1. Introduction

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 2E*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):

(1.1)

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., [2, 3]. 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:

(1.2)

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 [5–7].

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 [8–10].

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

(1.3)

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

(1.4)

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

(1.5)

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.

2. Preliminaries

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

(2.1)

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

(2.2)

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 supn→∞||x n || ≤ d, lim supn→∞||y n || ≤ d and limn→∞||α n x n + (1 - α n )y n || = d. Then, limn→∞||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

  1. (i)

    for every ;

  2. (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 → 2X 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 → 2X 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) ≠ ∅.

3. Existence results of gep

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 → 2C as follows:

(3.1)

Then, the following statements hold:

  1. (1)

    for each x ∈ E, S r (x) ≠ ∅;

  2. (2)

    S r is single-valued;

  3. (3)

    〈S r (x) - S r (y), J(S r x - x)〉 ≤ 〈S r (x) - S r (y), J(S r y - y)〉 for all x, y ∈ E;

  4. (4)

    F (S r ) = GEP (f);

  5. (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 → 2E by

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

  1. (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.

  2. (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,

    (3.2)

    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.

  3. (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

(3.3)

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 → 2E 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

(3.4)

Similarly, we also have

(3.5)

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.

4. Strong convergence theorem

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

(4.1)

where {t n } and {r n } are sequences which satisfy the following conditions:

(C1) {t n } ⊂ (0, 1) and limn→∞t n = 0;

(C2) {r n } ⊂ (0, 1) and lim infn→∞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:

(4.2)

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 ⊂ Ck - 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

(4.3)

Since {x n } is bounded, there exists a subsequence of {x n } such that . Since xn+2∈ Dn+1⊂ D n and , we have

Since {x n - x0} is bounded, we have limn→∞||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 limn →∞||x n - xn+1|| = 0.

Next, we show that . Since xn+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 = supn≥0||x n - w||. It follows from (C1) that limn →∞||x n - T n x 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 xn+1∈ D n , we obtain

as n → ∞. From (C2), we also have

(4.4)

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 = PΩx0. 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.

5. Deduced theorems

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

(5.1)

If {t n } ⊂ (0, 1) and limn→∞t n = 0, then {x n } converges strongly to PΩx0.

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

(5.2)

If {r n } ⊂ (0, 1) and lim infn→∞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

(5.3)

where {t n } and {r n } are sequences which satisfy the conditions:

(C1) {t n } ⊂ (0, 1) and limn→∞t n = 0;

(C2) {r n } ⊂ (0, 1) and lim infn→∞r n > 0.

Then, the sequence {x n } converges strongly to PΩx0.