Viscosity Approximation Methods for Equilibrium Problems and a Finite Family of Nonspreading Mappings in a Hilbert Space

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1 Π46fiΠ2ffl μ ff ρ Vol. 46, No ffi3ß ADVANCES IN MATHEMATICS (CHINA) Mar., 2017 doi: /sxjz b Viscosity Approximation Methods for Equilibrium Problems and a Finite Family of Nonspreading Mappings in a Hilbert Space HUO Xiaoyan, ZHOU Haiyun, HE Jiangyan (Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang, Hebei, , P. R. China) Abstract: In this paper, we introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a finite family of nonspreading mappings in a real Hilbert space. We obtain a strong convergence theorem for the sequences generated by this iterative scheme. Keywords: viscosity approximation method; equilibrium problem; fixed point; nonspreading mapping MR(2010) Subject Classification: 47B38; 47G10 / CLC number: O Document code: A Article ID: (2017) Introduction Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let F be a bifunction of C C into R, wherer is the set of real numbers. The equilibrium problem for F : C C R is to find x C such that F (x, y) 0, y C. (0.1) The set of solutions of (0.1) is denoted by EP(F ). Given a mapping T : C H, let F (x, y) = Tx,y x for all x, y C. Then z EP(F ) if and only if Tz,y z 0 for all y C, i.e., z is a solution of the variational inequality. For solving the equilibrium problem for a bifunction F : C C R, let us assume that 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 all x, y C; (A3) For each x, y, z C, lim t 0 F (tz +(1 t)x, y) F (x, y); (A4) For each x C, y F (x, y) is convex and lower semicontinous. A mapping S : C C is called nonexpansive if Sx Sy x y, x, y C. Received date: Foundation item: Supported by NSFC (No ). huoxiaoyan2006@163.com

2 292 ADVANCES IN MATHEMATICS (CHINA) Vol. 46 Apointx C is called a fixed point of S if Sx = x. We denote by F (S) thesetoffixed points of S. IfC H is bounded, closed and convex and S is a nonexpansive mapping of C into itself, then F (S) isnonempty. A mapping S : C C is called quasi-nonexpansive if Sx z x z, x, y C, z Fix(S). A mapping S : C C is called nonspreading if 2 Sx Sy 2 Sx y 2 + x Sy 2, x, y C (0.2) or Sx Sy 2 x y 2 +2 x Sx,y Sy, x, y C. (0.3) Remark When F (S), a nonspreading mapping is a quasi-nonexpansive mapping. It is clear that every nonexpansive mapping with a nonempty set of fixed points is quasi-nonexpansive. Let {T i } N i=1 be a finite family of nonspreading mappings. Assume throughout the rest of the paper that N Fix(T i ). i=1 For n>n, T n is understood as T n mod N with the mod function taking values in {1, 2,,N}, see [1]. Let λ n,1,λ n,2,,λ n,n (0, 1], n N. Given the mappings T 1,T 2,,T N, following [9] one can define, for each n, mappings U n,1,u n,2,,u n,n by U n,1 = λ n,1 T 1 +(1 λ n,1 I), U n,2 = λ n,2 T 2 U n,1 +(1 λ n,2 I),. U n,n 1 = λ n,n 1 T N 1 U n,n 2 +(1 λ n,n 1 I), W n U n,n = λ n,n T N U n,n 1 +(1 λ n,n I). (0.4) Such a mapping W n is called the W n -mapping generated by T 1,T 2,,T N and λ n,1,λ n,2,,λ n,n. In 2010, Maingé [6] proposed a new analysis of viscosity approximation method in some framework which takes into account the wide class of demicontractive operators, and established the strong convergences of the sequence given by x n+1 = α n Cx n +(1 α n )T ω x n, where {α n } is a slow vanishing sequence, ω (0, 1], T ω := (1 ω)i + ωt, with some conditions on T. And at the same time, Kurokawa and Takahashi [4] proved the following strong convergence theorem for nonspreading mappings in a Hilbert space. Theorem 0.1 [4] Let C be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonspreading mapping of C onto itself. Let u C and define two sequences {x n } and

3 No. 2 Huo X. Y., et al.: Viscosity Approximation Methods for Equilibrium Problems 293 {z n } in C as follows: x 1 = x C and x n+1 = α n u +(1 α n )z n, z n = 1 n 1 T k x n n n=0 for all n =1, 2,,where0 α n 1, α n 0, and n=1 α n =. If F (T ) is nonempty, then {x n } and {z n } converge strongly to Pu,whereP is the metric projection of H onto F (T ). On the other hand, given any r>0, it was shown [10] that under suitable hypotheses on F (to be stated precisely in Section 1), the mapping T r : H C defined by T r (x) = {z C : F (z,y)+ 1r } y z,z x 0, y C (0.5) is single-valued and firmly nonexpansive and satisfies Fix(T r )=EP(F). Using this result, Takahashi [8] obtained the following result for generalized hybrid mappings of C onto H with equilibrium problem in a Hilbert space. Theorem 0.2 [8] Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. Let S be a generalized hybrid mapping of C onto H. Let F be a bifunction of C C onto R satisfying (A1) (A4). Let 0 <k<1, and let g be a k-contraction of H onto itself. Let V be a γ-strongly monotone and L-Lipschitzian continuous of H into itself with γ >0and L>0. Take μ, γ R as follows: 0 <μ< 2 γ L L 2, γ 2 0 <γ<. k Suppose that F (S) EP(F ). Letx 1 = x H, and let {x n } H be a sequence generated by F (u n,y)+ 1 y u n,u n x n 0, y C, r n x n+1 = β n x n +(1 β n ){α n γg(x n )+(I α n V )Su n } for all n N, where{βn} (0, 1), {α n } (0, 1) and {r n } (0, ) satisfying 2 μ lim α n =0, α n =, n=1 lim inf r n > 0, 0 < lim inf β n lim sup β n < 1. Then the sequence {x n } converges strongly to z 0 F (S) EP(F ), where z 0 = P F (S) EP(F ) (I V + γg)z 0. Using (0.5), Colao et al. [1] introduced a viscosity approximation method for finding a common element of EP(F )andfix(w n ), where is defined by (0.4) for a finite family of nonexpansive mappings. Starting with an arbitrary element x 1 H, they defined the sequences {u n } and {x n } recursively by F (u n,y)+ 1 y u n,u n x n 0, y H, r n x n+1 = ε n γf(x n )+βx n + ((1 β)i ε n A)W n u n.

4 294 ADVANCES IN MATHEMATICS (CHINA) Vol. 46 They proved that under certain appropriate hypotheses both sequences {u n } and {x n } converge stronglytoapointx F which is an equilibrium point for F and the unique solution of the variational inequality (A γf)x,x x 0, x F EP(F ). Finding an optimal point in the intersection N i=1 Fix(T i) of the fixed points set of a finite family of nonexpansive mappings with equilibrium problems in a Hilbert space is a frequent problem interest in various branches of science; see [2, 5, 7]. In this paper, we propose a new iteration algorithm and study viscosity approximation methods for equilibrium problems and a finite family of nonspreading mappings in a Hilbert space and then prove a strong convergence theorem. 1 Preliminaries Let H be a real Hilbert space with inner product, and norm. When {x n } is a sequence in H, x n xmeans that {x n } converges weakly to x, andx n x means the strong convergence. For a bounded sequence {x n } H, w w (x n )={x H : {x nj } {x n }, s.t. x nj x} denotes the weak w-limit set of {x n }. In a Hilbert space, the following equalities are well known: λx +(1 λ)y 2 = λ x 2 +(1 λ) y 2 λ(1 λ) x y 2, (1.1) 2 x y, z w = x w 2 + y z 2 x z 2 y w 2, (1.2) x, y = 1 2 x y x y 2 (1.3) for all x, y, z, w H and λ R. Let C be a nonempty closed convex subset of H. Then for any x H, there exists a unique nearest point in C, denoted by P C (x), such that x P C (x) x y, y C. (1.4) Such a P C is called the metric projection of H onto C. We know that P C is nonexpansive. Furthermore, for x H and z C, z = P C (x) x z,z y 0, y C. We also know that for any sequence {x n } H with x n x, the following inequality holds: lim inf x n x < lim inf x n y, y H with x y. Lemma 1.1 Let {Γ n } be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence {Γ nj } j 0 of {Γ n } which satisfies Γ nj < Γ nj+1 for all j 0. Also consider the sequence of integers {τ(n)} n n0 defined by τ(n) =max{k n Γ k < Γ k+1 }. Then {τ(n)} n n0 is a nondecreasing sequence verifying lim τ(n) = and for all n n 0,it holds that Γ τ (n) Γ τ (n)+1 and we have Γ n Γ τ (n)+1.

5 No. 2 Huo X. Y., et al.: Viscosity Approximation Methods for Equilibrium Problems 295 Lemma 1.2 [10] Let C be a nonempty closed convex subset of Hilbert space H and F be a bifunction of C C R satisfying (A1) (A4). Then for any given r>0andx H, there exists z C such that F (z,y)+ 1 y z,z x 0, y C. r Lemma 1.3 [10] Assume that F : C C R satisfies (A1) (A4). For r>0andx H, define a mapping T r : H C as follows: T r (x) = {z C : F (z,y)+ 1r } y z,z x 0, y C for all x H. Then the following statements hold: (1) T r is single-valued; (2) T r is firmly nonexpansive, i.e., for any x, y H, T r x T r y 2 T r x T r y, x y. This implies that T r x T r y x y, x, y H, i.e., T r is a nonexpansive mapping; (3) F (T r )=EP(F), r >0; (4) EP(F ) is closed and convex. Lemma 1.4 [3] Let C be a nonempty closed convex subset of Hilbert space H, T : C C a nonspreading mapping and F (T ). Then I T is demiclosed at θ, i.e., for any sequence {x n } C, such that x n zand x n Tx n 0, then z F (T ). Lemma 1.5 [11] Let C be a nonempty closed convex subset of Hilbert space H and T : C C a quasi-nonexpansive mapping. Then the fixed points F (T )oft is a nonempty closed convex subset of C. Lemma 1.6 [2] For all x, y H, there holds the inequality x + y 2 x 2 +2 y, x + y. Lemma 1.7 Let C and Ω be two nonempty closed convex subsets of a real Hilbert space H such that Ω C. Letz be a solution of the following variational inequality Let {x n } be a bounded sequence in C. Ifw w (x n ) Ω, then (I f)x, y x 0, y Ω. (VI) lim inf (I f)z,x n z 0. Proof Since C is a nonempty closed convex subset and {x n } is a bounded sequence in C, we know that {x nk } {x n } such that lim inf (I f)z,x n z = lim k (I f)z,x n k z. (1.5) Without loss of generality, we can assume that x nk ν (k ), then ν Ω. (VI) and (1.5) obviously lead to lim inf (I f)z,x n z = (I f)z,ν z 0, which is the desired result.

6 296 ADVANCES IN MATHEMATICS (CHINA) Vol. 46 Lemma 1.8 Let W n be defined by (0.4), and T i (i =1, 2,,N) be a finite family of nonspreading mappings. If N i=1 F (T i), thenwehave (1) N i=1 F (T i)=f (W n ); (2) For n N, W n is quasi-nonexpansive; (3) For n N,W n is demi-closed. Proof (1) First we show that N i=1 F (T i) F (W n ). When x N i=1 F (T i), we have T i x = x (i =1, 2,,N), then U n,1 x = λ n,1 (T 1 x x)+x = x, U n,2 x = λ n,2 (T 2 U n,1 x x)+x = λ n,2 (T 2 x x)+x = x. Suppose when i = N 1, we have U n,n 1 x = x. Then when i = N, weobtain W n x = U n,n x = λ n,n (T N U n,n 1 x x)+x = λ n,n (T N x x)+x = x. So we have x F (W n ). Next we show that F (W n ) N i=1 F (T i). When N =1,U n,1 x = λ n,1 (T 1 x x) +x = x, then T 1 x = x. Suppose when n = N 1, T N 1 x = x. Then when n = N, wehave W n x = U n,n x = λ n,n (T N U n,n 1 x x)+x = λ n,n (T N x x)+x = x. Thus T N x = x. So x N i=1 F (T i), which completes the proof. (2) q N i=1 F (T i), by Remark, when N =1,wehave U n,1 x q = λ n,1 T 1 x +(1 λ n,1 )x q = λ n,1 (T 1 x q)+(1 λ n,1 )(x q) λ n,1 T 1 x q +(1 λ n,1 ) x q λ n,1 x q +(1 λ n,1 ) x q = x q. So U n,1 is quasi-nonexpansive. Suppose that U n,n 1 is quasi-nonexpansive, i.e., U n,n 1 x q x q. Next we show that U n,n is quasi-nonexpansive. W n x q = U n,n x q = λ n,n T N U n,n 1 x +(1 λ n,n )x q = λ n,n (T N U n,n 1 x q)+(1 λ n,n )(x q) λ n,n U n,n 1 x q +(1 λ n,n ) x q λ n,n x q +(1 λ n,1 ) x q = x q. So W n is quasi-nonexpansive. (3) We only need to show when x n x,w n x n x n 0, then x F (W n ), i.e., x N i=1 F (T i). We prove it by induction. When N =1, U n,1 x n x n = λ n,1 (T 1 x n x n ) 0,

7 No. 2 Huo X. Y., et al.: Viscosity Approximation Methods for Equilibrium Problems 297 T 1 is a nonspreading mapping, so x Fix(T 1 ). Since x n x,then U n,1 x n x.when N =2, U n,2 x n x n = λ n,2 (T 2 U n,1 x n x n ) 0 T 2 U n,1 x n x n 0, T 2 U n,1 x n U n,1 x n T 2 U n,1 x n x n + U n,1 x n x n 0, so x Fix(T 2 ), i.e., U n,2 is demi-closed. Suppose that U n,n 1 is demi-closed, next we show that W n is demi-closed. W n x n x n = λ n,n (T N U n,n 1 x n x n ) 0, T N U n,n 1 x n U n,n 1 x n T N U n,n 1 x n x n + U n,n 1 x n x n 0. Because of U n,n 1 x n x,wehavex Fix(T N ), that is to say W n is demi-closed. 2MainResults Theorem 2.1 Let C be a nonempty closed convex subset of Hilbert space H, T i : C C, i =1, 2,,N be a finite family of nonspreading mappings such that N i=1 F (T i), f : C C be a contraction on C with a constant ρ (0, 1), and let λ n,1,λ n,2,,λ n,n be a sequence in [a, b] with0<a b<1, for every n N, W n be the mapping generated by T 1,T 2,,T N and λ n,1,λ n,2,,λ n,n. Let F be a bifunction from C C R satisfying (A1) (A4). Assume that N i=1 F (T i) EP(F ). Let x 1 H, {x n } and {u n } be sequences defined by F (u n,y)+ 1 y u n,u n x n 0, y C, r n (2.1) x n+1 = β n x n +(1 β n )[α n f(x n )+(1 α n )W n u n ], where {α n } [0, 1] and {r n } (0, ) satisfying (i) lim α n (iii) 0 < lim inf β n lim sup β n < 1. Then both {x n } and {u n } converge strongly to z N = 0; (ii) n=1 α n = ; i=1 F (T i) EP(F ), where z = P N i=1 F (T i) EP(F ) f(z). Proof We split the proof into five steps. Step 1: Show that the variational inequality (VI) has a unique solution in N i=1 F (T i) EP(F ). Since T i : C C (i =1, 2,,N) is a nonspreading mapping with N i=1 F (T i), by Remark, we know that T i is a quasi-nonexpansive mapping, and hence it follows from Lemma 1.5 that N i=1 F (T i) is a nonempty closed and convex subset of C. By Lemma 1.3, we know that EP(F ) is also closed and convex. Consequently, N i=1 F (T i) EP(F ) is nonempty, closed and convex, and then the metric projection P N i=1 F (T i) EP(F ) is well defined. Note that the fact that z is a solution of variational inequality (VI) is equivalent to that z is a fixed point of P N i=1 F (T i) EP(F ) f. Since f : C C is a contraction and P N i=1 F (T i) EP(F ) : C C is nonexpansive, we have P N i=1 F (Ti) EP(F )f : C C is a contraction. By the Banach contraction mapping principle, we know that P N i=1 F (Ti) EP(F )f has a unique fixed point z in N i=1 F (T i) EP(F ), which implies that (I f)z,y z 0, Step 2: Show that {x n } is bounded. N y F (T i ) EP(F ). i=1

8 298 ADVANCES IN MATHEMATICS (CHINA) Vol. 46 Let v N i=1 F (T 1 i) EP(F ), M =max{ x 1 v, 1 ρ f(v) v }. Then from (0.3) and Lemma 1.3, Lemma 1.8, noting that u n = T rn x n,wehave Then by (2.1) (2.3) we have u n v = T rn x n T rn v x n v, n N, (2.2) W n u n v 2 u n v 2 x n v 2. (2.3) x n+1 v = β n x n +(1 β n )[α n f(x n )+(1 α n )W n u n ] v = β n (x n v)+(1 β n )α n (f(x n ) v)+(1 β n )(1 α n )(W n u n v) β n x n v +(1 β n )α n ρ x n v +(1 β n )α n f(v) v +(1 β n )(1 α n ) W n u n v 1 (1 (1 β n )(1 ρ)α n ) x n v +(1 β n )α n (1 ρ) f(v) v 1 ρ M. So we have x n v M for any n N and hence {x n } is bounded. We also obtain that {u n }, {W n u n }, {W n x n } and {f(x n )} are all bounded. Step 3: Show that W n u n u n 0. First show that W n u n x n 0. Let z be the solution of (VI). By the definition of {x n } we have x n+1 x n +(1 β n )α n (x n f(x n )) = (1 β n )(1 α n )(W n u n x n ), (2.4) so x n+1 x n +(1 β n )α n (x n f(x n )),x n z = (1 β n )(1 α n ) x n W n u n,x n z. (2.5) From (1.2) and (2.3), we have 2 x n W n u n,x n z = x n z 2 + W n u n x n 2 W n u n z 2 W n u n x n 2, (2.6) which together with (2.5) yields x n x n+1,x n z (1 β n )α n (I f)x n,x n z 1 2 (1 β n)(1 α n ) W n u n x n 2. (2.7) Furthermore, using (1.3) we have x n x n+1,x n z = 1 2 x n+1 z x n x n x n z 2 (1 β n )α n (I f)x n,x n z 1 2 (1 β n)(1 α n ) W n u n x n 2. Denote Γ n = 1 2 x n z 2.Then Γ n+1 Γ n 1 2 x n x n+1 2 (1 β n )α n (I f)x n,x n z 1 2 (1 β n)(1 α n ) W n u n x n 2. (2.8)

9 No. 2 Huo X. Y., et al.: Viscosity Approximation Methods for Equilibrium Problems 299 Noting that x n+1 x n = (1 β n )α n (f(x n ) x n )+(1 β n )(1 α n )(W n u n x n ) (1 β n )(α n f(x n ) x n +(1 α n ) W n u n x n ), we have x n+1 x n 2 (1 β n ) 2 [α n f(x n ) x n +(1 α n ) W n u n x n ] 2 =(1 β n ) 2 α 2 n f(x n ) x n 2 +(1 β n ) 2 (1 α n ) 2 W n u n x n 2 +2(1 β n ) 2 α n (1 α n ) f(x n ) x n W n u n x n. By (2.8), we obtain Γ n+1 Γ n (1 β n)(1 α n )(1 (1 β n )(1 α n )) W n u n x n 2 ( 1 (1 β n )α n 2 (1 β n)α n f(x n ) x n 2 ) +(1 α n )(1 β n ) f(x n ) x n W n u n x n (I f)x n,x n z. (2.9) Then we divide the proof into two cases: Case 1: Suppose 0 Γ n+1 Γ n, n 1, i.e., {Γ n } is nonincreasing. In this case, {Γ n } is convergent, so that lim (Γ n+1 Γ n ) = 0. By using the conditions (i), (iii) and the boundedness of {x n },weobtain lim W nu n x n =0. (2.10) Next we show that x n u n 0. Let v F (W n ) EP(F ), from (1.2) we have u n v 2 = T rn x n T rn v 2 T rn x n T rn v, x n v = u n v, x n v = 1 2 ( u n v 2 + x n v 2 x n u n 2 ), i.e., u n v 2 x n v 2 x n u n 2, (2.11) which together with (2.3) yields W n u n v 2 u n v 2 x n v 2 x n u n 2. (2.12) Because of the boundedness of {x n } and {W n u n }, we put M =sup n 1 { x n v + W n u n v }, then x n u n 2 x n v 2 W n u n v 2 M W n u n x n. (2.13) By (2.10), we obtain lim x n u n =0. (2.14) Noting that W n u n u n W n u n x n + x n u n, we also have lim W nu n u n =0. (2.15)

10 300 ADVANCES IN MATHEMATICS (CHINA) Vol. 46 Case 2: Suppose there exists a subsequence {Γ nk } k 0 {Γ n }, such that Γ nk < Γ nk +1, k 0. In this case, we define τ : N N by τ(n) =max{k n Γ k < Γ k+1 }, then it follows that Γ τ (n)+1 Γ τ (n) 0. By Lemma 1.1 and (2.9), we have 1 2 (1 β τ (n))(1 α τ (n) )(1 (1 β τ (n) )(1 α τ (n) )) W n u τ (n) x τ (n) 2 (1 β τ (n) )α τ (n) ((1 β τ (n) )α τ (n) f(x τ (n) ) x τ (n) 2 +(1 α τ (n) )(1 β τn ) f(x τ (n) ) x τ (n) W n u τ (n) x τ (n) (2.16) (I f)x τ (n),x τ (n) z ). Hence, by the boundedness of {x n }, the conditions (i) and (iii), we immediately obtain lim W nu τ (n) x τ (n) 2 =0, lim (Γ τ (n) Γ τ (n)+1 =0). (2.17) With similar arguments of (2.11) (2.14), we have lim x τ (n) u τ (n) =0, (2.18) then lim W nu τ (n) u τ (n) =0. (2.19) Step 4: Assume that there exists a subsequence {x nk } of {x n } which converges weakly to w, w w w (x n ), show that w w (x n ) F (W n ) EP(F ). By Lemma 1.8, we know that W n is demi-closed, so w F (W n ). Next we show w EP(F ). By u n = T rn x n,wehave From the monotonicity of F,wehave F (u n,y)+ 1 r n y u n,u n x n 0, y C. 1 r n y u n,u n x n F (u n,y) F (y, u n ), y C and hence 1 y u nk, (u nk x nk ) F (y, u nk ), y C. r nk Since lim inf r n a>0, we have u n k x nk r nk 0andu nk w.from(a4)wehave F (y, w) 0, y C. For 0 t 1andy C, lety t = ty +(1 t)w. Since y, w C, wehavey t C and hence F (y t,w) 0. So from (A1) and (A4) we have 0=F (y t,ty+(1 t)w) tf (y t,y)+(1 t)f (y t,w) tf (y t,y)

11 No. 2 Huo X. Y., et al.: Viscosity Approximation Methods for Equilibrium Problems 301 and hence F (y t,y) 0. From (A3) we have F (w, y) 0, y C and hence w EP(F ). Therefore w F (W n ) EP(F ), and w w (x n ) F (W n ) EP(F ). Step 5: Finally, we show that both {x n } and {u n } converge strongly to z, where z = P F (Wn) EP(F )f(z). In Case 1, from (2.9) and (2.10), we obtain [ ] 1 (1 β n )α n 2 (1 β n)α n f(x n ) x n 2 (I f)x n,x n z Γ n Γ n+1, then by the condition (ii), we deduce that ( lim inf 1 ) 2 (1 β n)α n f(x n ) x n 2 + (I f)x n,x n z 0, or equivalently (as (1 β n )α n f(x n ) x n 2 0), lim inf (I f)x n,x n z 0. (2.20) Put F = I f, then Fx Fy,x y (1 ρ) x y 2,andso (I f)x n (I f)z,x n z (1 ρ) x n z 2 =2(1 ρ)γ n, which by (2.20) entails lim inf (2(1 ρ)γ n + (I f)z,x n z ) 0. Invoking Lemma 1.7, Steps 1 and 4 and recalling that lim Γ n exists, we have 0 2(1 ρ) lim Γ n lim inf (I f)z,x n z 0. So lim Γ n =0, i.e., {x n } converges strongly to z. In Case 2, similarly, from (2.16) we can deduce 2(1 ρ) lim sup Γ τ (n) lim inf (I f)z,x τ (n) z. Invoking Lemma 1.7 and Steps 1 and 4, we have lim inf (I f)z,x τ (n) z 0. So lim Γ τ (n) = 0 and lim Γ τ (n)+1 =0 by (2.17). Recalling that Γ τ (n)+1 Γ n (by Lemma 1.1), we get lim Γ n =0, so that x n z strongly. As a direct consequence of Theorem 2.1, we obtain the following result.

12 302 ADVANCES IN MATHEMATICS (CHINA) Vol. 46 Corollary 2.1 Let C be a nonempty closed convex subset of Hilbert space H, S : C C a nonspreading mapping such that F (S), and f : C C a contraction on C with a constant ρ (0, 1). Assume that F (S). Let x 1 C, {x n } be a sequence defined by x n+1 = β n x n +(1 β n )[α n f(x n )+(1 α n )Sx n ], where {α n } [0, 1] satisfying (i) lim α n = 0; (ii) n=1 α n = ; (iii) 0 < lim inf β n lim sup β n < 1. Then {x n } converges strongly to z F (S), where z = P F (S) f(z). References [1] Colao, V., Marino, G. and Xu, H.K., An iterative method for finding common solutions of equilibrium and fixed point problems, J. Math. Anal. Appl., 2008, 344(1): [2] Hu, C.S. and Cai, K., Viscosity approximation schemes for fixed point problems and equilibrium problems and variational inequality problem, Nonlinear Anal., 2010, 72(3/4): [3] Ingarden, R.S., Über die Einbetting eines Finslerschen Rammes in einan Minkowskiischen Raum, Bull. Acad. Polon. Sci., 1954, 2: (in German). [4] Kurokawa, Y. and Takahashi, W., Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces, Nonlinear Anal., 2010, 73(6): [5] Li, H.Y. and Su, Y.F., Strong convergence theorem by a new hybrid method for equilibrium problems and variational inequality problems, Nonlinear Anal., 2010, 72(2): [6] Maingé, P.-E., The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces, Comput. Math. Appl., 2010, 59(1): [7] Qin, X.L., Cho, S.Y. and Kang, S.M., Some results on generalized equilibrium problems involving a family of nonexpansive mappings, Appl. Math. Comput., 2010, 217(7): [8] Takahashi, W., Wong, N.C. and Yao, J.C., Iterative common solutions for monotone inclusion problems, fixed point problems and equilibrium problems, Fixed Point Theory Appl., 2012, 182(2): [9] Yao, Y.H., A general iterative method for a finite family of nonexpansive mappings, Nonlinear Anal., 2007, 66(12): [10] Yao, Y.H., Noor, M.A. and Liou, Y.C., On iterative methods for equilibrium problems, Nonlinear Anal., 2009, 70(1): [11] Zhang, D.K. and Zhou, H.Y., Another iterative algorithm on fixed points for closed and quasi-nonexpansive mappings in Hilbert spaces, J. Hebei Normal Univ. Nat. Sci. Ed., 2009, 33(5): (in Chinese). 0ψfl-*&:)%/.541#,83+χ79fi("! =?@, C;B, <>A ( Ψfffl»±Λ οφξ Ων, Ψff,, ) 62 xhnpfk^z, recqupkmhti, aty D KRQ_D ]Xm lk[yvuosgzsfkgol[ykvww, EJIcqUdjbN`. $'ffi y D RQ; ]Xml; GOL; Sgzsf