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Strong convergence theorems for system of equilibrium problems and asymptotically strict pseudocontractions in the intermediate sense
Fixed Point Theory and Applications volume 2011, Article number: 13 (2011)
Abstract
Let be N uniformly continuous asymptotically λ_{ i } strict pseudocontractions in the intermediate sense defined on a nonempty closed convex subset C of a real Hilbert space H. Consider the problem of finding a common element of the fixed point set of these mappings and the solution set of a system of equilibrium problems by using hybrid method. In this paper, we propose new iterative schemes for solving this problem and prove these schemes converge strongly.
MSC: 47H05; 47H09; 47H10.
1. Introduction
Let H be a real Hilbert space and let C be a nonempty closed convex subset of H.
A nonlinear mapping S : C → C is a self mapping of C. We denote the set of fixed points of S by F(S) (i.e., F(S) = {x ∈ C : Sx = x}). Recall the following concepts.

(1)
S is uniformly Lipschitzian if there exists a constant L > 0 such that

(2)
S is nonexpansive if

(3)
S is asymptotically nonexpansive if there exists a sequence k_{ n } of positive numbers satisfying the property lim_{n→∞} k_{ n } = 1 and

(4)
S is asymptotically nonexpansive in the intermediate sense [1] provided S is continuous and the following inequality holds:

(5)
S is asymptotically λstrict pseudocontractive mapping [2] with sequence {γ_{ n } } if there exists a constant λ ∈ [0, 1) and a sequence {γ_{ n } } in [0, ∞) with lim_{n→∞} γ_{ n } = 0 such that
for all x, y ∈ C and n ∈ ℕ.

(6)
S is asymptotically λstrict pseudocontractive mapping in the intermediate sense [3, 4] with sequence {γ_{ n } } if there exists a constant λ ∈ [0, 1) and a sequence {γ_{ n } } in [0, ∞) with lim_{n→∞} γ_{ n } = 0 such that
(1.1)
for all x, y ∈ C and n ∈ ℕ.
Throughout this paper, we assume that
Then, c_{ n } ≥ 0 for all n ∈ N, c_{ n } → 0 as n → ∞ and (1.1) reduces to the relation
for all x, y ∈ C and n ∈ ℕ.
When c_{ n } = 0 for all n ∈ N in (1.2), then S is an asymptotically λstrict pseudocontractive mapping with sequence {γ_{ n } }. We note that S is not necessarily uniformly LLipschitzian (see [4]), more examples can also be seen in [3].
Let {F_{ k } } be a countable family of bifunctions from C × C to ℝ, where ℝ is the set of real numbers. Combettes and Hirstoaga [5] considered the following system of equilibrium problems:
where Γ is an arbitrary index set. If Γ is a singleton, then problem (1.3) becomes the following equilibrium problem:
The solution set of (1.4) is denoted by EP(F).
The problem (1.3) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games and others; see, for instance, [6, 7] and the references therein. Some methods have been proposed to solve the equilibrium problem (1.3), related work can also be found in [8–11].
For solving the equilibrium problem, let us assume that the bifunction F satisfies the following conditions:
(A1) F(x, x) = 0 for all x ∈ C;
(A2) F is monotone, i.e.F(x, y) + F(y, x) ≤ 0 for any x, y ∈ C;
(A3) for each x, y, z ∈ C, lim sup_{t→0}F(tz + (1  t)x, y) ≤ F(x, y);
(A4) F(x,·) is convex and lower semicontionuous for each x ∈ C.
Recall Mann's iteration algorithm was introduced by Mann [12]. Since then, the construction of fixed points for nonexpansive mappings and asymptotically strict pseudocontractions via Mann' iteration algorithm has been extensively investigated by many authors (see, e.g., [2, 6]).
Mann's iteration algorithm generates a sequence {x_{ n } } by the following manner:
where α_{ n } is a real sequence in (0, 1) which satisfies certain control conditions.
On the other hand, Qin et al. [13] introduced the following algorithm for a finite family of asymptotically λ_{ i } strict pseudocontractions. Let x_{0} ∈ C and be a sequence in (0, 1). The sequence {x_{ n } } by the following way:
It is called the explicit iterative sequence of a finite family of asymptotically λ_{ i } strict pseudocontractions {S_{1}, S_{2},..., S_{ N } }. Since, for each n ≥ 1, it can be written as n = (h  1)N + i, where i = i(n) ∈ {1, 2,..., N}, h = h(n) ≥ 1 is a positive integer and h(n) → ∞, as n → ∞. We can rewrite the above table in the following compact form:
Recently, Sahu et al. [4] introduced new iterative schemes for asymptotically strict pseudocontractive mappings in the intermediate sense. To be more precise, they proved the following theorem.
Theorem 1.1. Let C be a nonempty closed convex subset of a real Hilbert space H and T: C → C a uniformly continuous asymptotically κstrict pseudocontractive mapping in the intermediate sense with sequence γ_{ n } such that F(T) is nonempty and bounded. Let α_{ n }be a sequence in [0, 1] such that 0 < δ ≤ α_{ n } ≤ 1  κ for all n ∈ N. Let {x_{ n } } ⊂ C be sequences generated by the following (CQ) algorithm:
where θ_{ n } = c_{ n } + γ_{ n } Δ _{ n }and Δ _{ n } = sup {x_{ n }  z: z ∈ F(T)} < ∞. Then, {x_{ n } } converges strongly to P_{F(T)}(u).
Very recently, Hu and Cai [3] further considered the asymptotically strict pseudocontractive mappings in the intermediate sense concerning equilibrium problem. They obtained the following result in a real Hilbert space.
Theorem 1.2. Let C be a nonempty closed convex subset of a real Hilbert space H and N ≥ 1 be an integer, ϕ : C → C be a bifunction satisfying (A1)(A4) and A : C → H be an αinversestrongly monotone mapping. Let for each 1 ≤ i ≤ N, T_{ i } : C → C be a uniformly continuous k_{ i } strictly asymptotically pseudocontractive mapping in the intermediate sense for some 0 ≤ k_{ i } < 1 with sequences {γ_{ n,i } } ⊂ [0, ∞) such that lim_{n→∞}γ_{ n,i } = 0 and {c_{ n,i } } ⊂ [0, ∞) such that lim_{n→∞}c_{ n,i } = 0. Let k = max{k_{ i } : 1 ≤ i ≤ N}, γ_{ n } = max{γ_{ n,i } : 1 ≤ i ≤ N} and c_{ n } = max{c_{ n,i } : 1 ≤ i ≤ N}. Assume thatis nonempty and bounded. Let {α_{ n } } and {β_{ n } } be sequences in [0, 1] such that 0 < a ≤ α_{ n } ≤ 1, 0 < δ ≤ β_{ n } ≤ 1  k for all n ∈ N and 0 < b ≤ r_{ n } ≤ c < 2α. Let {x_{ n } } and {u_{ n } } be sequences generated by the following algorithm:
where, as n → ∞, where ρ_{ n } = sup{x_{ n }  v: v ∈ F} < ∞. Then, {x_{ n } } converges strongly to P_{F(T)}x_{0}.
Motivated by Hu and Cai [3], Sahu et al. [4], and Duan [8], the main purpose of this paper is to introduce a new iterative process for finding a common element of the fixed point set of a finite family of asymptotically λ_{ i } strict pseudocontractions and the solution set of the problem (1.3). Using the hybrid method, we obtain strong convergence theorems that extend and improve the corresponding results [3, 4, 13, 14].
We will adopt the following notations:

1.
⇀ for the weak convergence and → for the strong convergence.

2.
denotes the weak ωlimit set of {x_{ n } }.
2. Preliminaries
We need some facts and tools in a real Hilbert space H which are listed below.
Lemma 2.1. Let H be a real Hilbert space. Then, the following identities hold.
(i) x  y^{2} = x^{2}  y^{2}  2〈x  y, y〉, ∀x, y ∈ H.
(ii) tx +(1  t)y^{2} = tx^{2}+(1  t)y^{2}  t(1  t)x  y^{2}, ∀t ∈ [0, 1], ∀x, y ∈ H.
Lemma 2.2. ([10]) Let H be a real Hilbert space. Given a nonempty closed convex subset C ⊂ H and points x, y, z ∈ H and given also a real number a ∈ ℝ, the set
is convex (and closed).
Lemma 2.3. ([15]) Let C be a nonempty, closed and convex subset of H. Let {x_{ n } } be a sequence in H and u ∈ H. Let q = P_{ C }u. Suppose that {x_{ n } } is such that ω_{ w } (x_{ n } ) ⊂ C and satisfies the following condition
Then, x_{ n } → q.
Lemma 2.4. ([4]) Let C be a nonempty closed convex subset of a real Hilbert space H and T : C → C a continuous asymptotically κstrict pseudocontractive mapping in the intermediate sense. Then I  T is demiclosed at zero in the sense that if {x_{ n } } is a sequence in C such that x_{ n } ⇀ x ∈ C and lim sup_{m→∞}lim sup_{n→∞}x_{ n }  T^{m}x_{ n }  = 0, then (I  T)x = 0.
Lemma 2.5. ([4]) Let C be a nonempty subset of a Hilbert space H and T : C → C an asymptotically κ  strict pseudocontractive mapping in the intermediate sense with sequence {γ_{ n } }. Then
for all x, y ∈ C and n ∈ N.
Lemma 2.6. ([6]) Let C be a nonempty closed convex subset of H, let F be bifunction from C × C to ℝ satisfying (A1)(A4) and let r > 0 and x ∈ H. Then there exists z ∈ C such that
Lemma 2.7. ([5]) For r > 0, x ∈ H, define a mapping T_{ r } : H → C as follows:
for all x ∈ H. Then, the following statements hold:

(i)
T_{ r } is singlevalued;

(ii)
T_{ r } is firmly nonexpansive, i.e., for any x, y ∈ H,

(iii)
F(T_{ r } ) = EP(F);

(iv)
EP(F) is closed and convex.
3. Main result
Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H and N ≥ 1 be an integer, let F_{ k }, k ∈ {1, 2, ... M}, be a bifunction from C × C to ℝ which satisfies conditions (A1)(A4). Let, for each 1 ≤ i ≤ N, S_{ i } : C → C be a uniformly continuous asymptotically λ_{ i } strict pseudocontractive mapping in the intermediate sense for some 0 ≤ λ_{ i } < 1 with sequences {γ_{ n,i } } ⊂ [0, ∞) such that lim_{n→∞}γ_{ n,i } = 0 and {c_{ n,i } } ⊂ [0, ∞) such that lim_{n→∞}c_{ n,i } = 0. Let λ = max{λ_{ i } : 1 ≤ i ≤ N}, γ_{ n } = max{γ_{ n,i } : 1 ≤ i ≤ N} and c_{ n } = max{c_{ n,i } : 1 ≤ i ≤ N}. Assume thatis nonempty and bounded. Let {α_{ n } } and {β_{ n } } be sequences in [0, 1] such that 0 < a ≤ α_{ n } ≤ 1, 0 < δ ≤ β_{ n } ≤ 1  λ for all n ∈ ℕ and {r_{ k,n } } ⊂ (0, ∞) satisfies lim inf_{n→∞}r_{ k,n } > 0 for all k ∈ {1, 2, ... M}. Let {x_{ n } } and {u_{ n } } be sequences generated by the following algorithm:
where , as n → ∞, where ρ_{ n } = sup{x_{ n }  v : v ∈ Ω} < ∞. Then {x_{ n } } converges strongly to P_{Ω}x_{1}.
Proof. Denote for every k ∈ {1, 2,..., M} and for all n ∈ ℕ. Therefore . The proof is divided into six steps.
Step 1. The sequence {x_{ n } } is well defined.
It is obvious that C_{ n } is closed and Q_{ n } is closed and convex for every n ∈ ℕ. From Lemma 2.2, we also get that C_{ n } is convex.
Take p ∈ Ω, since for each k ∈ {1, 2,..., M}, is nonexpansive, and , we have
It follows from the definition of S_{ i } and Lemma 2.1(ii), we get
By virtue of the convexity of ·^{2}, one has
Substituting (3.2) and (3.3) into (3.4), we obtain
It follows that p ∈ C_{ n } for all n ∈ ℕ. Thus, Ω ⊂ C_{ n } .
Next, we prove that Ω ⊂ Q_{ n } for all n ∈ ℕ by induction. For n = 1, we have Ω ⊂ C = Q_{1}. Assume that Ω ⊂ Q_{ n } for some n ≥ 1. Since , we obtain
As Ω ⊂ C_{ n } ⋂ Q_{ n } by induction assumption, the inequality holds, in particular, for all z ∈ Ω. This together with the definition of Q_{n+1}implies that Ω ⊂ Q_{n +1}.
Hence Ω ⊂ Q_{ n } holds for all n ≥ 1. Thus Ω ⊂ C_{ n } ⋂ Q_{ n } and therefore the sequence {x_{ n } } is well defined.
Step 2. Set q = P_{Ω}x_{1}, then
Since Ω is a nonempty closed convex subset of H, there exists a unique q ∈ Ω such that q = P_{Ω}x_{1}.
From , we have
Since q ∈ Ω ⊂ C_{ n } ⋂ Q_{ n } , we get (3.6).
Therefore, {x_{ n } } is bounded. So are {u_{ n } } and {y_{ n } }.
Step 3. The following limits hold:
From the definition of Q_{ n } , we have , which together with the fact that x_{n+1}∈ C_{ n } ⋂ Q_{ n } ⊂ Q_{ n } implies that
This shows that the sequence {x_{ n }  x_{1}} is nondecreasing. Since {x_{ n } } is bounded, the limit of {x_{ n }  x_{1}} exists.
It follows from Lemma 2.1(i) and (3.7) that
Noting that lim_{n→∞}x_{ n }  x_{1} exists, this implies
It is easy to get
Since x_{n+1}∈ C_{ n } , we have
So, we get lim_{n→∞}y_{ n }  x_{n+1} = 0. It follows that
Next we will show that
Indeed, for p ∈ Ω, it follows from the firmly nonexpansivity of that for each k ∈ {1, 2,..., M}, we have
Thus we get
which implies that for each k ∈ {1, 2,..., M},
Therefore, by the convexity of ·^{2}, (3.5) and the nonexpansivity of , we get
It follows that
From (3.10) and (3.13), we obtain (3.11). Then, we have
Combining (3.8) and (3.14), we have
It follows that
Step 4. Show that u_{ n }  S_{ i }u_{ n }  → 0, x_{ n }  S_{ i }x_{ n }  → 0, as n → ∞; ∀i ∈ {1, 2,..., N}.
Since, for any positive integer n ≥ N, it can be written as n = (h(n)  1) N + i(n), where i(n) ∈ {1, 2,..., N}. Observe that
From (3.10), (3.14), the conditions 0 < a ≤ α_{ n } ≤ 1 and 0 < δ ≤ β_{ n } ≤ 1  λ, we obtain
Next, we prove that
It is obvious that the relations hold: h(n) = h(n  N) + 1, i(n) = i(n  N).
Therefore,
Applying Lemma 2.5 and (3.16), we get (3.19). Using the uniformly continuity of S_{ i } , we obtain
this together with (3.17) yields
We also have
for any i = 1, 2, ... N, which gives that
Moreover, for each i ∈ {1, 2, ... N}, we obtain that
Step 5. The following implication holds:
We first show that . To this end, we take ω ∈ ω_{ w } (x_{ n } ) and assume that as j → ∞ for some subsequence of x_{ n } .
Note that S_{ i } is uniformly continuous and (3.23), we see that , for all m ∈ ℕ. So by Lemma 2.4, it follows that and hence .
Next we will show that . Indeed, by Lemma 2.6, we have that for each k = 1, 2, ..., M,
From (A2), we get
Hence,
From (3.11), we obtain that as j → ∞ for each k = 1, 2, ..., M (especially, ). Together with (3.11) and (A4) we have, for each k = 1, 2, ..., M, that
For any, 0 < t ≤ 1 and y ∈ C, let y_{ t } = ty + (1  t)ω. Since y ∈ C and ω ∈ C, we obtain that y_{ t } ∈ C and hence F_{ k } (y_{ t } , ω) ≤ 0. So, we have
Dividing by t, we get, for each k = 1, 2, ..., M, that
Letting t → 0 and from (A3), we get
for all y ∈ C and ω ∈ EP(F_{ k } ) for each k = 1, 2, ..., M, i.e., .
Hence (3.24) holds.
Step 6. Show that x_{ n } → q = P_{Ω}x_{1}.
From (3.6), (3.24) and Lemma 2.3, we conclude that x_{ n } → q, where q = P_{Ω}x_{1}. □
Corollary 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H and N ≥ 1 be an integer, let F be a bifunction from C × C to ℝ which satisfies conditions (A1)(A4). Let, for each 1 ≤ i ≤ N, S_{ i } : C → C be a uniformly continuous λ_{ i }strict asymptotically pseudocontractive mapping in the intermediate sense for some 0 ≤ λ _{ i } < 1 with sequences {γ_{n,i}} ⊂ [0, ∞) such that lim_{n→∞}γ_{n,i}= 0 and {c_{n,i}} ⊂ [0, ∞) such that lim_{n→∞}c_{ n } ,_{ i }= 0. Let λ = max{λ_{ i } : 1 ≤ i ≤ N}, γ_{ n } = max{γ_{n,i}: 1 ≤ i ≤ N} and c_{ n } = max{c_{n,i}: 1 ≤ i ≤ N}. Assume thatis nonempty and bounded. Let {α_{ n } } and {β_{ n } } be sequences in [0, 1] such that 0 < a ≤ α_{ n } ≤ 1,0 < δ ≤ β_{ n } ≤ 1  λ for all n ∈ N and {r_{ n } } ⊂ (0,∞) satisfies lim inf_{n→∞}r_{ n } > 0 for all k ∈ {1, 2, ... M}.
Let {x_{ n } } and {u_{ n } } be sequences generated by the following algorithm:
where, as n → ∞, where ρ_{ n } = sup{x_{ n }  v : v ∈ Ω} < ∞. Then {x_{ n } } converges strongly to P_{Ω}x_{1}.
Proof. Putting M = 1, we can draw the desired conclusion from Theorem 3.1.
□
Remark 3.3. Corollary 3.2 extends the theorem of Tada and Takahashi [14] from a nonexpansive mapping to a finite family of asymptotically λ_{ i } strict pseudocontractive mappings in the intermediate sense.
Corollary 3.4. Let C be a nonempty closed convex subset of a real Hilbert space H and N ≥ 1 be an integer, let, for each 1 ≤ i ≤ N, S_{ i } : C → C be a uniformly continuous λ_{ i }strict asymptotically pseudocontractive mapping in the intermediate sense for some 0 ≤ λ _{ i } < 1 with sequences {γ_{n,i}} ⊂ [0, ∞) such that lim_{n→∞}γ_{n,i}= 0 and {c_{ n,i } } ⊂ [0, ∞) such that lim_{n→∞}c_{ n,i } = 0. Let λ= max{λ_{ i } : 1 ≤ i ≤ N}, γ_{ n } = max{γ_{n,i}: 1 ≤ i ≤ N} and c_{ n } = max{c_{ n,i } : 1 ≤ i ≤ N}. Assume thatis nonempty and bounded. Let {α_{ n } } and {β_{ n } } be sequences in [0, 1] such that 0 < a ≤ α_{ n } ≤ 1, 0 <δ ≤ β_{ n } ≤ 1  λ for all n ∈ ℕ. Let {x_{ n } } and {u_{ n } } be sequences generated by the following algorithm:
where, as n → ∞, where ρ_{ n } = sup{x_{ n }  v : v ∈ Ω} < ∞. Then {x_{ n }} converges strongly to P_{Ω}x_{1}.
Proof. If F_{ k } (x, y) = 0, α _{ n } = 1 in Theorem 3.1, we can draw the conclusion easily. □
Remark 3.5. Corollary 3.4 extends the Theorem 4.1 of [4] and Theorem 2.2 of [13], respectively.
4. Numerical result
In this section, in order to demonstrate the effectiveness, realization and convergence of the algorithm in Theorem 3.1, we consider the following simple example ever appeared in the reference [4]:
Example 4.1. Let x = R and C = [0, 1] For each x ∈ C, we define
where 0 < k < 1.
Set C_{1} : = [0, 1/2] and C_{2} : = (1/2, 1]. Hence,
and
For x ∈ C_{1} and y ∈ C_{2}, we have
Thus
for all x, y ∈ C, n ∈ ℕ and some K > 0. Therefore, T is an asymptotically kstrict pseudocontractive mapping in the intermediate sense.
In the algorithm (3.1), set . We apply it to find the fixed point of T of Example 4.1.
Under the above assumptions, (3.1) is simplified as follows:
In fact, in one dimensional case, the C_{ n } ⋂ Q_{ n } is an closed interval. If we set [a_{ n } , b_{ n } ] := C_{ n } ⋂ Q_{ n } , then the projection point x_{n+1}of x_{1} ∈ C onto C_{ n } ⋂ Q_{ n } can be expressed as:
Since the conditions of Theorem 3.1 are satisfied in Example 4.1, the conclusion holds, i.e., x_{ n } → 0 ∈ F (T).
Now we turn to realizing (3.1) for approximating a fixed point of T. Take the initial guess x_{1} = 1/2, 1/5 and 5/8, respectively. All the numerical results are given in Tables 1, 2 and 3. The corresponding graph appears in Figure 1a,b,c.
References
Bruck RE, Kuczumow T, Reich S: Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform opial property. Colloq Math 1993, 65: 169–179.
Kim TH, Xu HK: Convergence of the modified Mann's iteration method for asymptotically strict pseudocontractions. Nonlinear Anal 2008, 68: 2828–2836. 10.1016/j.na.2007.02.029
Hu CS, Cai G: Convergence theorems for equilibrium problems and fixed point problems of a finite family of asymptotically k strict pseudocontractive mappings in the intermediate sense. Comput Math Appl 2010.
Sahu DR, Xu HK, Yao JC: Asymptotically strict pseudocontractive mappings in the intermediate sense. Nonlinear Anal 2009, 70: 3502–3511. 10.1016/j.na.2008.07.007
Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. J Nonlinear Convex Anal 2005, 6: 117–136.
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math Stud 1994, 63: 123–145.
Colao V, Marino G, Xu HK: An iterative method for finding common solutions of equilibrium and fixed point problems. J Math Anal Appl 2008, 344: 340–352. 10.1016/j.jmaa.2008.02.041
Duan PC: Convergence theorems concerning hybrid methods for strict pseudocontractions and systems of equilibrium problems. J Inequal Appl 2010.
Flam SD, Antipin AS: Equilibrium programming using proximallike algorithms. Math Program 1997, 78: 29–41.
Marino G, Xu HK: Weak and srong convergence theorems for strict pseudocontractions in Hlibert spaces. J Math Anal Appl 2007, 329: 336–346. 10.1016/j.jmaa.2006.06.055
Takahashi S, Takahashi W: Strong convergence theorems for a generalized equilibrium problems and a nonexpansive mapping in a Hlibert space. Nonlinear Anal 2008, 69: 1025–1033. 10.1016/j.na.2008.02.042
Mann WR: Mean value methods in iteration. Proc Am Math Soc 1953, 4: 506–510. 10.1090/S00029939195300548463
Qin XL, Cho YJ, Kang SM, Shang M: A hybrid iterative scheme for asymptotically k strictly pseudocontractions in Hlibert spaces. Nonlinear Anal 2009, 70: 1902–1911. 10.1016/j.na.2008.02.090
Tada A, Takahashi W: Weak and strong convergence theorems for a nonexpansive mappingand a equilibrium problem. J Optim Theory Appl 2007, 133: 359–370. 10.1007/s109570079187z
MatinezYanes C, Xu HK: Srong convergence of the CQ method for fixed point processes. Nonlinear Anal 2006, 64: 2400–2411. 10.1016/j.na.2005.08.018
Acknowledgements
The authors would like to thank the reviewers for their good suggestions. This research is supported by Fundamental Research Funds for the Central Universities (ZXH2011C002).
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PD carried out the proof of convergence of the theorems and realization of numerical examples. JZ carried out the check of the manuscript. All authors read and approved the final manuscript.
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Duan, P., Zhao, J. Strong convergence theorems for system of equilibrium problems and asymptotically strict pseudocontractions in the intermediate sense. Fixed Point Theory Appl 2011, 13 (2011). https://doi.org/10.1186/16871812201113
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DOI: https://doi.org/10.1186/16871812201113
Keywords
 asymptotically strict pseudocontraction in the intermediate sense
 system of equilibrium problem
 hybrid method
 fixed point