Let E be a real Banach space with the dual E* and C be a nonempty closed convex subset of E. We denote by and the sets of positive integers and real numbers, respectively. Also, we denote by J the normalized duality mapping from E to 2^E* defined by

where 〈·,·〉 denotes the generalized duality pairing. We know that if E is smooth, then J is single-valued and if E is uniformly smooth, then J is uniformly norm-to-norm continuous on bounded subsets of E. We shall still denote by J the single-valued duality mapping. Let be a bifunction and A : C → E* be a nonlinear mapping. We consider the following generalized equilibrium problem (GEP):

The set of such u ∈ C is denoted by GEP (f), i.e.,

Whenever E = H a Hilbert space, the problem (1.1) was introduced and studied by Takahashi and Takahashi 1. Similar problems have been studied extensively recently. In the case of A ≡ 0, GEP (f) is denoted by EP (f). In the case of f ≡ 0, EP is also denoted by VI(C, A). Problem (1.1) is very general in the sense that it includes, as spacial cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others; see, e.g., 23. A mapping T : C → E is called nonexpansive if ||Tx - Ty|| ≤ ||x - y|| for all x, y ∈ C. Denote by F (T ) the set of fixed points of T , that is, F (T ) = {x ∈ C : Tx = x}. A mapping A : C → E* is called α-inverse-strongly monotone, if there exists an α > 0 such that

It is easy to see that if A : C → E* is an α-inverse-strongly monotone mapping, then it is 1/α- Lipschitzian.

In 1953, Mann 4 introduced the following iterative procedure to approximate a fixed point of a nonexpansive mapping T in a Hilbert space H:

where the initial point x₀ is taken in C arbitrarily and {α_n} is a sequence in [0, 1].

However, we note that Manns iteration process (1.2) has only weak convergence, in general; for instance, see 567.

Let C be a nonempty, closed, and convex subset of a Banach space E and {T_n} be sequence of mappings of C into itself such that . Then, {T_n} is said to satisfy the NST-condition if for each bounded sequence {z_n} ⊂ C,

implies , where ω_w(z_n) is the set of all weak cluster points of {z_n}; see 8910.

In 2008, Takahashi et al. 11 has adapted Nakajo and Takahashi's 12 idea to modify the process (1.2) so that strong convergence has been guaranteed. They proposed the following modification for a family of nonexpansive mappings in a Hilbert space: x₀ ∈ H, C₁ = C, and

where 0 ≤ α_n ≤ a < 1 for all . They proved that if {T_n} satisfies the NST-condition, then {u_n} generated by (1.3) converges strongly to a common fixed point of T_n.

Recently, motivated by Nakajo and Takahashi 12 and Xu 13, Matsushita and Takahashi 14 introduced the iterative algorithm for finding fixed points of nonexpansive mappings in a uniformly convex and smooth Banach space: x₀ = x ∈ C and

where denotes the convex closure of the set D, {t_n} is a sequence in (0,1) with t_n → 0, and is the metric projection from E onto C_n ∩ D_n. They proved that {x_n} generated by (1.4) converges strongly to a fixed point of T .

Very recently, Kimura and Nakajo 15 investigated iterative schemes for finding common fixed points of a family of nonexpansive mappings and proved strong convergence theorems by using the Mosco convergence technique in a uniformly convex and smooth Banach space. In particular, they proposed the following algorithm: x₁ = x ∈ C and

where {t_n} is a sequence in (0,1) with t_n → 0 as n → ∞. They proved that if {T_n} satisfies the NST-condition, then {x_n} converges strongly to a common fixed point of T_n.

Motivated and inspired by Nakajo and Takahashi 12, Takahashi et al. 11, Xu 13, Masushita and Takahashi 14, and Kimura and Nakajo 15, we introduce a hybrid projection algorithm for finding a common element in the solution set of a GEP and the common fixed point set of a family of nonexpansive mappings in a Banach space setting.

Let E be a real Banach space and let U = {x ∈ E : ||x|| = 1} be the unit sphere of E. A Banach space E is said to be strictly convex if for any x, y ∈ U,

It is also said to be uniformly convex if for each ε ∈ (0, 2], there exists δ > 0 such that for any x, y ∈ U,

It is known that a uniformly convex Banach space is reflexive and strictly convex. Define a function δ: [0, 2] → [0, 1] called the modulus of convexity of E as follows:

Then, E is uniformly convex if and only if δ(ε) > 0 for all ε ∈ (0, 2]. A Banach space E is said to be smooth if the limit

exists for all x, y ∈ U. Let C be a nonempty, closed, and convex subset of a reflexive, strictly convex and smooth Banach space E. Then, for any x ∈ E, there exists a unique point x₀ ∈ C such that

The mapping P_C : E → C defined by P_C × = x₀ is called the metric projection from E onto C. Let x ∈ E and u ∈ C. Then, it is known that u = P_C × if and only if

for all y ∈ C; see 16 for more details. It is well known that if P_C is a metric projection from a real Hilbert space H onto a nonempty, closed, and convex subset C, then P_C is nonexpansive. However, in a general Banach space, this fact is not true.

In the sequel, we will need the following lemmas.

Lemma 2.1. 17 Let E be a uniformly convex Banach space, {α_n} be a sequence of real numbers such that 0 < b ≤ α_n ≤ c < 1 for all n ≥ 1, and {x_n} and {y_n} be sequences in E such that lim sup_n→∞ ||x_n|| ≤ d, lim sup_n→∞ ||y_n|| ≤ d and lim_n→∞ ||α_nx_n + (1 - α_n)y_n|| = d. Then, lim_n→∞ ||x_n - y_n|| = 0.

Lemma 2.2. 18 Let C be a bounded, closed, and convex subset of a uniformly convex Banach space E. Then, there exists a strictly increasing, convex, and continuous function γ : [0, ∞) → [0, ∞) such thatγ (0) = 0 and

for all , {x₁, x₂,..., x_n} ⊂ C, {λ₁, λ₂,..., λ_n} ⊂ [0, 1] with and nonexpansive mapping T of C into E.

Following Bruck's 19 idea, we know the following result for a convex combination of nonexpansive mappings which is considered by Aoyama et al. 20 and Kimura and Nakajo 15.

Lemma 2.3. 15 Let C be a nonempty, closed, and convex subset of a uniformly convex Banach space E and {S_n} be a family of nonexpansive mappings of C into itself such that . Let be a family of nonnegative numbers with indices n, with k ≤ n such that

(i) for every ;

(ii) for every

and let for all , where {α_n} ⊂ [a, b] for some a, b ∈ (0, 1) with a ≤ b. Then, {T_n} is a family of nonexpansive mappings of C into itself with and satisfies the NST-condition.

Now, let us turn to following well-known concept and result.

Definition 2.4. Let B be a subset of topological vector space X. A mapping G : B → 2^X is called a KKM mapping if for x_i ∈ B and i = 1, 2,..., m, where coA denotes the convex hull of the set A.

Lemma 2.5. 21 Let B be a nonempty subset of a Hausdorff topological vector space × and let G : B → 2^X be a KKM mapping. If G(x) is closed for all × ∈ B and is compact for at least one x ∈ B, then ⋂_x∈BG(x) ≠ ∅.

Motivated by Takahashi and Zembayashi 22, and Ceng and Yao 23, we next prove the following crucial lemma concerning the GEP in a strictly convex, reflexive, and smooth Banach space.

Theorem 3.1. Let C be a nonempty, bounded, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space E, let f be a bifunction from C × C to satisfying (A1)-(A4), where

(A1) f(x, x) = 0 for all x ∈ C;

(A2) f is monotone, i.e. f(x, y) + f(y, x) ≤ 0 for all x, y ∈ C;

(A3) for all y ∈ C, f(., y) is weakly upper semicontinuous;

(A4) for all x ∈ C, f(x,.) is convex.

Let A be α-inverse strongly monotone of C into E*. For all r > 0 and × ∈ E, define the mapping S_r : E → 2^C as follows:

Then, the following statements hold:

(1) for each x ∈ E, S_r(x) ≠ ∅;

(2) S_r is single-valued;

(3) 〈S_r(x) - S_r(y), J(S_rx - x)〉 ≤ 〈S_r(x) - S_r(y), J(S_ry - y)〉 for all x, y ∈ E;

(4) F (S_r) = GEP (f);

(5) GEP(f) is nonempty, closed, and convex.

Proof. (1) Let x₀ be any given point in E. For each y ∈ C, we define the mapping G : C → 2^E by

It is easily seen that y ∈ G(y), and hence G(y). ≠ ∅

(a) First, we will show that G is a KKM mapping. Suppose that there exists a finite subset {y₁, y₂,..., y_m} of C and α_i > 0 with such that for all i = 1, 2,..., m. It follows that

By (A1) and (A4), we have

which is a contradiction. Thus, G is a KKM mapping on C.

(b) Next, we show that G(y) is closed for all y ∈ C. Let {z_n} be a sequence in G(y) such that z_n → z as n → ∞. It then follows from z_n ∈ G(y) that,

By (A3), the continuity of J, and the lower semicontinuity of || · ||², we obtain from (3.2) that

This shows that z ∈ G(y), and hence G(y) is closed for all y ∈ C.

(c) We prove that G(y) is weakly compact. We now equip E with the weak topology. Then, C, as closed, bounded convex subset in a reflexive space, is weakly compact. Hence, G(y) is also weakly compact.

Using (a), (b), and (c) and Lemma 2.5, we have ⋂_x∈CG(y) ≠ ∅. It is easily seen that

Hence, s_r(x₀) ≠ ∅. Since x₀ is arbitrary, we can conclude that s_r(x) ≠ ∅ for all x ∈ E.

(2) We prove that S_r is single-valued. In fact, for x ∈ C and r > 0, let z₁, z₂ ∈ S_r(x). Then,

and

Adding the two inequalities and from the condition (A2) and monotonicity of A, we have

and hence,

Hence,

Since J is monotone and E is strictly convex, we obtain that z₁ - x = z₂ - x and hence z₁ = z₂.

Therefore S_r is single-valued.

(3) For x, y ∈ C, we have

and

Again, adding the two inequalities, we also have

It follows from monotonicity of A that

(4) It is easy to see that

Hence, F (S_r) = GEP (f).

(5) Finally, we claim that GEP (f) is nonempty, closed, and convex. For each y ∈ C, we define the mapping Θ : C → 2^E by

Since y ∈ Θ (y), we have Θ(y) ≠ ∅ We prove that Θ is a KKM mapping on C. Suppose that there exists a finite subset {z₁, z₂,..., z_m} of C and α_i > 0 with such that for all i = 1, 2,..., m. Then,

From (A1) and (A4), we have

which is a contradiction. Thus, Θ is a KKM mapping on C.

Next, we prove that Θ (y) is closed for each y ∈ C. For any y ∈ C, let {x_n} be any sequence in Θ (y) such that x_n → x₀. We claim that x₀ ∈ Θ (y). Then, for each y ∈ C, we have

By (A3), we see that

This shows that x₀ ∈ Θ (y) and Θ(y) is closed for each y ∈ C. Thus, is also closed.

We observe that Θ (y) is weakly compact. In fact, since C is bounded, closed, and convex, we also have Θ(y) is weakly compact in the weak topology. By Lemma 2.5, we can conclude that .

Finally, we prove that GEP (f) is convex. In fact, let u, v ∈ F (S_r) and z_t = tu+(1 - t)v for t ∈ (0, 1). From (3), we know that

This yields that

Similarly, we also have

It follows from (3.4) and (3.5) that

Hence, z_t ∈ F (S_r) = GEP (f) and hence GEP (f) is convex. This completes the proof.

In this section, we prove a strong convergence theorem using a hybrid projection algorithm in a uniformly convex and smooth Banach space.

Theorem 4.1. Let E be a uniformly convex and smooth Banach space and C be a nonempty, bounded, closed, and convex subset of E. Let f be a bifunction from C × C to satisfying (A1)-(A4), A an α-inverse strongly monotone mapping of C into E* and a sequence of nonexpansive mappings of C into itself such that and suppose that satisfies the NST-condition. Let {x_n} be the sequence in C generated by

where {t_n} and {r_n} are sequences which satisfy the following conditions:

(C1) {t_n} ⊂ (0, 1) and lim_n→∞ t_n = 0;

(C2) {r_n} ⊂ (0, 1) and lim inf_n→∞ r_n > 0.

Then, the sequence {x_n} converges strongly to P_F x₀.

Proof. First, we rewrite the algorithm (4.1) as the following:

where S_r is the mapping defined by (3.1) for all r > 0. We first show that the sequence {x_n} is well defined. It is easy to verify that C_n ∩ D_n is closed and convex and Ω ⊂ C_n for all n ≥ 0. Next, we prove that Ω ⊂ C_n ∩ D_n. Since D₀ = C, we also have Ω ⊂ C₀ ∩ D₀. Suppose that Ω ⊂ C_{k - 1} ∩ D_{k - 1} for k ≥ 2. It follows from Lemma (3) that

for all u ∈ Ω. This implies that

for all u ∈ Ω. Hence, Ω ⊂ D_k. By the mathematical induction, we get that Ω ⊂ C_n ∩ D_n for each n ≥ 0 and hence {x_n} is well defined. Let w = P_F x₀. Since Ω ⊂ C_n ∩ D_n and , we have

Since {x_n} is bounded, there exists a subsequence of {x_n} such that . Since x_n+2 ∈ D_n+1 ⊂ D_n and , we have

Since {x_n - x₀} is bounded, we have lim_n→∞ ||x_n - x₀|| = d for some a constant d. Moreover, by the convexity of D_n, we also have and hence

This implies that

By Lemma 2.1, we have lim_{n →∞} ||x_n - x_n+1|| = 0.

Next, we show that . Since x_n+1 ∈ C_n and t_n > 0, there exists , {λ₀, λ₁,..., λ_m} ⊂ [0, 1] and {y₀, y₁,..., y_m} ⊂ C such that

for each i = 0, 1,..., m. Since C is bounded, by Lemma 2.2, we have

where M = sup_n≥0 ||x_n - w||. It follows from (C1) that lim_{n →∞} ||x_n - T_nx_n|| = 0. Since {T_n} satisfies the NST-condition, we have .

Next, we show that v ∈ GEP (f). By the construction of D_n, we see from (2.2) that . Since x_n+1 ∈ D_n, we obtain

as n → ∞. From (C2), we also have

as n → ∞. Since {x_n} is bounded, it has a subsequence which weakly converges to some v ∈ E.

By (4.4), we also have . By the definition of , for each y ∈ C, we obtain

By (A3) and (4.4), we have

This shows that v ∈ GEP (f) and hence .

Note that w = P_Ωx₀. Finally, we show that x_n → w as n → ∞. By the weakly lower semicontinuity of the norm, it follows from (4.3) that

This shows that

and v = w. Since E is uniformly convex, we obtain that . It follows that . Hence, we have x_n → w as n → w. This completes the proof.

If we take f ≡ 0 and A ≡ 0 in Theorem 4.1, then we obtain the following result.

Theorem 5.1. Let E be a uniformly convex and smooth Banach space, C a nonempty, bounded, closed, and convex subset of E and a sequence of nonexpansive mappings of C into itself such that and suppose that satisfies the NST-condition. Let {x_n} be the sequence in C generated by

If {t_n} ⊂ (0, 1) and lim_n→∞ t_n = 0, then {x_n} converges strongly to P_Ωx₀.

Remark 5.2. By Lemma 2.3, if we define for all n ≥ 0 in Theorems 3.1 and 5.1, then the theorems also hold.

If we take T_n ≡ I, the identity mapping on C, for all n ≥ 0 in Theorem 4.1, then we obtain the following result.

Theorem 5.3. Let E be a uniformly convex and smooth Banach space, C a nonempty, bounded, closed, and convex subset of E. Let f be a bifunction from C × C to satisfying (A1)-(A4) and A an α-inverse strongly monotone mapping of C into E*. Let {x_n} be the sequence in C generated by

If {r_n} ⊂ (0, 1) and lim inf_n→∞ r_n > 0, then {x_n} converges strongly to P_{GEP (f)}x₀.

If we take A ≡ 0 in Theorem 4.1, then we obtain the following result concerning an equilibrium problem in a Banach space setting.

Theorem 5.4. Let E be a uniformly convex and smooth Banach space and C be a nonempty, bounded, closed, and convex subset of E. Let f be a bifunction from C × C to satisfying (A1)-(A4) and let be a sequence of nonexpansive mappings of C into itself such that and suppose that satisfies the NST-condition. Let {x_n} be the sequence in C generated by

where {t_n} and {r_n} are sequences which satisfy the conditions:

(C1) {t_n} ⊂ (0, 1) and lim_n→∞ t_n = 0;

(C2) {r_n} ⊂ (0, 1) and lim inf_n→∞ r_n > 0.

Then, the sequence {x_n} converges strongly to P_Ωx₀.

GEP: generalized equilibrium problem.