Abstract
We first prove the existence of a solution of the generalized equilibrium problem (GEP) using the KKM mapping in a Banach space setting. Then, by virtue of this result, we construct a hybrid algorithm for finding a common element in the solution set of a GEP and the fixed point set of countable family of nonexpansive mappings in the frameworks of Banach spaces. By means of a projection technique, we also prove that the sequences generated by the hybrid algorithm converge strongly to a common element in the solution set of GEP and common fixed point set of nonexpansive mappings.
AMS Subject Classification: 47H09, 47H10
Keywords:
Banach space; Fixed point; Metric projection; Generalized equilibrium problem; Nonexpansive mapping1. 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 2^{E* }defined by
where 〈·,·〉 denotes the generalized duality pairing. We know that if E is smooth, then J is singlevalued and if E is uniformly smooth, then J is uniformly normtonorm continuous on bounded subsets of E. We shall still denote by J the singlevalued 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., [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 αinversestrongly monotone, if there exists an α > 0 such that
It is easy to see that if A : C → E* is an αinversestrongly 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 x_{0 }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 [57].
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 NSTcondition 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 [810].
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: x_{0 }∈ H, C_{1 }= C, and
where 0 ≤ α_{n }≤ a < 1 for all . They proved that if {T_{n}} satisfies the NSTcondition, 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: x_{0 }= 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: x_{1 }= 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 NSTcondition, 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
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 x_{0 }∈ C such that
The mapping P_{C }: E → C defined by P_{C }× = x_{0 }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_{n}x_{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 , {x_{1}, x_{2},..., x_{n}} ⊂ C, {λ_{1}, λ_{2},..., 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 NSTcondition.
Now, let us turn to following wellknown 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∈B}G(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 αinverse 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 singlevalued;
(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) F (S_{r}) = GEP (f);
(5) GEP(f) is nonempty, closed, and convex.
Proof. (1) Let x_{0 }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 {y_{1}, y_{2},..., 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∈C}G(y) ≠ ∅. It is easily seen that
Hence, s_{r}(x_{0}) ≠ ∅. Since x_{0 }is arbitrary, we can conclude that s_{r}(x) ≠ ∅ for all x ∈ E.
(2) We prove that S_{r }is singlevalued. In fact, for x ∈ C and r > 0, let z_{1}, z_{2 }∈ 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 z_{1 } x = z_{2 } x and hence z_{1 }= z_{2}.
Therefore S_{r }is singlevalued.
(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 {z_{1}, z_{2},..., 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 }→ x_{0}. We claim that x_{0 }∈ Θ (y). Then, for each y ∈ C, we have
By (A3), we see that
This shows that x_{0 }∈ 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.
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 αinverse strongly monotone mapping of C into E* and a sequence of nonexpansive mappings of C into itself such that and suppose that satisfies the NSTcondition. 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 }x_{0}.
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 D_{0 }= C, we also have Ω ⊂ C_{0 }∩ D_{0}. 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 }x_{0}. 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 } x_{0}} is bounded, we have lim_{n→∞ }x_{n } x_{0} = 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 , {λ_{0}, λ_{1},..., λ_{m}} ⊂ [0, 1] and {y_{0}, y_{1},..., 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_{n}x_{n} = 0. Since {T_{n}} satisfies the NSTcondition, 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_{Ω}x_{0}. 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 NSTcondition. 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_{Ω}x_{0}.
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 αinverse 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)}x_{0}.
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 NSTcondition. 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_{Ω}x_{0}.
Abbreviations
GEP: generalized equilibrium problem.
Acknowledgements
U. Kamraksa was supported by grant from under the program "Strategic Scholarships for Frontier Research Network for the Ph.D." Program Thai Doctoral degree from the Office of the Higher Education Commission, Thailand. The project was supported by the "Centre of Excellence in Mathematics" under the Commission on Higher Education, Ministry of Education, Thailand and the grant from under the program Strategic Scholarships for Frontier Research Network for the Ph.D. Program Thai Doctoral degree from the Office of the Higher Education Commission.
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