Piri Fixed Point Theory and Applications 2012, 2012:99 http://www.fixedpointtheoryandapplications.com/content/2012/1/99 RESEARCH Open Access Approximating fixed points of amenable semigroup and infinite family of nonexpansive mappings and solving systems of variational inequalities and systems of equilibrium problems Hossein Piri Correspondence: hossein_piri1979@yahoo.com Department of Mathematics, University of Bonab 55517-61167 Bonab, Iran Abstract We introduce an iterative scheme for finding a common element of the set of solutions for systems of equilibrium problems and systems of variational inequalities and the set of common fixed points for an infinite family and left amenable semigroup of nonexpansive mappings in Hilbert spaces The results presented in this paper mainly extend and improved some well-known results in the literature Mathematics Subject Classification (2000): 47H09; 47H10; 47H20; 43A07; 47J25 Keywords: common fixed point, strong convergence, amenable semigroup, explicit iterative, system of equilibrium problem Introduction Let H be a real Hilbert space and let C be a nonempty closed convex subset of H Let A: C ® H be a nonlinear mapping The classical variational inequality problem is to fined x Ỵ C such that Ax, y − x ≥ 0, ∀y ∈ C (1) The set of solution of (1) is denoted by VI(C, A), i.e., VI(C, A) = {x ∈ C : Ax, y − x ≥ 0, ∀y ∈ C} (2) Recall that the following definitions: (1) A is called monotone if Ax − Ay, x − y ≥ 0, ∀x, y ∈ C (2) A is called a-strongly monotone if there exists a positive constant a such that Ax − Ay, x − y ≥ α x − y , ∀x, y ∈ C © 2012 Piri; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Piri Fixed Point Theory and Applications 2012, 2012:99 http://www.fixedpointtheoryandapplications.com/content/2012/1/99 Page of 22 (3) A is called μ-Lipschitzian if there exist a positive constant μ such that Ax − Ay ≤ μ x − y , ∀x, y ∈ C (4) A is called a-inverse strongly monotone, if there exists a positive real number a >0 such that Ax − Ay, x − y ≥ α Ax − Ay , ∀x, y ∈ C It is obvious that any a-inverse strongly monotone mapping B is α1 -Lipschitzian (5) A mapping T : C ® C is called nonexpansive if ∥ Tx - Ty ∥≤∥ x - y ∥ for all x, y Ỵ C Next, we denote by Fix(T) the set of fixed point of T (6) A mapping f : C ® C is said to be contraction if there exists a coefficient a Ỵ (0, 1) such that f (x) − f (y) ≤ α x − y , ∀x, y ∈ C (7) A set-valued mapping U : H ® 2H is called monotone if for all x, y Ỵ H, f Ỵ Ux and g ỴUy imply 〈x - y, f - g〉 ≥ (8) A monotone mapping U : H ® 2H is maximal if the graph G(U) of U 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) ẻ H ì H, x - y, f - g〉 ≤ for every (y, g) Ỵ G(U) implies that f Ỵ Ux Let B be a monotone mapping of C into H and let NCx be the normal cone to C at x Ỵ C, that is, NCx = {y Ỵ H : 〈x - z, y〉 ≤ 0, ∀z Ỵ C} and define Ux = Bx + NC x, ∅ x ∈ C, x ∈ C Then U is the maximal monotone and Ỵ Ux if and only if x Ỵ VI(C, B); see [1] Let F be a bi-function of C×C into ℝ, where ℝ is the set of real numbers The equilibrium problem for F : C ì C đ ℝ is to determine its equilibrium points, i.e the set EP(F) = {x ∈ C : F(x, y) ≥ 0, ∀y ∈ C} Let J = {Fi }i∈I be a family of bi-functions from C × C into ℝ The system of equilibrium problems for J = {Fi }i∈I is to determine common equilibrium points for J = {Fi }i∈I , i.e the set EP(J ) = {x ∈ C : Fi (x, y) ≥ 0, ∀y ∈ C, ∀i ∈ I} (3) Numerous problems in physics, optimization, and economics reduce into finding some element of EP(F) Some method have been proposed to solve the equilibrium problem; see, for instance [2-5] The formulation (3), extend this formalism to systems of such problems, covering in particular various forms of feasibility problems [6,7] Piri Fixed Point Theory and Applications 2012, 2012:99 http://www.fixedpointtheoryandapplications.com/content/2012/1/99 Page of 22 Given any r >0 the operator JrF : H → C defined by JrF (x) = {z ∈ C : F(z, y) + y − z, z − x ≥ 0, ∀y ∈ C}, r is called the resolvent of F, see [3] It is shown [3] that under suitable hypotheses on F (to be stated precisely in Sect 2), JrF : H → C is single- valued and firmly nonexpansive and satisfies ∀r > Fix(JrF ) = EP(F), Using this result, in 2007, Yao et al [8], proposed the following explicit scheme with respect to W-mappings for an infinite family of nonexpansive mappings: xn+1 = αn f (xn ) + βn xn + γn Wn JrFn xn (4) They proved that if the sequences {an}, {bn}, {gn} and {rn} of parameters satisfy appropriate conditions, then, the sequences {xn} and {JrFn xn } both converge strongly to the x ∗ ∈ P∩ ∞ f (x∗ ) Their results extend unique x∗ ∈ ∩∞ i=1 Fix(Ti ) ∩ EP(F) , where i=1 Fix(Ti )∩EP(F) and improve the corresponding results announced by Combettes and Hirstoaga [3] and Takahashi and Takahashi [5] Very recently, Jitpeera et al [9], introduced the iterative scheme based on viscosity and Cesàro mean ⎧ φ(un , y) + ϕ(y) − ϕ(un ) + r1n y − un , un − xn ≥ 0, ∀y ∈ C, ⎪ ⎪ ⎨ yn = δn un + (1 − δn )PC (un − λn Bun ), ⎪ n ⎪ ⎩x = α γ f (x ) + β x + ((1 − β )I − α A) T i y , ∀n ≥ 0, n+1 n n n n n n n+1 i=0 n where B : C ® H is b-inverse strongly monotone, : C ® ℝ ∪ {∞} is a proper lower semi-continuous and convex function, Ti : C ® C is a nonexpansive mapping for all i = 1, 2, , n, {an}, {bn}, {δn} ⊂ (0, 1), {ln} ⊂ (0, 2b) and {rn} ⊂ (0, ∞) satisfy the following conditions (i) limn®∞ an = 0, ∞ n=1 αn = ∞ , (ii) limn®∞ δn = (iii) (9) Lemma 2.4 [28]Let {xn} and {yn} be bounded sequences in a Banach space E and let {an} be a sequence in [0, 1] with < lim inf αn ≤ lim sup αn < Suppose xn+1 = anxn n→∞ n→∞ +(1-an)yn for all integers n ≥ and lim sup( yn+1 − yn − xn+1 − xn ) ≤ n→∞ lim yn − xn Then, n→∞ = Let F : C ì C đ be a bi-function Given any r >0, the operator JrF : H → C defined by JrF x = z ∈ C : F(z, y) + y − z, z − x ≥ 0, ∀y ∈ C r Piri Fixed Point Theory and Applications 2012, 2012:99 http://www.fixedpointtheoryandapplications.com/content/2012/1/99 Page of 22 is called the resolvent of F, see [3] The equilibrium problem for F is to determine its equilibrium points, i.e., the set EP(F) = {x ∈ C : F(x, y) ≥ 0, ∀y ∈ C} Let J = {Fi }i∈I be a family of bi-functions from C × C into ℝ The system of equilibrium problems for J is to determine common equilibrium points for J = {Fi }i∈I i.e, the set EP(J ) = {x ∈ C : Fi (x, y) ≥ 0, ∀y ∈ C, ∀i ∈ I} Lemma 2.5 [3]Let C be a nonempty closed convex subset of H and F : C ì C đ satisfy (A1) F (x, x) = for all x Ỵ C, (A2) F is monotone, i.e, F(x, y) + F(y, x) ≤ for all x, y Ỵ C, (A3) for all x, y, z ẻ C, limtđ0 F(tz + (1 - t)x, y) ≤ F (x, y), (A4) for all x ẻ C, y đ F(x, y) is convex and lower semi-continuous Given r >0, define the operator JrF : H → C , the resolvent of F, by JrF (x) = {z ∈ C : F(z, y) + y − z, z − x ≥ 0, ∀y ∈ C} r Then, (1) JrF is single valued, (2) JrF is firmly nonexpansive, i.e, JrF x − JrF y ≤ JrF x − JrF y, x − y for all x, y Ỵ H, (3) Fix(JrF ) = EP(F) , (4) EP(F) is closed and convex Let T1, T2, be an infinite family of mappings of C into itself and let l1, l2, be a real numbers such that ≤ li 0 is arbitrary, we get (20) Let t Ỵ S and >0 Then, there exists δ >0, which satisfies (21) From condition (B1), (20) and Step 6, there exists N1 Ỵ N such that αn < D and xn − Tμn Wn yn < δ δ 4M0 , Tμn y ∈ Fδ (Tt , D) for all y Ỵ for all n ≥ N1 We note that αn f (Tμn Wn yn ) − Tμn Wn yn ≤ αn [ f (Tμn Wn yn ) − f (p) + f (p) − p + p − Tμn Wn yn ] δ ≤ αn [α yn − p + f (p) − p + p − yn ] ≤ 2M0 αn ≤ , for all n ≥ N1 Therefore, we have xn+1 = Tμn Wn yn + αn (f (Tμn Wn yn ) − Tμn Wn yn ) + βn (xn − Tμn Wn yn ) ∈ Fδ (Tt ; D) + B δ + B δ ⊂ Fδ (Tt ; D) + Bδ ⊂ Fε (Tt ; D), 2 for all n ≥ N1 This shows that xn − Tt xn ≤ ε, ∀n ≥ N1 Since >0 is arbitrary, we get limn®∞ ∥ xn - Tt(xn) ∥ = Step The weak ω-limit set of {xn}, ωω{xn}, is a subset of F Proof of Step Let z Î ωω{xn} and let {xnm } be a subsequence of {xn} weakly converging to z, we need to show that z ∈ F Noting Step 5, with no loss of generality, we may assume that Jnkm xnm z, ∀k ∈ {1, 2, , M} At first, note that by (A2) and given y Ỵ C and k Ỵ {1, 2, , M}, we have y − Jnk+1 xnm , m rk+1,nm (Jnk+1 xnm − Jnkm xnm ) ≥ Fk+1 (y, Jnk+1 xnm ) m m Step and condition(B5) imply that Jnk+1 xnm − Jnkm xnm m → rk+1,nm Since Jnkm xnm z , from the lower semi-continuity of Fk+1 on the second variable, we have Fk+1(y, z) ≤ for all y Î C and for all k Î {0, 1, 2, , M - 1} For t with < t ≤ and y Ỵ C, let yt = ty + (1 - t)z Since y Ỵ C and z Î C, we have yt Î C and hence Fk +1(yt, z) ≤ So from the convexity of Fk+1 on second variable, we have = Fk+1 (yt , yt ) ≤ tFk+1 (yt , y) + (1 − t)Fk+1 (yt , z) ≤ tFk+1 (yt , y) ≤ Fk+1 (yt , y) hence Fk+1(yt, y) ≥ therefore, we have Fk+1(z, y) ≥ for all y Ỵ C and k Ỵ {0, 1, 2, , M-1} Therefore z ∈ ∩M k=1 EP(Fk ) = EP(J ) z , it follows by Step and Lemma 2.2 that z Ỵ Fix(Tt) for all t Ỵ S Since xnm Therefore, z Ỵ Fix( ) We will show z Ỵ Fix(W) Assume z ∉ Fix(W) Since Piri Fixed Point Theory and Applications 2012, 2012:99 http://www.fixedpointtheoryandapplications.com/content/2012/1/99 Page 15 of 22 z ∈ Fix(ϕ) ∩ EP(J ) , by our assumption, we have Tiz Ỵ Fix( ),∀i Ỵ N and then Wnz Ỵ Fix( ) Hence by Lemma 2.1, Tμn Wn z = Wn z , therefore by Lemma 2.5, we get Tμn Wn JnM z = Wn z, ∀n ∈ N (25) Now, by (25), Step 6, Lemma 2.6 and Opial’s condition, we have lim inf xnm − z n→∞ < lim inf xnm − Wz n→∞ ≤ lim inf n→∞ xnm − Tμnm Wnm JnMm xnm + Tμnm Wnm JnMm xnm − Tμnm Wnm JnMm z + Tμnm Wnm JnMm z − Wz ≤ lim inf n→∞ xnm − Tμnm Wnm JnMm xnm + xnm − z + Wnm z − Wz ≤ lim inf xnm − z n→∞ ∞ This is a contradiction So we get z ∈ Fix(W) = Fix(Ti ) i=1 Now, let us show that z Î VI(C, A) ∩ VI(C, B) Observe that, xn+1 − p 2 ≤ αn f (Tμn Wn yn ) − p + β n xn − p + γn Tμn Wn JnM yn − p ≤ αn f (Tμn Wn yn ) − p + β n xn − p + γ n yn − p = αn f (Tμn Wn yn ) − p + β n xn − p + γn ηn PC (zn − ζn Azn ) + γn ηn [PC (zn − ζn Azn ) +(1 − ηn )PC (zn − δn Bzn ) − p = αn f (Tμn Wn yn ) − p (26) + β n xn − p −PC (p − ζn Ap)] + (1 − ηn )[PC (zn − δn Bzn ) − PC (p − δn Bp)] From (26), we have xn+1 − p 2 ≤ αn f (Tμn Wn yn ) − p + β n xn − p +γn [ηn (zn − p) − ζn (Azn − Ap) = αn f (Tμn Wn yn ) − p +γn (1 − ηn ) zn − p 2 2 + (1 − ηn ) zn − p ] + β n xn − p + γn ηn [ zn − p + ζn2 Azn − Ap 2 + ζn2 Azn − Ap −2ζn Azn − Ap, zn − p ] ≤ αn f (Tμn Wn yn ) − p +γn (1 − ηn ) zn − p + β n xn − p + γn ηn [ zn − p −2ζn β Azn − Ap, zn − p ] = αn f (Tμn Wn yn ) − p +γn zn − p ≤ xn − p 2 + β n xn − p + ζn (ζn − 2β) Azn − Ap + αn [ f (Tμn Wn yn ) − p +ζn (ζn − 2β) Azn − Ap , 2 − xn − p ] Piri Fixed Point Theory and Applications 2012, 2012:99 http://www.fixedpointtheoryandapplications.com/content/2012/1/99 Page 16 of 22 which implies that −ζn (ζn − 2β) Azn − Ap ≤ [ xn − p + xn+1 − p ] xn − xn+1 +αn [ f (Tμn Wn yn ) − p 2 − xn − p ] Therefore, from step and condition B1, we obtain lim Azn − Ap = (27) n→∞ On the other hand from (26), we have xn+1 − p + β n xn − p +(1 − ηn ) (zn − p) − δn (Bzn − Bp) ≤ αn f (Tμn Wn yn ) − p = αn f (Tμn Wn yn ) − p +(1 − ηn )( zn − p ≤ αn f (Tμn Wn yn ) − p +(1 − ηn )( zn − p = αn f (Tμn Wn yn ) − p 2 xn − p + γn ηn zn − p 2 − 2δn Bzn − Bp, zn − p + δn2 Bzn − Bp ) 2 + β n xn − p + γn ηn zn − p − 2δn γ Bzn − Bp 2 + β n xn − p +δn (δn − 2γ )γn (1 − ηn ) Bzn − Bp ≤ + β n xn − p + γn ηn zn − p + αn [ f (Tμn Wn yn ) − p +δn (δn − 2γ ) Bzn − Bp 2 + δn2 Bzn − Bp ) + γ n zn − p 2 2 − xn − p ] which implies that −δn (δn − 2γ ) Bzn − Bp ≤ [ xn − p + xn+1 − p ] xn − xn+1 +αn [ f (Tμn Wn yn ) − p 2 − xn − p ] Therefore, from step and condition B1, we obtain lim Bzn − Bp = (28) n→∞ From (6) and (12), we have − p = PC (zn − ζn Azn ) − PC (p − ζn Ap) ≤ (zn − ζn Azn ) − (p − ζn Ap), − p (zn − ζn Azn ) − (p − ζn Ap) + − p = − (zn − ζn Azn ) − (p − ζn Ap) − (vn − p) = zn − p + − p − zn − 2 +2ζn zn − , Azn − Ap − ζn2 Azn − Ap Piri Fixed Point Theory and Applications 2012, 2012:99 http://www.fixedpointtheoryandapplications.com/content/2012/1/99 Page 17 of 22 So we obtain − p ≤ zn − p − zn − 2 + 2ζn zn − , Azn − Ap − ζn2 Azn − Ap (29) By using the same method as (29), we have wn − p ≤ zn − p − z n − wn 2 + 2δn zn − wn , Bzn − Bp − δn2 Bzn − Bp (30) From (29), (30) and definition of yn, we have, yn − p ηn [PC (zn − ζn Azn ) − p] = +(1 − ηn )[PC (zn − δn Bzn ) − p] ηn (vn − p) + (1 − ηn )(wn − p) = ≤ ηn − p ≤ ηn [ zn − p −ζn2 + (1 − ηn ) wn − p − zn − Azn − Ap ] + (1 − ηn )[ zn − p zn − p −ζn2 + 2ζn zn − , Azn − Ap +2δn zn − wn , Bzn − Bp − ≤ 2 + ηn [− zn − δn2 2 − z n − wn Bzn − Bp ] + 2ζn zn − Azn − Ap ] + (1 − ηn )[− zn − wn +2δn zn − wn (31) Azn − Ap 2 Bzn − Bp − δn2 Bzn − Bp ] By (31), we have ||xn+1 − p||2 ≤ αn ||f (Tμn Wn yn ) − p||2 + βn ||xn − p||2 + γn ||Tμn Wn JnM yn − p||2 ≤ αn ||f (Tμn Wn yn ) − p||2 + βn ||xn − p||2 + γn ||zn − p||2 +γn ηn [−||zn − ||2 + 2ζn ||zn − || ||Azn − Ap|| −ζn2 ||Azn − Ap||2 ] + γn (1 − ηn )[−||zn − wn ||2 +2δn || ||zn − wn || ||Bzn − Bp|| − δn2 ||Bzn − Bp||2 ] ≤ αn ||f (Tμn Wn yn ) − p||2 + βn ||xn − p||2 + γn ||xn − p||2 −γn ηn ||zn − ||2 + γn ηn [2ζn ||zn − || ||Azn − Ap|| −ζn2 ||Azn − Ap||2 ] − γn (1 − ηn )||zn − wn ||2 +γn (1 − ηn )[2δn || ||zn − wn || ||Bzn − Bp|| − δn2 ||Bzn − Bp||2 ] = ||xn − p||2 + αn [||f (Tμn Wn yn ) − p||2 − ||xn − p||2 ] −γn ηn ||zn − ||2 + γn ηn [2ζn ||zn − || ||Azn − Ap|| −ζn2 ||Azn − Ap||2 ] − γn (1 − ηn )||zn − wn ||2 +γn (1 − ηn )[2δn || ||zn − wn || ||Bzn − Bp|| − δn2 ||Bzn − Bp||2 ], Piri Fixed Point Theory and Applications 2012, 2012:99 http://www.fixedpointtheoryandapplications.com/content/2012/1/99 Page 18 of 22 which implies that γn ηn ||zn − ||2 ≤ [||xn − p|| + ||xn+1 − p||]||xn+1 − xn || +αn [||f (Tμn Wn yn ) − p||2 − ||xn − p||2 ] +γn ηn [2ζn ||zn − || ||Azn − Ap|| − ζn2 ||Azn − Ap||2 ] +γn (1 − ηn )[2δn || ||zn − wn || ||Bzn − Bp|| − δn2 ||Bzn − Bp||2 ], and γn (1 − ηn )||zn − wn ||2 ≤ [||xn − p|| + ||xn+1 − p||]||xn+1 − xn || +γn ηn [2ζn ||zn − || ||Azn − Ap|| − ζn2 ||Azn − Ap||2 ] +γn (1 − ηn )[2δn || ||zn − wn || ||Bzn − Bp|| − δn2 ||Bzn − Bp||2 ] Therefore, from