báo cáo hóa học:" Research Article Iterative Methods for Finding Common Solution of Generalized Equilibrium Problems and Variational Inequality Problems and Fixed Point Problems of a Finite Family of Nonexpansive Mappings" pdf

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 836714, 29 pages doi:10.1155/2010/836714 Research Article Iterative Methods for Finding Common Solution of Generalized Equilibrium Problems and Variational Inequality Problems and Fixed Point Problems of a Finite Family of Nonexpansive Mappings Atid Kangtunyakarn Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand Correspondence should be addressed to Atid Kangtunyakarn, beawrock@hotmail.com Received October 2010; Accepted November 2010 Academic Editor: T D Benavides Copyright q 2010 Atid Kangtunyakarn This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited We introduce a new method for a system of generalized equilibrium problems, system of variational inequality problems, and fixed point problems by using S-mapping generated by a finite family of nonexpansive mappings and real numbers Then, we prove a strong convergence theorem of the proposed iteration under some control condition By using our main result, we obtain strong convergence theorem for finding a common element of the set of solution of a system of generalized equilibrium problems, system of variational inequality problems, and the set of common fixed points of a finite family of strictly pseudocontractive mappings 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, and let F : C × C → R be a bifunction A mapping T of H into itself is called nonexpansive if T x − T y ≤ x − y for all x, y ∈ H We denote by F T the set of fixed points of T i.e., F T {x ∈ H : T x x} Goebel and Kirk showed that F T is always closed convex, and also nonempty provided T has a bounded trajectory A bounded linear operator A on H is called strongly positive with coefficient γ if there is a constant γ > with the property Ax, x ≥ γ x 1.1 Fixed Point Theory and Applications The equilibrium problem for F is to find x ∈ C such that F x, y ≥ 0, ∀y ∈ C 1.2 The set of solutions of 1.2 is denoted by EP F Many problems in physics, optimization, and economics are seeking some elements of EP F , see 2, Several iterative methods have been proposed to solve the equilibrium problem, see, for instance, 2–4 In 2005, Combettes and Hirstoaga introduced an iterative scheme of finding the best approximation to the initial data when EP F is nonempty and proved a strong convergence theorem The variational inequality problem is to find a point u ∈ C such that v − u, Au ≥ ∀ v ∈ C 1.3 The set of solutions of the variational inequality is denoted by VI C, A , and we consider the following generalized equilibrium problem Find z ∈ C such that F z, y Az, y − z ≥ 0, ∀y ∈ C 1.4 The set of such z ∈ C is denoted by EP F, A , that is, EP F, A z ∈ C : F z, y Az, y − z ≥ 0, ∀y ∈ C 1.5 In the case of A ≡ 0, EP F, A EP F Numerous problems in physics, optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games reduce to find element of 1.5 A mapping A of C into H is called inverse-strongly monotone, see , if there exists a positive real number α such that x − y, Ax − Ay ≥ α Ax − Ay 1.6 for all x, y ∈ C The problem of finding a common fixed point of a family of nonexpansive mappings has been studied by many authors The well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive mapping see 6, The ploblem of finding a common element of EP F, A and the set of all common fixed points of a family of nonexpansive mappings is of wide interdisciplinary interest and importance Many iterative methods are purposed for finding a common element of the solutions of the equilibrium problem and fixed point problem of nonexpansive mappings, see 8–10 Fixed Point Theory and Applications In 2008, S.Takahashi and W.Takahashi 11 introduced a general iterative method for finding a common element of EP F, A and F T They defined {xn } in the following way: u, x1 ∈ C, xn y − zn , zn − xn ≥ 0, λn Axn , y − zn F zn , y βn xn arbitrarily; − βn T an u − an zn , ∀y ∈ C, 1.7 ∀n ∈ N, where A is an α-inverse strongly monotone mapping of C into H with positive real number α, and {an } ∈ 0, , {βn } ⊂ 0, , {λn } ⊂ 0, 2α , and proved strong convergence of the scheme 1.7 to z ∈ N1 F Ti ∩ EP F, A , where z P N1 F Ti ∩EP F, A u in the framework of a Hilbert i i space, under some suitable conditions on {an }, {βn }, {λn } and bifunction F Very recently, in 2010, Qin, et al 12 introduced a iterative scheme method for finding a common element of EP F1 , A , EP F2 , B and common fixed point of infinite family of nonexpansive mappings They defined {xn } in the following way: x1 ∈ C, arbitrarily; F1 un , u Axn , u − un u − un , un − xn ≥ 0, r ∀u ∈ C, F2 , v Bxn , v − v − , − xn ≥ 0, s ∀v ∈ C, yn xn αn f xn δ n un βn xn 1.8 − δn , γn Wn xn , ∀n ∈ N, where f : C → C is a contraction mapping and Wn is W-mapping generated by infinite family of nonexpansive mappings and infinite real number Under suitable conditions of these parameters they proved strong convergence of the scheme 1.8 to z PF f z , where ∞ F i F Ti ∩ EP F1 , A ∩ EP F2 , B In this paper, motivated by 11, 12 , we introduced a general iterative scheme {xn } defined by F un , u Axn , u − un u − un , un − xn ≥ 0, rn G , v Bxn , v − v − , − xn ≥ 0, sn δn PC un − λn Aun − δn PC − ηn Bvn , yn xn αn f xn βn xn γn Sn yn , ∀n ≥ 0, 1.9 Fixed Point Theory and Applications where f : C → C and Sn is S-mapping generated by T0 , , Tn and αn , αn−1 , , α0 Under suitable conditions, we proved strong convergence of {xn } to z PF f z , and z is solution of Ax∗ , x − x∗ ≥ 0, 1.10 Bx∗ , x − x∗ ≥ Preliminaries In this section, we collect and give some useful lemmas that will be used for our main result in the next section Let C be closed convex subset of a real Hilbert space H, and let PC be the metric projection of H onto C, that is, for x ∈ H, PC x satisfies the property x − y x − PC x 2.1 y∈C The following characterizes the projection PC Lemma 2.1 see 13 Given x ∈ H and y ∈ C Then PC x inequality y if and only if there holds the x − y, y − z ≥ ∀z ∈ C 2.2 Lemma 2.2 see 14 Let {sn } be a sequence of nonnegative real numbers satisfying sn − αn sn ∀n ≥ βn , 2.3 where {αn }, {βn } satisfy the conditions {αn } ⊂ 0, , ∞ n αn ∞, lim supn → ∞ βn /αn ≤ Then limn → ∞ sn Lemma 2.3 see 15 Let C be a closed convex subset of a strictly convex Banach space E Let {Tn : n ∈ N} be a sequence of nonexpansive mappings on C Suppose that ∞ F Tn is nonempty n Let {λn } be a sequence of positive numbers with Σ∞ λn Then a mapping S on C defined by n S x Σ∞ λn Tn x n for x ∈ C is well defined, nonexpansive, and F S ∞ n 2.4 F Tn hold Lemma 2.4 see 16 Let E be a uniformly convex Banach space, C a nonempty closed convex subset of E, and S : C → C a nonexpansive mapping Then I − S is demiclosed at zero Fixed Point Theory and Applications Lemma 2.5 see 17 Let {xn } and {zn } be bounded sequences in a Banach space X, and let {βn } be a sequence in 0, with < lim infn → ∞ βn ≤ lim supn → ∞ βn < Suppose that xn βn xn − βn zn 2.5 for all integer n ≥ and lim supn → ∞ zn Then limn → ∞ xn − zn − zn − xn − xn ≤ 2.6 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 for all x ∈ C; A2 F is monotone, that is, F x, y F y, x ≤ 0, ∀x, y ∈ C, A3 for all x, y, z ∈ C, limt → F tz − t x, y ≤ F x, y , 2.7 A4 for all x ∈ C, y → F x, y is convex and lower semicontinuous The following lemma appears implicitly in Lemma 2.6 see Let C be a nonempty closed convex subset of H, and let F be a bifunction of C × C into R satisfying (A1)–(A4) Let r > and x ∈ H Then, there exists z ∈ C such that F z, y y − z, z − x r 2.8 for all x ∈ C Lemma 2.7 see Assume that F : C × C → R satisfies (A1)–(A4) For r > and x ∈ H, define a mapping Tr : H → C as follows: Tr x z ∈ C : F z, y y − z, z − x ≥ 0, ∀y ∈ C r 2.9 for all z ∈ H Then, the following hold: Tr is single-valued; Tr is firmly nonexpansive, that is, Tr x − Tr y F Tr EP F ; EP F is closed and convex ≤ Tr x − Tr y , x − y ∀x, y ∈ H; 2.10 Fixed Point Theory and Applications In 2009, Kangtunyakarn and Suantai 18 defined a new mapping and proved their lemma as follows Definition 2.8 Let C be a nonempty convex subset of real Banach space Let {Ti }N1 be a finite i j j j α1 , α2 , α3 ∈ family of nonexpansive mappings of C into itself For each j 1, 2, , N, let αj j I × I × I, where I ∈ 0, and α1 j j α2 We define the mapping S : C → C as follows: α3 U0 I, U1 α1 T1 U0 α1 U0 α1 I, U2 α2 T2 U1 α2 U1 α2 I, U3 α3 T3 U2 α3 U2 α3 I, 2.11 UN−1 S αN−1 TN−1 UN−2 αN−1 UN−2 αN TN UN−1 UN αN−1 I, αN UN−1 αN I This mapping is called S-mapping generated by T1 , , TN and α1 , α2 , , αN Lemma 2.9 Let C be a nonempty closed convex subset of strictly convex Let {Ti }N1 be a finite family i j j j of nonexpanxive mappings of C into itself with N1 F Ti / ∅, and let αj α1 , α2 , α3 ∈ I × I × I, i j j j j 1, 2, 3, , N, where I 0, , α1 α2 α3 1, α1 ∈ 0, for all j 1, 2, , N − 1,αN ∈ j j 1, 2, , N Let S be the mapping generated by T1 , , TN and 0, α2 , α3 ∈ 0, for all j N α1 , α2 , , αN Then F S i F Ti j Lemma 2.10 Let C be a nonempty closed convex subset of Banach space Let {Ti }N1 be a finite family i n,j n,j n,j j j j n of nonexpansive mappings of C into itself and αj α1 , α2 , α3 , αj α1 , α2 , α3 ∈ I × I × I, n,j n,j n,j j j j n,j j α2 α3 and α1 α2 α3 such that αi → αi ∈ 0, as n → where I 0, , α1 ∞ for i 1, and j 1, 2, 3, , N Moreover, for every n ∈ N, let S and Sn be the S-mappings n n n generated by T1 , T2 , , TN and α1 , α2 , , αN and T1 , T2 , , TN and α1 , α2 , , αN , respectively for every x ∈ C Then limn → ∞ Sn x − Sx Lemma 2.11 see 19 Let C be a nonempty closed convex subset of a Hilbert space H, and let G : C → C be defined by G x PC x − λAx , ∀x ∈ C, 2.12 with ∀λ > Then x∗ ∈ V I C, A if and only if x∗ ∈ F G Main Result Theorem 3.1 Let C be a nonempty closed convex subset of a Hilbert space H Let F and G be two bifunctions from C×C into R satisfying (A1)–(A4), respectively Let A : C → H a α-inverse strongly monotone mapping and B : C → H be a β-inverse strongly monotone mapping Let {Ti }N1 be finite i Fixed Point Theory and Applications N family of nonexpansive mappings with F i F Ti ∩ EP F, A ∩ EP G, B ∩ F G1 ∩ F G2 / ∅, PC x − λn Ax , G2 x PC x − ηn Bx , ∀x ∈ C where G1 , G2 : C → C are defined by G1 x Let f : C → C be a contraction with the coefficient θ ∈ 0, Let Sn be the S-mappings generated n,j n,j n,j n,j n n n n α1 , α2 , α3 ∈ I × I × I, I 0, , α1 by T1 , T2 , , TN and α1 , α2 , , αN , where αj n,j n,j n,j α3 and < η1 ≤ α1 ≤ θ1 < ∀n ∈ N, ∀j 1, 2, , N − 1, < ηN ≤ αn,N ≤ and α2 n,j n,j ≤ α2 , α3 ≤ θ3 < ∀n ∈ N, ∀j 1, 2, , N Let {xn }, {un }, {vn }, {yn } be sequences generated by x1 , u, v ∈ C F un , u Axn , u − un u − un , un − xn ≥ 0, rn G , v Bxn , v − v − , − xn ≥ 0, sn δn PC un − λn Aun − δn PC − ηn Bvn , yn xn αn f xn βn xn 3.1 ∀n ≥ 1, γn Sn yn , where {αn }, {βn }, {γn } ∈ 0, such that αn βn γn 1, rn ∈ a, b ⊂ 0, 2α , sn ∈ c, d ⊂ 0, 2β , λn ∈ e, f ⊂ 0, 2α , ηn ∈ g, h ⊂ 0, 2β Assume that and Σ∞ αn n i limn → ∞ n ∞, ii < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1, iii limn → ∞ δn δ ∈ 0, , iv Σ∞ |sn −sn |, Σ∞ |rn −rn |, Σ∞ |λn −λn |,Σ∞ |ηn −ηn |, Σ∞ |αn −αn |, Σ∞ |βn − n n n n n n βn | < ∞, n 1,j v |α1 n,j n 1,j − α1 | → 0, and |α3 n,j − α3 | → as n → ∞, for all j ∈ {1, 2, 3, , N} Then the sequence {xn }, {yn }, {un }, {vn } converge strongly to z PF f z , and z is solution of Ax∗ , x − x∗ ≥ 0, 3.2 Bx∗ , x − x∗ ≥ Proof First, we show that I − λn A , I − ηn B I − rn A and I − sn B are nonexpansive Let x, y ∈ C Since A is α-strongly monotone and λn < 2α for all n ∈ N, we have I − λn A x − I − λn A y x − y − λn Ax − Ay x−y − 2λn x − y, Ax − Ay ≤ x−y − 2αλn Ax − Ay x−y ≤ x−y 2 λ2 Ax − Ay n λ2 Ax − Ay n λn λn − 2α Ax − Ay 2 3.3 Fixed Point Theory and Applications Thus I − λn A is nonexpansive By using the same proof, we obtain that I − ηn B and I − sn B are nonexpansive We will divide our proof into steps I − rn A Step We will show that the sequence {xn } is bounded Since u − un , un − xn ≥ 0, rn Axn , u − un F un , u ∀u ∈ C, 3.4 then we have F un , u u − un , un − I − rn A xn ≥ rn 3.5 By Lemma 2.7, we have un Trn I − rn A xn By the same argument as above, we obtaine that Tsn I − sn B xn Let z ∈ F Then F z, y y − z, Az ≥ and G z, y y − z, Bz ≥ Hence F z, y y − z, z − z rn rn Az ≥ 0, G z, y y − z, z − z sn sn Bz ≥ 3.6 Again by Lemma 2.7, we have z Trn z − rn Az Tsn z − sn Bz Since z ∈ F, we have z PC I − λn A z PC I − ηn B z By nonexpansiveness of Trn , Tsn , I − rn A, I − sn B, we have xn − z ≤ αn f xn − z βn xn − z ≤ αn f xn − f z ≤ αn θ xn − z γn Sn yn − z αn f z − z αn f z − z βn xn − z γn yn − z βn xn − z γn δn PC un − λn Aun − z − δn PC − ηn Bvn − z ≤ αn θ xn − z αn f z − z βn xn − z αn θ xn − z αn f z − z βn xn − z γn δn un − z − δn − z γn δn Trn I − rn A xn − Trn I − rn A z − δn Tsn I − sn B xn − Tsn I − sn B z ≤ αn θ xn − z αn f z − z αn θ xn − z αn f z − z − αn − θ ≤ max xn − z , xn − z βn xn − z γn xn − z − αn xn − z αn f z − z f z −z 1−θ 3.7 Fixed Point Theory and Applications By induction we can prove that {xn } is bounded and so are {un }, {vn }, {yn }, {Sn yn } Without of generality, assume that there exists a bounded set K ⊂ C such that {un }, {vn }, yn , Sn yn ∈ K 3.8 Step We will show that limn → ∞ xn − xn xn − βn xn / − βn , we have Putting kn xn − βn kn ∀n ≥ βn xn , 3.9 From definition of kn , we have kn xn − kn − βn xn − βn αn f xn αn f xn αn 1 − βn ≤ αn 1 − βn ≤ αn 1 − βn 1 xn − βn xn − βn γn Sn yn − βn 1 1 − Sn yn αn f xn γn Sn yn − βn − − βn − αn − βn 1 f xn − Sn yn − − αn f xn αn f xn − Sn yn − βn f xn − Sn yn αn f xn − Sn yn − βn f xn − Sn yn 1 − βn − αn Sn yn − βn Sn yn − Sn yn αn f xn − Sn yn − βn Sn yn − Sn yn xn Sn yn − Sn yn − xn 3.10 10 Fixed Point Theory and Applications By definition of Sn , for k ∈ {2, 3, , N}, we have Un 1,k yn − Un,k yn αn 1,k Tk Un 1,k−1 yn αn 1,k Un αn 1,k−1 yn 1,k yn −αn,k Tk Un,k−1 yn − αn,k Un,k−1 yn − αn,k yn αn 1,k Tk Un 1,k−1 yn − Tk Un,k−1 yn αn αn − Un,k−1 yn αn − αn,k yn αn ≤ αn 1,k 1,k − αn,k Tk Un,k−1 yn − αn,k Un,k−1 yn 1,k Un 1,k 1,k Un 1,k−1 yn αn − Un,k−1 yn 1,k−1 yn αn 1,k − αn,k yn αn 1,k − αn,k αn Un 1,k 1,k 1,k − αn,k Tk Un,k−1 yn Un,k−1 yn αn αn 1,k 1,k−1 yn αn 1,k − αn,k 1,k − αn,k 1,k−1 yn − Un,k−1 yn − Un,k−1 yn Tk Un,k−1 yn αn Un Un,k−1 yn ≤ Un αn 1,k−1 yn 1,k − − αn,k Un αn ≤ Un αn 1,k−1 yn 1,k 1,k Un αn 1,k−1 yn 1,k − αn αn yn 1,k yn Tk Un,k−1 yn αn 1,k 1,k − αn,k αn 1,k αn,k − αn 1 Tk Un,k−1 yn 1,k αn,k − αn 3 − αn,k Tk Un,k−1 yn 1,k 1,k Un,k−1 yn Un,k−1 yn Un,k−1 yn − Un,k−1 yn − αn,k − αn,k − αn,k αn,k − αn 1 yn − Un,k−1 yn 1,k 1,k 1,k Un,k−1 yn − Un,k−1 yn − αn,k αn,k − αn 3 yn αn,k − αn,k 1,k−1 yn αn − Un,k−1 yn − αn,k αn yn αn 1,k − αn,k Un,k−1 yn Tk Un,k−1 yn Un,k−1 yn 3.11 Fixed Point Theory and Applications 15 By nonexpansiveness of Trn , Tsn , I − λn A, I − ηn B and 3.23 , we have yn − z 2 ≤ δn PC un − λn Aun − PC I − λn A z − δn ≤ δn I − λn A un − I − λn A z ≤ δ n un − z δn xn − rn Axn − z rn Az δn xn − z xn − z 2 rn δn Axn − Az 2 2 2 2 − 2rn xn − z, Axn − Az 2 − 2sn xn − zBxn − Bz − 2δn rn xn − z, Axn − Az − 2sn − δn xn − zBxn − Bz − 2δn rn α Axn − Az − 2sn − δn β Bxn − Bz 2 rn δn Axn − Az sn Bz xn − z − sn Bxn − Bz s2 − δn Bxn − Bz n s2 − δn Bxn − Bz n xn − z − δn s2 Bxn − Bz n I − sn B xn − I − sn B z − δn xn − sn Bxn − z rn Axn − Az − δn xn − z ≤ xn − z − δn Tsn I − sn B xn − δn δn xn − z − rn Axn − Az − δn 2 ≤ δn I − rn A xn − I − rn A z xn − z 2 − δn − z −Tsn I − sn B z I − ηn B − I − ηn B z − δn δn Trn I − rn A xn − Trn I − rn A z δn PC − ηn Bvn − PC I − ηn B z − δn rn 2α − rn Axn − Az − sn − δn 2β − sn Bxn − Bz 3.28 By 3.28 , we have xn −z βn xn − z αn f xn − z γn Sn yn − z ≤ αn f xn − z βn xn − z γn Sn yn − z ≤ αn f xn − z βn xn − z γn yn − z ≤ αn f xn − z βn xn − z − δn rn 2α − rn Axn − Az γn xn − z −sn − δn 2β − sn αn f xn − z Bxn − Bz βn xn − z 2 γn xn − z − δn γn rn 2α − rn Axn − Az 16 Fixed Point Theory and Applications − sn γn − δn 2β − sn ≤ αn f xn − z xn − z − sn γn − δn 2β − sn Bxn − Bz 2 − δn γn rn 2α − rn Axn − Az Bxn − Bz 3.29 By 3.29 , we have δn γn rn 2α − rn Axn − Az 2 ≤ αn f xn − z xn − z − sn γn − δn 2β − sn ≤ αn f xn − z − xn xn 2 Bxn − Bz xn − z −z 1 −z 3.30 xn − xn From 3.17 and conditions i – iii , we have lim Axn − Az 3.31 lim Bxn − Bz 3.32 lim yn − xn 3.33 n→∞ By using the same method as 3.31 , we have n→∞ Step We will show that n→∞ Putting Mn PC un − λn Aun and Nn lim un − xn n→∞ lim − xn n→∞ PC − ηn Bvn , we will show that lim Mn − un lim Nn − n→∞ n→∞ − δn N n − z 3.34 Let z ∈ F; by 3.28 , we have yn − z ≤ δn Mn − z ≤ δn un − z − δn − z 3.35 Fixed Point Theory and Applications 17 By nonexpansiveness of I − rn A, we have un − z 2 Trn xn − rn Axn − Trn z − rn Az ≤ xn − rn Axn − z − rn Az , un − z xn − rn Axn − z − rn Az un − z − xn − rn Axn − z − rn Az − un − z ≤ xn − z un − z − xn − z un − z − xn − un 2 3.36 xn − un − rn Axn − Az 2 2rn xn − un , Axn − Az − rn Axn − Az This implies un − z ≤ xn − z − xn − un 2rn xn − un , Axn − Az − rn Axn − Az 3.37 By using the same method as 3.37 , we have − z 2 ≤ xn − z − xn − 2sn xn − , Bxn − Bz − s2 Bxn − Bz n 3.38 Substituting 3.37 and 3.38 into 3.35 , we have yn − z ≤ δ n un − z ≤ δn xn − z − δn ≤ δn xn − z − δn − z xn − z 2 2 2rn xn − un , Axn − Az − rn Axn − Az − xn − − δn xn − un − − δn xn − xn − z − xn − un 2 − δn xn − un 2sn − δn xn − 2 2sn xn − , Bxn − Bz − s2 Bxn − Bz n 2δn rn xn − un 2sn − δn xn − 2 2δn rn xn − un Axn − Az − δn xn − z Bxn − Bz Axn − Az − − δn xn − Bxn − Bz 3.39 18 Fixed Point Theory and Applications By 3.39 , we have −z ≤ αn f xn − z βn xn − z ≤ αn f xn − z xn βn xn − z γn xn − z − δn xn − un 2δn rn xn − un 2 γn xn − z − γn δn xn − un Axn − Az − − δn γn xn − 3.40 Bxn − Bz xn − z 2γn δn rn xn − un Bxn − Bz 2sn γn − δn xn − ≤ αn f xn − z βn xn − z 2γn δn rn xn − un Axn − Az − − δn xn − 2sn − δn xn − αn f xn − z γn yn − z − γn δn xn − un Axn − Az − − δn γn xn − 2sn γn − δn xn − Bxn − Bz It follows that γn δn xn − un ≤ αn f xn − z 2γn δn rn xn − un 2γn δn rn xn − un − xn −z Axn − Az − − δn γn xn − 2sn γn − δn xn − ≤ αn f xn − z xn − z Bxn − Bz xn − z xn Axn − Az 3.41 −z xn − xn 2sn γn − δn xn − Bxn − Bz By conditions i – iii , 3.41 , 3.31 , 3.32 , and 3.17 , we have lim xn − un n→∞ 3.42 3.43 By using the same method as 3.42 , we have lim xn − n→∞ Fixed Point Theory and Applications 19 By nonexpansiveness of Trn I − rn A , we have Mn − z 2 PC un − λn Aun − PC z − λn Az ≤ un − αn Aun − z − αn Az , Mn − z un − αn Aun − z − αn Az Mn − z un − z Trn I − rn A xn − Trn I − rn A z Mn − z − xn − z Mn − z −α2 Aun − Az n un − αn Aun un − Mn − αn Aun − Az Mn − z 2αn un − Mn , Aun − Az − α2 Aun − Az n ≤ − − z − αn Az − Mn − z ≤ − un − Mn 2 − un − Mn 3.44 2 2αn un − Mn , Aun − Az Hence, we have Mn − z ≤ xn − z − un − Mn 2αn un − Mn , Aun − Az 3.45 − α2 Aun − Az n By using the same method as 3.45 , we have Nn − z 2 ≤ xn − z − − Nn 2ηn − Nn , Bvn − Bz − ηn Bvn − Bz 3.46 Substituting 3.45 and 3.46 into 3.35 , we have yn − z ≤ δn Mn − z ≤ δn xn − z − δn ≤ δn xn − z − δn N n − z − un − Mn xn − z 2 2αn un − Mn , Aun − Az − α2 Aun − Az n − − Nn − δn un − Mn − δn xn − z xn − z 2 2 2 2ηn − Nn , Bvn − Bz − ηn Bvn − Bz 2δn αn un − Mn − − δn − Nn − δn un − Mn − δn ηn − Nn 2 Aun − Az − δn ηn − Nn 2δn αn un − Mn Bvn − Bz Aun − Az − − δn − Nn Bvn − Bz 3.47 20 Fixed Point Theory and Applications By 3.47 , we have −z ≤ αn f xn − z βn xn − z ≤ αn f xn − z xn βn xn − z γn xn − z − δn un − Mn 2δn αn un − Mn 2 2 − δn γn un − Mn 2 Bvn − Bz xn − z 2δn γn αn un − Mn γn xn − z Aun − Az − − δn γn − Nn − δn γn ηn − Nn ≤ αn f xn − z Bvn − Bz βn xn − z 2δn γn αn un − Mn Aun − Az − − δn − Nn − δn ηn − Nn αn f xn − z γn yn − z − δn γn un − Mn Aun − Az − − δn γn − Nn − δn γn ηn − Nn Bvn − Bz 3.48 It follows that δn γn un − Mn ≤ αn f xn − z xn − z 2δn γn αn un − Mn − xn −z Aun − Az − − δn γn − Nn − δn γn ηn − Nn ≤ αn f xn − z Bvn − Bz xn − z 2δn γn αn un − Mn xn Aun − Az −z xn − xn − δn γn ηn − Nn Bvn − Bz 3.49 From 3.17 , 3.26 , 3.27 , and conditions i – iii , we have lim un − Mn n→∞ 3.50 3.51 By using the same method as 3.50 , we have lim − Nn n→∞ Fixed Point Theory and Applications 21 By 3.42 and 3.50 , we have lim Mn − xn 3.52 lim Nn − xn 3.53 n→∞ By 3.43 and 3.51 , we have n→∞ Since Mn PC un − λn Aun and Nn yn − xn PC − ηn Bvn , we have δn Mn − xn − δn Nn − xn 3.54 By 3.52 and 3.53 , we obtain lim yn − xn n→∞ 3.55 Note that xn − Sn xn ≤ xn − Sn yn Sn yn − Sn xn ≤ xn − Sn yn yn − xn lim xn − Sn xn 3.56 From 3.20 and 3.55 , we have n→∞ 3.57 Step We will show that lim sup f z − z, xn − z ≤ 0, n→∞ where z 3.58 PF f z To show this inequality, take subsequence {xni } of {xn } such that lim sup f z − z, xn − z n→∞ lim sup f z − z, xni − z i→∞ 3.59 Since {xni } is bounded, there exists a subsequence {xnij } of {xni } which converges weakly q Since C is closed convex, C is to q Without loss of generality, we can assume that xni N weakly closed So, we have q ∈ C Let us show that q ∈ F i F Ti ∩ EP F, A ∩ EP G, B ∩ F G1 ∩ F G2 We first show that q ∈ EP F, A ∩ EP G, B ∩ F G1 ∩ F G2 From 3.42 , we q Since un Trn I − rn A xn , for any y ∈ C, we have have uni F un , y Axn , y − un y − un , un − xn ≥ rn 3.60 22 Fixed Point Theory and Applications From A2 , we have y − un , un − xn ≥ F y, un rn 3.61 y − uni , uni − xni ≥ F y, uni rni 3.62 Axn , y − un This implies that Axni , y − uni Put zt have ty − t q for all t ∈ 0, and y ∈ C Then, we have zt ∈ C So, from 3.62 , we zt − uni , Azt ≥ zt − uni , Azt − zt − uni , Axni − zt − uni , zt − uni , Azt − Auni − zt − uni , uni − xni rni zt − uni , Auni − Axni uni − xni rni F zt , uni 3.63 F zt , uni Since uni − xni → 0, we have Auni − Axni → Further, from monotonicity of A, we have zt − uni , Azt − Auni ≥ So, from A4 , we have zt − q, Azt ≥ F zt , q as i −→ ∞ 3.64 From A1 , A4 , and 3.64 , we also have F zt , zt ≤ tF zt , y ≤ tF zt , y − t F zt , q − t zt − q, Azt 3.65 − t t y − q, Azt tF zt , y Thus ≤ F zt , y − t y − q, Azt 3.66 y − q, Aq 3.67 Letting t → 0, we have, for each y ∈ C, ≤ F q, y This implies that q ∈ EP F, A 3.68 Fixed Point Theory and Applications 23 Tsn I − sn B xn , for any y ∈ C, we have q Since From 3.43 , we have vni Bxn , y − G , y y − , − xn ≥ sn 3.69 From A2 , we have y − , − xn ≥ G y, sn 3.70 y − vni , vni − xni ≥ G y, vni sni Bxn , y − 3.71 This implies that Bxni , y − vni Put zt ty − t q for all t ∈ 0, and y ∈ C Then, we have zt ∈ C So, from 3.71 we have zt − vni , Bzt ≥ zt − vni , Bzt − zt − vni , Bxni − zt − vni , zt − vni , Bzt − Bvni vni − xni sni G zt , vni zt − vni , Bvni − Bxni − zt − vni , vni − xni sni 3.72 G zt , vni Since vni − xni → 0, we have Bvni − Bxni → Further, from monotonicity of B, we have zt − vni , Bzt − Bvni ≥ So, from A4 , we have zt − q, Bzt ≥ G zt , q 3.73 From A1 , A4 , and 3.64 , we also have G zt , zt ≤ tG zt , y ≤ tG zt , y tG zt , y − t G zt , q − t zt − q, Bzt 3.74 − t t y − q, Bzt , hence ≤ G zt , y − t y − q, Bzt 3.75 y − q, Bq 3.76 Letting t → 0, we have, for each y ∈ C, ≤ G q, y 24 Fixed Point Theory and Applications This implies that q ∈ EP G, B 3.77 Define a mapping Q : C → C by Qx − δ PC I − ηn B x, δPC I − λn A x ∀x ∈ C, 3.78 δ ∈ 0, From Lemma 2.3, we have that Q is nonexpansive with where limn → ∞ δn F Q F P C I − ηn B F P C I − λn A 3.79 Next, we show that lim xn − Qxn n→n 3.80 By nonexpansiveness of I − ηn B and I − λn A, we have xn − Qxn ≤ xn − yn xn − yn yn − Qxn δn PC un − λn Aun − δn PC − ηn Bvn − δPC I − λn A xn − − δ PC I − ηn B xn xn − yn δn PC I − λn A un − δn PC I − λn A xn δn PC I − λn A xn − δn PC I − ηn B − − δn PC I − ηn B xn − δn PC I − ηn B xn − δPC I − λn A xn − − δ PC I − ηn B xn xn − yn δn PC I − λn A un − PC I − λn A xn δn − δ PC I − λn A xn − δn PC I − ηn B − PC I − ηn B xn δ − δn PC I − ηn B xn ≤ xn − yn − δn ≤ xn − yn δn PC I − λn A un − PC I − λn A xn PC I − ηn B − PC I − ηn B xn δn un − xn |δn − δ| PC I − λn A xn |δn − δ| PC I − ηn B xn |δn − δ| PC I − λn A xn − δn − xn |δn − δ| PC I − ηn B xn ≤ xn − yn δn un − xn 2|δn − δ|M1 − δn − xn , 3.81 Fixed Point Theory and Applications 25 PC I − ηn B xn } From 3.17 , 3.42 , 3.43 , 3.55 , where M1 supn≥0 { PC I − λn A xn Since xni q, it follows from 3.80 that, and condition iii , we have limn → n xn − Qxn By Lemma 2.4, we obtain that limi → ∞ xni − Qxni q∈F Q ∩ F P C I − ηn B F P C I − λn A F G1 ∩ F G2 3.82 Assume that q / Sq Using Opial s, property, 3.57 and Lemma 2.10 we have lim inf xni − q < lim inf xni − Sq i→∞ i→∞ ≤ lim inf xni − Sni xni Sni xni − Sni q i→∞ Sni q − Sq 3.83 ≤ lim inf xni − q i→∞ This is a contradiction, so we have N q∈ F Ti 3.84 F S i From 3.68 , 3.77 3.82 , and 3.84 , we have q ∈ F Since PF f is contraction with the coefficient θ ∈ 0, , PF has a unique fixed point Let z be a fixed point of PF f, that is q and q ∈ F, we have z PF f z Since xni lim sup f z − z, xn − z n→∞ lim sup f z − z, xni − z i→∞ 3.85 f z − z, q − z ≤ Step Finally, we will show that xn → z as n → ∞ By nonexpansiveness of Trn , Tsn , I − λn A, I − ηn B, I − rn A, I − sn B, we can show that yn − z ≤ xn − z Then xn −z βn xn − z αn f xn − z αn f xn − z, xn −z ≤ αn f xn − f z , xn γn Sn yn − z xn ≤ αn f xn − f z γn yn − z xn 1 βn xn − z, xn −z −z αn f z − z, xn −z γn Sn yn − z, xn −z −z βn xn − z xn −z −z xn γn Sn yn − z , xn −z −z αn f z − z, xn −z βn xn − z xn −z 26 Fixed Point Theory and Applications ≤ αn θ xn − z xn −z γn xn − z xn −z − αn − θ ≤ − αn − θ ≤ − αn − θ αn f z − z, xn xn − z xn xn − z −z xn xn −z βn xn − z xn αn f z − z, xn −z xn − z −z 2 1 −z −z αn f z − z, xn αn f z − z, xn 1 −z −z ; 3.86 we have xn −z ≤ − αn − θ xn − z 2αn f z − z, xn By Step 5, 3.87 , and Lemma 2.2, we have limn → ∞ xn z, where z that sequences {yn }, {un }, and {vn } converge strongly to z PF f z −z 3.87 PF f z It easy to see Application Using our main theorem Theorem 3.1 , we obtain the following strong convergence theorems involving finite family of κ-strict pseudocontractions To prove strong convergence theorem in this section, we need definition and lemma as follows Definition 4.1 A mapping T : C → C is said to be a κ-strongly pseudo contraction mapping, if there exist κ ∈ 0, such that Tx − Ty ≤ x−y κ I −T x− I −T y , ∀ x, y ∈ C 4.1 Lemma 4.2 see 20 Let C be a nonempty closed convex subset of a real Hilbert space H and T : C → C a κ-strict pseudo contraction Define S : C → C by Sx αx − α T x for each x ∈ C Then, as α ∈ κ, S is nonexpansive such that F S F T Theorem 4.3 Let C be a nonempty closed convex subset of a Hilbert space H Let F and G be two bifunctions from C × C into R satisfying (A1)–(A4), respectively Let A : C → H is a α-inverse strongly monotone mapping and B : C → H be a β-inverse strongly monotone mapping Let {Ti }N1 i N be a finite family of κi -psuedo contractions with F i F Ti ∩ EP F, A ∩ EP G, B ∩ F G1 ∩ PC x − λn Ax , G2 x PC x − F G2 / ∅, where G1 , G2 : C → C are defined by G1 x ηn Bx , for all x ∈ C Define a mapping Tκi by Tκi κi x 1−κi Ti x, for all x ∈ C, i ∈ {1, 2, , N} Let f : C → C be a contraction with the coefficient θ ∈ 0, Let Sn be the S-mappings generated n,j n,j n,j n n n n α1 , α2 , α3 ∈ I × I × I, I 0, , by Tκ1 , Tκ2 , , TκN and α1 , α2 , , αN , where αj Fixed Point Theory and Applications n,j n,j n,j 27 n,j α2 α3 and < η1 ≤ α1 ≤ θ1 < for all n ∈ N, for all j 1, 2, , N − 1, < ηN ≤ α1 n,j n,j n,N α1 ≤ and ≤ α2 , α3 ≤ θ3 < for all n ∈ N, for all j 1, 2, , N Let {xn }, {un }, {vn }, {yn } be sequences generated by x1 , u, v ∈ C F un , u Axn , u − un u − un , un − xn ≥ 0, rn G , v Bxn , v − v − , − xn ≥ 0, sn δn PC un − λn Aun − δn PC − ηn Bvn , yn xn αn f xn βn xn γn Sn yn , 4.2 ∀n ≥ 1, where {αn }, {βn }, {γn } ∈ 0, such that αn βn γn 1, r n ∈ a, b ⊂ 0, 2α , sn ∈ c, d ⊂ 0, 2β , λn ∈ e, f ⊂ 0, 2α , ηn ∈ g, h ⊂ 0, 2β Assume that i limn → ∞ αn and Σ∞ αn n ∞, ii lim infn → ∞ βn ≤ lim supn → ∞ βn < 1, iii limn → ∞ δn δ ∈ 0, , iv Σ∞ |sn − sn |, Σ∞ |rn − rn |, Σ∞ |λn n n n αn |, Σ∞ |βn − βn | < ∞, n n 1,j v |α1 n,j n 1,j − α1 | → and |α3 − λn |, Σ∞ |ηn n − ηn |, Σ∞ |αn n − n,j − α3 | → as n → ∞, for all j ∈ {1, 2, 3, , N} Then the sequence {xn }, {yn }, {un }, {vn } converges strongly to z Ax∗ , x − x∗ ≥ 0, Bx∗ , x − x∗ ≥ PF f z , and z is solution of 4.3 Proof For every i ∈ {1, 2, , N}, by Lemma 4.2, we have Tκi is nonexpansive mappings From Theorem 3.1, we can concluded the desired conclusion Theorem 4.4 Let C be a nonempty closed convex subset of a Hilbert space H Let F and G be two bifunctions from C × C into R satisfying (A1)–(A4), respectively Let A : C → H be a α-inverse strongly monotone mapping Let {Ti }N1 be a finite family of κi -strict pseudo contractions with F i N F Ti ∩EP F, A ∩F G1 / ∅, where G1 : C → C defined by G1 x PC x−λn Ax , for all x ∈ i κi x − κi Ti x, for all x ∈ C, i ∈ N Let f : C → C a C Define a mapping Tκi by Tκi contraction with the coefficient θ ∈ 0, Let Sn be the S-mappings generated by Tκ1 , Tκ2 , , TκN n,j n,j n,j n,j n,j n,j n n n n and α1 , α2 , , αN , where αj α1 , α2 , α3 ∈ I × I × I, I 0, , α1 α2 α3 n,j and < η1 ≤ α1 ≤ θ1 < for all n ∈ N, for all j 1, 2, , N − 1, < ηN ≤ αn,N ≤ and 28 Fixed Point Theory and Applications n,j n,j ≤ α2 , α3 ≤ θ3 < for all n ∈ N, for all j generated by x1 , u, ∈ C αn f xn where {αn }, {βn }, {γn } ∈ 0, such that αn 0, 2α Assume that i limn → ∞ αn and Σ∞ αn n 4.4 PC un − λn Aun , yn xn u − un , un − xn ≥ 0, rn Axn , u − un F un , u 1, 2, , N Let {xn }, {un }, {yn } be sequences βn xn βn γn Sn yn , γn ∀n ≥ 1, 1, rn ∈ a, b ⊂ 0, 2α , λn ∈ e, f ⊂ ∞, lim infn → ∞ βn ≤ lim sup n → ∞ βn < 1, ∞ ∞ Σn |rn − rn |, Σn |λn − λn |, Σ∞ |αn − αn |, Σ∞ |βn − βn | < ∞, n n n 1,j n,j n 1,j n,j |α1 − α1 | → and |α3 − α3 | → as n → ∞, for all j ∈ {1, 2, 3, , N} ii < iii iv Then the sequence {xn }, {yn }, {un } converges strongly to z Ax∗ , x − x∗ ≥ PF f z , and z is solution of 4.5 Proof For every i ∈ {1, 2, , N}, by Lemma 4.2, we have that Tκi is nonexpansive mappings, putting F ≡ G, A ≡ B, sn rn , λn ηn , and un From Theorem 3.1, we can conclude the desired conclusion Acknowledgment The authors would like to thank Professor Dr Suthep Suantai for his suggestion in doing and improving this paper References K Goebel and W A Kirk, Topics in Metric Fixed Point Theory, vol 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990 E Blum and W Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol 63, no 1–4, pp 123–145, 1994 P L Combettes and S A Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol 6, no 1, pp 117–136, 2005 A Moudafi and M Th´ ra, “Proximal and dynamical approaches to equilibrium problems,” in Ille Posed Variational Problems and Regularization Techniques, vol 477 of Lecture Notes in Econom and Math Systems, pp 187–201, Springer, Berlin, Germany, 1999 H Iiduka and W Takahashi, “Weak convergence theorems by Ces´ ro means for nonexpansive a mappings and inverse-strongly-monotone mappings,” Journal of Nonlinear and Convex Analysis, vol 7, no 1, pp 105–113, 2006 H H Bauschke and J M Borwein, “On projection algorithms for solving convex feasibility problems,” SIAM Review, vol 38, no 3, pp 367–426, 1996 P L Combettes, “The foundations of set theoretic estimation,” Proceedings of the IEEE, vol 81, pp 182–208, 1993 Fixed Point Theory and Applications 29 A Kangtunyakarn and S Suantai, “A new mapping for finding common solutions of equilibrium problems and fixed point problems of finite family of nonexpansive mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol 71, no 10, pp 4448–4460, 2009 V Colao, G Marino, and H.-K Xu, “An iterative method for finding common solutions of equilibrium and fixed point problems,” Journal of Mathematical Analysis and Applications, vol 344, no 1, pp 340– 352, 2008 10 S Takahashi and W Takahashi, “Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol 331, no 1, pp 506–515, 2007 11 S Takahashi and W Takahashi, “Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space,” Nonlinear Analysis: Theory, Methods & Applications, vol 69, no 3, pp 1025–1033, 2008 12 X Qin, Y J Cho, and S M Kang, “Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications,” Nonlinear Analysis: Theory, Methods & Applications, vol 72, no 1, pp 99–112, 2010 13 W Takahashi, Nonlinear Functional Analysis: Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, Japan, 2000 14 H.-K Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society Second Series, vol 66, no 1, pp 240–256, 2002 15 R E Bruck Jr., “Properties of fixed-point sets of nonexpansive mappings in Banach spaces,” Transactions of the American Mathematical Society, vol 179, pp 251–262, 1973 16 F E Browder, “Nonlinear operators and nonlinear equations of evolution in Banach spaces,” in Proceedings of Symposia in Pure Mathematics, pp 1–308, Amer Math Soc., Providence, RI, USA, 1976 17 T Suzuki, “Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Applications, vol 305, no 1, pp 227–239, 2005 18 A Kangtunyakarn and S Suantai, “Hybrid iterative scheme for generalized equilibrium problems and fixed point problems of finite family of nonexpansive mappings,” Nonlinear Analysis: Hybrid Systems, vol 3, no 3, pp 296–309, 2009 19 W Takahashi, Introduction to Nonlinear and Convex Analysis, Yokohama Publishers, Yokohama, Japan, 2009 20 H Zhou, “Convergence theorems of fixed points for κ-strict pseudo-contractions in Hilbert spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol 69, no 2, pp 456–462, 2008 ... integrals,” Journal of Mathematical Analysis and Applications, vol 305, no 1, pp 227–239, 2005 18 A Kangtunyakarn and S Suantai, “Hybrid iterative scheme for generalized equilibrium problems and. .. iterative method for finding common solutions of equilibrium and fixed point problems, ” Journal of Mathematical Analysis and Applications, vol 344, no 1, pp 340– 352, 2008 10 S Takahashi and W Takahashi,... Combettes and S A Hirstoaga, ? ?Equilibrium programming in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol 6, no 1, pp 117–136, 2005 A Moudafi and M Th´ ra, “Proximal and dynamical approaches