throughout this paper, unless other stated, ℝ and are denoted by the set of the real numbers and the set {1, 2,..., N}, respectively, where N is any given positive integer. Let E be a real Banach space with the norm || · ||, E* be the dual of E, and 〈·,·〉 be the pairing between E and E*. Suppose that C is a nonempty closed convex subset of E.

Let be N mappings and be N bifunctions. For each , the generalized equilibrium problem for f_k and A_k is to seek such that

The common solution problem (P1) of generalized equilibrium problems for and is to seek an element in , where and G(k) is the set of solutions of (1.1). We write G instead of in the case of N = 1.

Let be a family of mappings. The common fixed-point problem (P2) for is to seek an element in , where and F (S_i) is the set of fixed points of S_i.

Motivated by the works in 891011, in this paper we will produce a new iterative sequence approximating a common solution of (P1) and (P2) (i.e., some point belonging to ), and show a strong convergence theorem in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property, where in (P2) is a family of uniformly quasi-ϕ-asymptotically nonexpansive mappings and for each i ≥ 1, S_i is locally uniformly Lipschitz continuous or uniformly Hölder continuous with order Θ_i .

Let E be a real Banach space, and {x_n} be a sequence in E. We denote by x_n → x and x_n ⇀ x the strong convergence and weak convergence of {x_n}, respectively. The normalized duality mapping J : E → 2^E* is defined by

By the Hahn-Banach theorem, Jx ≠ ∅ for each x ∈ E.

A Banach space E is said to be strictly convex if for all x, y ∈ U = {u ∈ E : ||u|| = 1} with x ≠ y; to be uniformly convex if for each ε ∈ (0, 2], there exists γ > 0 such that for all x, y ∈ U with ||x - y|| ≥ ε; to be smooth if the limit

exists for every x, y ∈ U; to be uniformly smooth if the limit (2.1) exists uniformly for all x, y ∈ U.

Remark 2.1. The basic properties below hold (see 12).

(i) If E is a real uniformly smooth Banach space, then J is uniformly continuous on each bounded subset of E.

(ii) If E is a strictly convex reflexive Banach space, then J^-1 is hemicontinuous, that is, J^-1 is norm-to-weak*-continuous.

(iii) If E is a smooth and strictly convex reflexive Banach space, then J is single-valued, one-to-one and onto.

(iv) Each uniformly convex Banach space E has the Kadec-Klee property, that is, for any sequence {x_n} ⊂ E, if x_n ⇀ x ∈ E and ||x_n|| → ||x||, then x_n → x.

(v) A Banach space E is uniformly smooth if and only if E* is uniformly convex.

(vi) A Banach space E is strictly convex if and only if J is strictly monotone, that is,

(vii) Both uniformly smooth Banach spaces and uniformly convex Banach spaces are reflexive.

Now let E be a smooth and strictly convex reflexive Banach space. As Alber 13 and Kamimura and Takahashi 14 did, the Lyapunov functional ϕ : E × E → ℝ⁺ is defined by

It follows from 15 that ϕ(x, y) = 0 if and only if x = y, and that

Further suppose that C is a nonempty closed convex subset of E. The generalized projection (see 13) Π_C: E→C is defined by for each x ∈ E,

A mapping A : C → E* is said to be δ-inverse-strongly monotone, if there exists a constant δ > 0 such that

A mapping S : C → C is said to be closed if for each {x_n} ⊂ C, x_n → x and Sx_n → y imply Sx = y; to be quasi-ϕ-asymptotically nonexpansive (see 16) if F(S) ≠ ∅, and there exists a sequence {l_n} ⊂ [1, ∞) with l_n → 1 such that

It is easy to see that if A : C → E* is δ-inverse-strongly monotone, then A is -Lipschitz continuous. The class of quasi-ϕ-asymptotically nonexpansive mappings contains properly the class of relatively nonexpansive mappings (see 17) as a subclass.

Definition 2.1 (see 11). Let be a sequence of mappings. is said to be a family of uniformly quasi-ϕ-asymptotically nonexpansive mappings, if and there exists a sequence {l_n} ⊂ [1, ∞) with l_n → 1 such that for each i ≥ 1,

Now we introduce the following concepts.

Definition 2.2. A mapping S : C → C is said

(1) to be locally uniformly Lipschitz continuous if for any bounded subset D in C, there exists a constant L_D > 0 such that

(2) to be uniformly Hölder continuous with order Θ (Θ > 0) if there exists a constant L > 0 such that

Remark 2.2. It is easy to see that any uniformly Lipschitz continuous mapping (see 11) is locally uniformly Lipschitz continuous, and is also uniformly Hölder continuous with order Θ = 1. However, the converse is not true.

Example 2.1. Suppose that S : ℝ → ℝ is defined by

Then S is locally uniformly Lipschitz continuous. In fact, for any bounded subset D in ℝ, setting M = 1 + sup{|x| : x ∈ D}, we have |Sⁿx - Sⁿy| ≤ 2M |x - y|, x, y ∈ D, ∀n ≥ 1. But S fails to be uniformly Lipschitz continuous.

Example 2.2. Suppose that S : ℝ - ℝ is defined by

S is uniformly Hölder continuous with order , since , ∀x, y ∈ ℝ, ∀n ≥ 1. But S fails to be uniformly Lipschitz continuous.

Lemma 2.1 (see 1314). If C is a nonempty closed convex subset of a smooth and strictly convex reflexive Banach space E, then

(1) ϕ(x, Π_C(y)) + ϕ(Π_C(y), y) ≥ ϕ(x, y), ∀x ∈ C, y ∈ E;

(2) for × ∈ E and u ∈ C, one has

□

Lemma 2.2. Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property, {x_n} and{y_n} be two sequences of E, and . If and ϕ(x_n, y_n) → 0, then .

Proof. We complete this proof by two steps.

Step 1. Show that there exists a subsequence of {y_n} such that .

In fact, since ϕ(x_n, y_n) → 0, by (2.2) we have ||x_n|| - ||y_n|| → 0. It follows from that

and so

Then {Jy_n} is bounded in E*. It follows from Remark 2.1(v) and (vii) that E* is reflexive. Hence there exist a point f₀ ∈ E* and a subsequence of {Jy_n} such that

It follows from Remark 2.1(vii) and (iii) that there exists a point x ∈ E such that Jx = f₀. By the definition of ϕ, we obtain

By weak lower semicontinuity of norm || · ||, we have

which implies that and . It follows from Remark 2.1(iv) and (v) that E* has the Kadec-Klee property, and so by (2.4) and (2.5). By Remark 2.1(vii) and (ii), we have , which implies that by (2.3) and the Kadec-Klee property of E.

Step 2. Show that .

In fact, suppose that . For some given number ε₀ > 0, there exists a positive integer sequence {n_k} with n₁ < n₂ < · · · < n_k < · · ·, such that

Replacing {y_n} by in Step 1, there exists a subsequence of such that , which contradicts (2.6). □

Lemma 2.3. Let C be a nonempty closed convex subset of a smooth and strictly convex reflexive Banach space E, and let A : C → E* be a δ-inverse-strongly monotone mapping and f : C × C → ℝ be a bifunction satisfying the following conditions

(B₁) f(z, z) = 0, ∀z ∈ C;

(B₂) ;

(B₃) for any z ∈ C, the function y α f(z, y) is convex and lower semicontinuous;

(B₄) for some β ≥ 0 with β ≤ δ,

Then the following conclusions hold:

(1) For any r > 0 and u ∈ E, there exists a unique point z ∈ C such that

(2) For any given r > 0, define a mapping K_r : E → C as follows: ∀u ∈ E,

We have (i) F(K_r) = G and G is closed convex in C, where

(ii) ϕ(z, K_ru) + ϕ(K_ru, u) ≤ ϕ(z, u), ∀z ∈ F(K_r).

(3) For each n ≥ 1, r_n > a > 0 and u_n ∈ C with , we have

Proof. (1) We consider the bifunction instead of f. It follows from the proof of Lemma 2.5 in 10 that satisfies (B₁)-(B₃). Since A is δ-inverse-strongly monotone, by (B₄), we have

which implies is monotone. By Blum amd Oettli 18, for any r > 0 and u ∈ E, there exists z ∈ C such that (2.7) holds. Next we show that (2.7) has a unique solution. If for any given r > 0 and u ∈ E, z₁ and z₂ are two solutions of (2.7), then

and

Adding these two inequalities, we have

It follows from (2.8) that

which implies that z₁ = z₂ by Remark 2.1(vi).

(2) Since satisfies (B₁)-(B₃) and is monotone, the conclusion (2) follows from Lemmas 2.8 and 2.9 in 9.

(3) Since

we have

by the monotonicity of . It follows from . r_n > a > 0 and Remark 2.1(i) that

Since is convex and lower semicontinuous, it is also weakly lower semicontinuous. Letting n → ∞ in (2.9), we have , ∀y ∈ C. For any t ∈ (0, 1] and y ∈ C, setting , we have y_t ∈ C and , which together with (B₁) implies that

Thus f(y_t, y) + 〈y - y_t, Ay_t〉 ≥ 0, ∀y ∈ C, ∀t ∈ (0, 1]. Letting t ↓ 0, since z α f(z, y) + 〈y - z, Az〉 satisfies (B₂), we have , ∀y ∈ C.

Remark 2.3. If β = 0 in (B₄), that is, f is monotone, then the conclusions (1) and (2) in Lemma 2.3 reduce to the relating results of Lemmas 2.5 and 2.6 in 10, respectively.

Next we give an example to show that there exist the mapping A and the bifunction f satisfying the conditions of Lemma 2.3. However, f is not monotone.

Example 2.3. Define A : ℝ → ℝ and f : ℝ × ℝ → ℝ by ∈ ∀x ∈ ℝ and , ∀(x, y) ∈ ℝ × ℝ, respectively. It is easy to see that A is -inverse-strongly monotone, f satisfies (B₁)-(B₃), and , ∀(x, y) : ℝ × ℝ with .

Lemma 2.4 (see 12). Let C be a nonempty closed convex subset of a real uniformly smooth and strictly convex Banach space E with the Kadec-Klee property, S : C → C be a closed and quasi-ϕ-asymptotically nonexpansive mapping with a sequence {l_n} ⊂ [1, ∞), l_n → 1. Then F(S) is closed convex in C.

Lemma 2.5 (see 11). Let E be a uniformly convex Banach space, η > 0 be a positive number and B_η(0) be a closed ball of E. Then, for any given sequence and for any given with , there exists a continuous, strictly increasing and convex function g : [0, 2η) → [0, ∞) with g(0) = 0 such that for any positive integers i, j with i < j,

□

In this section, let C be a nonempty closed convex subset of a real uniformly smooth and strictly convex Banach space E with the Kadec-Klee property.

Theorem 3.1. Suppose that

(C₁) for each , the mapping A_k : C → E* is δ_k-inverse-strongly monotone, the bifunction f_k : C × C → ℝ satisfies (B₁)-(B₃), and for some β_k ≥ 0 with β_k ≤ δ_k,

(C₂) is a family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {l_n} ⊂ [1, ∞), l_n → 1;

(C₃) for each i ≥ 1, S_i is either locally uniformly Lipschitz continuous or uniformly Hölder continuous with order Θ_i (Θ_i > 0), and is bounded in C.

(C₄) . Take the sequence generated by

where for each , with some a > 0, , and . If , ∀n ≥ 0 and lim inf_n→∞ α_n,0 α_{n, i} > 0, ∀i ≥ 1, then .

Proof. We shall complete this proof by seven steps below.

Step 1. Show that , , H_n and W_n for all n ≥ 0 are closed convex.

In fact, is closed convex since for each i ≥ 1, F(S_i) is closed convex by (C₂) and Lemma 2.4. is closed convex since for each , G(k) is closed convex by (C₁) and Lemma 2.3(2)(i). H₀ = C is closed convex. Since ϕ(v,u_N,n) ≤ ϕ(v,x_n) + ξ_n is equivalent to

we know that H_n(n ≥ 0) are closed convex. Finally, W_n is closed convex by its definition. Thus and are well defined.

Step 2. Show that {x_n} and are bounded.

From , ∀n ≥ 0 and Lemma 2.1(1), we have

which implies that {ϕ(x_n, x₀)} is bounded, and so is {x_n} by (2.2). It follows from (C₂) that for all , i ≥ 1, n ≥ 1,

Hence for all i ≥ 1, is uniformly bounded, and so is by (2.2). Obviously,

Step 3. Show that , ∀n ≥ 0.

Since Banach space E is uniformly smooth, E* is uniformly convex, by Remark 2.1(v). For any given , any n ≥ 1 and any positive integer j, by (C₂) and Lemma 2.5, we have

Put , , ∀n ≥ 0. It follows from (3.3) and Lemma 2.3(2)(ii) that

which implies that if , then p ∈ H_n, ∀n ≥ 0. Hence, , ∀n ≥ 0. By induction, now we prove that , ∀n ≥ 0. In fact, it follows from W₀ = C that . Suppose that for some m ≥ 0. By the definition of and Lemma 2.1(2), we have

and so

which shows z ∈ W_m+1, so .

Step 4. Show that there exists such that .

Without loss of generalization, we can assume that , since {x_n} is bounded and E is reflexive. Moreover, it follows that , ∀n ≥ 0 from H_n+1 ∩ W_n+1 ⊂ H_n ∩ W_n and the closeness and convexity of H_n ∩ W_n. Noting that

we have

by (3.1). It follows that

and so by . Hence,

by the Kadec-Klee property of E, and so

by Remark 2.1(i).

Step 5. Show that .

Since x_n+1 ∈ C, setting u = x_n+1 in (3.1), we have

By (3.5),

By x_n+1 ∈ H_n+1, (3.2) and (3.8), we have

which together with (3.6) and Lemma 2.2 implies that

For any j ≥ 1 and any given , it follows from (3.2)-(3.4) and (3.9) that

which implies that

since , ∀i ≥ 1. We obtain

since g(0) = 0 and g is strictly increasing and continuous. By (3.7) and (3.11), we have and for all j ≥ 1. It follows from Remark 2.1(ii) that , which implies

by the uniform boundedness of and the Kadec-Klee property of E. Thus

By (C₃) and (3.6), we have

Hence, for each j ≥ 1,

By (3.12) and the closeness of S_j, we have for all j ≥ 1 and so .

Step 6. Show that .

In fact, it is easy to see that for each , and , the sequence {ϕ(p, u_k,n)} is bounded by (3.2), (3.4) and the boundedness of {x_n} and , which implies that {u_k,n} is bounded in C by (2.2). Since , by (3.2), (3.3), (3.5) and (3.10), we have

It follows from Lemma 2.2 that

Furthermore, it follows from (3.4) and Lemma 2.3(2)(ii) that for any given ,

which implies

by Remark 2.1(i), (3.9) and (3.13). Then by (3.13) and Lemma 2.2. Similarly, we also obtain . Hence, together with (3.9) and (3.13), for each ,

For each , since , we have

which together with (3.14) and Lemma 2.3(3) implies that , ∀y ∈ C. Therefore and so .

Step 7. Show that .

In fact, letting , by and , we have

It follows from (3.6) that

Hence, , and so . □

Setting N = 1, u₀,_n = y_n and u_N,n = u_n in Theorem 3.1, we can obtain the following result.

Corollary 3.1 Suppose that

(D₁) the mapping A : C → E* is a mapping with δ -inverse-strongly monotone, the bifunction f : C × C → ℝ satisfies (B₁)-(B₃) and for some β > 0 with β ≤ δ,

(D₂) both (C₂) and (C₃) hold, and Take the sequence generated by

where , for some a > 0 and . If , ∀_n ≥ 0 and lim inf_n→∞ α_n,0α_n,i > 0, ∀i ≥ 1, then . □

Furthermore, if S_i = S, i ≥ 1 in Corollary 3.1, the following corollary can be obtained immediately.

Corollary 3.2. Suppose that, besides (D1),

(E₁) S : C → C is closed and quasi-ϕ-asymptotically nonexpansive with {l_n} ⊂ [1, ∞), l_n → 1;

(E₂) S is either locally uniformly Lipschitz continuous or uniformly Hölder continuous with order Θ (Θ > 0), F(S) is bounded in C and F(S) ∩ G ≠ ∅. Take the sequence generated by

where , for some a > 0 and ξ = sup_u∈F(S)(l_n -1)ϕ(u, x_n) . If lim inf_n→∞ α_n(1- α_n) > 0, then . □

The authors declare that they have no completing interests.

All the authors read and approved the final manuscript.