Journal of the Egyptian Mathematical Society (2015) 23, 362–370 Egyptian Mathematical Society Journal of the Egyptian Mathematical Society www.etms-eg.org www.elsevier.com/locate/joems ORIGINAL ARTICLE A viscosity Cesa`ro mean approximation method for split generalized vector equilibrium problem and fixed point problem K.R Kazmi *, S.H Rizvi, Mohd Farid Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India Received 23 August 2013; revised January 2014; accepted May 2014 Available online 13 June 2014 KEYWORDS Split generalized vector equilibrium problem; Fixed-point problem; Nonexpansive mapping; Viscosity cesa`ro mean approximation method Abstract In this paper, we introduce and study an explicit iterative method to approximate a common solution of split generalized vector equilibrium problem and fixed point problem for a finite family of nonexpansive mappings in real Hilbert spaces using the viscosity Cesa`ro mean approximation We prove a strong convergence theorem for the sequences generated by the proposed iterative scheme Further we give a numerical example to justify our main result The results presented in this paper generalize, improve and unify the previously known results in this area 2010 MATHEMATICS SUBJECT CLASSIFICATION: 49J30; 47H10; 47H17; 90C99 ª 2014 Production and hosting by Elsevier B.V on behalf of Egyptian Mathematical Society Introduction Throughout the paper unless otherwise stated, let H1 and H2 be real Hilbert spaces with inner product hÁ; Ái and norm k Á k Let C and Q be nonempty closed convex subsets of H1 and H2 , respectively Let Y be a Hausdorff topological space and P be a pointed, proper, closed and convex cone of Y with intP – ; * Corresponding author E-mail addresses: krkazmi@gmail.com (K.R Kazmi), shujarizvi07@ gmail.com (S.H Rizvi), mohdfrd55@gmail.com (Mohd Farid) Peer review under responsibility of Egyptian Mathematical Society Production and hosting by Elsevier In 1994, Blum and Oettli [1] introduced and studied the following equilibrium problem (in short, EP): Find x C such that F1 ðx; yÞ P 0; 8y C; ð1:1Þ where F1 : C  C ! R is a bifunction We denote the solution set of EP(1.1) by sol(EP(1.1)) In the last two decades, EP(1.1) has been generalized and extensively studied in many directions due to its importance; see for example [2–10] for the literature on the existence and iterative approximation of solution of the various generalizations of EP(1.1) Recently, Kazmi and Rizvi [11] considered the following pair of equilibrium problems in different spaces, which is called split equilibrium problem (in short, SEP): Let F1 : C  C ! R and F2 : Q  Q ! R be nonlinear bifunctions and let A : H1 ! H2 be a bounded linear operator then the split equilibrium problem (SEP) is to find xà C such that 1110-256X ª 2014 Production and hosting by Elsevier B.V on behalf of Egyptian Mathematical Society http://dx.doi.org/10.1016/j.joems.2014.05.001 A viscosity Cesa`ro mean approximation method for split generalized vector equilibrium problem F1 ðxà ; xÞ P 0; 8x C; ð1:2Þ The fixed point problem (in short, FPP) for a nonexpansive mapping T is: ð1:3Þ Find x C and such that yà ¼ Axà Q solves F2 ðyà ; yÞ P 0; 8y Q: They introduced and studied some iterative methods for finding the common solution of SEP(1.2) and (1.3), variational inequality and fixed point problems We denote the solution set of SEP(1.2) and (1.3) by sol(SEP(1.2) and (1.3)) :ẳ fp solEP1:2ịị : Ap solðEPð1:3ÞÞg For related work, see [12,14] In this paper, we introduce and study the following class of split generalized vector equilibrium problems (in short, SGVEP): Let F1 : C  C ! Y and F2 : Q  Q ! Y be nonlinear bimappings and let /1 : C ! Y; /2 : Q ! Y be nonlinear mappings, then SGVEP is to find xà C such that F1 x ; xị ỵ /1 xị /1 ðxÃ Þ P; 8x C; ð1:4Þ and such that yà ¼ Axà Q solves F2 ðyà ; yị ỵ /2 yị /2 y ị P; 8y Q: ð1:5Þ When looked separately, (1.4) is the generalized vector equilibrium problem (GVEP) and we denote its solution set by sol(GVEP(1.4)) The SGVEP(1.4) and (1.5) constitutes a pair of generalized vector equilibrium problems which have to be solved so that the image yà ¼ Axà under a given bounded linear operator A, of the solution xà of the GVEP(1.4) in H1 is the solution of another GVEP(1.5) in another space H2 , we denote the solution set of GVEP(1.5) by sol(GVEP(1.5)) The solution set of SGVEP(1.4) and (1.5) is denoted by C ẳ fp sol GVEP1:4ịị : Ap solðGVEPð1:5ÞÞg GVEP(1.4) has been studied by Kazmi and Farid [19] in Banach spaces SGVEP(1.4) and (1.5) generalize multiple-sets split feasibility problem It also includes as special case, the split variational inequality problem [15] which is the generalization of split zero problems and split feasibility problems, see for detail [33,34,15–17] If /1 ¼ /2 ¼ 0, then SGVEP(1.4) and (1.5) reduces to the split vector equilibrium problem (in short, SVEP): Find xà C such that F1 ðxà ; xÞ P; 8x C; ð1:6Þ and such that yà ¼ Axà Q solves F2 ðyà ; yÞ P; 8y Q; ð1:7Þ which appears to be new and is the vector version of SEP(1.2) and (1.3) [11] Further, if H1 ¼ H2 ; C ¼ Q, and F1 ¼ F2 , then SVEP(1.6) and (1.7) reduces to the strong vector equilibrium problem (in short, VEP) of finding xà C such that F1 ðxà ; xÞ P; 8x C; ð1:8Þ which has been studied by Kazmi and Khan [18] In recent years, the vector equilibrium problem has been intensively studied by many authors (see, for example [2–4,18] and the references therein) Next, we recall that a mapping T : C ! C is said to be contraction if there exists a constant a ð0; 1Þ such that kTx À Tyk akx À yk; 8x; y C If a ¼ 1, T is called nonexpansive on C such that x FixðTÞ; ð1:9Þ where FixðTÞ is the fixed point set of the nonexpansive mapping T It is well known that FixðTÞ is closed and convex In 1997, using Cesa`ro mean approximation, Shimizu and Takahashi [20] established a strong convergence theorem for a finite family of nonexpansive mappings fTi g ði ¼ 0;1; 2; ;NÞ in a real Hilbert space For further related work, see [21] Very recently, Colao et al [23] introduced and studied the following iterative method to obtain a strong convergence theorem for FPP(1.9) of a nonexpansive semigroup fTðsÞ : s < 1g in the presence of the error sequence fen g in Hilbert space: & x0 C; xnỵ1 ẳ an cfxn ị ỵ bn xn ỵ bn ịI an BịTsịxn ỵ en ; where f : H1 ! H1 is a contraction mapping with constant a; T : C ! C is a nonexpansive mapping, and B : H1 ! H1 is a strongly positive linear bounded operator, i.e., if there exists a constant c > such that hBx; xi P ckxk2 ; 8x H1 ; c a with < c < and t ð0; 1Þ and proved that the sequence fxn g converges strongly to the unique solution of the variational inequality hðB À cf Þz; x À zi P 0; 8x FixðTÞ; which is the optimality condition for the minimization problem x2FixðTÞ hBx; xi À hðxÞ; where h is the potential function for cf We note that in spite of the fact that the fixed point iterative methods are designed for numerical purposes, and hence the consideration of errors is of both theoretical and practical importance, however, the condition which implies the errors tend to zero, is not suitable for the randomness of the occurrence of errors in practical computations, see [24] Motivated by the work of Shimizu and Takahashi [20], Colao et al [23], Shan and Haung [26] and Kazmi and Rizvi [11,12,14] and by the on going research in this direction, we introduce and study the strong convergence of an explicit iterative method for approximating a common solution of SGVEP(1.4) and (1.5) and FPP(1.9) for a finite family of nonexpansive mappings in real Hilbert spaces using viscosity Cesa`ro mean approximation in Hilbert spaces The results presented in this paper generalize, improve and unify many previously known results in this research area, see instance [5,10–13,22,23] Preliminaries We recall some concepts and results which are needed in sequel For every point x H1 , there exists a unique nearest point in C denoted by PC x such that kx À PC xk kx À yk; 8y C: ð2:1Þ 364 K.R Kazmi et al PC is called the metric projection of H1 onto C It is well known that PC is nonexpansive mapping and is characterized by the following property: hx À PC x; y À PC xi 0: ð2:2Þ Further, it is well known that every nonexpansive operator T : H1 ! H1 satisfies, for all ðx; yÞ H1  H1 , the inequality hðx À TðxÞÞ À ðy À TðyÞÞ; TðyÞ À TðxÞi ð1=2ÞkðTðxÞ À xÞ À ðTðyÞ À yÞk2 ; ð2:3Þ and therefore, we get, for all ðx; yÞ H1  FixðTÞ, hx À TðxÞ; y À TðxÞi ð1=2ÞkTðxÞ À xk2 ; ð2:4Þ see, e.g [27, Theorem 3.1] It is also known that H1 satisfies Opial’s condition [28], i.e., for any sequence fxn g with xn * x the inequality lim inf kxn À xk < lim inf kxn À yk n!1 ð2:5Þ n!1 holds for every y H1 with y – x Lemma 2.4 The following inequality hold in real Hilbert space H1 : kx ỵ yk2 kxk2 ỵ 2hy; x ỵ yi; 8x; y H1 : Definition 2.3 [26,31] Let X and Y be two Hausdorff topological spaces, and let E be a nonempty, convex subset of X and P be a pointed, proper, closed, convex cone of Y with intP – ; Let be the zero point of Y; Uð0Þ be the neighborhood set of 0; Uðx0 Þ be the neighborhood set of x0 , and f : E ! Y be a mapping (i) If for any V Uð0Þ in Y, there exists U Uðx0 Þ such that fðxÞ fx0 ị ỵ V ỵ P or fxị fx0 ị ỵ V Pị; 8x U \ E; then f is called upper P-continuous at x0 If f is upper P-continuous (lower P-continuous) for all x E, then f is called upper P-continuous (lower P-continuous) on E; (ii) If for any x; y E and t ½0; 1, the mapping f satisfies Definition 2.1 A mapping T : H1 ! H1 is said to be rmly nonexpansive, if fxị ftx ỵ tịyị ỵ P hTx Ty; x yi P kTx À Tyk2 ; then f is called proper P-quasiconvex; (iii) If for any x1 ; x2 E and t ½0; 1, the mapping f satisfies 8x; y H1 : Definition 2.2 A mapping T : H1 ! H1 is said to be averaged if and only if it can be written as the average of the identity mapping and a nonexpansive mapping, i.e., T :¼ ð1 aịI ỵ aS; where a 0; 1ị and S : H1 ! H1 is nonexpansive and I is the identity operator on H1 We note that the averaged mappings are nonexpansive Further, the firmly nonexpansive mappings are averaged Further for some key properties of averaged operators, see for instance [16] Lemma 2.1 [29] Let fxn g and fyn g be bounded sequences in a Banach space X and fbn g be a sequence in ½0; 1 with < lim infn!1 bn lim supn!1 bn < Suppose xnỵ1 ẳ bn ịyn ỵ bn xn , for all integers n P and lim supn!1 kynỵ1 yn k kxnỵ1 xn kị Then limn!1 kyn À xn k ¼ Lemma 2.2 [30] Let fan g be a sequence of nonnegative real numbers such that anỵ1 an ịan ỵ dn ; n P 0; where fan g is a sequence in ð0; 1Þ and fdn g is a sequence in R such that ðiÞ X an ¼ 1; ðiiÞ n¼1 dn or n!1 an lim sup or fyị ftx ỵ tịyị þ P; tfðx1 Þ þ ð1 À tÞfðx2 Þ ftx ỵ tịyị ỵ P; then f is called P-convex Lemma 2.5 [26,32] Let X and Y be two real Hausdorff topological spaces; let E be a nonempty, compact, convex subset of X, and let P be a pointed, proper, closed and convex cone of Y with intP – ; Assume that g : E  E ! Y and U : E ! Y are two mappings Suppose that g and U satisfy (i) gðx; xÞ P , for all x E, and gðÁ; yÞ is lower P-continuous for all y E; (ii) U is upper P-continuous on E, and gx; ị ỵ Uị is proper P-quasiconvex for all x E Then there exists a point x E satisfies Gðx; yÞ P n f0g; 8y E; where Gx; yị ẳ gx; yị ỵ Uyị Uxị; 8x; y E: Let F1 : C  C ! Y and /1 : C ! Y be two mappings For any z H1 , define a mapping G1z : C  C ! Y as follows: e G1z x; yị ẳ F1 x; yị ỵ /1 yị /1 xị ỵ hy À x; x À zi; r ð2:6Þ X jdn j < 1: where r is a positive number in R and e P n¼1 Assumption 2.1 Let G1z ; F1 ; /1 satisfy the following conditions: Then limn!1 an ¼ Lemma 2.3 [25] Assume that B is a strong positive linear bounded self adjoint operator on a Hilbert space H1 with coefficient c > and < q kBkÀ1 Then kI À qBk À qc (i) For all x C; F ðx; xÞ P; F is P-monotone, i.e., F x; yị ỵ F y; xị ÀP for all x; y C; F ðÁ; yÞ is continuous for all y 2, and F ðx; ÁÞ is weakly continuous and P-convex, i.e., A viscosity Cesa`ro mean approximation method for split generalized vector equilibrium problem jr2 À r1 j ðF1 ;/1 Þ ðF1 ;/1 Þ ;/1 Þ ðx2 Þ À TðF ðx1 Þ kx2 À x1 k ỵ x2 ị x2 : Tr2 Tr2 r1 r2 tF1 x; y1 ị ỵ tịF1 x; y2 ị F1 x; ty1 ỵ tịy2 ị ỵ P; 8x; y1 ; y2 C; 8t ẵ0; 1; (ii) G1z ; yị is lower P-continuous for all y C and z H , and G1z ðx; ÁÞ is proper P-quasiconvex for all x C and z H (iii) /1 ðÁÞ is P-convex and weakly continuous Lemma 2.6 [26] Assume that C # H1 and Q # H2 are nonempty, compact and convex sets Assume that F1 ; /1 and G1z are satisfying Assumption 2.1 For r > and for all x H1 , define a mapping TrðF1 ;/1 Þ : H1 ! C as follows: TrðF1 ;/1 ị xị ẳ fz C : F1 z; yị ỵ /1 yị /1 zị e ỵ hy À z; z À xi P; 8y Cg: r Then the following hold: ;/1 Þ (i) T ðF ðxÞ is nonempty for all x H r ;/1 Þ (ii) T ðF is single-valued and firmly nonexpansive r ;/1 Þ (iii) FixðT ðF Þ ¼ solðGVEPð1:4ÞÞ and solðGVEPð1:4ÞÞ is r closed and convex Further, assume that F2 : Q  Q ! Y; /2 : Q ! Y and G2z : Q  Q ! Y dened by e G1z u; vị ẳ F2 u; vị ỵ /2 vị /2 uị ỵ hv À u; u À wi; r are satisfying Assumption 2.1 For s > and for all w H2 , define a mapping TsðF2 ;/2 Þ : H2 ! Q as follows: n TsF2 ;/2 ị wị ẳ u Q : F2 u; vị ỵ /2 vị /2 uị o e ỵ hv u; u À wi P; 8v Q : s Then, we easily observe that TsðF2 ;/2 Þ ðwÞ is nonempty for each w H2 ; TsðF2 ;/2 Þ is single-valued and firmly nonexpansive; ;/2 Þ sol(GVEP(2.7))is closed and convex and FixTF ịẳ s solGVEP2:7ịị, where sol(GVEP(2.7)) is the solution set of the following GVEP: Find yà Q such that F2 y ; yị ỵ /2 yị /2 ðxÞ P; 8y Q: ð2:7Þ We observe that solðGVEPð1:5ÞÞ & solðGVEPð2:7ÞÞ Further, it is easy to prove that C is closed and convex set Notation Let fxn g be a sequence in H1 , then xn ! x (respectively, xn * x) denotes strong (respectively, weak) convergence of the sequence fxn g to a point x H1 Main result In this section, we prove a strong convergence theorem based on the proposed viscosity Cesa`ro mean approximation method for computing the approximate common solution of SGVEP(1.4) and (1.5) and FPP(1.9) for a finite family of nonexpansive mappings in real Hilbert spaces First, we have the following lemma The proof is similar to the proof given in [26], and hence omitted Lemma 3.1 Let F1 ; /1 and G1z satisfy Assumption 2.1 and let ;/1 Þ TðF be defined as in Lemma 2.6 for r > Let x1 ; x2 H1 r and r1 ; r2 > Then: Now, we prove the following main result We assume that C – ; Theorem 3.1 Let H1 and H2 be two real Hilbert spaces; let C # H1 and Q # H2 be nonempty, compact and convex subsets; let Y be a Hausdorff topological space and let P be a proper, closed and convex cone of Y with intP – ; Let A : H1 ! H2 be a bounded linear operator Assume that F1 : C  C ! Y; F2 : Q  Q ! Y, /1 : C ! Y and /2 : Q ! Y are nonlinear mappings satisfying Assumption 2.1 and F2 is upper semicontinuous in first argument Let Ti : C ! C be a nonexpansive T mapping for each i ¼ 0; 1; 2; ; n such that H ẳ niẳ1 FixTi ị\ C – ; Let f : H1 ! H1 be a contraction mapping with constant a ð0; 1Þ and B be a strongly positive bounded linear self adjoint operator on H1 with constant c > such that < c < ac < c ỵ 1a For a given x0 C arbitrarily, let the iterative sequences fun g and fxn g be generated by À À Á Á F1 ;/1 ị ;/2 ị xn ỵ dA TðF À I Axn ; > rn < un ¼ Trn > : xnỵ1 ẳ an cfxn ị ỵ bn xn ỵ bn ịI an Bị nỵ1 n X Ti un ỵ cn en ; iẳ0 ð3:1Þ where fen g is an bounded error sequence in H1 ; d ð0; 1=LÞ; L is the spectral radius of the operator Aà A and Aà is the adjoint of A and fan g, fbn g; fcn g are the sequences in ð0; 1Þ and rn & ð0; 1ị satisfying the following conditions: (i) limn!1 an ẳ and (ii) limn!1 acnn ¼ 0; P1 n¼0 an ¼ 1; (iii) < lim infn!1 bn lim supn!1 bn < 1; (iv) lim infn!1 rn > and limn!1 jrnỵ1 rn j ẳ Then the sequence fxn g converges strongly to z PH , where z ẳ PH I B ỵ cfịz Proof By using condition (i) and Lemma 2.3, we can observe that there exists a unique element z H1 such that z ẳ P\n FixTi ị\C I B ỵ cfịzị, see [12] i¼1 T Let p H :¼ ni¼0 FixðTi Þ \ C, i.e., p C, we have ;/1 ị ;/2 ị p ẳ TF p and Ap ẳ TF Apị Using the similar argurn rn ments used in proof of Theorem 3.1 [11], we have the following estimates: À 2 Á ;/2 Þ kun pk2 kxn pk2 ỵ dLd 1ị TðF À I Axn : rn ð3:2Þ À Á Since, d 0; L1 , we obtain kun À pk2 kxn À pk2 : ð3:3Þ Pn i Now, on setting tn :ẳ nỵ1 iẳ0 T , we can easily observe that the mapping tn is nonexpansive Since p H, we have tn p ¼ n n X X Ti p ¼ p ¼ p: n ỵ iẳ0 n ỵ iẳ0 3:4ị 366 n K.R Kazmi et al an kcfðxn Þ À Bpk ỵ bn kxn pk ỵ bn an cịkun pk ỵ cn ken k anỵ1 c kenỵ1 k klnỵ1 ln k kcfxnỵ1 ị Btnỵ1 unỵ1 k ỵ nỵ1 anỵ1 bnỵ1 an cn ken k kBtn un cfxn ịk ỵ ỵ ktnỵ1 unỵ1 tn un k ỵ an bn anỵ1 c kenỵ1 k ỵ kxnỵ1 xn k kcfxnỵ1 ị Btnỵ1 unỵ1 k ỵ nỵ1 bnỵ1 anỵ1 2 kun pk ỵ kpk ỵ ckAkrn ỵ dn ỵ nỵ2 nỵ2 an c ken k þ : kBtn un À cfðxn Þk þ n an bn an ckfxn ị fpịk ỵ an kcfpị Bpk ỵ bn kxn pk ỵ cn ken k Therefore, we obtain Since fen g is bounded, using condition (ii), we obtain that o is bounded Then, there exists a nonnegative real num- cn ken k an ber K such that kcfpị Bpk ỵ cn ken k K; an ð3:5Þ for all n P 0: Further, it follows by (3.1), (3.3) and (3.5) that kxnỵ1 pk ẳ kan cfxn ị ỵ bn xn þ ðð1 À bn ÞI À an BÞtn un þ cn en pk ỵ bn an cịkxn pk anỵ1 c kenỵ1 k kcfxnỵ1 ị Btnỵ1 unỵ1 k ỵ nỵ1 bnỵ1 anỵ1 an cn ken k kBtn un cfxn ị ỵ ỵ bn an klnỵ1 ln k kxnỵ1 xn k 6 an cakxn pk ỵ an kcfpị Bpk ỵ an cịkxn pk ỵ cn ken k c caịan ịkxn pk ỵ an K & ' K max kxn À pk; ; nP0 c ca ỵ ckAkrn ỵ dn ỵ 2 kun pk ỵ kpk: nỵ2 nỵ2 Taking n ! and using the conditions (i)–(iv), we obtain lim sup klnỵ1 ln k kxnỵ1 xn kÞ & ' K max kx0 À pk; :z c À ca ð3:8Þ 0: n!1 ð3:6Þ Hence fxn g is bounded and consequently, we deduce that fun g; ftn un g and ffðxn Þg are bounded Next, it follows from Lemma 3.1 that kunỵ1 un k kxnỵ1 xn k ỵ dkAkrn ỵ dn ; From Lemma 2.1 and (3.8), we obtain limn!1 kln À xn k ẳ and kxnỵ1 xn k lim bn ịkln xn k ẳ 0: 3:9ị n!1 Since, we can write kxn À tn un k kxn xnỵ1 k ỵ kan cfxn ị ỵ bn xn ỵ bn ịI where rnỵ1 TF2 ;/2 ị Axn Axn ; rn ẳ 1 r rn an Bịtn un ỵ cn en tn un k kxn xnỵ1 k ỵ an kcfxn ị Btn un k ỵ bn kxn n tn un k ỵ cn ken k; À À Á Á rnỵ1 TF1 ;/1 ị xn ỵ dA TF2 ;/2 Þ À I Axn dn ¼ 1 À rn r rn and then n À À Á Á À xn ỵ dA TrFn ;/2 ị I Axn ; kxn À tn un k see [12] for details Next, we easily estimate that an c ken k : kxn xnỵ1 k ỵ kcfxn ị Btn un k ỵ n bn bn an Since an ! and kxnỵ1 À xn k ! as n ! 1, we obtain lim kxn À tn un k ¼ 0: 2 kun pk ỵ kpk: ktnỵ1 unỵ1 tn un k kunỵ1 un k ỵ n ỵ 2ị n ỵ 2ị n!1 3:10ị It follows from the above two inequalities that Again, since fxn g is bounded, we may assume a nonnegative real number K such that kxn À pk M It follows from (3.2) and Lemma 2.4 that ktnỵ1 unỵ1 tn un k kxnỵ1 xn k ỵ dkAkrn ỵ dn kxnỵ1 pk2 ẳ kan cfxn ị Bpị ỵ bn xn tn un ị ỵ 2 kun pk ỵ kpk: nỵ2 nỵ2 3:7ị Setting xnỵ1 ẳ bn ịln ỵ bn xn , then we have an cfxn ị ỵ bn ịI an Bịtn un ỵ cn en ln ẳ ; bn and anỵ1 c enỵ1 lnỵ1 ln ẳ cfxnỵ1 ị Btnỵ1 unỵ1 ỵ nỵ1 bnỵ1 anỵ1 an c en ỵ tnỵ1 unỵ1 tn un ỵ : Btn un cfxn ị n À bn an It follows from (3.7) that þ ð1 À an BÞðtn un À pÞ þ cn en k2 kð1 À an BÞðtn un À pÞ þ bn ðxn À tn un Þk2 þ 2han cfðxn ị Bp ỵ cn en ; xnỵ1 pi ẵk1 an Bịtn un pịk ỵ bn kxn tn un k2 ỵ 2an hcfxn ị Bp; xnỵ1 pi ỵ 2hcn en ;xnỵ1 pi ẵ1 ancịkun pk ỵ bn kxn tn un k2 ỵ 2an hcfxn ị Bp; xnỵ1 pi ỵ 2cn ken kM ẳ ancị2 kun pk2 ỵ b2n kxn tn un k2 þ 2ð1 À ancÞbn kun À pk  kxn À tn un k ỵ 2an hcfxn ị Bp; xnỵ1 pi ỵ 2cn ken kM 3:11ị A viscosity Cesa`ro mean approximation method for split generalized vector equilibrium problem ancị2 ẵkxn pk2 ỵ dLd 1ịkTFrn2 IịAxn k2 ỵ b2n kxn tn un k2 ỵ 21 ancịbn kun pkkxn tn un k ỵ 2an hcfxn ị Bp; xnỵ1 pi ỵ 2cn ken kM ;/2 ị Since an ! 0; kxnỵ1 xn k ! 0, TðF À I Axn ! and rn kxn À tn un k ! as n ! and from (i) and (iv), we obtain lim kun xn k ẳ 0: 3:13ị n!1 Using (3.10) and (3.13), we obtain kxn pk2 ỵ ancị2 ịkxn pk2 ỵ ancị2 dLd À 1Þk TFrn2 À I Axn k2 ktn un À un k ktn un xn k ỵ kxn À un k ! as n ! 1: Next, we show that lim supn!1 hðcf À BÞz; xn À zi 0, where z ¼ PH ðI À B þ cfÞz To show this inequality, we choose a subsequence funi g of fun g such that ỵ b2n kxn tn un k2 ỵ 21 ancịbn kun pkkxn À tn un k c ken kM : ỵ 2an hcfxn ị Bp;xnỵ1 pi ỵ n an lim sup hðcf À BÞz; un À zi ¼ lim hðcf À BÞz; uni À zi: i!1 n!1 Therefore, À Á ð1 À an cÞ2 dð1 À LdÞk TFrn2 À I Axn k2 2 kxn À pk kxnỵ1 pk ỵ b2n kxn tn un k 2 ỵ an c kxn pk ỵ 21 an cịbn kun pkkxn tn un k c ken kM ỵ 2an hcfxn ị Bp; xnỵ1 pi ỵ n an kxn pk ỵ kxnỵ1 pkịkxn xnỵ1 k ỵ b2n kxn tn un k2 ỵ an c2 kxn pk2 ỵ 21 an cịbn kun À pkkxn À tn un k c ken k M: ỵ 2an ckfxn ịk ỵ kBpk ỵ n an Since funi g is bounded, there exists a subsequence funij g of funi g which converges weakly to some w C Without loss of generality, we can assume that uni * w From ktn un À un k ! 0, we obtain tT n uni * w Now, we prove that w niẳ1 FixTi ị \ C Let us first show Pn i Assume that that w Fixtn ị ẳ nỵ1 iẳ0 FixT ị P n i w R nỵ1 iẳ0 FixT ị Since uni * w and tn w – w Form Opial’s condition (2.5), we have lim inf kuni À wk < lim inf kuni À tn wk i!1 i!1 Since dðLd 1ị > 0; kxnỵ1 xn k ! and kxn À tn un k ! as n ! and from (i) and (ii), we obtain À 2 Á lim TðF2 ;/2 Þ À I Axn ẳ 0: 3:12ị n!1 lim inf fkuni tn uni k ỵ ktn uni tn wkg i!1 lim inf kuni À wk; i!1 rn Next, we show that kxn À un k ! as n ! Since p H, we can obtain kun À pk2 kxn À pk2 À kun À xn k2 ỵ 2dkAun xn ịk TF2 ;/2 Þ À I Axn ; which is a P contradiction Thus, we obtain n i w Fixðtn ị ẳ nỵ1 iẳ0 FixT ị Next, we show that w solðGVEPð1:4ÞÞ Since ;/1 Þ ;/2 ị un ẳ TF dn where dn :ẳ xn þ dAà TðF À I Axn , we rn rn have rn see [11] It follows from (3.11) and (3.12) that 2 kxnỵ1 pk an cị kun pk ỵ b2n kxn tn un k ỵ 21 an cịbn kun pk 3:14ị F1 un ; yị ỵ /1 yị /1 un ị ỵ e hy un ; un À dn i P; rn 8y C; ð3:15Þ kxn tn un k ỵ 2an hcfxn ị Bp; xnỵ1 pi ỵ 2cn ken kM h À i Á 2 ð1 À an cị kxn pk kun xn k ỵ 2dkAðun À xn Þk TrðFn ;/2 Þ À I Axn ỵ b2n kxn tn un k þ 2ð1 À an cÞbn kun À pkkxn À tn un k ỵ 2an hcfxn ị Bp; xnỵ1 pi ỵ 2cn ken kM 2 kxn pk ỵ an cị kxn pk À an cÞ which implies that F1 ðy; un Þ À ð/1 ðyÞ À /1 ðun ÞÞ À e hy un ; un dn i ỵ P; rn 8y C: Let yt ¼ ð1 tịw ỵ ty for all t 0; Since y C and w C, we get yt C and now (3.15) shows that kun À xn k2 ỵ 21 an cị dkAðun À xn Þk TrðFn ;/2 Þ À I Axn 0 1+ TrðFn ;/2 Þ I Axni uni xni i @ A ỵ P: F1 ðyt ; uni Þ À ð/1 ðyt Þ À /1 ðuni ÞÞ À e yt À uni ; ỵ dA r ni rni * ỵ b2n kxn tn un k2 ỵ 21 an cịbn kun À pk kxn À tn un k c ken kM : ỵ 2an hcfxn ị Bp; xnỵ1 pi ỵ n an 3:16ị Since A is bounded linear, it follows from (3.12) and (3.13) uni Àxni and lim inf rn > that !0 and rni Therefore, ð1 À an cÞ2 kun xn k2 6kxn pk2 kxnỵ1 pk2 þ b2n kxn À tn un k2 ðF ;/2 Þ þ an c2 kxn À pk2 þ 2ð1 À an cÞbn kun À pkkxn À tn un k À ;/2 ị I Axn ỵ 21 À an cÞ2 dkAðun À xn Þk TðF rn c ken kM ỵ 2an hcfxn ị Bp; xnỵ1 pi ỵ n an A @ Trn i ÀI Axni rni A ! 0, and so F1 ðyt ; wÞ À ð/1 ðyt Þ À /1 wịị ỵ P: 3:17ị kxn pk þ kxnþ1 À pkÞkxn À xnþ1 k þ b2n kxn tn un k ỵ 21 an cÞ2 dkAðun À xn Þk TðF2 ;/2 Þ À I Axn rn ỵ an c2 kxn pk2 þ 2ð1 À an cÞbn kun À pkkxn À tn un k c ken k ỵ 2an ckfxn ịk ỵ kBpk ỵ n M: an It follows from Assumption 2.1 (i) and (iii) that tF1 ðyt ; yÞ þ ð1 À tÞF1 ðyt ; wÞ þ t/1 ðyÞ þ ð1 À tÞ/1 ðwÞ À /1 ðyt Þ F1 yt ; yt ị ỵ /1 yt ị /1 yt ị ỵ P ẳ P; 368 K.R Kazmi et al which implies that tẵF1 yt ;yị ỵ /1 yị /1 yt ị tịẵF1 yt ;wị ỵ /1 wị /1 yt ị P: 3:18ị ỵ an hcfzị Bz;xnỵ1 zi ỵ cn ken kM From (3.17) and (3.18), we get tẵF1 yt ; yị ỵ /1 yị /1 yt ị tịẵF1 yt ; wị ỵ /1 ðwÞ À /1 ðyt Þ À P ÀP an c caị kxn zk2 ỵ kxnỵ1 À zk2 Þ À an ðc À caÞ kxn zk2 ỵ kxnỵ1 zk2 2 þ an hcfðzÞ À Bz;xnþ1 À zi þ cn ken kM: and so tẵF1 yt ; yị ỵ /1 yị À /1 ðyt Þ ÀP: This implies that It follows that kxnỵ1 zk2 ẵ1 an c caịkxn zk2 cn ken kM ỵ 2an hcfzị Bz; xnỵ1 zi ỵ an F1 yt ; yị ỵ /1 yị /1 yt ị P: Letting t ! 0, we obtain ẳ ẵ1 an c caịkxn zk2 ỵ 2an Mn : F1 w; yị ỵ /1 yị /1 wị P; 8y C: This implies that w solðGVEPð1:4ÞÞ Next, we show that Aw solðGVEPð1:5ÞÞ Since kun À xn k ! 0; un * w as n ! and fxn g is bounded, there exists a subsequence fxnk g of fxn g such that xnk * w and since A is a bounded linear operator so that Axnk * Aw Now setting vnk ¼ Axnk À TrðFn ;/2 Þ Axnk It follows that from k ;/2 ị (3.12) that limk!1 vnk ẳ and Axnk À vnk ¼ TðF Axnk rn k Therefore from Lemma 2.6, we have F2 ðAxnk À vnk ; zị ỵ /1 zị /1 unk ị ỵ ðAxnk À vnk Þ À Axnk i P; e hz À ðAxnk À vnk Þ; rnk Since F2 is upper semicontinuous in first argument and P is closed, taking lim sup to above inequality as k ! and using condition (iii), we obtain 8z Q; which means that Aw solðGVEPð1:5ÞÞ and hence w C Next, we claim that lim supn!1 hðcf À BÞz; xn À zi 0, where z ¼ PH ðI À B þ cfÞz Now from (2.2), we have lim sup hðcf Bịz; xn zi ẳ lim sup hcf BÞz; tn un À zi n!1 n!1 lim suphðcf Bịz; tn uni zi i!1 ẳ hcf BÞz; w À zi 0: ð3:19Þ Finally, we show that xn ! z It follows from (3.3) that kxnỵ1 zk2 ẳ an hcfxn ị Bz; xnỵ1 zi ỵ bn hxn z;xnỵ1 zi ỵ h1 bn ịI an Bịtn un zị ỵ cn en ;xnỵ1 zi an chfxn ị fzị; xnỵ1 zi ỵ hcfzị Bz;xnỵ1 ziị þ bn kxn À zkkxnþ1 À zk þ kð1 À bn ịI an Bkktn un zkkxnỵ1 zk þ cn ken kM an ackxn À zkkxnþ1 À zk ỵ an hcfzị Bz;xnỵ1 zi ỵ bn kxn zkkxnỵ1 zk ỵ bn ancị kxn zkkxnỵ1 zk ỵ cn ken kM ẳ ẵ1 an c caịkxn zkkxnỵ1 zk ỵ cn ken kM ỵ an hcfzị Bz; xnỵ1 zi Since limn!1 an ẳ and nẳ0 an ¼ 1, it is easy to see that lim supn!1 Mn Hence, from (3.19) and (3.20) and Lemma 2.2, we deduce that xn ! z, where z ẳ PH I ỵ cf Bị This completes the proof h Remark 3.1 The method presented in this paper extend, improve and unify the methods considered in [11–14] Moreover, the algorithm and approach considered in Theorem 3.1 are different from those considered in [15,16] Numerical example 8z Q: F2 Aw; zị ỵ /1 zị /1 unk Þ P; ð3:20Þ P1 Now, we give a numerical example which justify Theorem 3.1 Example 4.1 Let H1 ¼ H2 ¼ R, the set of all real numbers, with the inner product defined by hx; yi ¼ xy; 8x; y R, and induced usual norm j Á j Let Y ẳ R, then P ẳ ẵ0; ỵ1ị Let C ẳ ẵ0; and C ẳ ẵ4; 0; let F1 : C  C ! R and F2 : Q Q ! R be defined by F1 ðx; yÞ ¼ ðx À 6Þðy À xÞ; 8x; y C and F2 u; vị ẳ u ỵ 1ịv uị; 8u; v Q; let /1 : C ! R and /2 : Q ! R be defined by /1 xị ẳ 4x; 8x C and /2 uị ẳ 3u; 8u Q, respectively, and let for each x R, we dene fxị ẳ 18 x; Axị ẳ 2x; Bxị ẳ 2x; en ẳ sinnị; 8n and let, for each x C; Txị ẳ x Then there exist unique sequences fxn g & R; fun g & C, and fzn g & Q generated by the iterative schemes ! zn ẳ TFrn2 Axn ị; un ẳ TFrn1 xn ỵ A zn Axn ị ; ! ! 1 1 I B un ỵ sinnị; xnỵ1 ẳ xn ỵ 0:1 ỵ xn ỵ 0:1 þ 4n n n n n ð4:1Þ ð4:2Þ where an ẳ 1n ; bn ẳ 0:1 ỵ n12 , cn ¼ n13 and rn ¼ Then fxn g converges strongly to 2 FixðTÞ \ C Proof It is easy to prove that the bifunctions F1 and F2 and mappings /1 and /2 satisfy the Assumption 2.1 and F2 is upper semicontinuous A is a bounded linear operator on RÀ with Á adjoint operator Ẫ and kAk ¼ kẪ k ¼ Hence d 0; 14 , so we can choose d ¼ 18 Further, f is contraction mapping with constant a ¼ 15 and B is a strongly positive bounded linear operator with constant c ¼ on R Therefore, we can choose c ¼ which satises < c < ac < c ỵ 1a Furthermore, it is easy A viscosity Cesa`ro mean approximation method for split generalized vector equilibrium problem 1.5 x n+1 0.5 −0.5 −1 −1.5 −2 10 20 30 40 50 60 70 No of iteration when initial guess x1=0.5 Figure Convergence of fxn g to observe that FixTị ẳ 0; 1ị, solGVEP1:4ị ẳ f2g; solGVEP1:5ịị ¼ fÀ4g Hence C :¼ f2g Consequently, FixðTÞ \ C ¼ f2g – ; After simplification, schemes (4.1) and (4.2) reduce to un ẳ 3xn ỵ 10ị; ! 3:5 4:5 15 xn ỵ ẳ ỵ ỵ sinnị: 8n 4n n zn ẳ xn ỵ 2ị; 4:3ị xnỵ1 4:4ị Following the proof of Theorem 3.1, we obtain that fzn g converges strongly to À4 solðGVEPð1:5ÞÞ and fxn g; fun g converge strongly to w ẳ 2 FixTị \ C as n ! 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