Applied Mathematics, 2011, 2, 12131220 doi:10.4236/am.2011.210169 Published Online October 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Strong Convergence of an Iterative Method for Generalized Mixed Equilibrium Problems and Fixed Point Problems Lijun Chen, Jianhua Huang* Institute of Mathematics and Computer, Fuzhou University, Fuzhou, China Email: chenlijun861010@163.com , *fjhjh57@yahoo.com.cn Received April 24, 2011; revised July 4, 2011; accepted July 11, 2011 Abstract In this paper, we introduce a hybrid iterative method for finding a common element of the set of common solutions of generalized mixed equilibrium problems and the set of common fixed points of an finite family of nonexpansive mappings. Furthermore, we show a strong convergence theorem under some mild condi tions. Keywords: Generalized Mixed Equilibrium Problem, Hybrid Iterative Scheme, Fixed Point, Nonexpansive Mapping, Strong Convergence 1. Introduction Equilibrium problems theory provides us with a natural, novel and unified framework for studying a wide class of problems arising in economics, finance, transportation, network and structural analysis, elasticity and optimiza tion. The ideas and techniques of this theory are being used in a variety of diverse areas and proved to be pro ductive and innovative. Let H be a Hilbert space with inner product , and norm . Let C be a nonempty closed convex subset of H and :2 TC a multivalued mapping. Let :CC R be a realvalued function and : CC R ,,wuv ,,=0wvu uC be an equilibriumlike function , i.e., for each . The generalized mixed equilibrium problem (for short, GMEP) is to find vHCC,,wu and such that wTu :,, ,,0,GMEPw uvvuu uvC . (1.1) in particular, if T is singlevalued mapping, this problem is equivalent to finding such that uC ,,,, 0,.Tu uvvuuuvC (1.2) Denote the set of solutions of GMEP by . Now, we recall the following definitions. A mapping : CC is said to be contractive if there exists a constant (0,1) such that xfy xy for any , yC. A mapping : CC is said to be firmly nonexpansive if 2, xgygxgyxy . A mapping :TC C is said to be nonexpansive if Tx Ty y for any , yC . The set of fixed points of T is denoted by T. Let =1 ii be a finite family of nonexpansive map pings of C into H and . Define the map pings T =1 Ø N i iFT ,1,1 1,1 ,2,2 2,1,2 ,1,11 ,2,1 ,,,1 , =1 =1 =1 == 1 nn n nnn n nNnN NnNnN nnNnNNnN nN UT I UTU I UTU WU TUI I (1.3) where ,0,1 N ni W map =1 i for all . Such a mapping n is called 1n Wping generated by 1,, TT and ,=1 ni i . 2. Preliminaries Let C be a nonempty closed convex subset of a real Hil bert space H. Then, for any H C P, there exists a unique nearest point in C, denoted by , such that x C Px xy for all C . Such a is called th e metric proj ection C P
1214 L. J. CHEN ET AL. of H into C. We know that is nonexpansive. What’s more, C P =,0, C xPx xxy . x yC Let C be a convex subset of a real Hilbert space H, :CC H and a Frechet differential function. Then k is said to be :kC R strongly convex if there exists a constant >0 such that 2 ,, 2 ,. kykxk xyxxy , Cxy If =0 , then k is said to be convex. In particular, if ,= xy x for all , xC , then k is said to be strongly convex. Let C be a nonempty subset of a real Hilbert space H. A bifunction is said to be skew symmetric if ,:CC R u uv ,0 ,,,, ,. uv vuuvv C 0, . It is easy to see that if the skewsymmetric bifunction is lin ear in both arguments, then , ,uvu C We denote for weak convergence and for strong convergence. A bifunction :CC R 00 , is called weakly sequentially continuous at yCC if 00nn ,, yx , nn y as for each sequence n yC in converging weakly to C 0 ,0 y. The function is called weakly sequentially con tinuous on if it is weakly sequentially continuous at each point of . , CCC C Let denote the set of nonempty closed bounded subset of X. For CB X , BCBX, define the Hausdorff metric as follows: ,=max , ,. bB bA B , aA a Bsupinfda supdba b inf In order to solve the generalized mixed equilibrium problems for an equilibriumlike bifunction : CC R :2 , we assume that satisfies the fol lowing conditions with respect to the multivalued map ping TC : 1 for each fixed vC , ,,wuvwu , is an upper semicontinuous function from C to R, that is, and imply n wwn uu ,v,, nv,wu limsup nn wu ; 2 for each fixed ,wv HC, ,,uwuv is a concave function; 3 for each fixed ,wuH C, ,,vwuv is a convex function; 4 12 ,, ,, rs sr wTxT ywT yTx 2 Tx Ty rs for all , yC and ,0,rs , where >0 , 1 wTx and 2 wTy :kC. Let be a differential function with Frechet derivative R kx at x satisfying the following: k 1 k is continuous from the weak topology to the strong topology; 2 k k is Lipschitz continuous with constant >0 . Let :CCH be a function satisfying the fol lowing: 1 ,,xy yx =0 for all , yC; 2 , is affine in the first coordinate variable; 3 for each fixed C , , Cxyx is se quentially continuous from the weak topology to the weak topology. Let C be a nonempty closed convex subset of a real Hilbert space and :2 TC a multivalued mapping. For C , let wTx. Let :CR be a real valued function satisfying the following: 1 , is skew symmetric; 2 for each fixed , , is convex and up per semicontinuous; 3 , is weakly continuous on . CC Recently WeiYou Zeng, NanJing Huang and Chang Wen Zhao [1] introduce and consider a new class of equilibrium problems, which is known as the generalized mixed equilibrium problems. Furthermore, they intro duce an iterative scheme (1.4) by the viscosity approxi mation method for finding a common element of the set of common solutions for generalized mixed equilibrium problems and the set of common fixed points of a se quence of nonexpansi ve m appings in Hilbert space. 11 1 1 1,; 1 ,,, ,0, =1 n n n nnnnn n nnn TxTx n vvu uukukxvuvC r x Wu ,, nn nn nn ww wu xf (2.1) Copyright © 2011 SciRes. AM
L. J. CHEN ET AL. Copyright © 2011 SciRes. AM 1215 Motivated and inspired by the research going on in this important field, we introduce the following hybrid iterative scheme (1.5) for finding a common element of the set of common solutions for generalized mixed equi librium problems and the set of common fixed points of a sequence of nonexpansive mappings. We show that the approximation solution converges strongly to a unique solution of a class of variational inequalities under some mild conditions. Results obtained in this paper can be viewed as an improvement and refinement of the recent results in this direction. Algorithm 1.1. Let C be a nonempty closed convex subset of a real Hilbert space H, :TC CBH be a multivalued mapping, f be a contraction of C into itself with coefficient 0,1 . Let be defined by (1.3), and . For given 1 :WCC n >0r C and 11 wTx, there exists sequences n , n u in C and n x: nn ww T in H such that for all , =1n,2, 11 1 1 1,; 1 ,,,,,,0, = nnn n nnn nnnnn nnnnnnnnn ww TxTx n wuvvuuukukxvuv r xafWxbxcWu C (2.2) where n a, and are three sequences in (0, 1) such that . n b ab n c =1 nnn It is easy to see that the iterative scheme (1.5) may be well defined. c Let r be a positive number. For a given point C and , consider the following auxiliary prob lem for GMEP: find such that x wTxuC ,, ,, 1,, 0, x wuv vu uu kukxvuvC r , (2.3) It is easy to see that if , then u is a solution of GMEP. =ux We need the following important results. Lemma 1.1. [2] Let C be a nonempty closed convex bounded subset of a real Hilbert space H and let :CC R be a realvalued function satisfying 1  3 . Let :2 TC be a multivalued mapping and : CC R :CC H be an equilibriumlike bifunction satisfying the conditions . Assume that 1 4 >0 is a Lipschitz function with lipschitz constant which satisfies the conditions 1  3 . Let be an :kC R strongly convex func tion with constant >0 which satisfies the conditions 1 k and 2 k. For each C, let x. For , define a mapping by wTx >0r: r TCC 1 =:,, ,,,,0, rx TxuCwuvvuuuku kxvuvC r (2.4) Then there hold the following: 1) the auxiliary problem (1.6) has a unique solution; 2) is singlevalued; r 3) if T1 , it follows that is firmly nonex pansive; r T 4) ; = r FT 5) is closed and convex. Lemma 1.2. [3] Let be a real Hilbert space and let C be a nonempty closed convex subset of H. Let =1 ii T be a finite family of nonexpansive mappings of C into H and =1 Ø N i iFT , and let ,=1 ni i be a se quence in 0,b b N ni for some . Then, 0,1 =1 i = WFT . Lemma 1.3. [4] If the sequences and n u n are bounded and n is defined by (1.3), then the following estimates hold: W 1111,, =1 2, N nnnnnnni ni i Wu WuuuMn 0 and 1111,, =1 2, N nnnnnnni ni i Wx WxxxMn 0
1216 L. J. CHEN ET AL. for some constant . >0M Lemma 1.4. [4] In a real Hilbert space H, ,, yz H and with , there holds the following equality: 123 ,, [0,1]ttt12 3 =1ttt 22 2 123123.txtytztxtytz 2 Lemma 1.5. [6] Let n and be bounded sequences in a Banach space X and let be a se quence in [0,1] with . Suppose n u limb n b sup0<< 1 nn nn liminf b nn 1=1 nnn bz bx for all integers and 0n 11 limsup 0. nnn n nzzxx Then, lim= 0. nn nzx Lemma 1.6. [5] Let is a sequence of nonnega tive real numbers such that n a 11,=1 nnnn aabn ,2, where n is a sequence in (0,1), =1= n n and limsupnn b0 n , then . limnn a = 0 Lemma 1.7. [2] Let n be a sequence in a normed space ,X such that 121,=1,2, nn nnnn xx xxsrn where (0,1) , and n and are sequences satisfying the following conditions: n r 1) and ; 1 n s =1 1< n ns 2) , and . 0 n r=1 < n nr Then n is a Cauchy sequence. Lemma 1.8. [7] Let , BCBX and a. Then for A >1 , there must exist a point such that . bB b ,,da AB Lemma 1.9. [5] In a real Hilbert space H, there holds the following equality: 22 2,, ,. yx yxyxyH 3. Main Results Theorem 2.1. Let C be a nonempty closed convex bounded subset of a real Hilbert space H and , be a multivalued Lipschitz continu ous mapping with constant , and let >0r CC R :TC CBH>0L: be a realvalued function satisfying 1  3 . and : CC :CC H R be an equilibriumlike function sat isfying the conditions . Assume that 1 4 is a Lipschitz function with lipschitz constant >0 which satisfies the conditions 1  3 . Let be an :kC R strongly convex func tion with constant >0 which satisfies the conditions 1 k and 2 k. with 1 . Let =1 ii T be a finite family of nonexpansive mappings on H such that =1i iØ NFT . Let f be a contraction of C into itself with coefficient 0,1 . Let n , n u, n w be sequences generated by (1.5), where n a, n b and n c are three sequences in (0,1) with satisfying the following conditions: =1 n c nn ab 1) , lim= 0 nn a =1 n na= and 1 =1n< nn aa ; 2) 0<n bb liminf nn limsup <1 n and 1 =1n< nn bb ; 3) 1, , =1 < nini i ; 4) 1 =1 < nn ncc . Then th e sequenc es n and n u converge strongly to =1 N ii xFT , and converges strongly n w to wTx , where FT ii =1 N = Pfx . To proof Theorem 2.1, we first establish the following lemma. Lemma 2.1. Let C be a nonempty closed convex bounded subset of a real Hilbert space H and , >0r CBH:TC be a multivalued Lipschitz continu ous mapping with constant , and let >0L:CC R be a realvalued function satisfying 1  3 . and : CC R be an equilibriumlike function sat isfying the conditions 1  4 . Assum e that :CC H >0 is a Lipschitz function with lipschitz constant which satisfies the conditions 1  3 . Let be an :kC R strongly convex func tion with constant >0 which satisfies the conditions 1 k and 2 k with 1 . Let =1 ii T be a finite family of nonexpansive mappings on H such that i FT =1 N iØ . Let f be a contraction of C into itself with coefficient . Let 0, 1 n , n u, n w be sequences generated by (1.5), where n a, n b and n c are three sequences in (0,1) with , satisfying the following conditions: =1 n c nn ab 1) , lim= 0 nn a =1 n na= and 1 =1n< nn aa ; 2) 0< liminfnnn bb limsup < 1 n ; 3) ,1 ,nnin lim= 0 i ; 4) 1 =1 < nn ncc . then 1) 1 lim=0 nnn uu , 1nn xx lim = n 0 ; 2) lim= 0 nnnn xWu , lim= 0 nnn Proof. 1) From the nonexpansity of , we have xu r T . Copyright © 2011 SciRes. AM
L. J. CHEN ET AL. Copyright © 2011 SciRes. AM 1217 111 = n nrnrnn n uuTxTxxx (3.1) and set 1 =1 nn nn n bx zb , we obtain 111111 211 1 111 11 11 11 11 11 11 == 11 11 =111 111 nnnnnnnnnnn nnnnnn nn nn nn nnn n nnnnn nnn nnn nn nn nnn afWxcWuafWx cWu xbxxbx zz bb bb aaa fW xfWxfWx bbb ccc Wu Wu bbb nn Wu n By Lemma 1.3, we arrive at 11 1 111 11 11 1 11 11,, =1 11 111,, =1 1 1111 2 111 2 1 nnnn nnn nnnnnnnn nnn nnnn N nnn nnn ininnnn i nnn N nnn nini i n aaac zzfWxfWxfWx WuWuWu bbbb aaa xx MfWxWu bbb cuu M b (3.2) Hence, it follows from (2.1) that 11 111,, =1 11 111,, =1 1 1 11, , =1 1 2 111 2 1 2 11 N nnn nnnnn ininnnn i nnn N nnn nini i n N nn nnnnnnn ini i nn aaa zzxxMfWxWu bbb cxx M b aa xxfWx WuM bb (3.3) It follows from conditions (a) and (c), we have 11 limsup 0. nnn n nzzxx Hence by Lemma 1.5, we can see that lim=0 nn nzx Consequently 1 lim=lim1= 0 nn nnn nn xx bzx (3.4) From (2.1), we get 1 lim= 0 nn nuu (3.5) 2) In view of (1.5), we conclude that 11 1 , nnn nnnnn nnnnn nn nn nn xWuxxxWu xafWxWu bx Wu that is 1 1 1 , 1 nnn nn n nnn nn n xWu xx b a Wx Wu b which implies that lim= 0 nnn nxWu (3.6) For =1 =N ii pFT , note that is firmly nonexpansive, we can see that r T 22 22 =, =, 1 =2 nrnrrnrn nn nnn upTx TpTx Tpxp upxp upxp ux 2 n and so 22 2 nnnnn upxpuxxp 2 (3.7) In view of Lemma 1.4, (2.6) and (2.7), we compute
L. J. CHEN ET AL. Copyright © 2011 SciRes. AM 1218 2 2 1 222 222 222 222 nnnnnnnnn nnnnn nnn nnnnn nn nnnnn nnnn nnnn nnn xpafWxbxcWup afWxpbx pcWu p afWxpbx pcup afWxpbx pcx pxu afWxpxpcx u 2 which follows that 2 2 11nn nnnnnnnn cx uxpxpxxafWxp and hence lim= 0 nn nxu This completes the proof. Proof of Theorem 2.1. We div ide our proof into 3 steps. Step 1. We prove that there exists C , such that n x , n ux and n as n, where ww wTx . From (1.5), (2.1) and Lemma 1.3, we compute 11111 11 111 11 1111 1111, =1 = 2 nnnnnnnnnn nnn nnnnn nnnn nnnnnnnn nnnnnnn nnnnn N nnnnnnnni i xxafWxbxcWu afWx bxcWu aafWxafWxfW xbbx bxxccWu cWuWu aafWx axxM 1111 ,1 1111 11,, =1 11 2 11 nin nn N nnnnnnn nnnnini i nnn n bb x bxxccWu cuuM axx r (3.8) where 11 =1 12 nn a , and =1 n s 111 =1 =2 N nnnnnnn nnnnnnini i raafWxbbxccWuM1,, . By Lemma 1.7 and conditions (a)(d), we conclude that n is a Cauchy sequence in C such that lim = nn ux , there exists an element C . On the other hand, lim= 0xu nnn implies that lim = nn ux . From (1.5), we have 11 11 1 1, 2, 2 nnn n nn nn ww TxTx n Tx TxLxx (3.9) and for , >1mn 11 11 == 2 mm mn iiii in in wwww Lxx (3.10) 11 1 11 == = 11 1 1 == == 11 1 == = = mm m iii ii in inin mm mm ii i nm in ininin mm ini in in axx ar ar aaa aar 1 i r Hence 1 1= 11 =11 m i min ii nn in r xx xx In view of (2.4) and (2.8) , we obtain , lim=0 mn mn ww (3.11) which implies that n w e exists an xt we ca is a Cauchy sequence in H and therefore ther element w in H such that . Nen see that lim = nn ww ,=inf,n bTx dwTxdwbw w , nn dwTx w wTx Hence, we derive that ,Tx nn ww Lx x (3.12) ,=dwTx 0, that is
L. J. CHEN ET AL. 1219 as Let . Then Q is a contrac wTx Step 2. tion of C into i Tx CBH . =1 =NFT ii QP tself. In fact, for all f , yC QxQyf xfyxy qC Therefore there exists a unique element such th at at =qQq . Noting thqC and , we N FT hat =1 Qq ii get t qF =1 en N ii T . Th =1 ,0,fqqp qp . N ii FT (3.13) Next, we show that . Since =1 N ii xFT n x 0, From and we kno (1 nnn .5) and ux ,w that ku k x 1 , we have ,,, 0wx vx ,vx x that is . We shall show x n FW . Assu me n FW , that is n Wx . Since n u is bounded, there exists a subsequence n u of which con n u verges weakly to . By Lemma 2.1, we conclude that 0 nnn Wu u . From Opial’s condition, we have liminf<liminf liminf liminf nnn jj jj nnn nnn jjj j nj j ux uWx uWu WuWx ux This is a contradiction. So, we get =1 =N ni i FW FT . Therefore =1 N ii xFT . Step 3. From (2.13) and n x , we obtain lim,=,0 n nfq qxqfq qx q (3.14) By Lemma 1.9, (1.5) and (2.7), we compute 2 1 2 11 11 2, 2, 2, 2, nnn nn nn nnnnnn nnnnn nnnnn nnn nn afWxqq Wuq xqcWuqa fWxqxq bxqcuq afWxfqxqafqqxq cxqqxqafqqx q Hence 2 1 2 = nn nn n xqbxc b 22 nn nn bx q ax 2 22 1 1 2 22 1 211 2 12, 11 12 2 =, 111 21 211 1, 11211 =1 , nn n nn n nn nn nn nn n nnn nn n n n nn nn nn aa a xqxq fqqxq aa aa aa xqxqfqqx q aaa aa aM qf aa xq qqxq where 2 1=sup: 1 n Mxqn , 21 =1 n nn a a and 11 1 =, 21 1 n nn aM fq qxq . It is easy to see that 0 n , =1 = n n , and limsup 0 nn . Hence, by sequence Lemma 1.6, the n converge obtain that s rongly owe canst to q. Cnsequently, n u = also converges strongly t, and so o q q . This com pletes th closed convex e Putting for all in Theorem 2.1, we obtain. Corolla Let Cnonempty bounded subset of a real Hilbert space H, proof. = i Txx ry 2.1.1i be a :TC CBH ous mapping with c be a multivalued Lipschitz continu onstant , and let >0L:CC R 1 be a realvalued function satisfying  3 . and : CC R isfying th e conditions be an equilibriumlike function sat 1  4 and Ø . As sume that :CC H >0 is a Lipschitz function with lipschitz constant which satisfies the conditions 1  3 . Let :kC R be an strongly convex function wnstith coant >0 which s tions atisfies the condi 1 k and 2 k with 1 . L be a con traction of C into itself with coefficient 0,1 . T the sequences et Fhen n , n u, and n wrated itera tively by gene Copyright © 2011 SciRes. AM
1220 L. J. CHEN ET AL. 1 ; 1 ,,,,, , = nn nnn nnnnnnn w uvvu uukukxvu r xafxbxcu converge strongly to , and 11 1, nnn n ww TxTx n 1 0, n nnvC (3.15) x n w converges strongly to , wh wTx ere = Pf x , and n a, n b and n c >0r are sequences i and 1) and n satisfying the following conditions: (0,1) with =1 n c, nn ab lim= 0 nn a , =1 = n na 1 =1 < nn naa ; 2) and 0<liminflimsup<1 nn nn bb 1 =1 < nn nbb ; 3) 1 =1 < nn ncc . 4. References J Hua C. Approximds for Generaed Equilib riumquence of Non expa lication Vol. [2] , N.Cong and J.COn Convergence Analysis of an Itive Algorithm for Finding Commo eneralized Mixed Equilibriulem oblems,” Mathematical ties and ce Th n Comdelli p. 14631471. doi:10.1016/S08957177(00)002181 [1] W.Y. Zeng, N.ng andW. Zhao, “Viscosity ation Metholized Mix Problems and Fixed Points of a Se nsive Mappings,” Fixed Point Theory Apps, 2008, 2008, Article ID 714939. D. R. Sahu. W. Yao, “ tera n s aSolution of Gm Prob Fixed Point PrInequli nd Applications, Article in Press. [3] W. Takahashi and K. Shimoji, “Convergeneorems for Nonexpansive Mappings and Feasibility Problems,” Mathematical ad puter Mong, Vol. 32, No. 11, 2000, p [4] L.C. Ceng and J.C. Yao, “A Hybrid Iterative Scheme for Mixed Equilibrium Problems and Fixed Point Prob lems,” Journal of Computational and Applied Mathemat ics, Vol. 214, No. 1, 2008, pp. 18 doi:10.1016/j.cam.2007.02.0226201. [5] H. K. Xu, “Viscosity Approximation for Nonexpansive Mappings,” Journal of Mathematical Analysis and Ap plications, Vol. 298, No. 1, 2004, pp. 279291. doi:10.1016/j.jmaa.2004.04.059 [6] T. Suzuki, “Strong Convergence of Krasnoselskii and Mann’s Type Sequences for OneParameter Nonexpan sive Semigroups without Bochner Integrals,” Journal of Mathematical Analysis and Applications, Vol. 305, No. 1, 2005, pp. 227239. doi:10.1016/j.jmaa.2004.11.017 [7] S. B. Nadle Jr., “MultiValued Contraction Mappings,” Pacific Journal of Mathematics, Vol. 30, 1969, pp. 475 488. Copyright © 2011 SciRes. AM
