RESEARC H Open Access Strong convergence theorem for amenable semigroups of nonexpansive mappings and variational inequalities Hossein Piri * and Ali Haji Badali * Correspondence: h. piri@bonabetu.ac.ir Department of Mathematics, University of Bonab, Bonab 55517- 61167, Iran Abstract In this paper, using strongly monotone and lipschitzian operator, we introduce a general iterative process for finding a common fixed point of a semigroup of nonexpansive mappings, with respect to strongly left regular sequence of means defined on an appropriate space of bounded real-valued functions of the semigroups and the set of solutions of variational inequality for b-inverse strongly monotone mapping in a real Hilbert space. Under suitable conditions, we prove the strong convergence theorem for approximating a common element of the above two sets. Mathematics Subject Classification 2000: 47H09, 47H10, 43A07, 47J25 Keywords: projection, common fixed point, amenable semigroup, iterative process, strong convergence, variational inequality 1 Introduction Throughout this paper, we assume that H is a real Hilbert space with inner product and norm are denoted by 〈.,.〉 and || . ||, respectively, and let C be a nonempty closed convex subset of H.AmappingT of C into itself is called nonexpansive if || Tx - Ty ||≤|| x - y ||, for all x, y Î H.ByFix(T), we denote the set of fixed points of T (i.e., Fix (T)={x Î H : Tx = x}), it is well known that Fix(T) is closed and convex. Recall that a self-mapping f : C ® C is a contraction on C if there exists a constant a Î [0 , 1) such that || f(x)-f(y)||≤ a || x - y || for all x, y Î C. Let B : C ® H be a mapping. The variational inequality problem, denoted by VI(C, B), is to fined x Î C such that  Bx, y − x  ≥ 0 , (1) for all y Î C. The variational inequality problem has been extensively studied in lit- erature. See, for example, [1,2] and the references therein. Definition 1.1 Let B : C ® H be a mapping. Then B (1) is called h-strongly monotone if there exists a positive constant h such that  Bx − By, x − y  ≥ η  x − y  2 , ∀x, y ∈ C , Piri and Badali Fixed Point Theory and Applications 2011, 2011:55 http://www.fixedpointtheoryandapplications.com/content/2011/1/55 © 2011 Piri and Badali; licens ee Springer. This is an Open Access articl e distributed under the terms of the Creative Commons Attribution License (http://crea tivecommons.org/licens es/by/2.0), which permits unrestricted use, distribution, and re prod uction in any medium, provide d the origin al work is properly cited. (2) is called k-Lipschitzian if there exist a positive constant k such that  Bx −−B y ≤ k  x − y , ∀x, y ∈ C , (3) is called b-inverse strongly monotone if there exists a positive real number b >0 such that  Bx − By, x − y  ≥ β  Bx − By  2 , ∀x, y ∈ C . It is obvious that any b-inverse strongly monotone mapping B is 1 β -Lipschitzian. Moudafi [3] introduced the viscosity approximation method for fixed point of nonex- pansive mappings (see [4] for further developments in both Hilbert and Banach spaces). Starting with an arbitrary initial x 0 Î H, define a sequence {x n } recursively by x n+1 = ( 1 − α n ) Tx n + α n f ( x n ) , n ≥ 0 , (2) where a n is sequence in (0, 1). Xu [4,5] proved that under certain appropriate condi- tions on {a n }, the sequences {x n } generated by (2) strongly converges to the unique solution x* in Fix(T) of the variational inequality:  ( f − I ) x ∗ , x − x ∗ ≤0, ∀x ∈ Fix ( T ). Let A is strongly positive operator on C. That is, there is a constant ¯ γ > 0 with the property that Ax, x≥ ¯ γ  x  2 , ∀x ∈ C . In [5], it is proved that the sequence {x n } generated by the iterative method bellow with initial guess x 0 Î H chosen arbitrarily, x n+1 = ( I − α n A ) Tx n + α n u, n ≥ 0 , (3) converges strongly to the unique solution of the minimization problem min x∈Fix ( T ) 1 2  Ax, x  −  x, b  , where b is a given point in H. Combining the iterati ve method (2) and (3), M arino and Xu [6] consider the fo llow- ing iterative method: x n+1 = ( I − α n A ) Tx n + α n γ f ( x n ) , n ≥ 0 , (4) it is proved that if the sequence {a n } of parameters satisfies the following conditions: (C 1 ) a n ® 0, (C 2 ) ∞  n=0 α n = ∞ , C 3 ) either ∞  n = 0 | α n+1 − α n | < ∞ or lim n →∞ α n+1 α n = 1 . then, the sequence {x n } generated by (4) converges strongly, as n ® ∞, to the unique solution of the variational inequality:  (γ f − A)x ∗ , x − x ∗  ≤ 0, ∀x ∈ Fix(T) , Piri and Badali Fixed Point Theory and Applications 2011, 2011:55 http://www.fixedpointtheoryandapplications.com/content/2011/1/55 Page 2 of 16 which is the optimality condition for minimization problem min x∈Fix ( T ) 1 2  Ax, x  − h(x) , where h is a potential function for gf (i.e., h’(x)=gf(x), for all x Î H). Some people also study the application of the iterative method (4) [7,8]. Yamada [9] introduce the following hybrid iterative method for solving the varia- tional inequality: x n+1 = Tx n − μα n F ( Tx n ) , n ∈ N , (5) where F is k-Lipschitzian and h-strongly monotone ope rator with k>0, h >0, 0 <μ< 2η k 2 , then he proved that if {a n } satisfying appropriate conditions, then {x n } generated by (5) converges strongly to the unique solution of the variational inequality:  Fx ∗ , x − x ∗  ≥ 0, ∀x ∈ Fix(T) . In 2010, Tian [ 10] combined the iterative (4) with the iterative method (5) and con- sidered the iterative methods: x n+1 = ( I − μα n F ) Tx n + α n γ f ( x n ) , n ≥ 0 , (6) andheprovethatifthesequence{a n } of parameters satisfies the con ditions (C 1 ), (C 2 ), and (C 3 ), then the sequences {x n } generated by (6) converges strongly to the unique solution x* Î Fix(T) of the variational inequality:  ( μF − γ f ) x ∗ , x − x ∗ ≥0, ∀x ∈ Fix ( T ). In this paper motivated and inspired by Atsushiba and Takahashi [11], Ceng and Yao [12], Kim [13], Lau et al. [14], Lau et al [15], Marino and Xu [6], Piri and Vaezi [16], Tian [10], Xu [5] and Yamada [9], we introduce the following general iterative algo- rithm: Let x 0 Î C and  y n = β n x n +(1− β n )P C (x n − δ n Bx n ), x n+1 = α n γ f (x n )+(I − α n μF)T μ n y n , n ≥ 0 . (7) where P C isametricprojectionofHontoC,Bisb-inverse strongly monotone,  = {T t : t Î S} is a nonexpansive semigroup on H such that the set F = Fix ( ϕ ) ∩ VI ( C, B ) = ∅ , , X isasubspaceofB(S)suchthat1Î X and the mapping t ® 〈T t x, y〉 is an element of X for each x, y Î H, and {μ n } is a sequence of means on X. Our purpose in this paper is to introduce this general iterative algorithm for approxi- mating a common element of the set of co mmon fixed point of a semigroup of nonex- pansive mappings and the set of solutions of variational inequality for b-inverse strongly monotone mapping which solves some variatio nal inequality. We will prove that if {μ n } is left regular and the sequences {a n }, {b n }, and {δ n } of parameters satisfies appropriate conditions, then the sequences {x n }and{y n } generated by (7) conve rges strongly to the unique solution x ∗ ∈ F of the variational inequalities:   (μF − γ f )x ∗ , x − x ∗  ≥ 0, ∀x ∈ F ,  Bx ∗ , y − x ∗  ≥ 0 ∀y ∈ C. Piri and Badali Fixed Point Theory and Applications 2011, 2011:55 http://www.fixedpointtheoryandapplications.com/content/2011/1/55 Page 3 of 16 2 Preliminaries Let S be a semigroup and let B(S) be the space of all bounded real-valued functions defined on S with supremum norm. For s Î S and f Î B(S), we define elements l s f and r s f in B(S)by ( l s f )( t ) = f ( st ) , ( r s f )( t ) = f ( ts ) , ∀t ∈ S . Let X beasubspaceofB(S)containing1andletX* be its topological dual. An ele- ment μ of X* issaidtobeameanonX if || μ || = μ(1) = 1. We often write μ t (f(t)) instead of μ(f)forμ Î X*andf Î X.LetX be left invariant (resp. right invariant), i.e., l s (X) ⊂ X (resp. r s (X) ⊂ X) for each s Î S.Ameanμ on X is said to be left invariant (resp. right invariant) if μ(l s f)=μ(f)(resp.μ(r s f)=μ(f)) for each s Î S and f Î X. X is said to be left (resp. right) amenable if X has a left (resp. right) invariant mean. X is amenable if X is both left and right amenable. As is well known, B(S)isamenable when S is a co mmutative semigroup, see [ 15]. A net {μ a } of means on X is said to be strongly left regular if lim α  l s ∗ μ α − μ α =0, for each s Î S, where l ∗ s is the adjoint operator of l s . Let S be a se migroup and let C be a nonempty closed and conve x subset of a reflex- ive Banach space E.Afamily ={T t : t Î S}ofmappingfromC into itself is said to be a nonexpansive semigroup on C if T t is nonexpansive and T ts = T t T s for each t, s Î S.ByFix(), we denote the set of common fixed points of , i.e., Fix(ϕ)=  t∈S {x ∈ C : T t x = x} . Lemma 2.1 [15]Let S be a semigroup and C be a nonempty closed convex subset of a reflexive Banach s pace E. Let  ={T t : t Î S} be a nonexp ansive semigroup on H such that {T t x : t Î S} is bounded for some x Î C, let X be a s ubspace of B(S) such that 1 Î X and the mapping t ® 〈T t x, y*〉 is an element of X for each x Î C and y* Î E*, and μ is a mean on X. If we write T μ x instead of  T t xdμ(t ) , then the followings hold . (i) T μ is non-expansive mapping from C into C. (ii) T μ x = x for each x Î Fix(). (iii) T μ x ∈ co{T t x : t ∈ S } for each x Î C. Let C be a nonempty subset of a Hilbert space H and T : C ® H a mapping. Then T is said to be demiclosed at v Î H if, for any sequ ence {x n }inC, the following implica- tion holds: x n → u ∈ C, Tx n → vim p l y Tu = v , where ® (resp. ⇀) denotes strong (resp. weak) convergence. Lemma 2.2 [17]LetCbeanonemptyclosedconvexsubsetofaHilbertspaceHand suppose that T : C ® H is nonexpansive. Then, the mapping I - T is demiclosed at zero. Lemma 2.3 [18]For a given x Î H, y Î C, y = P C x ⇔  y − x, z − y  ≥ 0, ∀z ∈ C . Piri and Badali Fixed Point Theory and Applications 2011, 2011:55 http://www.fixedpointtheoryandapplications.com/content/2011/1/55 Page 4 of 16 It is well known that P C is a firmly nonexpansive mapping of H onto C and satisfies  P C x − P C y  2 ≤  P C x − P C y, x − y  , ∀x, y ∈ H (8) Moreover, P C is characterized by the following properties: P C x Î C and for all x Î H, y Î C,  x − P C x, y − P C x  ≤ 0 . (9) It is easy to see that (9) is equivalent to the following inequality  x − y  2 ≥ x − P C x  2 +  y − P C x  2 . (10) Using Lemma 2.3, one can see that the variational inequality (24) is equivalent to a fixed point problem. It is easy to see that the following is true: u ∈ VI ( C, B ) ⇔ u = P C ( u − λBu ) , λ>0 . (11) A set-valued mapping U : H ® 2 H is call ed monotone if for all x, y Î H, f Î Ux and g Î Uy imply 〈x-y, f - g〉 ≥ 0. A monotone mapping U : H ® 2 H is maximal if the graph of G(U)ofU is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping U is maximal if and only if for (x, f) Î H×H, 〈x - y, f - g〉 ≥ 0forevery(y, g) Î G(U) implies that f Î Ux.LetB beamono- tone mapping of C into H and let N C x be the normal cone to C at x Î C, that is, N C x ={y Î H : 〈z - x, y〉 ≤ 0, ∀z Î C} and define Ux =  Bx + N C x, x ∈ C , ∅ x /∈ C . (12) Then U is the maximal monotone and 0 Î Ux if and only if x Î VI(C, B); see [19]. The following lemma is well known. Lemma 2.4 Let H be a real Hilbert space. Then, for all x, y Î H  x − y  2 ≤x  2 +2  y, x + y  , . Lemma 2.5 [5]Let {a n } be a sequence of nonnegative real numbers such that a n+1 ≤ ( 1 − b n ) a n + b n c n , n ≥ 0 , where {b n } and {c n } are sequences of real numbers satisfying the following conditions: (i) {b n } ⊂ (0, 1), ∞  n = 0 b n = ∞ , (ii) either lim sup n →∞ c n ≤ 0 or ∞  n = 0 | b n c n | < ∞ . Then, lim n →∞ a n =0 . As far as we know, the following lemma has been used implicitly in some papers; for the sake of completeness, we include its proof. Lemma 2.6 LetHbearealHilbertspaceandFbeak-Lipschitzianandh-strongly monotone operator with k >0, h >0. Let 0 <μ< 2η k 2 and τ = μ(η − μk 2 2 ) . Then for t ∈ (0, min{1, 1 τ } ) , I - t μ F is contraction with constant 1-tτ. Piri and Badali Fixed Point Theory and Applications 2011, 2011:55 http://www.fixedpointtheoryandapplications.com/content/2011/1/55 Page 5 of 16 Proof. Notice that  (I − tμF)x − (I − tμF)y  2 =  (I − tμF)x − (I − tμF)y,(I − tμF)x − (I − tμF)y  =  x − y  2 + t 2 μ 2  Fx − Fy  2 − 2tμ  x − y, Fx − Fy  ≤x − y  2 + t 2 μ 2 k 2  x − y  2 − 2tμη  x − y  2 ≤x − y  2 + tμ 2 k 2  x − y  2 − 2tμη  x − y  2 =  1 − 2tμ  η − μk 2 2   x − y  2 =(1− 2tτ )  x − y  2 ≤ ( 1 − tτ ) 2  x − y  2 . It follows that  ( I − tμF ) x − ( I − tμF ) y ≤ ( 1 − tτ )  x − y  , and hence I - tμF is contractive due to 1 - tτ Î (0, 1). □ Notation Throughout the rest of this paper, F will denote a k-Lipschitzian and h- strongly monotone operator of C into H with k>0, h >0, f is a contraction on C with coefficient 0 < a <1. We will al so always use g to mean a number in (0, τ α ) ,where τ = μ(η − μk 2 2 ) and 0 <μ< 2 η k 2 . The open ball of radius r centered at 0 is denoted by B r and for a subset D of H,by co D , we denote the closed convex hull of D. For ε >0 and a mapping T : D ® H,weletF ε (T; D) be the set of ε-approximate fixed points of T, i.e., F ε (T; D)={x Î D :||x - T x || ≤ ε }. Weak convergence is denoted by ⇀ and strong convergence is denoted by ®. 3 Main results Theorem 3.1 Let S be a semigroup, C a nonempty closed convex subset of real Hilbert space H an d B : C ® Hbeab-inverse strongly monotone. Let  ={T t : t Î S} be a nonexpansive semigroup of C into itself such that F = Fix ( ϕ ) ∩ VI ( C, B ) = ∅ , , Xaleft invariant subspace of B(S) such that 1 Î X, and the function t ® 〈T t x, y〉 is an element of X fo r each x Î CandyÎ H,{μ n } a left regular sequence of means on X such that  ∞ n =1  μ n+1 − μ n < ∞ . Let {a n } and {b n } be sequences in (0, 1) and {δ n } be a sequence in [a, b], where 0 <a<b<2b. Suppose the following conditions are satisfied. (B 1 ) lim n®∞ a n = 0, lim n®∞ b n =0, (B 2 )  ∞ n =1 α n = ∞ , (B 3 )  ∞ n =1 | α n+1 − α n | < ∞ ,  ∞ n =1 | β n+1 − β n | < ∞ ,  ∞ n =1 | δ n+1 − δ n | < ∞ . If {x n } and {y n } be generated by x 0 Î C and  y n = β n x n +(1− β n )P C (x n − δ n Bx n ), x n+1 = α n γ f (x n )+(I − α n μF)T μ n y n , n ≥ 0 . Then,{x n } and {y n } converge strongly, as n ® ∞, to x ∗ ∈ F , which is a unique solution of the variational inequalities: Piri and Badali Fixed Point Theory and Applications 2011, 2011:55 http://www.fixedpointtheoryandapplications.com/content/2011/1/55 Page 6 of 16   (μF − γ f )x ∗ , x − x ∗  ≥ 0, ∀x ∈ F ,  Bx ∗ , y − x ∗  ≥ 0 ∀y ∈ C. Proof.Since{a n } satisfies in condition (B 1 ), we may assume, with no loss of general- ity, that α n < min{1, 1 τ } .SinceB is b-inverse strongly monotone and δ n <2b,forany x, y Î C, we have  (I − δ n B)x − (I − δ n B)y  2 =  (x − y) − δ n (Bx − By)  2 =  x − y  2 − 2δ n  x − y, Bx − By  + δ 2 n  Bx − By  2 ≤x − y  2 − 2δ n β  Bx − By  2 + δ 2 n  Bx − By  2 =  x − y  2 + δ n (δ n − 2β)  Bx − By  2 ≤x − y  2 . It follows that  ( I − δ n B ) x − ( I − δ n B ) y ≤  x − y  . (13) Let p ∈ F , in the context of the variational inequality problem, the characterization of projection (11) implies that p = P C (p - δ n B p ). Using (13), we get  y n − p  = β n x n +(1− β n )P C (x n − δ n Bx n ) − p  =  β n [x n − p]+(1− β n )[P C (x n − δ n Bx n ) − P C (p − δ n Bp)]  ≤ β n  x n − p  +(1− β n )  P C (x n − δ n Bx n ) − P C (p − δ n Bp)  ≤ β n  x n − p  + ( 1 − β n )  x n − p  =  x n − p  . (14) We claim that {x n } is bounded. Let p ∈ F , using Lemma 2.6 and (14), we have  x n+1 − p  =  α n γ f (x n )+(I − α n μF)T μ n y n − p  =  α n γ f (x n )+(I − α n μF)T μ n y n − (I − α n μF)p − α n μFp  ≤ α n  γ f (x n ) − μFp  +  (I − α n μF)T μ n y n − (I − α n μF)p  ≤ α n  γ f (x n ) − γ f (p)  + α n  γ f (p) − μFp  +(1− α n τ )  T μ n y n − p  ≤ α n γα  x n − p  + α n  γ f (p) − μFp  +(1− α n τ )  y n − p  ≤ α n γα  x n − p  + α n  γ f (p) − μFp  +(1− α n τ )  x n − p  =(1− α n (τ − γα))  x n − p  + α n  γ f (p) − μFp  ≤ max{ x n − p , ( τ − γα ) −1  γ f ( p ) − μFp }. By induction we have,  x n − p ≤ max{ ( τ − γα ) −1  γ f ( p ) − μFp ,  x 0 − p } = M 0 . Hence, {x n } is bounded and also {y n }and{f(x n )} are bounded. Set D ={y Î H :||y - p||≤ =M 0 }. We remark that D is -invariant bounded closed convex set and {x n }, {y n } ⊂ D. Now we claim that lim sup n→∞ sup y ∈D  T μ n y − T t T μ n y  =0, ∀t ∈ S . (15) Let ε >0. By [[20], Theorem 1.2], there exists δ >0 such that coF δ ( T t ; D ) + B δ ⊂ F ε ( T t ; D ) , ∀t ∈ S . (16) Piri and Badali Fixed Point Theory and Applications 2011, 2011:55 http://www.fixedpointtheoryandapplications.com/content/2011/1/55 Page 7 of 16 Also by [[20], Corollary 1.1], there exists a natural number N such that      1 N +1 N  i=0 T t i s y − T t  1 N +1 N  i=0 T t i s y       ≤ δ , (17) for all t, s Î S and y Î D. Let t Î S. Since {μ n } is strongly left regular, there exists N 0 Î N such that  μ n − l ∗ t i μ n ≤ δ (M 0 +p) for n ≥ N 0 and i = 0, 1, 2, , N. Then we have sup y∈D      T μ n y −  1 N +1 N  i=0 T t i s ydμ n s      =sup y∈D sup z=1      T μ n y, z−   1 N +1 N  i=0 T t i s ydμ n s, z       =sup y∈D sup z=1      1 N +1 N  i=0 (μ n ) s T s y, z− 1 N +1 N  i=0 (μ n ) s T t i s y, z      ≤ 1 N +1 N  i=0 sup y∈D sup z=1 | (μ n ) s T s y, z−(l ∗ t i μ n ) s T s y, z| ≤ max i=1 , 2 , N  μ n − l ∗ t i μ n  (M 0 +  p ) ≤ δ, ∀n ≥ N 0 . (18) By Lemma 2.1, we have  1 N +1 N  i=0 T t i s ydμ n s ∈ co  1 N +1 N  i=0 T t i (T s y):s ∈ s  . (19) It follows from (16), (17), (18), and (19) that T μ n (y) ∈ co  1 N +1 N  i=0 T t i s (y):s ∈ S  + B δ ⊂ coF δ ( T t ; D ) + B δ ⊂ F ε ( T t ; D ) , for all y Î D and n ≥ N 0 . Therefore, lim sup n→∞ sup y ∈D  T t (T μ n y) − T μ n y ≤ε . Since ε > 0 is arbitrary, we get (15). In this stage, we will show lim n → ∞  x n − T t x n  =0, ∀t ∈ S . (20) Let t Î S and ε > 0. Then, there exists δ > 0, which satisiies (16). From lim n®∞ a n = 0 and (15) there exists N 1 Î N such that α n ≤ δ ( τ +μk ) M 0 and T μ n y n ∈ F δ (T t ; D ) ,foralln ≥ N 1 . By Lemma 2.6 and (14), we have α n γ f (x n ) − μFT μ n y n  ≤ α n (γ  f (x n ) − f (p)  +  γ f (p) − μFp  +  μFp − μFT μ n y n  ) ≤ α n (γα  x n − p  +  γ f (p) − μFp  +μk  y n − p ) ≤ α n (γαM 0 +(τ − γα)M 0 + μkM 0 ) ≤ α n ( τ + μk ) M 0 ≤ δ, Piri and Badali Fixed Point Theory and Applications 2011, 2011:55 http://www.fixedpointtheoryandapplications.com/content/2011/1/55 Page 8 of 16 for all n ≥ N 1 . Therefore, we have x n+1 = T μ n y n + α n [γ f (x n )+μF(T μ n y n ) ] ∈ F δ ( T t ; D ) + B δ ⊂ F ε ( T t ; D ) , for all n ≥ N 1 . This shows that  x n − T t x n  ≤ ε, ∀n ≥ N 1 . Since ε > 0 is arbitrary, we get (20). Let Q = P F . Then Q(I - μF + g f) is a contraction of H into itself. In fact, we see that  Q(I − μF + γ f )x − Q(I − μF + γ f)y  ≤ (I − μF + γ f )x − (I − μF + γ f )y  ≤ (I − μF)x − (I − μF)y  + γ  f (x) − f(y)  = lim n→∞      I −  1 − 1 n  μF  x −  I −  1 − 1 n  μF  y     + γ  f (x) − f(y)  ≤ lim n→∞ (1 − (1 − 1 n )τ )  x − y  + γα  x − y  = ( 1 − τ )  x − y  +γα  x − y , and hence Q(I - μF + g f) is a contraction due to (1 - (τ -ga)) Î (0, 1). Therefore, by Banachs contraction principal, P F ( I − μF + γ f ) has a unique fixed point x*. Then using (9), x* is the unique solution of the variational inequality:  ( γ f − μF ) x ∗ , x − x ∗ ≤0, ∀x ∈ F . (21) We show that lim sup n →∞ γ f (x ∗ ) − μFx ∗ , x n − x ∗ ≤0 . (22) Indeed, we can choose a subsequence { x n i } of {x n } such that lim sup n →∞ γ f (x ∗ ) − μFx ∗ , x n − x ∗  = lim i→∞ γ f (x ∗ ) − μFx ∗ , x n i − x ∗  . (23) Because {x n i } is bounded, we may assume that x n i → z . In terms of Lemma 2.2 and (20), we conclude that z Î Fix (). Now, let us show that z Î VI (C, B). Let w n = P C (x n - δ n Bx n ), it follows from the definition of {y n } that y n+1 − y n  =  β n+1 x n+1 +(1− β n+1 )P C (x n+1 − δ n+1 Bx n+1 ) − β n x n − (1 − β n )P C (x n − δ n Bx n )  =  β n+1 (x n+1 − x n )+(β n+1 − β n )x n +(1− β n+1 )P C (x n+1 − δ n+1 Bx n+1 ) − (1 − β n+1 )P C (x n − δ n+1 Bx n )+(1− β n+1 )P C (x n − δ n+1 Bx n ) − (1 − β n )P C (x n − δ n Bx n )  ≤ β n+1  x n+1 − x n  + | β n+1 − β n | x n  +(1− β n+1 )  P C (x n+1 − δ n+1 Bx n+1 ) − P C (x n − δ n+1 Bx n )  +  P C (x n − δ n+1 Bx n ) − P C (x n − δ n Bx n )  +  β n P C (x n − δ n Bx n ) − β n+1 P C (x n − δ n+1 Bx n ) ] ≤ β n+1  x n+1 − x n  + | β n+1 − β n | x n  +(1− β n+1 )  x n+1 − x n  + | δ n+1 − δ n | Bx n  +  β n P C (x n − δ n Bx n ) − β n P C (x n − δ n+1 Bx n ) + β n P C (x n − δ n+1 Bx n ) − β n+1 P C (x n − δ n+1 Bx n )  ≤ β n+1  x n+1 − x n  + | β n+1 − β n | x n  +(1− β n+1 )  x n+1 − x n  + | δ n+1 − δ n | Bx n  +β n | δ n+1 − δ n | Bx n  + | β n+1 − β n | P C (x n − δ n+1 Bx n )  = x n+1 − x n  + | β n+1 − β n | x n  +(1+β n ) | δ n+1 − δ n | Bx n  + | β n+1 − β n | P C ( x n − δ n+1 Bx n )  . Piri and Badali Fixed Point Theory and Applications 2011, 2011:55 http://www.fixedpointtheoryandapplications.com/content/2011/1/55 Page 9 of 16 Using the last inequality, we get  x n+1 − x n  =  α n γ f(x n )+(I − α n μF)T μ n y n − α n−1 γ f(x n−1 ) − (I − α n−1 μF)T μ n−1 y n−1  =  α n γ f(x n ) − α n γ f(x n−1 )+(α n − α n−1 )γ f(x n−1 ) +(I − α n μF)T μ n y n − (I − α n μF)T μ n−1 y n−1 +(I − α n μF)T μ n−1 y n−1 − (I − α n−1 μF)T μ n−1 y n−1  ≤ α n γα  x n − x n−1  + | α n − α n−1 | γ  f (x n−1 )  +(1− α n τ )  T μ n y n − T μ n−1 y n−1  + | α n − α n−1 | μ  FT μ n−1 y n−1  ≤ α n γα  x n − x n−1  + | α n − α n−1 | γ  f (x n−1 )  +(1 − α n τ )  y n − y n−1  +(1− α n τ )  T μ n y n−1 − T μ n−1 y n−1  + | α n − α n−1 | μ  FT μ n−1 y n−1  ≤ α n γα  x n − x n−1  + | α n − α n−1 | γ  f (x n−1 )  +(1 − α n τ )  x n − x n−1  +(1− α n τ ) | β n − β n−1 | x n−1  +(1 − α n τ )(1 + β n−1 ) | δ n − δ n−1 | Bx n−1  +(1− α n τ ) | β n − β n−1 | P C (x n−1 − δ n Bx n−1 )  +(1− α n τ )  T μ n y n−1 − T μ n−1 y n−1  + | α n − α n−1 | μ  FT μ n−1 y n−1  . Thus, for some large enough constant M > 0, we have  x n+1 − x n ≤(1 − α n (τ − γα))  x n − x n−1  + [ | α n − α n−1 | + | β n − β n−1 | + | δ n − δ n−1 | +  μ n − μ n−1  ] M . Therefore, using condition B 3 and Lemma 2.5, we get lim n → ∞  x n+1 − x n  =0 . (24) Let p ∈ F , from (11) and deiinition of {y n }, we have y n − p 2 =  β n x n +(1− β n )P C (x n − δ n Bx n ) − p 2 =  β n (x n − p)+(1− β n )(P C (x n − δ n Bx n ) − P C (p − δ n Bp)) 2 ≤ β n  x n − p 2 +(1− β n )  (x n − p) − δ n (Bx n − Bp)) 2 = β n  x n − p 2 +(1− β n )  x n − p 2 + δ 2 n (1 − β n )  Bx n − Bp 2 − 2δ n (1 − β n )x n − p, Bx n − Bp ≤x n − p 2 + δ 2 n (1 − β n )  Bx n − Bp 2 − 2δ n (1 − β n )β  Bx n − Bp 2 =  x n − p 2 + δ n ( 1 − β n )( δ n − 2β )  Bx n − Bp 2 . (25) Using (25), we have x n+1 − p 2 =  α n γ f (x n )+(I − α n μF)T μ n y n − p 2 =  α n (γ f (x n ) − μFT μ n y n )+(T μ n y n − p) 2 = α 2 n  γ f (x n ) − μFT μ n y n  2 +  T μ n y n − p  +2α n γ f (x n ) − μFT μ n y n , T μ n y n − p ≤ α 2 n  γ f (x n ) − μFT μ n y n  2 +  y n − p 2 +2α n γ f (x n ) − μFT μ n y n , T μ n y n − p 2 ≤ α 2 n  γ f (x n ) − μFT μ n y n  2 +  x n − p 2 + δ n (1 − β n )(δ n − 2β)  Bx n − Bp 2 +2α n γ f (x n ) − μFT μ n y n , T μ n y n − p = α 2 n  γ f (x n ) − μFT μ n y n  2 +  x n − p 2 + δ n (δ n − 2β n )  Bx n − Bp 2 − δ n β n (δ n − 2β n )  Bx n − Bp 2 +2α n γ f (x n ) − μFT μ n y n , T μ n y n − p. (26) Piri and Badali Fixed Point Theory and Applications 2011, 2011:55 http://www.fixedpointtheoryandapplications.com/content/2011/1/55 Page 10 of 16 [...]... H, Takahashi, W: Strong convergence theorems for asymptotically nonexpansive mappings in general Banach spaces, Dynamics of Continuous Discret Impuls Syst Ser A Math Anal 13, 621–640 (2006) doi:10.1186/1687-1812-2011-55 Cite this article as: Piri and Badali: Strong convergence theorem for amenable semigroups of nonexpansive mappings and variational inequalities Fixed Point Theory and Applications 2011... Existence of nonexpansive retractions for amenable semigroups of nonexpansive mappings and nonlinear ergodic theorems in Banach spaces J Funct Anal 161, 62–75 (1999) doi:10.1006/ jfan.1998.3352 16 Piri, H, Vaezi, H: Strong Convergence of a Generalized Iterative Method for Semigroups of Nonexpansive Mappings in Hilbert Spaces Fixed Point Theory and Applications 2010, 16 (2010) Article ID 907275 Page 15 of 16... Then {xn} and {yn} convergence strongly to x* which is the unique solution of the systems of variational inequalities: Piri and Badali Fixed Point Theory and Applications 2011, 2011:55 http://www.fixedpointtheoryandapplications.com/content/2011/1/55 Page 14 of 16 (μF − γ f )x∗ , x − x∗ ≥ 0, ∀x ∈ F , Bx∗ , y − x∗ ≥ 0 ∀y ∈ C, Proof Take λn = n−1 n , for n Î N, we define μn (f ) = 1 λn λn 0 f (t)dt for each... doi:10.1016/j.na.2008.02.009 9 Yamada, I: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, Inherently parallel algorithms in feasibility and optimization and their applications (Haifa, 2000) Stud Comput Math 8, 473–504 (2001) 10 Tian, M: A general iterative algorithm for nonexpansive mappings in Hilbert spaces Nonlinear Anal 73, 689–694... 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(2006) doi:10.1186/1687-1812-2011-55 Cite this article as: Piri and Badali: Strong convergence theorem for amenable semigroups of nonexpansive mappings and variational inequalities. Fixed Point Theory and Applications 2011 2011:55. Submit. RESEARC H Open Access Strong convergence theorem for amenable semigroups of nonexpansive mappings and variational inequalities Hossein Piri * and Ali Haji Badali * Correspondence:. point of a semigroup of nonexpansive mappings, with respect to strongly left regular sequence of means defined on an appropriate space of bounded real-valued functions of the semigroups and the