New Iterative Method for Variational Inclusion and Fixed Point Problems

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1 Proceedigs of the World Cogress o Egieerig 04 Vol II, WCE 04, July - 4, 04, Lodo, U.K. Ne Iterative Method for Variatioal Iclusio ad Fixed Poit Problems Yaoaluck Khogtham Abstract We itroduce a iterative method for fidig a commo elemet of the set solutios of equilibrium problems, the set of solutios of a variatioal iclusio problems for a iverse-strogly mootoe mappig ad set-valued maximal mootoe mappig, ad the set of fixed poits of a oexpasive mappig i a real Hilbert space. The, e prove a strog covergece theorem of the proposed method ith suitable cotrol coditios. Idex Terms Fixed poit, variatioal iequality, optimizatio problem, oexpasive mappig T I. INTRODUCTION HROUGHOUT this paper, e alays assume that H be a real Hilbert space ith ier product ad orm, are deoted by, ad, respectively ad let K be a oempty closed covex subset of H. Let G be a bifuctio of KK R, here R is the set of real umbers. The equilibrium problem for a bifuctio G : KK R is to fid uksuch that G(u,v) 0, v K. (.) The set of solutios of (.) is deoted by EP(G). Numerous problems i Physics, optimizatio, ad ecoomics reduce to fid a solutio of (.). let A: K H be a oliear map. The classical variatioal iequality hich is deoted by VI(K,A) is to fid uksuch that Au,v u 0, v K. We have ko from Blum ad Oettli [] that the equilibrium problem cotais the fixed poit problem, optimizatio problem, saddle poit problem, variatioal iequality problem ad Nash equilibrium problem as its special case. Give a mappig T : K H, Let G(u,v) Tu,v u, u,v K. The z EP(G) if ad oly if Tz, v z 0, v K, i.e., z is a solutio of the variatioal iequality. A mappig S of K ito itself is called oexpasive if Su Sv u v, u, v K. We deoted by F(S) the set of fixed poits of S (see [4], [5]). A mappig A of K ito H is called iverse-strogly mootoe (see [3], [8]) if there exists a positive real umber such Mauscript received February, 04; revised April 7, 04. This ork as supported i part by Maejo Uiversity, Chiag Mai, Thailad, 5090, uder Grat OT Y. Khogtham is ith Faculty of Sciece, Maejo Uiversity, Chiag Mai, Thailad, 509 (PHONE: ; Fax: ; yaoa.k@mju.ac.th). that Au Av Au Av, u, v K. Recall that a mappig f : K K is said to be cotractive ith coefficiet (0,), if f(u) f(v) u v, u,v K. Let B be a strogly positive bouded liear operator o H: that is, there is a costat 0 ith property Bx,x x, x H. Let A: H H be a sigle-valued oliear mappig ad let M : H variatioal iclusio, hich is to fid u H such that H be a set-valued mappig. We cosider the A(u) M(u), (.) here is the zero vector i H. The set of solutio of problem (.) is deote by I(A,M). It is ko that (.) provides a coveiet i the frameork for the uified study of optimal solutios i may optimizatio related areas icludig mathematical programmig, complemetarity, variatioal iequalities, optimal cotrol, mathematical ecoomics, equilibria, ad game theory (see [8] ad the referece therei). If M, here K is a oempty closed covex subset of H ad : H [0, ] K is the idicator fuctio of K, the the variatioal iclusio problem (.) is equivalet to variatioal iequality problem. Recall the resolvet operator JM, associated ith M ad J as M, K (u) (I M) (u), u H, here M is maximal mootoe mappig ad is a positive umber. The resolvet operator JM, is sigle-valued, mootoe ad - iverse-strogly mootoe, ad that a solutio of problem (.) is a fixed poit of the operator J M, (I A) for all 0, see for example [8]. Some methods have bee proposed to solve the equilibrium problem, variatioal iequality ad fixed poit problem of oexpasive mappig (see []-[4], [7], [9], [0], ad the referece therei). Very recetly, Jug [3] itroduced a e geeral composite iterative scheme for fidig a commo poit of the set of solutios of the variatioal iequality problem ad the set of fixed poit of a oexpasive mappig i Hilbert space. Startig ith x x K, y (u f(x )) (I (I B))SP K(x Ax ), (.3) x ( )y SP (y Ay ),. K They proved that uder certai appropriate coditios imposed o { },{ }, ad { } of parameters, the the sequece{x } coverges strogly to qf(s) VI(K, A), hich is a solutio of the optimizatio problem: ISBN: ISSN: (Prit); ISSN: (Olie) WCE 04

2 Proceedigs of the World Cogress o Egieerig 04 Vol II, WCE 04, July - 4, 04, Lodo, U.K. mi Bx, x x u h(x), (.4) xf(s) VI(K,A) here h is a potetial fuctio for f. I this paper motivated by the iterative scheme that proposed by Jug [3]. We ill itroduce a e iterative method for a commo elemet of the set solutio of equilibrium problem, variatioal iclusio ad the set of fixed poit of a oexpasive mappig hich ill preset i the mai result sectio. II. PRELIMINARIES Let K be a oempty closed covex subset of a real Hilbert space H. It ell ko that H satisfies the Opial s coditio (see [6]), that is, for ay sequece {x }ith {x } coverges eakly to x (deote by x x), the iequality: limif x x limif x y holds for every y H ith y x. The folloig lemmas are useful for provig our theorem. Lemma. (See [3].) I a real Hilbert space H, there holds the iequality x y x y,x y, x,y H. Moreover, let F : H K be a mappig defied by for all x H. The, the folloig hold: () F is a sigle value; F (x) z K : G(z,y) y z,z x 0, y K, () F is firmly oexpasive; that is, for ay x,yh, F x F y F x F y,x y ; (3) F(F ) EP(G); (4) EP(G) is closed ad covex. Lemma.5 (See [3].) Let C be a bouded oempty closed covex subset of a real Hilbert space H, ad let g : C R be a proper loer semicotiuous differetiable covex fuctio. If problem g(x ) Iparticular, if the u f (I xc x is a solutio to the miimizatio if g(x), the g (x),x x 0,x C. x solves the optimizatio problem mi Bx, x x u h(x), xc potetial fuctio for f. B))x,x x 0,x C, here h is a Lemma. (See [7].) Assume A is a strogly positive liear bouded operator o a Hilbert space H ith coefficiet 0 ad 0 A. The I A. Lemma.3 (See [4].) Assume a is a sequece of oegative real umbers such that a ( )a, 0, here R such that () ; () limsup 0 or. The lim a 0. is a sequece i (0,) ad is a sequece i For solvig the equilibrium problem for a bifuctio G : KK R here R is the set of real umbers, let us assume that G satisfies the folloig coditios: (A) G(x, x) 0 for all x K; (A) G is mootoe, that is, G(x, y) G(y, x) 0for all x,y K; (A3) for each x,y,z K, limt0g(tz ( t) x, y) G(x,y); (A4) for each x K, y G(x,y) is covex ad loer semicotiuous. Lemma.4 (see [8].) Let K be a covex closed subset of a Hilbert spaces H. Let G : KK R, is a bifuctio satisfyig (A)-(A4). Let 0 ad x H. The. There exists z K such that G(z, y) y z, z x 0, y K. III. MAIN RESULT I this sectio, e prove a strog covergece theorem. Theorem 3.. Let K be a oempty closed covex subset of a real Hilbert space H such that K K K, let G : KK R is a bifuctio satisfyig (A)-(A4), ad H M : H be a maximal mootoe mappig. Let A be a iverse-strogly mootoe mappig of K ito H ad S a oexpasive mappigs of K ito itself such that : F(S) EP(G) I(A, M). Let f be a cotractive of K ito itself ith costat (0,) ad let B be a strogly positive bouded liear operator o K ith costat (0,). Assume that 0ad 0 ( ) /. Let {x } be a sequece geerated by x x K, F(u,y) r y u,u x 0, y K, (3.) y (u f(x )) (I (I B))SJ (u Au ), M, x ( )y SJ (y Ay ),, M, here u F r x, { } [0,),{ } [0, ],{r } (r, ), r 0, ad { } [0,] satisfy : i) lim 0; ; ii) [0,d) for all 0ad for some d (0,); iii) [a,b] for all 0ad for some a, b ith 0 a b ; iv),, r r, ad. ISBN: ISSN: (Prit); ISSN: (Olie) WCE 04

3 Proceedigs of the World Cogress o Egieerig 04 Vol II, WCE 04, July - 4, 04, Lodo, U.K. The {x }coverges strogly to zf(s) EP(G) I(A,M), hich is a solutio of the optimizatio problem mi Bx, x x u h(x), (3.) xf(s) EP(G) I(A,M) here h is a potetial fuctio for f. Proof. From the coditio i), e may assume that ( B ). Applyig Lemma., e obtai I (I A)) ( ). Let v. Sice u F r x e have u v F x F v x v, r r N. Let M, z J (u u ) ad v J M, (y y ), N. As I A is oexpasive ad v, e have M, M, z v J (u Au ) J (v Av) u v, N. Similarly, e have Similarly, e have v v y y Ay (3.9) Usig (3.7) ad (3.8), e obtai z z x x r r u x r Au From (3.) ad (3.0), e have y y ( u f(x ) Sz ) (3.0) ( (( ) ) ) x x (3.) r r r u x Au. It follos from (3.3)ad (3.), e obtai The e obtai v v y v, N. (3.3) x x ( (( ) ) ) x x G G G G r r, 3 4 r (3.) From the coditio i) ad (3.), e have z v x v, N. (3.4) y Sz u f(x ) Sz 0,. (3.5) For v, ad let (I B), e have y v u f(x ) v I Sz v The e have ( (( ) ) ) x v f (v) v u ( ) (( ) ),. f (v) v u ( ) x v max x v,,. (3.6) It follos from (3.6) ad iductio that f (v) v u ( ) x v max x v,,. Hece {x }is bouded, so are { Sz }. {u },{y },{f(x )},{Sz },{Sv },{Ay },{Au }, ad Next e sho that lim x x 0. We observe that u u F x F x r r x x r r ( u x ), r r 0, N. Moreover, e ca ote that z z J (u Au ) J (u Au ) M, M, u u Au. (3.7) (3.8) here Au By :, G3 sup Sv y : G4 sup u x :. G sup u f(x ) Sz :, G sup, ad ad iv), e have The, from the coditio i) lim x x 0. (3.3) By usig the coditio ii), e ca sho that d x y (d) x x Sz y. (3.4) Combiig (3.5) ad (3.3), e get the folloig We ca also get that lim x y 0. (3.5) Next, e sho lim x u 0. Sice lim x y 0. (3.6) u u v x v x u, e have v u v x v x u. (3.7) It follos from (3.5) ad usig z v u v, N, e have x v u f(x ) v x v ISBN: ISSN: (Prit); ISSN: (Olie) WCE 04

4 Proceedigs of the World Cogress o Egieerig 04 Vol II, WCE 04, July - 4, 04, Lodo, U.K. The e also have ( ( ) x u u f(x ) v z v. (3.8) Moreover, by the coditio i) ad (3.6), e have lim y Su 0. (3.7) ( ( ) x u u f(x ) v x v x v u f(x ) v z v u f(x ) v x v x v x x u f(x ) v z v. By the coditio i) ad usig (3.), e have (3.9) lim x u 0. (3.0) We ote from (3.) ad the coditio iii) that y p u f(x ) v x v Hece, e obtai u f(x ) v z v ( ( ) )a(b ) Au Av. ( ( ) )a(b ) Au Av u f(x ) v x v y v x y u f(x ) v z v. Usig (3.6), (3.), ad the coditio i), e have (3.) (3.) lim Au Av 0. (3.3) Furthermore, applyig Lemma., e obtai z p x p u z (3.4) u z,au Av Au Av. The e obtai ( ( ) ) u z u f(x ) v x v y v x y ( ) u z,au Av ( ( ) )c(d ) Au Av u f(x ) v z v. (3.5) Usig (3.6), (3.3), (3.5), ad the coditio i), e have lim u z 0. (3.6) It follos from y u y x x u ad usig (3.6) ad (3.0), e obtai lim y u 0. (3.8) Usig (3.7), (3.8), ad this iequality Su u Su y y u, e have Next, e sho that lim Su u 0. (3.9) limsup u ( f )x,y x 0, here x is a solutio of (3.). To sho this iequality, e first sho that limsup u ( f )x,su x 0. Sice {u } is bouded, e choose a subsequece i i {u i } of {u } such that limsup u ( f ) x,su x limsup u ( f )x,s u i x Without loss of geerality, e ca assume that u z. From (3.4), e have i follos by (3.) ad (A) that Sice u i u i r i 0 (as i ) ad y i z. It u i u i i r i i y u, u i z, G(y,u ). it follos by (A4) that 0 G(y, z) for all y H. For t ith 0 t ad y H, let yt ty ( t) z. Sice yh ad z H, e have yt Had hece G(y t,z) 0. From (A) ad (A4), e have 0 G(y t,y t) tg(y t,y) ( t)g(y t,z) t(y t,y), ad 0 G(y t,y). From (A3), e have 0 G(z,y) for all yh ad Lemma.4, e have zep(g). By the same argumet as i proof of Theorem 3. of Plubtieg ad Sriprad [8], e have zf(s) I(A, M). The e have z. It follos from Lemma.5 ad (3.9) that limsup u ( f )x,su x limsup u ( f ) x,su i i i x u ( f )x,u x u ( f )x,z x 0. We ca ote that limsup u ( f )x,y Su limsup u ( f )x,su x limsup u ( f )x,su x limsup u ( f )x y Su limsup u ( f ) x, Su x. It follos from (3.7) ad (3.9), e obtai that limsup u ( f (I B))x,y x 0. Fially, e sho that lim x x 0, here x is a uique solutio of (3.). Usig Lemma., e ca ote that x x ( (( ) )) x x ISBN: ISSN: (Prit); ISSN: (Olie) WCE 04

5 Proceedigs of the World Cogress o Egieerig 04 Vol II, WCE 04, July - 4, 04, Lodo, U.K. (( ) ) x x x x y x u ( f )x,y x. (3.30) Applyig Lemma.3 to (3.30), e have lim x x 0, that is, {x } coverges strogly to x. This completes the proof. IV. CONCLUSION We proposed a iterative method ad proved that the sequece of the proposed iterative method coverges to a poit of solutios of above three sets. This iterative method ad covergece theorem are improved ad exteded from Theorem 3. of Jug [3]. ACKNOWLEDGMENT I ould like to thak the aoymous referee for valuable commets. REFERENCES [] E. Blum ad Oettli, "From optimizatio ad variatioal iequalities to equilibrium problems," The Mathematics Studet. vol. 63, pp. 3-45, 994. [] Q. L. Dog ad B. C. Deg, "Strog covergece theorem by hybrid method for equilibrium problems, variatioal iequality problems ad maximal mootoe operators," Noliear Aalysis: Hybrid Systems. vol. 4, pp , 00. [3] J. S. Jug, "A geeral iterative approach to variatioal iequality problems ad optimizatio problems," Fixed Poit Theory Appl. (0), Article ID 84363, doi:0.55/0/ [4] Y. Khogtham ad S. Plubtieg, "A geeral iterative for equilibrium problems of a coutable family of oexpasive mappigs i Hilbert spaces," Far East J. Math. Sci. (FJMS), vol. 30, pp , 009. [5] G. Mario ad H.-K. Xu, "A geeral iterative method for oexpasive mappigs i Hilbert spaces," J.Math. Aal. Appl., vol. 38, pp. 43-5, 006. [6] Z. Opial, "Weak covergece of the sequece of successive approximatio for oexpasive mappigs," Bull. Amer. Math. Soc., vol. 73, pp , 967. [7] S. Plubtieg ad R. Pupaeg, "A geeral iterative method for equilibrium problems ad fixed poit problem i Hilbert spaces," J. Math. Aal. Appl., vol. 336, pp , 007. [8] S. Plubtieg ad W. Sriprad, "A viscosity approximatio method for fidig commo solutios of variatioal iclusios, equilibrium problems, ad fixed poit problems i Hilbert spaces," Fixed Poit Theory Appl. (009), Article ID56747, doi:0.55/009/ [9] T. Suzuki, "Strog covergece of Krasoselskii ad Ma s type sequeces for oe-parameter oexpasive semigroups ithout Bocher itegrals," J. Math. Aal. Appl., vol. 305, pp. 7-39, 005. [0] W. Takahashi ad M. Toyoda, "Weak covergece theorems for oexpasive mappigs ad mootoe mappigs," J. Optim. Theory Appl., vol., pp , 003. ISBN: ISSN: (Prit); ISSN: (Olie) WCE 04