A General Iterative Scheme for Variational Inequality Problems and Fixed Point Problems

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1 A Geeral Iterative Scheme for Variatioal Iequality Problems ad Fixed Poit Problems Wicha Khogtham Abstract We itroduce a geeral iterative scheme for fidig a commo of the set solutios of variatioal iequality problems for a iverse-strogly mootoe mappig ad the set of commo fixed poits of a coutable family of oexpasive mappigs i a real Hilbert space. We sho that the sequece coverges strogly to a commo elemet of the above to sets uder some parameters cotrollig coditios. The results preseted i this paper improve ad exted the correspodig results aouced by may others. Idex Terms Fixed poit, variatioal iequality, optimizatio problem, oexpasive mappig L I. INTRODUTION ET H be a real Hilbert space ith ier product ad orm, are deoted by, ad, respectively. Let be a oempty closed covex subset of H, ad let B: H be a oliear map. The classical variatioal iequality hich is deoted by VI(,B) is to fid vsuch that Bv,u v 0, u. The variatioal iequality has bee extesively studied i literature. See, for example, [6], [7], [9], ad the refereces therei. A mappig A of ito H is called iverse-strogly mootoe, see []-[3], if there exists a positive real umber such that Au Av Au Av, u,v. A mappig T of ito itself is called oexpasive if Tu Tv u v, u, v. We deoted by F(T) the set of fixed poits of T. A mappig f : is said to be cotractive ith coefficiet (0,), if f(u) f(v) u v, u,v. Let G be a strogly positive bouded liear operator o H: that is, there is a costat 0 ith property Gx, x x, x H. Recetly, may authors proposed some e iterative schemes for fidig elemet i F(S) VI(, B), see []-[3], [5], [8], [3], ad referece therei. Moreover, Jug [4] itroduced the folloig iterative scheme as the folloig. Let a oempty closed covex subset of a real Hilbert space H such that. Let A be a iversestrogly mootoe mappig of ito H ad S be a Mauscript received February 3, 04; revised April 3, 04. This ork as supported i part by Maejo Uiversity, hiag Mai, Thailad, 5090,uder Grat OT W. Khogtham is ith Faculty of Sciece, Maejo Uiversity, hiag Mai, Thailad, 509 (PHONE: ; Fax: ; icha_k@mju.ac.th). oexpasive mappig of ito itself such that F(S) VI(, A). Let u ad let B be a strog positive bouded liear operator o ith costat (0,) ad f be a cotractive of ito itself ith costat k (0,). Assume that 0ad 0 ( ) / k. Let {x } be a sequece geerated by x x, y (u f(x )) (I (I B))SP (x Ax ), x ( )y SP (y Ay ),. They proved that uder certai appropriate coditios imposed o { }, { }, ad { } of parameters, the the sequece {x } coverges strogly to qf(s) VI(, A), hich is a solutio of the optimizatio problem: mi Bx, x x u h(x), xf(s) VI(,A) here h is a potetial fuctio for f. I this paper motivated by the iterative scheme proposed by Jug [4], e ill itroduce a geeral iterative for a commo elemet of the set solutio of variatioal iequality problem for a iversestrogly mootoe mappig ad the set of commo fixed poits of a coutable family of oexpasive mappigs hich ill preset i the mai result. II. PRELIMINARIES Let be a oempty closed covex subset of a real Hilbert space H. It ell ko that H satisfies the Opial s coditio, 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. For every poit x H, there exist a uique earest poit i, deoted by P x, such that x P x x y for all y. P is called the metric projectio of H oto. It ell ko that P is a oexpasive mappig of H oto ad satisfies x y,p x P y P x P y, x,y H. Moreover, Px is characterized by the folloig properties: P x ad x y x P x y P y, x H,y. It is easy to see that uvi(, A) u P (u Au), 0. Propositio. (See [4 ].) Let b e a bouded oempty closed covex subset of a real Hilbert space H ad let B be a iverse-strogly mootoe mappig of ito H. The, VI(,B) is oempty. H A set-valued mappig M : H is called mootoe if for all x,yh, f Mx ad gmy imply x y,f g 0. ISBN: ISSN: (Prit); ISSN: (Olie) WE 04

2 H A mootoe mappig M : H is maximal if the graph G(T) of T is ot properly cotaied i the graph of ay other mootoe mappig. It ell ko that a mootoe mappig T is maximal if ad oly if for (x,f ) H H, x y,f g 0 for every (y,g) G(T) implies f Tx. Let B be a iverse-strogly mootoe mappig ito H ad let Nv be the ormal coe to at v, that is, Nv { H : v u, 0, for all u}, ad defie Bv Nv, v, Tv, v. The T is maximal mootoe ad 0Tv if ad oly if v VI(, B) (see [9], [0], []). The folloig Lemmas ill be useful for provig our theorem i the ext sectio. Lemma. (See [9].) 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 Lemma. (See [].) Let K be a oempty closed covex subset of a Baach space ad let T be a sequece of mappigs of K ito itself. Suppose that sup T z T z : z K. The, for each y K, Tycoverges strogly to some poit of K. Moreover, let T be a mappig of K ito itself defied by Ty lim Ty for all y K. The lim sup Tz T z : zk 0. Lemma.3 (See [4 ].) I a real Hilbert space H, there holds the iequality z y z y, z y. Lemma.4 (See [4 ].) Let be a bouded oempty closed covex subset of a real Hilbert space H, ad let g : R be a proper loer semicotiuous differetiable covex fuctio. If problem g(x ) if g(x), the g (x),x particular, if the u f (I x x is a solutio to the miimizatio x 0,x. x solves the optimizatio problem mi Bx, x x u h(x), x potetial fuctio for f. I B))x,x x 0,x, here h is a Lemma.5 (See [9].) Assume A is a strogly positive liear bouded operator o a Hilbert space H ith coefficiet 0 ad 0 A. The I A. III. MAIN RESULT I this sectio, e prove a strog covergece theorem. Theorem 3.. Let be a oempty closed covex subset of a real Hilbert space H such that. Let B be a iverse-strogly mootoe mappig of ito H ad {T }be a sequece of oexpasive mappigs of ito itself such that : F(T ) VI(, B). Let u ad let A be a strogly positive bouded liear operator o ith costat (0,) ad f be a cotractive of ito itself ith costat (0,). Assume that 0ad 0 ( ) /. Let {x } be a sequece geerated by x x, y (u f(x )) (I (I A))T P (x Bx ), (3.) x ( )y T P (y By ),, here { } [0,),{ } [0, ], ad { } [0,] satisfy the folloig coditios: i) lim 0; ; ii) [0, b) for all 0ad for some b (0,); iii) [r,s] for all 0ad for some r, s ith 0 r s ; iv),, ad. Suppose that sup{ T z Tz : z D} for ay bouded subset D of. Let T be mappig of H ito itself defied by Tx lim T x, for all x ad suppose that F(T) F(T ). The, {x } coverges strogly to F(T) VI(,B), hich is a solutio of the optimizatio problem mi Ax, x x u h(x), (3.) xf(t) VI(,B) here h is a potetial fuctio for f. Proof. From the coditio i), e may assume that ( A ). Applyig Lemma.5 ad by the same argumet as that i the proof of Jug ([4], Theorem 3., pp. 6-7),e have that (I (I A))u,u Au, u 0, I (I A)) ( ), t v x v, ad v v y v, here v, P (x Bx ), ad v P (y By ). Let (I A). It follos that y v u ( f(x ) v) (I )(T v) t ( ( ) ) t v u x v f(v) v f (v) v u ( ) (( ) ), ad x v ( )(y v) (T v T v) ISBN: ISSN: (Prit); ISSN: (Olie) WE 04

3 f (v) v u max x v, ( ). f (v) v u By iductio that ( ) x v max x v,,. Hece {x }is bouded, so are{y },{t },{v }, {f(x )}, {By },{Bx }, {T t },{ T t }, ad {T v }. Moreover, e observe that t t x x Bx (3.3) ad v v x x Bx. (3.4) It follos from the assumptio ad usig (3.), (3.3), ad (3.4), e have y y ( u f(x ) T t ) The, e obtai x x ( ( ) ) ) x x ( ( ) ) Bx sup{ T z T z : z {t }}. x x ( ( ) ) x x G G G, 3 here G sup Bx By :, ad G y : sup{ T z T z : z {v }}. G sup ) T t :, lemma. to (3.6), e have sup T v 3 Applyig By usig (3.3) ad (3.5), e have ad From (3.), e ote that (3.5) (3.6) lim x x 0. (3.7) lim t t 0 (3.8) lim y y 0. (3.9) y T t (u f(x ) T t 0, (3.0) ad v t y x. (3.) Moreover, by (3.), (3.), ad the coditio ii), e have b x y ( b) x x Tt y. (3.) From (3.) ad usig (3.7) ad (3.0), e obtai We apply that x y 0 as. (3.3) x y x x x y 0 as. (3.4) Let p. By the same argumet as i [4 ] ( Theorem 3., pp. -), e ca sho that y p ) p x p The, e obtai u f(x ) t p ( ( ) )r(s ) Bx Bp. ( ( ) )r(s ) Bx Bp ) p ( x p y p ) x y ) p ( x p y p ) x y ) p t p. It follos from the coditio i), e have (3.5) (3.6) Bx Bp 0as (3.7) Similarly, e ca sho that t p x p x t x t,bx Bp The, e obtai Bx Bp. ( ( ) ) x t ) p ( x p y p ) x y ( ( ) ) x t,bx Bp ( ( ) ) Bx Bp u f(x ) p t p. (3.8) (3.9) It follos from (3.4), (3.7), ad the coditio i), e obtai ad so (3.0) lim x t 0 y t y x x t 0 as. (3.) For p, e defie a subset D of H by f (p) p u K, here K max p x, ( ). D y : y p learly, D is ISBN: ISSN: (Prit); ISSN: (Olie) WE 04

4 bouded, closed covex subset of H, T(D) D ad {t } D. By our assumptio, here sup T z T z : z D ad Lemma., e have lim sup Tz T z : z D 0. The e have lim sup Tz Tz : z {t } lim sup Tz Tz : z D 0. This implies that lim Tz T z 0. (3.) From (3.), the coditio i), ad usig (3.0), (3.9), e ote that T t t T t y y t 0 (3.3) ad T t x (u f(x ) T t y T t (3.4) 0 as. Usig (3.3), ad (3.9), e have x t 0, as. The e have Tt t Tt T t T t x x t 0. The, from (3.) ad (3.5), e obtai (3.5) y Tt y t t T t 0 as. (3.6) Next e sho that limsup u ( f ),y 0, here is a solutio of the optimizatio (3.). First e prove that limsup u ( f ),Tt 0. Sice {t } is bouded, e ca choose a subsequece {t i } of {t }such that limsup u ( f ),Tt lim u ( f ),T t. i Without loss of geerality, e may assume that i t i (3.7) z, here z. We ill sho that z. First, let us sho z F(T) F(T ). Assume that zf(t). Sice t i z, z T z, ad (.3), it follos by the Opial s coditio that limif t z limif t Tz i i i i i limif t Tt Tt Tz limif Tt Tz i i i i i (3.8) limif t z. i i This is a cotradictio. Hece zf(t). From the property of the maximal mootoe, B is a iverse-strogly mootoe, ad (3.0), e obtai z VI(, B). Therefore, z. By Lemma.4 ad (3.5), e have limsup u ( f ),Tt 0. (3.9) Hece, by (3.6) ad (3.9), e obtai limsup u ( f ),y limsup u ( f ),y Tt limsup u ( f ),Tt limsup u ( f ) y Tt limsup u ( f ),Tt 0. Fially, e prove that lim x 0, here is a solutio of (3.). We observe that x ( (( ) ) ) x (( ) ) x x y x u ( f ),y (3.30) ad applyig Lemma.,.3, ad.4 to (3.30), e have lim x 0. This completes the proof. IV. ONLUSION We itroduced a iterative scheme for fidig a commo elemet of the set solutios of variatioal iequality problems ad the set of commo fixed poit of a coutable family of oexpasive. The, e proved that the sequece of the proposed iterative scheme coverges strogly to a commo elemet of the above to sets, hich is a solutio of a certai optimizatio problems. Theorem 3. improve ad exteds Theorem 3. of Jug [4] ad referece therei i the sese that our iterative scheme ad covergece theorem are for the more geeral class of oexpasive mappigs. AKNOWLEDGMENT We ould like to thak the aoymous referee for valuable commets. REFERENES [] K. Aoyama, Y. Kimura, W. Takahashi, ad M. Toyoda, "Approximatio of commo fixed poits of a coutable family of oexpasive mappigs i a Baach space," Noliear Aal., vol. 67 pp , 007. ISBN: ISSN: (Prit); ISSN: (Olie) WE 04

5 [] F. E. Broder ad W. V. Petryshy, "ostructio of fixed poits of oliear mappigs i Hilbert space," J. Math. Aal. Appl., vol. 0, pp. 97-5, 967. [3] H. Iiduka ad W. Takahashi, "Strog covergece theorems for oexpasive mappigs ad iverse-strogly mootoe mappigs," Noliear Aal., vol. 6, pp , 005. [4] 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/ [5] 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. [6] 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. [7] N. Nadezhkia ad W. Takahashi, "Weak covergece theorem by a extragradiet method for oexpasive mappigs ad mootoe mappigs," J. Optim. Theory Appl., vol. 8, pp. 9-0, 006. [8] Z. Opial, "Weak covergece of the sequece of successive approximatio for oexpasive mappigs, Bull. Amer. Math. Soc., vol. 73, pp , 967. [9] 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. [0] R.T. Rockafellar, "Mootoe operators ad the proximal poit algorithm," SIAM Joural o otrol ad Optimizatio, vol. 4 pp , 976. [] 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, 005. [] W. Takahashi ad M. Toyoda, "Weak covergece theorems for oexpasive mappigs ad mootoe mappigs," J. Optim. Theory Appl., vol., pp , 003. [3] H.-K. Xu, "Viscosity approximatio methods for oexpasive mappigs," J. Math. Aal. Appl., vol. 98, pp. 79-9, 004. ISBN: ISSN: (Prit); ISSN: (Olie) WE 04