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A hybrid iteration scheme for equilibrium problems and common fixed point problems of generalized quasiϕasymptotically nonexpansive mappings in Banach spaces
Fixed Point Theory and Applications volume 2012, Article number: 33 (2012)
Abstract
In this article, we introduce an iterative algorithm for finding a common element of the set of common fixed points of a finite family of closed generalized quasiϕasymptotically nonexpansive mappings and the set of solutions of equilibrium problem in Banach spaces. Then we study the strong convergence of the algorithm. Our results improve and extend the corresponding results announced by many others.
Mathematics Subject Classification (2000): 47H09; 47H10; 47J05; 54H25.
1. Introduction and preliminary
Let E be a Banach space with the dual E*. Let C be a nonempty closed convex subset of E and f :C × C → ℝ a bifunction, where ℝ is the set of real numbers. The equilibrium problem for f is to find \widehat{x}\in C such that
for all y ∈ C. The set of solutions of (1.1) is denoted by EP(f). Given a mapping T :C → E*, let f(x, y) = 〈Tx, y  x〉 for all x,y ∈ C. Then \widehat{x}\in EP\left(f\right) if and only if \u27e8T\widehat{x},y\widehat{x}\u27e9\ge 0 for all y ∈ C, i.e., \widehat{x} is a solution of the variational inequality. Numerous problems in physics, optimization, engineering and economics reduce to find a solution of (1.1). Some methods have been proposed to solve the equilibrium problem; see, for example, BlumOettli [1] and Moudafi [2]. For solving the equilibrium problem, let us assume that f satisfies the following conditions:
(A 1) f(x, x) = 0 for all x ∈ C;
(A 2) f is monotone, that is, f(x, y) + f(y, x) ≤ 0 for all x, y ∈ C;
(A 3) for each x, y, z ∈ C, lim_{t→0}f(tz + (1  t)x, y) ≤ f(x, y);
(A 4) for each x ∈ C, the function y ↦ f(x, y) is convex and lower semicontinuous.
Let E be a Banach space with the dual E*. 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 uniformly smooth, strictly convex, and reflexive, then the normalized duality mapping J is singlevalued, onetoone, onto and uniformly normtonorm continuous on each bounded subset of E. Moreover, if E is a reflexive and strictly convex Banach space with a strictly convex dual, then J^{1} is singlevalued, onetoone, surjective, and it is the duality mapping from E* into E and thus JJ^{1} = I_{E*}and J^{1}J = I_{ E }(see, [3]). It is also well known that if E is uniformly smooth if and only if E* is uniformly convex.
Let C be a nonempty closed convex subset of a Banach space E and T : C → C a mapping. A point x ∈ C is said to be a fixed point of T provided Tx = x. In this article, we use F(T) to denote the fixed point set and use → to denote the strong convergence. Recall that a mapping T : C → C is called nonexpansive if
A mapping T: C → C is called asymptotically nonexpansive if there exists a sequence {k_{ n }} of real numbers with k_{ n }→ 1 as n → ∞ such that
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [4] in 1972. They proved that, if C is a nonempty bounded closed convex subset of a uniformly convex Banach space E, then every asymptotically nonexpansive selfmapping T of C has a fixed point. Further, the set F(T) is closed and convex. Since 1972, a host of authors have studied the weak and strong convergence problems of the iterative algorithms for such a class of mappings (see, e.g., [4–6] and the references therein).
It is well known that if C is a nonempty closed convex subset of a Hilbert space H and P_{ C }: H → C is the metric projection of H onto C, then P_{ C }is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. In this connection, Alber [7] recently introduced a generalized projection operator Π_{ C }in a Banach space E which is an analogue of the metric projection in Hilbert spaces.
Next, we assume that E is a smooth Banach space. Consider the functional defined by
Following Alber [7], the generalized projection Π_{ C }: E → C is a mapping that assigns to an arbitrary point x ∈ E the minimum point of the functional ϕ(y, x), that is, {\prod}_{C}x=\stackrel{\u0304}{x}, where \stackrel{\u0304}{x} is the solution to the following minimization problem:
It follows from the definition of the function ϕ that
If E is a Hilbert space, then ϕ(y, x) = ∥y  x∥^{2} and Π_{ C }= P_{ C }is the metric projection of H onto C.
Remark 1.1[8, 9] If E is a reflexive, strictly convex and smooth Banach space, then for x, y ∈ E, ϕ(x, y) = 0 if and only if x = y.
Let C be a nonempty, closed and convex subset of a smooth Banach E and T a mapping from C into itself. The mapping T is said to be ϕnonexpansive if ϕ(Tx, Ty) ≤ ϕ(x, y), ∀x, y ∈ C. The mapping T is said to be quasiϕnonexpansive if F\left(T\right)\ne \mathrm{0\u0338}, ϕ(p, Tx) ≤ ϕ(p, x), ∀x ∈ C, p ∈ F(T). The mapping T is said to be ϕasymptotically nonexpansive if there exists some real sequence {k_{ n }} with k_{ n }≥ 1 and k_{ n }→ 1 as n → ∞ such that ϕ(T^{n}x, T^{n}y) ≤ k_{ n }ϕ(x,y), ∀x, y ∈ C. The mapping T is said to be quasiϕasymptotically nonexpansive if F\left(T\right)\ne \mathrm{0\u0338} and there exists some real sequence {k_{ n }} with k_{ n }≥1 and k_{ n }→ 1 as n → ∞ such that ϕ(p, T^{n}x) ≤ k_{ n }ϕ(p, x), ∀x ∈ C, p ∈ F(T). The mapping T is said to be generalized quasiϕasymptotically nonexpansive if F\left(T\right)\ne \mathrm{0\u0338} and there exist nonnegative real sequences {k_{ n }} and {c_{ n }} with k_{ n }≥ 1, lim_{n→∞}k_{ n }= 1 and lim_{n→∞}c_{ n }= 0 such that ϕ(p, T^{n}x) ≤ k_{ n }ϕ(p, x) + c_{ n }, ∀x ∈ C, p ∈ F(T). The mapping T is said to be asymptotically regular on C if, for any bounded subset K of C, lim sup_{n→∞}{∥T^{n+1}x  T^{n}x∥: x ∈ K} = 0. The mapping T is said to be closed on C if, for any sequence {x_{ n }} such that lim_{n→∞}x_{ n }= x_{0} and lim_{n→∞}Tx_{ n }= y_{0}, then Tx_{0} = y_{0}.
We remark that a ϕasymptotically nonexpansive mapping with a nonempty fixed point set F(T) is a quasiϕasymptotically nonexpansive mapping, but the converse may be not true. The class of generalized quasiϕasymptotically nonexpansive mappings is more general than the class of quasiϕasymptotically nonexpansive mappings and ϕasymptotically nonexpansive mappings. The following example shows that the inclusion is proper. Let K=\left[\frac{1}{\pi},\frac{1}{\pi}\right] and define (see [10]) Tx=\frac{x}{2}\text{sin}\left(\frac{1}{x}\right) if x ≠ 0 and Tx = 0 if x = 0. Then T^{n}x → 0 uniformly but T is not Lipschitzian. It should be noted that F(T) = {0}. For each fixed n, define f_{ n }(x) = ∥T^{n}x∥^{2}  ∥x∥^{2} for x ∈ K. Set c_{ n }= sup_{x∈K}{f_{ n }(x), 0}. Then lim_{n→∞}c_{ n }= 0 and
This show that T is a generalized quasiϕasymptotically nonexpansive but it is not quasiϕasymptotically nonexpansive and ϕasymptotically nonexpansive. Recently, many authors studied the problem of finding a common element of the set of fixed points of nonexpansive or quasiϕasymptotically nonexpansive mappings and the set of solutions of an equilibrium problem in the frame work of Hilbert spaces and Banach spaces respectively; see, for instance, [11–15] and the references therein.
In 2009, Cho, Qin and Kang [16] introduced the following iterative scheme on a closed quasiϕasymptotically nonexpansive mapping:
Strong convergence theorems of fixed points are established in a uniformly smooth and uniformly convex Banach space.
Recently, Takahashi and Zembayashi [17] introduced the following iterative process:
where f:C × C → ℝ is a bifunction satisfying (A 1)(A 4), J is the normalized duality mapping on E and S : C → C is a relatively nonexpansive mapping. They proved the sequences {x_{ n }} defined by (1.2) converge strongly to a common point of the set of solutions of the equilibrium problem (1.1) and the set of fixed points of S provided the control sequences {α_{ n }} and {r_{ n }} satisfy appropriate conditions in Banach spaces.
In this article, inspired and motivated by the works mentioned above, we introduce an iterative process for finding a common element of the set of common fixed points of a finite family of closed generalized quasiϕasymptotically nonexpansive mappings and the solution set of equilibrium problem in Banach spaces. In the meantime, the method of the proof is different from the original one. The results presented in this article improve and generalize the corresponding results announced by many others.
Let C_{ n }be a sequence of nonempty closed convex subsets of a reflexive Banach space E. We denote two subsets s  Li_{ n }C_{ n }and w  Ls_{ n }C_{ n }as follows: x ∈ s  Li_{ n }C_{ n }if and only if there exists {x_{ n }} ⊂ E such that {x_{ n }} converges strongly to x and that x_{ n }∈ C_{ n }for all n ≥ 0. Similarly, y ∈ w  Ls_{ n }C_{ n }if and only if there exists a subsequence \left\{{C}_{{n}_{i}}\right\} of {C_{ n }} and a sequence {y_{ i }} ⊂ E such that {y_{ i }} converges weakly to y and that {y}_{i}\in {C}_{{n}_{i}} for all i ≥ 0. We define the Mosco convergence [18] of {C_{ n }} as follows: If C_{0} satisfies that C_{0} = s  Li_{ n }C_{ n }= w  Ls_{ n }C_{ n }, it is said that {C_{ n }} converges to C_{0} in the sense of Mosco and we write C_{0} = M  lim_{ n }_{→∞} _{ C }_{ n }. For more detail, see [19].
In order to obtain the main results of this paper, we need the following lemmas.
Lemma 1.2[20]Let E be a smooth and uniformly convex Banach space and let {x_{ n }} and {y_{ n }} be sequences in E such that either {x_{ n }} or {y_{ n }} is bounded. If lim_{n→∞}ϕ(x_{ n },y_{n}) = 0, then lim_{n→∞}∥x_{ n } y_{ n }∥ = 0.
Lemma 1.3[21]Let E be a smooth, strictly convex and reflexive Banach space having the KadecKlee property. Let {K_{ n }} be a sequence of nonempty closed convex subsets of E. If K_{0} = Mlim_{n→∞}K_{ n }exists and is nonempty, then\left\{{\prod}_{{K}_{n}}x\right\}converges strongly to\left\{{\prod}_{{K}_{0}}x\right\}for each x ∈ C.
Lemma 1.4[8, 22]Let E be a uniformly convex Banach space, s > 0 a positive number and B_{ s }(0) a closed ball of E. Then there exists a strictly increasing, continuous, and convex function g: [0, ∞) → [0, ∞) with g(0) = 0 such that
for any k, l ∈ {0, 1,. .., N}, for all x_{0}, x_{1}, .. ., x_{ N }∈ B_{ s }(0) = {x ∈ E : ∥x∥ ≤ s} and α_{0}, α_{1},...,α_{ n }∈ [0, 1] such that{\sum}_{i=0}^{N}{\alpha}_{i}=1.
Lemma 1.5[1]Let C be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E, let f be a bifunction from C × C to ℝ satisfying (A 1)(A 4), and letr > 0 and x ∈ E. Then, there exists z ∈ C such that
Lemma 1.6[17]Let C be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space E. Let f be a bifunction from C × C to ℝ satisfying (A 1)(A 4). For r > 0 and x ∈ E, define a mapping T_{ r }: E → C as follows:
for all x ∈ E. Then, the following hold:

(1)
T _{ r } is singlevalued;

(2)
T _{ r } is firmly nonexpansive, i.e., for any x, y ∈ E,
\u27e8{T}_{r}x{T}_{r}y,J{T}_{r}xJ{T}_{r}y\u27e9\le \u27e8{T}_{r}x{T}_{r}y,JxJy\u27e9; 
(3)
F(T _{ r }) = EP(f);

(4)
EP(f) is closed and convex;

(5)
ϕ(q,T _{ r } x) + ϕ(T _{ r } x, x) ≤ ϕ ( q, x ), ∀q ∈ F(T _{ r }).
Lemma 1.7 Let E be a uniformly convex and uniformly smooth Banach space, C a nonempty, closed and convex subset of E and T a closed generalized quasiϕasymptotically nonexpansive mapping from C into itself. Then F(T) is a closed convex subset of C.
Proof. We first show that F(T) is closed. To see this, let {p_{ n }} be a sequence in F(T) with p_{ n }→ p as n → ∞, we shall prove that p ∈ F(T). By using the definition of T, we have
which implies that ϕ(p_{ n }, T^{n}p) → 0 as n → ∞. It follows from Lemma 1.2 that p_{ n }T^{n}p → 0 as n → ∞ and hence T^{n}p → p as n → ∞. We have T(T^{n}p) = T^{n+1}p → p as n → ∞. It follows from the closedness of T that Tp = p. We next show that F(T) is convex. To prove this, for arbitrary p, q ∈ F(T), t ∈ (0, 1), we set w = tp + (1  t)q. By (1.3), we have
which implies that ϕ(w, T^{n}w) → 0 as n → ∞. By Lemma 1.2, we obtain T^{n}w → w as n → ∞, and hence T(T^{n}w) = T^{n+1}w → w as n → ∞. Since T is closed, we see that w = Tw. This completes the proof.
2. Main results
Theorem 2.1 Let C be a nonempty, closed and convex subset of a uniformly convex and uniformly smooth real Banach space E and let T_{ i }: C → C be a closed and generalized quasiϕasymptotically nonexpansive mapping with real sequences {k_{ n,i }} ⊂ [1, ∞) and {c_{ n,i }} ⊂ [0, ∞) such that lim _{n→∞}k_{ n,i }= 1 and lim_{n→∞}c_{ n,i }= 0 for each 1 ≤ i ≤ N. Let f be a bifunction from C × C to ℝ satisfying (A 1)(A 4). Assume that T_{ i }is asymptotically regular on C for each 1 ≤ i ≤ N andF=\left({\bigcap}_{i=1}^{N}F\left({T}_{i}\right)\right)\bigcap EP\left(f\right)\ne \mathrm{0\u0338}. Let k_{ n }= max_{1≤i≤N}{k_{ n, i }} and c_{ n }= max_{1≤i≤N}{c_{ n,i }}. Define a sequence {x_{ n }} in C in the following manner:
for every n ≥ 1, where {r_{ n }} is a real sequence in [a, ∞) for some a > 0, J is the normalized duality mapping on E. Assume that the control sequences {α_{n,0}}, {α_{n,1}}, ⋯, {α_{ n,N }} are real sequences in (0,1) satisfy{\sum}_{i=0}^{N}{\alpha}_{n,i}=1and lim inf_{n→∞}α_{n,0}α_{ n,i }> 0 for each i ∈ {1, 2, · · ·, N}. Then the sequence {x_{ n }} converges strongly to ∏_{ F }x_{1}, where Π_{ F }is the generalized projection from C into F.
Proof. Firstly, by Lemma 1.7, we know that F(T_{ i }) is a closed convex subset of C for every 1 ≤ i ≤ N. Hence, F=\left({\bigcap}_{i=1}^{N}F\left({T}_{i}\right)\right)\bigcap EP\left(f\right)\ne \mathrm{0\u0338} is a nonempty closed convex subset of C and Π_{ F }x_{1} is well defined for x_{1} ∈ C. Now we show that C_{ n }is closed and convex for each n ≥ 1. From the definition of C_{ n }, it is obvious that C_{ n }is closed for each n ≥ 1. We show that C_{ n }is convex for each n ≥ 1. It is obvious that C_{1} = C is convex. Suppose that C_{ n }is convex for some integer n. Observe that the set
can be written to
For z_{1}, z_{2} ∈ C_{n+1}⊂ C_{ n }and t ∈ (0,1), denote z = tz_{1} + (1  t)z_{2}, we have z ∈ C_{ n }. Setting A = ∥u_{ n }∥^{2}  k_{ n }∥x_{ n }∥^{2} c_{ n }and B = Ju_{ n }k_{ n }Jx_{ n }, by noting that ∥ · ∥^{2} is convex, we have
So we obtain
which implies that z ∈ C_{n+1}, so we get C_{n+1}is convex. Thus, C_{ n }is closed and convex for each n ≥ 1.
Secondly, we prove that F ⊂ C_{ n }for all n ≥ 1. We do this by induction. For n = 1, we have F ⊂ C = C_{1}. Suppose that F ⊂ C_{ n }for some n ≥ 1. Let p ∈ F ⊂ C. Putting {u}_{n}={T}_{{r}_{n}}{y}_{n} for all n ≥ 1, we have that {T}_{{r}_{n}} is quasiϕnonexpansive from Lemma 1.6. Since T_{ i }is generalized quasiϕasymptotically nonexpansive, by noting that ∥ · ∥^{2} is convex, we have
which infers that p ∈ C_{n+1}, and hence F ⊂ C_{n+1}. This proves that F ⊂ C_{ n }for all n ≥ 1.
Thirdly, we show that \underset{n\to \infty}{\text{lim}}{x}_{n}={x}^{*}={\prod}_{\overline{C}}{x}_{1}, where \overline{C}={\cap}_{n=1}^{\infty}{C}_{n}. Indeed, since {C_{ n }} is a decreasing sequence of closed convex subsets of E such that \overline{C}={\cap}_{n=1}^{\infty}{C}_{n} is nonempty, it follows that
By Lemma 1.3, \left\{{x}_{n}\right\}=\left\{{\Pi}_{{C}_{n}}{x}_{1}\right\} converges strongly to \left\{{x}^{*}\right\}=\left\{{\Pi}_{\overline{C}}{x}_{1}\right\} and {x_{ n }} is bounded.
Fourthly, we prove that x* ∈ F.
Since {x}_{n+1}={\Pi}_{{C}_{n+1}}{x}_{1}\in {C}_{n+1}, from the definition of C_{n+1}, we get
From lim_{n → ∞}x_{ n }= x*, one obtain ϕ(x_{n+1},x_{ n }) → 0 as n → ∞, and it follows from lim_{n → ∞}c_{ n }= 0 we have
Thus, lim_{n→∞}∥x_{n+1} u_{ n }∥ = 0 by Lemma 1.2. It should be noted that
for all n ≥ 1. It follows that
which implies that u_{ n }→ x* as n → ∞. Since J is uniformly normtonorm continuous on bounded sets, from (2.3), we have
Let s=\text{sup}\left\{\u2225{x}_{n}\u2225,\u2225\underset{1}{\overset{n}{T}}{x}_{n}\u2225,\u2225\underset{2}{\overset{n}{T}}{x}_{n}\u2225,\cdots \phantom{\rule{0.3em}{0ex}},\u2225\underset{N}{\overset{n}{T}}{x}_{n}\u2225:n\in \mathbb{N}\right\}. Since E is uniformly smooth Banach space, we know that E* is a uniformly convex Banach space. Therefore, from Lemma 1.4 we have, for any p ∈ F, that
Therefore, we have
On the other hand, we have
It follows from (2.3) and (2.4) that
Since lim_{n→∞}k_{ n }= 1, lim_{n→∞}c_{ n }= 0 and lim inf_{n→∞}α_{n,0}α_{n, 1}> 0, from (2.5) and (2.6) we have
Therefore, from the property of g, we obtain
Since J^{1} is uniformly normtonorm continuous on bounded sets, we have
and hence {T}_{1}^{n}{x}_{n}\to {x}^{*} as n → ∞. Since \u2225{T}_{1}^{n+1}{x}_{n}{x}^{*}\u2225\le \u2225{T}_{1}^{n+1}{x}_{n}{T}_{1}^{n}{x}_{n}\u2225+\u2225{T}_{1}^{n}{x}_{n}{x}^{*}\u2225, it follows from the asymptotical regularity of T_{1} that
That is, {T}_{1}\left({T}_{1}^{n}{x}_{n}\right)\to {x}^{*} as n → ∞. From the closedness of T_{1}, we get T_{1}x* = x*. Similarly, one can obtain that T_{ i }x* = x* for i = 2,..., N. So, {x}^{*}\in {\cap}_{i=1}^{N}F\left({T}_{i}\right).
Now we show x* ∈ EP(f) = F(T_{ r }). Let p ∈ F. From {u}_{n}={T}_{{r}_{n}}{y}_{n}, (2.2) and Lemma 1.6, we obtain that
It follows from (2.6), k_{ n }→ 1 and c_{ n }→ 0 that ϕ(u_{ n }, y_{ n }) → 0 as n → ∞. Now, by Lemma 1.2, we have that ∥u_{ n } y_{ n }∥ → 0 as n → ∞, and hence, ∥Ju_{ n } Jy_{ n }∥ → 0 as n → ∞. Since u_{ n }→ x* as n → ∞, we obtain that y_{ n }→ x*. From the assumption r_{ n }> a, we get
Noting that {u}_{n}={T}_{{r}_{n}}{y}_{n}, we obtain
From (A 2), we have
Letting n → ∞, we have from u_{ n }→ x*, (2.8) and (A 4) that f(y, x*) ≤ 0(∀y ∈ C). For t with 0 < t ≤ 1 and y ∈ C, let y_{ t }= ty + (1  t)x*. Since y ∈ C and x* ∈ C, we have y_{ t }∈ C and hence f(y_{ t }, x*) ≤ 0. Now, from (A 1) and (A 4), we have
and hence f(y_{ t },y) ≥ 0. Letting t → 0, from (A 3), we have f(x*, y) ≥ 0. This implies that x* ∈ EP(f). Thus, x* ∈ F.
Finally, since {x}^{*}={\Pi}_{\overline{C}}{x}_{1}\in F and F is a nonempty closed convex subset of \overline{C}={\cap}_{n=1}^{\infty}{C}_{n}, we conclude that x* = Π_{ F }x_{1}. This completes the proof.
In Hilbert spaces, Theorem 2.1 reduces to the following theorem.
Theorem 2.2 Let C be a nonempty, closed and convex subset of a Hilbert space H and let T_{ i }:C → C be a closed and generalized quasiϕasymptotically nonexpansive mapping with real sequences {k_{ n,i }} ⊂ [1, ∞) and {c_{ n,i }} ⊂ [0, ∞) such that lim_{n→∞}k_{ n,i }= 1 and lim_{n→∞}c_{ n,i }= 0 for each 1 ≤ i ≤ N. Let f be a bifunction from C × C to ℝ satisfying (A 1)(A 4). Assume that T_{ i }is asymptotically regular on C for each 1 ≤ i ≤ N andF=\left({\bigcap}_{i=1}^{N}F\left({T}_{i}\right)\right)\bigcap EP\left(f\right)\ne \mathrm{0\u0338}. Let k_{ n }= max_{1≤i≤N}{k_{ n,i }} and c_{ n }= max_{1≤i≤N}{c_{ n,i }}. Define a sequence {x_{ n }} in C in the following manner:
for every n ≥ 1, where {r_{ n }} is a real sequence in [a, ∞) for some a > 0. Assume that the control sequences {α_{n,0}}, {α_{n,1}}, .. ., {α_{ n,N }} are real sequences in (0,1) satisfy{\sum}_{i=0}^{N}{\alpha}_{n,i}=1αnd lim inf_{n→∞}α_{n,0}α_{ n,i }> 0 for each i ∈ {1, 2, .. ., N}. Then the sequence {x_{ n }} converges strongly to P_{ F }x_{1}.
Remark 2.3 Theorems 2.1 and 2.2 extend the main results of [16] from quasiϕnonexpansive mappings to more general generalized quasiϕasymptotically nonexpansive mappings.
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Acknowledgements
The research was supported by the science research foundation program in Civil Aviation University of China (2011kys02), it was also supported by Fundamental Research Funds for the Central Universities (Program No. ZXH2009D021 and No. ZXH2011D005).
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ZJ carried out the algorithm design and drafted the manuscript. HS conceived of the study and helped to draft the manuscript. All authors read and approved the final manuscript.
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Zhao, J., He, S. A hybrid iteration scheme for equilibrium problems and common fixed point problems of generalized quasiϕasymptotically nonexpansive mappings in Banach spaces. Fixed Point Theory Appl 2012, 33 (2012). https://doi.org/10.1186/16871812201233
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Published:
DOI: https://doi.org/10.1186/16871812201233
Keywords
 equilibrium problem
 generalized quasiϕasymptotically nonexpansive mapping
 strong convergence
 common fixed point
 Banach space