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Some extragradient methods for common solutions of generalized equilibrium problems and fixed points of nonexpansive mappings
Fixed Point Theory and Applications volumeÂ 2011, ArticleÂ number:Â 12 (2011)
Abstract
In this article, we introduce some new iterative schemes based on the extragradient method (and the hybrid method) for finding a common element of the set of solutions of a generalized equilibrium problem, and the set of fixed points of a family of infinitely nonexpansive mappings and the set of solutions of the variational inequality for a monotone, Lipschitzcontinuous mapping in Hilbert spaces. We obtain some strong convergence theorems and weak convergence theorems. The results in this article generalize, improve, and unify some wellknown convergence theorems in the literature.
1. Introduction
Let H be a real Hilbert space with inner product ã€ˆ.,.ã€‰ and induced norm Â·. Let C be a nonempty closed convex subset of H. Let F be a bifunction from C Ã— C to R and let B : C â†’ H be a nonlinear mapping, where R is the set of real numbers. Moudafi [1], Moudafi and Thera [2], Peng and Yao [3, 4], Takahashi and Takahashi [5] considered the following generalized equilibrium problem:
The set of solutions of (1.1) is denoted by GEP(F, B). If B = 0, the generalized equilibrium problem (1.1) becomes the equilibrium problem for F : C Ã— C â†’ R, which is to find x âˆˆ C such that
The set of solutions of (1.2) is denoted by EP(F).
The problem (1.1) 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 [1â€“7].
Recall that a mapping S : C â†’ C is nonexpansive if there holds that
We denote the set of fixed points of S by Fix(S).
Let the mapping A : C â†’ H be monotone and kLipschitzcontinuous. The variational inequality problem is to find x âˆˆ C such that
for all y âˆˆ C. The set of solutions of the variational inequality problem is denoted by V I(C, A).
Several algorithms have been proposed for finding the solution of problem (1.1). Moudafi [1] introduced an iterative scheme for finding a common element of the set of solutions of problem (1.1) and the set of fixed points of a nonexpansive mapping in a Hilbert space, and proved a weak convergence theorem. Moudafi and Thera [2] introduced an auxiliary scheme for finding a solution of problem (1.1) in a Hilbert space and obtained a weak convergence theorem. Peng and Yao [3, 4] introduced some iterative schemes for finding a common element of the set of solutions of problem (1.1), the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for a monotone, Lipschitzcontinuous mapping and obtain both strong convergence theorems, and weak convergence theorems for the sequences generated by the corresponding processes in Hilbert spaces. Takahashi and Takahashi [5] introduced an iterative scheme for finding a common element of the set of solutions of problem (1.1) and the set of fixed points of a nonexpansive mapping in a Hilbert space, and proved a strong convergence theorem.
Some methods also have been proposed to solve the problem (1.2); see, for instance, [8â€“19] and the references therein. Takahashi and Takahashi [9] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a nonexpansive mapping, and proved a strong convergence theorem in a Hilbert space. Su et al. [10] introduced and researched an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for an Î±inversestrongly monotone mapping in a Hilbert space. Tada and Takahashi [11] introduced two iterative schemes for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a nonexpansive mapping in a Hilbert space, and obtained both strong convergence and weak convergence theorems. Plubtieng and Punpaeng [12] introduced an iterative processes based on the extragradient method for finding the common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem and the set of solutions of variational inequality problem for an Î±inversestrongly monotone mapping. Chang et al. [13] introduced an iterative processes based on the extragradient method for finding the common element of the set of solutions of an equilibrium problem, the set of common fixed point for a family of infinitely nonexpansive mappings and the set of solutions of variational inequality problem for an Î±inversestrongly monotone mapping. Yao et al. [14] and Ceng and Yao [15] introduced some iterative viscosity approximation schemes for finding the common element of the set of solutions of problem (1.2) and the set of fixed points of a family of infinitely nonexpansive mappings in a Hilbert space. Colao et al. [16] introduced an iterative viscosity approximation scheme for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a family of finitely nonexpansive mappings in a Hilbert space. We observe that the algorithms in [13â€“16] involves the Wmapping generated by a family of infinitely (finitely) nonexpansive mappings which is an effective tool in nonlinear analysis (see [20, 21]). However, the Wmapping generated by a family of infinitely (finitely) nonexpansive mappings is too completed to use for finding the common element of the set of solutions of problem (1.2) and the set of fixed points of a family of infinitely (finitely) nonexpansive mappings. It is natural to raise and to give an answer to the following question: Can one construct algorithms for finding a common element of the set of solutions of a generalized equilibrium problem (an equilibrium problem), the common set of fixed points of a family of infinitely nonexpansive mappings and the set of solutions of a variational inequality without the Wmapping generated by a family of infinitely (finitely) nonexpansive mappings? In this article, we will give a positive answer to this question.
Recently, OHaraa et al. [22] introduced and researched an iterative approach for finding a nearest point of infinitely many nonexpansive mappings in a Hilbert spaces without using the Wmapping generated by a family of infinitely (finitely) nonexpansive mappings. Inspired by the ideas in [1â€“6, 8â€“16, 22] and the references therein, we introduce some new iterative schemes based on the extragradient method (and the hybrid method) for finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of a family of infinitely nonexpansive mappings, and the set of solutions of the variational inequality for a monotone, Lipschitzcontinuous mapping without using the Wmapping generated by a family of infinitely (finitely) nonexpansive mappings. We obtain both strong convergence theorems and weak convergence theorems for the sequences generated by the corresponding processes. The results in this article generalize, improve, and unify some wellknown convergence theorems in the literature.
2. Preliminaries
Let H be a real Hilbert space with inner product ã€ˆÂ·,Â·ã€‰ and norm Â·. Let C be a nonempty closed convex subset of H. Let symbols â†’ and â‡€ denote strong and weak convergences, respectively. In a real Hilbert space H, it is well known that
for all x, y âˆˆ H and Î» âˆˆ [0, 1].
For any x âˆˆ H, there exists the unique nearest point in C, denoted by P_{ C } (x), such that x  P_{ C } (x) â‰¤ x  y for all y âˆˆ C. The mapping P_{ C } is called the metric projection of H onto C. We know that P_{ C } is a nonexpansive mapping from H onto C. It is also known that P_{ C }x âˆˆ C and
for all x âˆˆ H and y âˆˆ C.
It is easy to see that (2.1) is equivalent to
for all x âˆˆ H and y âˆˆ C.
A mapping A of C into H is called monotone if
for all x, y âˆˆ C. A mapping A of C into H is called Î±inversestrongly monotone if there exists a positive real number Î± such that
for all x, y âˆˆ C. A mapping A : C â†’ H is called kLipschitzcontinuous if there exists a positive real number k such that
for all x, y âˆˆ C. It is easy to see that if A is Î±inversestrongly monotone, then A is monotone and Lipschitzcontinuous. The converse is not true in general. The class of Î±inversestrongly monotone mappings does not contain some important classes of mappings even in a finitedimensional case. For example, if the matrix in the corresponding linear complementarity problem is positively semidefinite, but not positively definite, then the mapping A will be monotone and Lipschitzcontinuous, but not Î±inversestrongly monotone (see [23]).
Let A be a monotone mapping of C into H. In the context of the variational inequality problem, the characterization of projection (2.1) implies the following:
and
It is also known that H satisfies the Opial's condition [24], i.e., for any sequence {x_{ n } } âŠ‚ H with x_{ n } â‡€ x, the inequality
holds for every y âˆˆ H with x â‰ y.
A setvalued mapping T : H â†’ 2 ^{H} is called monotone if for all x, y âˆˆ H, f âˆˆ Tx and g âˆˆ Ty imply ã€ˆx  y, f  gã€‰ â‰¥ 0. A monotone mapping T : H â†’ 2 ^{H} is maximal if its graph G(T) of T is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping T is maximal if and only if for (x, f) âˆˆ H Ã— H, ã€ˆx  y, f  gã€‰ â‰¥ 0 for every (y, g) âˆˆ G(T) implies f âˆˆ Tx. Let A be a monotone, kLipschitzcontinuous mapping of C into H and N_{ C }v be normal cone to C at v âˆˆ C, i.e., N_{ C }v = {w âˆˆ H : ã€ˆv  u, wã€‰ â‰¥ 0, âˆ€u âˆˆ C}. Define
Then, T is maximal monotone and 0 âˆˆ Tv if and only if v âˆˆ V I(C, A) (see [25]).
For solving the equilibrium problem, let us assume that the bifunction F satisfies the following condition:
(A1) F(x, x) = 0 for all x âˆˆ C;
(A2) F is monotone, i.e., F(x, y) + F(y, x) â‰¤ 0 for any x, y âˆˆ C;
(A3) for each x, y, z âˆˆ C,
(A4) for each x âˆˆ C, y â†¦ F(x, y) is convex and lower semicontinuous.
We recall some lemmas which will be needed in the rest of this article.
Lemma 2.1.[7] Let C be a nonempty closed convex subset of H, let F be a bifunction from C Ã— C to R satisfying (A1)(A4). Let r > 0 and x âˆˆ H. Then, there exists z âˆˆ C such that
Lemma 2.2.[8] Let C be a nonempty closed convex subset of H, let F be a bifunction from C Ã— C to R satisfying (A1)(A4). For r > 0 and x âˆˆ H, define a mapping Tr : H â†’ C as follows:
for all x âˆˆ H. Then, the following statements hold:

(1)
T_{ r } is singlevalued;

(2)
T_{ r } is firmly nonexpansive, i.e., for any x, y âˆˆ H,

(3)
F(T_{ r } ) = EP (F);

(4)
EP(F) is closed and convex.
3. The main results
We first show a strong convergence of an iterative algorithm based on extragradient and hybrid methods which solves the problem of finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of a family of infinitely nonexpansive mappings, and the set of solutions of the variational inequality for a monotone, Lipschitzcontinuous mapping in a Hilbert space.
Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C Ã— C to R satisfying (A1)(A4). Let A be a monotone and kLipschitzcontinuous mapping of C into H and B be an Î±inversestrongly monotone mapping of C into H. Let S_{1}, S_{2},... be a family of infinitely nonexpansive mappings of C into itself such that . Assume that for all i âˆˆ {1, 2,...} and for any bounded subset K of C, thenthere holds
Let {x_{ n } }, {u_{ n } }, {y_{ n } } and {z_{ n } } be sequences generated by
for every n = 1, 2,... where {Î»_{ n } } âŠ‚ [a, b] for some , {r_{ n }} âŠ‚ [d, e] for some d, e âˆˆ (0, 2Î±), and {Î±_{ n } }, {Î²_{ n } }, {Î³_{ n } } are three sequences in [0, 1] satisfying the conditions:

(i)
Î±_{ n } + Î²_{ n } â‰¤ 1 for all n âˆˆ N;

(ii)
;

(iii)
;

(iv)
and for all n âˆˆ N;
Then, {x_{ n } }, {u_{ n } }, {y_{ n } } and {z_{ n } } converge strongly to w = P_{Î©}(x).
Proof. It is obvious that C_{ n } is closed, and Q_{ n } is closed and convex for every n = 1, 2,.... Since
we also have that C_{ n } is convex for every n = 1, 2,.... It is easy to see that ã€ˆx_{ n }  z, x  x_{ n } ã€‰ â‰¥ 0 for all z âˆˆ Q_{ n } and by (2.1), . Let t_{ n }= P_{ C }(u_{ n } Î»_{ n }Ay_{ n }) for every n = 1, 2,.... Let u âˆˆ Î© and let >be a sequence of mappings defined as in Lemma 2.2. Then . From and the Î±inverse strongly monotonicity of B, we have
From (2.2), the monotonicity of A, and u âˆˆ V I(C, A), we have
Further, Since y_{ n } = (1  Î³_{ n } )u_{ n } + Î³_{ n }P_{ C } (u_{ n }  Î»_{ n }Au_{ n } ) and A is kLipschitzcontinuous, we have
In addition, from the definition of P_{ C } , we have
It follows from , and (3.2) that
In addition, from u âˆˆ V I(C, A) and (3.2), we have
Therefore, from (3.2) to (3.4) and z_{ n } = (1  Î±_{ n }  Î²_{ n } )x_{ n } + Î±_{ n }y_{ n } + Î²_{ n }S_{ n }t_{ n } and u = S_{ n }u, we have
for every n = 1, 2,... and hence u âˆˆ C_{ n } . So, Î© âŠ‚ C_{ n } for every n = 1, 2,.... Next, let us show by mathematical induction that x_{ n } is well defined and Î© âŠ‚ C_{ n } âˆ© Q_{ n } for every n = 1, 2,.... For n = 1 we have x_{1} = x âˆˆ C and Q_{1} = C. Hence, we obtain Î© âŠ‚C_{1} âˆ© Q_{1}. Suppose that x_{ k } is given and Î© âŠ‚ C_{ k } âˆ© Q_{ k } for some k âˆˆ N. Since Î© is nonempty, C_{ k } âˆ© Q_{ k } is a nonempty closed convex subset of H. Hence, there exists a unique element x_{k+1}âˆˆ C_{ k }âˆ© Q_{ k }such that . It is also obvious that there holds ã€ˆx_{k+1} z, x  x_{k+1}ã€‰ â‰¥ 0 for every z âˆˆ C_{ k }âˆ© Q_{ k }. Since Î© âŠ‚ C_{ k } âˆ© Q_{ k } , we have ã€ˆx_{k+1} z, x  x_{k+1}ã€‰ â‰¥ 0 for every z âˆˆ Î© and hence Î© âŠ‚ Q_{k+1}. Therefore, we obtain Î© âŠ‚ C_{k+1}âˆ© Q_{k+1}.
Let l_{0} = P_{Î©}x. From and l_{0} v Î© âŠ‚ C_{ n } âˆ© Q_{ n } , we have
for every n = 1, 2,.... Therefore, {x_{ n } } is bounded. From (3.2) to (3.5) and the lipschitz continuity of A, we also obtain that {u_{ n } }, {y_{ n } }, {Au_{ n } }, {t_{ n } } and {z_{ n } } are bounded. Since x_{n+1}âˆˆ C_{ n } âˆ© Q_{ n } âŠ‚ C_{ n } and , we have
for every n = 1, 2,.... It follows from (3.6) that lim_{nâ†’âˆž}x_{ n } x exists.
Since and x_{n+1}âˆˆ Q_{ n } , using (2.2), we have
for every n = 1, 2,.... This implies that
Since x_{n+1}âˆˆ C_{ n }, we have z_{ n } x_{n+1}^{2} â‰¤ x_{ n } x_{n+1}^{2} + (3  3Î³_{ n }+ Î±_{ n })b^{2}Au_{ n }^{2} and hence it follows from lim_{nâ†’âˆž}Î³_{ n }= 1 and lim_{nâ†’âˆž}Î±_{ n }= 0 that lim_{nâ†’âˆž}z_{ n } x_{n+1} = 0. Since
for every n = 1, 2,..., we have x_{ n }  z_{ n }  â†’ 0.
For u âˆˆ Î©, from (3.5), we obtain
Since lim_{nâ†’âˆž}Î³_{ n }= 1 and lim_{nâ†’âˆž}Î±_{ n }= 0, {x_{ n } }, {y_{ n } }, {Au_{ n } }, and {z_{ n } } are bounded, we have
By lim inf_{nâ†’âˆž}Î²_{ n }> 0, we get
From (3.3) and u = S_{ n }u, we have
Thus, lim_{nâ†’âˆž}t_{ n }  u^{2} x_{ n }  u^{2} = 0.
From (3.3) and (3.2), we have
It follows that
The assumptions on Î³_{ n } and Î»_{ n } imply that and . Consequently, lim_{nâ†’âˆž}u_{ n }  y_{ n }  = lim_{nâ†’âˆž}t_{ n }  y_{ n }  = 0. Since A is Lipschitzcontinuous, we have lim_{nâ†’âˆž}At_{ n }  Ay_{ n }  = 0. It follows from u_{ n }  t_{ n }  â‰¤ u_{ n }  y_{ n }  + t_{ n }  y_{ n }  that lim_{nâ†’âˆž}u_{ n }  t_{ n }  = 0.
We rewrite the definition of z_{ n } as
From lim_{nâ†’âˆž}z_{ n }  x_{ n }  = 0, lim_{nâ†’âˆž}Î±_{ n }= 0, the boundedness of {x_{ n } }, {y_{ n }} and lim inf_{nâ†’âˆž}Î²_{ n }> 0 we infer that lim_{nâ†’âˆž}S_{ n }t_{ n } x_{ n } = 0.
By (3.2)(3.5), we have
Hence, we have
Since lim_{nâ†’âˆž}Î±_{ n }= 1, lim inf_{nâ†’âˆž}Î²_{ n }> 0, lim_{nâ†’âˆž}Î³_{ n }= 1, x_{ n }  z_{ n }  â†’ 0 and the sequences {x_{ n } } and {z_{ n } } are bounded, we obtain Bx_{ n }  B_{ u }  â†’ 0.
For u âˆˆ Î©, we have, from Lemma 2.2,
Hence,
Then, by (3.5), we have
Hence,
Since lim_{nâ†’âˆž}Î±_{ n }= 0, lim inf_{nâ†’âˆž}Î²_{ n }> 0, lim_{nâ†’âˆž}Î³_{ n }= 1, x_{ n } z_{ n } â†’ 0, Bx_{ n } Bu â†’ 0 and the sequences {x_{ n }}, {u_{ n }} and {z_{ n }} are bounded, we obtain x_{ n } u_{ n } â†’ 0. From z_{ n }  t_{ n }  â‰¤ z_{ n }  x_{ n } +x_{ n }  u_{ n } +u_{ n }  t_{ n } , we have z_{ n }  t_{ n }  â†’ 0.
From t_{ n }  x_{ n }  â‰¤ t_{ n }  u_{ n }  + x_{ n }  u_{ n } , we also have t_{ n }  x_{ n }  â†’ 0.
Since z_{ n } = (1  Î±_{ n }  Î²_{ n } )x_{ n } + Î±_{ n }y_{ n } + Î²_{ n }S_{ n }t_{ n } , we have Î²_{ n } (S_{ n }t_{ n }  t_{ n } ) = (1  Î±_{ n }  Î²_{ n } )(t_{ n }  x_{ n } ) + Î±_{ n } (t_{ n }  y_{ n } ) + (z_{ n }  t_{ n } ). Then
and hence S_{ n }t_{ n }  t_{ n }  â†’ 0. At the same time, observe that for all i âˆˆ {1, 2,...},
It follows from (3.8) and the condition (*) that for all i âˆˆ {1, 2,...},
As {x_{ n } } is bounded, there exists a subsequence of {x_{ n } } such that x_{ ni } â‡€ w. From x_{ n }  u_{ n }  â†’ 0, we obtain that u_{ ni } â‡€ w. From u_{ n }  t_{ n }  â†’ 0, we also obtain that t_{ ni } â‡€ w. Since {u_{ ni } } âŠ‚ C and C is closed and convex, we obtain w âˆˆ C.
First, we show w âˆˆ GEP(F, B). By , we know that
It follows from (A2) that
Hence,
For t with 0 < t â‰¤ 1 and y âˆˆ C, let y_{ t } = t_{ y } + (1  t)w. Since y âˆˆ C and w âˆˆ C, we obtain y_{ t } âˆˆ C. So, from (3.10) we have
Since , we have . Further, from the inversestrongly monotonicity of B, we have . Hence, from (A4), and , we have
as i â†’ âˆž. From (A1), (A4) and (3.11), we also have
and hence
Letting t â†’ 0, we have, for each y âˆˆ C,
This implies that w âˆˆ GEP(F, B).
We next show that . Assume . Since and for some i_{0} âˆˆ {1, 2,...} from the Opial condition, we have
This is a contradiction. Hence, we get .
Finally we show w âˆˆ V I(C, A). Let
where N_{ C }v is the normal cone to C at v âˆˆ C. We have already mentioned that in this case the mapping T is maximal monotone, and 0 âˆˆ Tv if and only if v âˆˆ V I(C, A). Let (v, g) âˆˆ G(T). Then Tv = Av + N_{ C }v and hence g  Av âˆˆ N_{ C }v.
Hence, we have ã€ˆv  t, g  Avã€‰ â‰¥ 0 for all t âˆˆ C. On the other hand, from t_{ n } = P_{ C } (u_{ n }  Î»_{ n }Ay_{ n } ) and v âˆˆ C, we have
and hence
Therefore, we have
Hence, we obtain ã€ˆv  w, gã€‰ â‰¥ 0 as i â†’ âˆž. Since T is maximal monotone, we have w âˆˆ T^{1}0 and hence w âˆˆ V I(C, A). This implies that w âˆˆ Î©.
From l_{0} = P_{Î©}x, w âˆˆ Î© and (3.6), we have
Hence, we obtain
From , we have , and hence . Since and l_{0} âˆˆ Î© âŠ‚ C_{ n } âˆ© Q_{ n } âŠ‚ Q_{ n } , we have
As i â†’ âˆž, we obtain  l_{0} w^{2} â‰¥ ã€ˆl_{0} w, x  l_{0}ã€‰ â‰¥ 0 by l_{0} = P_{Î©}x and w âˆˆ Î©. Hence, we have w = l_{0}. This implies that x_{ n } â†’ l_{0}. It is easy to see u_{ n } â†’ l_{0}, y_{ n } â†’ l_{0} and z_{ n } â†’ l_{0}. The proof is now complete.
By combining the arguments in the proof of Theorem 3.1 and those in the proof of Theorem 3.1 in [3], we can easily obtain the following weak convergence theorem for an iterative algorithm based on the extragradient method which solves the problem of finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of a family of infinitely nonexpansive mappings and the set of solutions of the variational inequality for a monotone, Lipschitzcontinuous mapping in a Hilbert space.
Theorem 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C Ã— C to R satisfying (A1)(A4). Let A be a monotone, and kLipschitzcontinuous mapping of C into H and B be an Î±inversestrongly monotone mapping of C into H. Let S_{1}, S_{2},... be a family of infinitely nonexpansive mappings of C into itself such that . Assume that for all i âˆˆ {1, 2,...} and for any bounded subset K of C, thenthere holds
Let {x_{ n } }, {u_{ n } } and {y_{ n } } be the sequences generated by
for every n = 1, 2,.... If {Î»_{ n } } âŠ‚ [a, b] for some , {Î²_{ n } } âŠ‚ [Î´, Îµ] for some Î´, Îµ âˆˆ (0, 1) and {r_{ n } } âŠ‚ [d, e] for some d, e âˆˆ (0, 2Î±). Then, {x_{ n } }, {u_{ n } } and {y_{ n } } converge weakly to w âˆˆ Î©, where w = lim_{nâ†’âˆž}P_{Î©}x_{ n }.
4. Applications
By Theorems 3.1 and 3.2, we can obtain many new and interesting convergence theorems in a real Hilbert space. We give some examples as follows:
Let A = 0, by Theorems 3.1 and 3.2, respectively, we obtain the following results.
Theorem 4.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from CÃ—C to R satisfying (A1)(A4). Let B be an Î±inversestrongly monotone mapping of C into H. Let S_{1}, S_{2},... be a family of infinitely nonexpansive mappings of C into itself such that . Assume that for all i âˆˆ {1, 2,...} and for any bounded subset K of C, thenthere holds
Let {x_{ n } }, {u_{ n } } {y_{ n } }, and {z_{ n } } be the sequences generated by
for every n = 1, 2,.... where {r_{ n } } âŠ‚ [d, e] for some d, e âˆˆ (0, 2Î±), and {Î±_{ n } }, {Î²_{ n } } are sequences in [0, 1] satisfying the conditions:

(i)
Î±_{ n } + Î²_{ n } â‰¤ 1 for all n âˆˆ N;

(ii)
;

(iii)
for all n âˆˆ N;
Then, {x_{ n } }, {u_{ n } }, and {z_{ n } } converge strongly to w = P_{âˆ‘}(x).
Theorem 4.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from CÃ—C to R satisfying (A1)(A4). Let B be an Î±inversestrongly monotone mapping of C into H. Let S_{1}, S_{2},... be a family of infinitely nonexpansive mappings of C into itself such that . Assume that for all i âˆˆ {1, 2,...} and for any bounded subset K of C, thenthere holds
Let {x_{ n } } and {u_{ n } } be sequences generated by
for every n = 1, 2,.... If {Î²_{ n } } âŠ‚ [Î´, Îµ] for some Î´, Îµ âˆˆ (0, 1) and {r_{ n } } âŠ‚ [d, e] for some d, e âˆˆ (0, 2Î±). Then, {x_{ n } } and {u_{ n } } converge weakly to w âˆˆ âˆ‘, where w = lim_{nâ†’âˆž}P_{âˆ‘}x_{ n } .
Theorem 4.3. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from CÃ—C to R satisfying (A1)(A4). Let A be a monotone and kLipschitzcontinuous mapping of C into H and B be an Î±inversestrongly monotone mapping of C into H. Let S_{1}, S_{2},... be a family of infinitely nonexpansive mappings of C into itself such that . Assume that for all i âˆˆ {1, 2,...} and for any bounded subset K of C, thenthere holds
Let {x_{ n } }, {u_{ n } }, {y_{ n } }, and {z_{ n } } be sequences generated by
for every n = 1, 2,... where {Î»_{ n } } âŠ‚ [a, b] for some , {r_{ n } } âŠ‚ [d, e] for some d, e âˆˆ (0, 2Î±), and {Î²_{ n } } is a sequence in [0, 1] satisfying . Then, {x_{ n } }, {u_{ n } }, {y_{ n } }, and {z_{ n } } converge strongly to w = P_{Î©}(x).
Proof. Putting Î³_{ n } = 1 and Î±_{ n } = 0, by Theorem 3.1, we obtain the desired result.
Let B = 0, by Theorems 3.1, 3.2, and 4.3, we obtain the following results.
Theorem 4.4. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from CÃ—C to R satisfying (A1)(A4). Let A be a monotone and kLipschitzcontinuous mapping of C into H. Let S_{1}, S_{2},... be a family of infinitely nonexpansive mappings of C into itself such that . Assume that for all i âˆˆ {1, 2,...}, and for any bounded subset K of C, there holds
Let {x_{ n } }, {u_{ n } }, {y_{ n } }, and {z_{ n } } be the sequences generated by
for every n = 1, 2,.... where {Î»_{ n } } âŠ‚ [a, b] for some , {r_{ n } } âŠ‚ [d, +âˆž) for some d > 0, and {Î±_{ n } }, {Î²_{ n } }, {Î³_{ n } } are three sequences in [0, 1] satisfying the following conditions:

(i)
Î±_{ n } + Î²_{ n } â‰¤ 1 for all n âˆˆ N;

(ii)
;

(iii)
;

(iv)
and for all n âˆˆ N;
Then, {x_{ n } }, {u_{ n } }, {y_{ n } } and {z_{ n } } converge strongly to w = P_{Î›}(x).
Theorem 4.5. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from CÃ—C to R satisfying (A1)(A4). Let A be a monotone and kLipschitzcontinuous mapping of C into H. Let S_{1}, S_{2},... be a family of infinitely nonexpansive mappings of C into itself such that . Assume that for all i âˆˆ {1, 2,...} and for any bounded subset K of C, thenthere holds
Let {x_{ n } }, {u_{ n } }, and {y_{ n } } be the sequences generated by
for every n = 1, 2,.... If {Î»_{ n } } âŠ‚ [a, b] for some ,{Î²_{ n } } âŠ‚ [Î´, Îµ], for some Î´, Îµ âˆˆ (0, 1) and {r_{ n } } âŠ‚ [d, +âˆž] for some d > 0, then {x_{ n } }, {u_{ n } } and {y_{ n } } converge weakly to w âˆˆ Î›, where w = lim_{nâ†’âˆž}P_{Î›}x_{ n } .
Theorem 4.6. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C Ã— C to R satisfying (A1)(A4). Let A be a monotone and kLipschitzcontinuous mapping of C into H. Let S_{1}, S_{2},... be a family of infinitely nonexpansive mappings of C into itself such that . Assume that for all i âˆˆ {1, 2,...} and for any bounded subset K of C, thenthere holds
Let {x_{ n } }, {u_{ n } } {y_{ n } }, and {z_{ n } } be the sequences generated by
for every n = 1, 2,.... where {Î»_{ n } } âŠ‚ [a, b] for some , {r_{ n } } âŠ‚ [d, +âˆž) and for some d > 0, and {Î²_{ n } } is a sequence in [0, 1] satisfying . Then, {x_{ n } }, {u_{ n } }, {y_{ n } }, and {z_{ n } } converge strongly to w = P_{Î›}(x).
Let B = 0 and F(x, y) = 0 for x, y âˆˆ C, by Theorems 3.1 and 4.3, we obtain the following results.
Theorem 4.7. Let C be a nonempty closed convex subset of a real Hilbert space H. Let A be a monotone and kLipschitzcontinuous mapping of C into H. Let S_{1}, S_{2},... be a family of infinitely nonexpansive mappings of C into itself such that . Assume that for all i âˆˆ {1, 2,...} and for any bounded subset K of C, thenthere holds
Let {x_{ n } }, {y_{ n } }, and {z_{ n } } be the sequences generated by
for every n = 1, 2,.... where {Î»_{ n } } âŠ‚ [a, b] for some , and {Î±_{ n } }, {Î²_{ n } }, {Î³_{ n } }are three sequences in [0, 1] satisfying the following conditions:

(i)
Î±_{ n } + Î²_{ n } â‰¤ 1 for all n âˆˆ N;

(ii)
;

(iii)
;

(iv)
and for all n âˆˆ N;
Then, {x_{ n } }, {y_{ n } }, and {z_{ n } } converge strongly to w = P_{Î“}(x).
Theorem 4.8. Let C be a nonempty closed convex subset of a real Hilbert space H. Let A be a monotone and kLipschitzcontinuous mapping of C into H. Let S_{1}, S_{2},... be a family of infinitely nonexpansive mappings of C into itself such that . Assume that for all i âˆˆ {1, 2,...} and for any bounded subset K of C, thenthere holds
Let {x_{ n } }, {y_{ n } }, and {z_{ n } } be the sequences generated by
for every n = 1, 2,.... where {Î»_{ n } } âŠ‚ [a, b] for some , and {Î²_{ n } } is a sequence in [0, 1] satisfying . Then, {x_{ n } }, {y_{ n } }, and {z_{ n } } converge strongly to w = P_{Î“}(x).
Let F(x, y) = 0 for x, y âˆˆ C, then by Theorem 3.2 and the proof of Theorem 4.7 in [3], we obtain the following result.
Theorem 4.9. Let C be a nonempty closed convex subset of a real Hilbert space H. Let A be a monotone and kLipschitzcontinuous mapping of C into H and B be an Î±inversestrongly monotone mapping of C into H. Let S_{1}, S_{2},... be a family of infinitely nonexpansive mappings of C into itself such that . Assume that for all i âˆˆ {1, 2,...} and for any bounded subset K of C, thenthere holds
Let {x_{ n } }, {u_{ n } }, and {y_{ n } } be the sequences generated by
for every n = 1, 2,.... if {Î»_{ n } } âŠ‚ [a, b] for some , {Î²_{ n } } âŠ‚ [Î´, Îµ] for some Î´, Îµ âˆˆ (0, 1) and {r_{ n } } âŠ‚ [d, e] for some d, e âˆˆ (0, 2Î±). Then, {x_{ n } } and {u_{ n } } converge weakly to w âˆˆ Îž, where w = lim_{nâ†’âˆž}P_{Îž}x_{ n } .
Remark 4.1.

(i)
For all n â‰¥ 1, let S_{ n } = S be a nonexpansive mapping, by Theorems 3.2, 4.2, 4.7, 4.8, and 4.9 we recover Theorem 3.1 in [5], Theorem 3.1 in [1], Theorem 5 in [26], Theorem 3.1 in [23], and Theorem 4.7 in [3]. In addition, let A = 0, by Theorems 4.6 and 4.5, respectively, we recover Theorems 3.1 and 4.1 in [11].

(ii)
For all n â‰¥ 1, let S_{ n } = S be a nonexpansive mapping, by Theorems 3.1, 4.3, and 4.4, respectively, we recover Theorems 4.3, 4.4, and 4.7 in [4] with some modified conditions on F.

(iii)
Theorems 3.1, 3.2, 4.34.7 also improve the main results in [10, 12, 13] because the inverse strongly monotonicity of A has been replaced by the monotonicity and Lipschitz continuity of A.
The following result illustrates that there are the nonexpansive mappings S_{1}, S_{2} ,... satisfying the condition (*).
Lemma 4.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonexpansive mapping of C into itself such that Fix(T) â‰ âˆ…. If we define for n âˆˆ {1, 2,...}, and x âˆˆ C, then the following results hold:

(a)
For any bounded subset K of C, there holds

(b)
.

(c)
for all i âˆˆ {1, 2,...} and for any bounded subset K of C, there holds
Proof.

(a)
It is due to Bruck [27, 28] (please also see Lemma 3.1 in [22]).

(b)
It follows from (a) that .
Moreover, it is obvious that . Hence, .

(c)
It can be proved by mathematical induction. In fact, it is clear that this conclusion holds for i = 1. Assume that the conclusion holds for i = m, that is, for any bounded subset K of C, there holds
(4.1)
We now prove that the conclusion also holds for i = m + 1. In fact, we observe that
It is easy to verify that S_{1}, S_{2},... are nonexpansive mappings. It follows from (4.1) and (4.2) that for any bounded subset K of C, there holds
From Lemma 4.1, we know that by Theorems 3.1 and 3.2, respectively, we can obtain the following results.
Theorem 4.10. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C Ã— C to R satisfying (A1)(A4). Let A be a monotone and kLipschitzcontinuous mapping of C into H and B be an Î±inversestrongly monotone mapping of C into H. Let T be a nonexpansive mapping of C into itself such that Î˜ = Fix(T)âˆ©VI(C, A)âˆ©GEP(F, B) â‰ âˆ…. Let {Î»_{ n } } âŠ‚ [a, b] for some , {r_{ n } } âŠ‚ [d, e] and for some d, e âˆˆ (0, 2Î±), and {Î±_{ n } }, {Î²_{ n } }, and {Î³_{ n } } be three sequences in [0, 1] satisfying the following conditions:

(i)
Î±_{ n } + Î²_{ n } â‰¤ 1 for all n âˆˆ N;

(ii)
;

(iii)
;

(iv)
and for all n âˆˆ N; If we define for n âˆˆ {1, 2,...}, and x âˆˆ C, then the sequences {x_{ n } }, {u_{ n } }, {y_{ n } }, and {z_{ n } } generated by algorithm (3.1) converge strongly to w = P _{Î˜}(x).
Theorem 4.11. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C Ã— C to R satisfying (A1)(A4). Let A be a monotone and kLipschitzcontinuous mapping of C into H and B be an Î±inversestrongly monotone mapping of C into H, and T be a nonexpansive mapping of C into itself such that Î˜ = Fix(T)âˆ©VI(C, A)âˆ©GEP(F, B) â‰ âˆ…. Assume that {Î»_{ n } } âŠ‚ [a, b] for some {Î²_{ n } } âŠ‚ [Î´, Îµ] for some Î´, Îµ âˆˆ (0, 1), and {r_{ n } } âŠ‚ [d, e] some d, e âˆˆ (0, 2Î±). If we define for n âˆˆ {1, 2,...} and x âˆˆ C, then the sequences {x_{ n } }, {u_{ n } }, and {y_{ n } } generated by algorithm (3.12) converge weakly to w âˆˆ Î˜, where w = lim_{nâ†’âˆž}P_{Î˜}x_{ n } .
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Acknowledgements
This research was supported by the National Natural Science Foundation of China, the Natural Science Foundation of Chongqing (Grant No. CSTC, 2009BB8240), and the Special Fund of Chongqing Key Laboratory (CSTC). The author is grateful to the referees for their detailed comments and helpful suggestions, which have improved the presentation of this article.
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Peng, JW. Some extragradient methods for common solutions of generalized equilibrium problems and fixed points of nonexpansive mappings. Fixed Point Theory Appl 2011, 12 (2011). https://doi.org/10.1186/16871812201112
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DOI: https://doi.org/10.1186/16871812201112