Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 756492, 22 pages doi:10.1155/2010/756492 Research Article Strong Convergence for Mixed Equilibrium Problems of Infinitely Nonexpansive Mappings Jintana Joomwong Division of Mathematics, Faculty of Science, Maejo University, Chiang Mai 50290, Thailand Correspondence should be addressed to J intana Joomwong, jintana@mju.ac.th Received 29 March 2010; Accepted 24 May 2010 Academic Editor: Tomonari Suzuki Copyright q 2010 Jintana Joomwong. 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 iterative scheme for finding a common element of infinitely nonexpansive mappings, the set of solutions of a mixed equilibrium problems, and the set of solutions of the variational inequality for an α-inverse-strongly monotone mapping in a Hilbert Space. Then, the strong converge theorem is proved under some parameter controlling conditions. The results of this paper extend and improve the results of Jing Zhao and Songnian He2009 and many others. Using this theorem, we obtain some interesting corollaries. 1. Introduction Let H be a real Hilbert space with norm · and inner product ·, ·. And let C be a nonempty closed convex subset of H.Letϕ : C → R be a real-valued function and let Θ : C × C → R be an equilibrium bifunction, that is, Θu, u0 for each u ∈ C. Ceng and Yao 1 considered the following mixed equilibrium problem. Find x ∗ ∈ C such that Θ  x ∗ ,y   ϕ  y  − ϕ  x ∗  ≥ 0, ∀y ∈ C. 1.1 The set of solutions of 1.1 is denoted by MEPΘ,ϕ. It is easy to see that x ∗ is the solution of problem 1.1 and x ∗ ∈ dom ϕ  {x ∈ ϕx < ∞}. In particular, if ϕ ≡ 0, the mixed equilibrium problem 1.1 reduced to the equilibrium problem. Find x ∗ ∈ C such that Θ  x ∗ ,y  ≥ 0, ∀y ∈ C. 1.2 2 Fixed Point Theory and Applications The set of solutions of 1.2 is denoted by EPΘ. If ϕ ≡ 0andΘx, yAx, y − x for all x, y ∈ C, where A is a mapping from C to H, then the mixed equilibrium problem 1.1 becomes the following variational inequality. Find x ∗ ∈ C such that  Ax ∗ ,y− x ∗  , ∀y ∈ C. 1.3 The set of solutions of 1.3 is denoted by VIA, C. The variational inequality and the mixed equilibrium problems which include fixed point problems, optimization problems, variational inequality problems have been extensively studied in literature. See, for example, 2–8. In 1997, Combettes and Hirstoaga 9 introduced an iterative method for finding the best approximation to the initial data and proved a strong convergence theorem. Subsequently, Takahashi and Takahashi 7 introduced another iterative scheme for finding a common element of EPΘ and the set of fixed points of nonexpansive mappings. Furthermore,Yao et al. 8, 10 introduced an iterative scheme for finding a common element of EPΘ and the set of fixed points of finitely infinitely nonexpansive mappings. Very recently, Ceng and Yao 1 considered a new iterative scheme for finding a common element of MEPΘ,ϕ and the set of common fixed points of finitely many nonexpansive mappings in a Hilbert space and obtained a strong convergence theorem. Now, we recall that a mapping A : C → H is said to be i monotone if Au − Av, u − v≥0, for all u, v ∈ C, ii L-Lipschitz if there exists a constant L>0 such that Au − Av≤L u − v, for all u, v ∈ C, iii α-inverse strongly monotone if there exists a positive real number α such that Au− Av, u − v≥αAu − Av 2 , for all u, v ∈ C. It is obvious that any α-inverse strongly monotone mapping A is monotone and Lipscitz. A mapping S : C → C is called nonexpansive if Su − Sv≤u − v, for all u, v ∈ C. We denote by FS : {x ∈ C : Sx  x} the set of fixed point of S. In 2006, Yao and Yao 11 introduced the following iterative scheme. Let C be a closed convex subset of a real Hilbert space. Let A be an α-inverse strongly monotone mapping of C into H and let S be a nonexpansive mapping of C into itself such that FS ∩ VIA, C /  ∅. Suppose that x 1  u ∈ C and {x n } and {y n } are given by y n  P C  x n − λ n Ax n  , x n1  α n u  β n x n  γ n SP C  y n − λ n Ay n  , 1.4 where {α n }, {β n },and{γ n } are sequence in 0, 1 and {λ n } is a sequence in 0,2λ. They proved that the sequence {x n } defined by 1.4 converges strongly to a common element of FS ∩ VIA, C under some parameter controlling conditions. Moreover, Plubtieng and Punpaeng 12 introduced an iterative scheme 1.5 for finding a common element of the set of fixed point of nonexpansive mappings, the set of solutions of an equilibrium problems, and the set of solutions of the variational of inequality Fixed Point Theory and Applications 3 problem for an α-inverse strongly monotone mapping in a real Hilbert space. Suppose that x 1  u ∈ C and {x n }, {y n },and{u n } are given by Θ  u n ,y   1 r n  y − u n ,u n − x n  ≥ 0, ∀y ∈ C, y n  P C  u n − λ n Au n  , x n1  α n u  β n x n  γ n SP C  y n − λ n Ay n  , 1.5 where {α n }, {β n },and{γ n } are sequence in 0, 1, {λ n } is a sequence in 0,2λ,and{r n }⊂ 0, ∞. Under some parameter controlling conditions, they proved that the sequence {x n } defined by 1.5 converges strongly to P FS∩VIA,C∩EPΘ u. On the other hand, Yao et al. 8 introduced an iterative scheme 1.7 for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed point of infinitely many nonexpansive mappings in H.Let{T n } ∞ n1 be a sequence of nonexpansive mappings of C into itself and let {t n } ∞ n1 be a sequence of real number in 0, 1. For each n ≥ 1, define a mapping W n of C into itself as follows: U n,n1  I, U n,n  t n T n U n,n1   1 − t n  I, U n,n−1  t n−1 T n−1 U n,n   1 − t n−1  I, . . . U n,k  t k T k U n,k1   1 − t k  I, U n,k−1  t k−1 T k−1 U n,k   1 − t k−1  I, . . . U n,2  t 2 T 2 U n,3   1 − t 2  I, W n  U n,1  t 1 T 1 U n,2   1 − t 1  I. 1.6 Such a mapping W n is called the W-mapping generated by T n ,T n−1 , ,T 1 and t n ,t n−1 , ,t 1 . In 8,givenx 0 ∈ H arbitrarily, the sequences {x n } and {u n } are generated by Θ  u n ,x   1 r n  x − u n ,u n − x n  ≥ 0, ∀x ∈ C, x n1  α n f  x n   β n x n  γ n W n u n . 1.7 They proved that under some parameter controlling conditions, {x n } generated by 1.7 converges strongly to z ∈∩ ∞ n1 FT n  ∩ EPΘ, where z  P ∩ ∞ n1 FT n ∩EPΘ fz. 4 Fixed Point Theory and Applications Subsequently, Ceng and Yao 13 introduced an iterative scheme by the viscosity approximation method: Θ  u n ,x   1 r n  x − u n ,u n − x n  ≥ 0, ∀x ∈ C, y n   1 − γ n  x n  γ n W n u n , x n1   1 − α n − β n  x n  α n f  y n   β n W n y n , 1.8 where {α n }, {β n } and {γ n } are sequence in 0,1 such that α n  β n ≤ 1. Under some parameter controlling conditions, they proved that the sequence {x n } defined by 1.8 converges strongly to z ∈∩ ∞ n1 FT n  ∩ EPΘ, where z  P ∩ ∞ n1 FT n ∩EPΘ fz. Recently, Zhao and He 14 introduced the following iterative process. Suppose that x 1  u ∈ C, Θ  u n ,y   1 r n  y − u n ,u n − x n  ≥ 0, ∀y ∈ C, y n  s n P C  u n − λ n Au n    1 − s n  x n , x n1  α n u  β n x n  γ n W n  P C  y n − λ n Ay n  , 1.9 where {s n }, {α n }, {β n },and{γ n }∈0, 1 such that α n  β n  γ n  1. Under some parameter controlling conditions, they proved that the sequence {x n } defined by 1.9 converges strongly to z ∈∩ ∞ i1 FT i  ∩ VIA, C ∩ EPΘ, where z  P ∩ ∞ i1 FT i ∩VIA,C∩EPΘ u. Motivated by the ongoing research in this field, in this paper we suggest and analyze an iterative scheme for finding a common element of the set of fixed point of infinitely nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of the variational of inequality problem for an α-inverse strongly monotone mapping in a real Hilbert space. Under some appropriate conditions imposed on the parameters, we prove another strong convergence theorem and show that the approximate solution converges to a unique solution of some variational inequality which is the optimality condition for the minimization problem. The results of this paper extend and improve the results of Zhao and He 14 and many others. For some related works, we refer the readers to 15–22 and the references therein. 2. Preliminaries Let H be a real Hilbert space and let C be a closed convex subset of H. Then, for any x ∈ H, there exists a unique nearest point in C, denoted by P C x such that x − P C  x  ≤x − y, ∀y ∈ C. 2.1 P C is called the metric projection of H onto C. It is well known that P C is nonexpansive mapping and satisfies  x − y, P C x − P C y  ≥   P C x − P C y   2 , ∀x, y ∈ H. 2.2 Fixed Point Theory and Applications 5 Moreover, P C is characterized by the following properties: P c x ∈ C and x − P C x, y − P C x≤0,   x − y   2 ≥  x − P C x  2    y − P C x   2 , ∀x ∈ H, y ∈ C. 2.3 It is clear that u ∈ VIA, C ⇔ u  P C u − λAu,λ>0. A space X is said to satisfy Opials condition if for each sequence {x n } in X which converges weakly to a point x ∈ X, we have lim inf n →∞ x n − x < lim inf n →∞ x n − y, ∀y ∈ X, y /  x. 2.4 The following lemmas will be useful for proving the convergence result of this paper. Lemma 2.1 see 23. Let {x n } and {y n } be bounded sequences in a Banach space X and let {β n } be a sequence in 0, 1 with 0 < lim inf n →∞ β n  lim sup n →∞ β n < 1. Suppose that x n1 1−β n y n  β n x n for all integer n ≥ 1 and lim sup n →∞ y n1 −y n −x n1 −x n   0. Then lim n →∞ y n −x n   0. Lemma 2.2 see 24. Let H be a real Hilbert space, let C be a closed convex subset of H, and let T : C → C be a nonexpansive mapping with FT /  ∅. If {x n } is a sequence in C weakly converging to x and if I − Tx n converge strongly to y,thenI − Tx  y. Lemma 2.3 see 25. Assume that {a n } is a sequence of nonnegative real numbers such that a n1 ≤  1 − α n  a n  δ n ,n≥ 0, 2.5 where {α n } is a sequence in 0, 1 and {δ n } is a sequence in R such that 1 lim n →∞ α n  0 and  ∞ n1 α n  ∞. 2 lim sup n →∞ δ n /α n  ≤ 0 or  ∞ n1 |δ n | < ∞. Then lim n →∞ a n  0. In this paper, for solving the mixed equilibrium problem, let us give the following assumptions for a bifunction Θ,ϕand the set C: A1Θx, x0 for all x ∈ C; A2Θis monotone, that is, Θx, yΘy, x ≤ 0 for any x, y ∈ C; A3Θis upper-hemicontinuous, that is, for each x, y, z ∈ C, lim t → 0  sup Θ  tz   1 − t  x, y  ≤ Θ  x, y  ; 2.6 A4Θx, · is convex and lower semicontinuous for each x ∈ C; 6 Fixed Point Theory and Applications B1 for each x ∈ H and r>0, there exists a bounded subset D x ⊂ C and y x ∈ C such that for any z ∈ C \ D x , Θ  z, y   ϕ  y x   1 r n  y x − z, z − x  <ϕ  z  , 2.7 B2 C is a bounded set. By a similar argument as in the proof of Lemma 2.3 in 26,wehavethefollowing result. Lemma 2.4. Let C be a nonempty closed convex subset of a real Hilbert space H.LetΘ be a bifunction from C × C → R that satisfies (A1)–(A4) and let ϕ : C → R ∪{∞}be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For r>0 and x ∈ H, define a mapping T r : H → C as follows: T r  x    z ∈ C : Θ  z, y   ϕ  y   1 r  y − z, z − x  ≥ ϕ  z  , ∀y ∈ C  2.8 for all x ∈ H. Then, the following conditions hold: 1 for each x ∈ H, T r x /  ∅; 2 T r is single-valued; 3 T r is firmly nonexpansive, that is, for any x, y ∈ H, T r x − T r y 2 ≤T r x − T r y, x − y; 4 FT r MEPΘ,ϕ; 5 MEPΘ,ϕ is closed and convex. Let {T n } ∞ n1 be a sequence of nonexpansive mappings of C into itself, where C is a nonempty closed convex subset of a real Hilbert space H. Given a sequence {t n } ∞ n1 in 0, 1, we define a sequence {W n } ∞ n1 of self-mappings on C by 1.6. Then We have the following result. Lemma 2.5 see 27. Let C be a nonempty closed convex subset of a real Hilbert space H.Let {T n } ∞ n1 be a sequence of nonexpansive self-mappings on C such that ∩ ∞ n1 FT n  /  ∅ and let {t n } be a sequence in 0,b for some b ∈ 0, 1. Then, for every x ∈ C and k ≥ 1, lim n →∞ U n,k x exists. Remark 2.6 see 8. It can be shown from Lemma 2.5 that if D is a nonempty bounded subset of C, then for >0, there exists n 0 ≥ k such that for all n>n 0 ,sup x∈D U n,k x − U k x≤, where U k x  lim n →∞ U n,k x. Remark 2.7 see 8. Using Lemma 2.5, we define a mapping W : C → C as follows: Wx  lim n →∞ W n x  lim n →∞ U n,1 x, for all x ∈ C. W is called the W-mapping generated by T 1 ,T 2 , and t 1 ,t 2 , Since W n is nonexpansive, W : C → C is also nonexpansive. Indeed, for all x, y ∈ C, W x − W y   lim n →∞ W n x − W n y≤x − y. If {x n } is a bounded sequence in C, then we put D  {x n : n ≥ 0}. Hence it is clear from Remark 2.6 that for any arbitrary >0, there exists n 0 ≥ 1 such that for all n>n 0 , W n x n − Wx n   U n,1 x n − U 1 x n ≤sup x∈D U n,1 x − U 1 x <. Fixed Point Theory and Applications 7 This implies that lim n →∞ W n x n − Wx n   0. Lemma 2.8 see 27. Let C be a nonempty closed convex subset of a real Hilbert space H.Let {T n } ∞ n1 be a sequence of nonexpansive self-mappings on C such that ∩ ∞ n1 FT n  /  ∅ and let {t n } be a sequence in 0,b for some b ∈ 0, 1.ThenFW∩ ∞ n1 FT n . 3. Main Results Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H.Letϕ : C → R ∪{∞} be a lower semicontinuous and convex function. Let Θ be a bifunction from C × C → R satisfying (A1)–(A4), let A be an α-inverse-strongly monotone mapping of C into H, and let {T n } ∞ n1 be a sequence of nonexpansive self-mapping on C such that ∩ ∞ n1 FT n  ∩ VIA, C ∩ MEPΘ,ϕ /  ∅. Suppose that {s n }, {α n }, {β n }, and {γ n } are sequences in 0, 1,{λ n } is a sequence in 0, 2α such that λ n ∈ a, b for some a, b with 0 <a<b<2α, and {r n }⊂0, ∞ is a real sequence. Suppose that the following conditions are satisfied: i α n  β n  γ n  1, ii lim n →∞ α n  0 and  ∞ n1 α n  ∞, iii 0 < lim inf n →∞ β n ≤ lim sup n →∞ β n < 1, iv 0 < lim inf n →∞ s n ≤ lim sup n →∞ s n < 1/2 and lim n →∞ |s n1 − s n |  0, v lim n →∞ |λ n1 − λ n |  0, vi lim inf n →∞ r n > 0 and lim n →∞ |r n1 − r n |  0. Let f be a contraction of C into itself with coefficient β ∈ 0, 1. Assume that either (B1) or (B2) holds. Let the sequences {x n }, {u n }, and {y n } be generated by, x 1 ∈ C and Θ  u n ,y   ϕ  y  − ϕ  u n   1 r n  y − u n ,u n − x n  ≥ 0, ∀y ∈ C, y n  s n P C  u n − λ n Au n    1 − s n  x n , x n1  α n f  x n   β n x n  γ n W n  P C  y n − λ n Ay n  , 3.1 for all n ∈ N,whereW n is defined by 1.6 and {t n } is a sequence in 0,b,forsomeb ∈ 0, 1.Then the sequence {x n } converges strongly to a point x ∗ ∈∩ ∞ n1 FT n  ∩ VIA, C ∩ MEPΘ,ϕ,where x ∗  P ∩ ∞ n1 FT n ∩VIA,C∩MEPΘ,ϕ fx ∗ . Proof. For any x, y ∈ C and λ n ∈ a, b ⊂ 0, 2α,wenotethat   I − λ n Ax − I − λ n Ay   2    x − y − λ n  Ax − Ay    2    x − y   2 − 2λ n  x − y, Ax − Ay   λ 2 n   Ax − Ay   2 ≤   x − y   2  λ n  λ n − 2α    Ax − Ay   2 ≤   x − y   2 , 3.2 which implies that I − λ n A is nonexpansive. 8 Fixed Point Theory and Applications Let {T r n } be a sequence of mappping defined as in Lemma 2.4 and let x ∗ ∈∩ ∞ n1 FT n  ∩ VIA, C ∩ MEPΘ,ϕ. Then x ∗  W n x ∗ and x ∗  P C x ∗ − λ n Ax ∗ T r n x ∗ .Putv n  P C y n − λ n Ay n .From3.2 we have v n − x ∗   P C  y n − λ n Ay n  − P C  x ∗ − λ n Ax ∗   ≤  y n − λ n Ay n  −  x ∗ − λ n Ax ∗   ≤y n − x ∗   s n P C  u n − λ n Au n    1 − s n  x n − s n P C  x ∗ − λ n Ax ∗  −  1 − s n  x ∗  ≤ s n P C  u n − λ n Au n  − P C  x ∗ − λ n Ax ∗     1 − s n  x n − x ∗  ≤ s n u n − x ∗    1 − s n  x n − x ∗   s n T r n x n − T r n x ∗    1 − s n  x n − x ∗  ≤ s n x n − x ∗    1 − s n  x n − x ∗   x n − x ∗ . 3.3 Hence, we obtain that x n1 − x ∗   α n f  x n  − β n x n − γ n W n v n − x ∗  ≤ α n f  x n  − x ∗   β n x n − x ∗   γ n W n v n − x ∗  ≤ α n f  x n  − f  x ∗    α n f  x ∗  − x ∗   β n x n − x ∗   γ n v n − x ∗  ≤ α n βx n − x ∗   α n f  x ∗  − x ∗   β n x n − x ∗   γ n x n − x ∗    1 − β  α n f  x ∗  − x ∗  1 − β   1 −  1 − β  α n  x n − x ∗  ≤ max  x n − x ∗ , f  x ∗  − x ∗  1 − β  ≤ max  x 0 − x ∗ , f  x ∗  − x ∗  1 − β  . 3.4 Therefore {x n } is bounded. Consequently, {fx n }, {u n }, {y n }, {v n }, {W n v n }, {Au n },and {Ay n } are also bounded. Next, we claim that lim n →∞ x n1 − x n   0. Fixed Point Theory and Applications 9 Indeed, setting x n1  β n x n 1 − β n z n , for all n ≥ 1, it follows that z n1 − z n  α n1 f  x n1   γ n1 W n1 v n1 1 − β n1 − α n f  x n   γ n W n v n 1 − β n  α n1 f  x n1   γ n1 W n1 v n1 1 − β n1 − γ n1 W n1 v n 1 − β n1  γ n1 W n1 v n 1 − β n1 − α n f  x n   γ n W n v n 1 − β n  α n1 f  x n1  1 − β n1 − α n f  x n  1 − β n  γ n1 1 − β n1  W n1 v n1 − W n1 v n   1 − β n1 − α n1 1 − β n1 W n1 v n − 1 − β n − α n 1 − β n W n v n  α n1 f  x n1  1 − β n1 − α n f  x n  1 − β n  γ n1 1 − β n1  W n1 v n1 − W n1 v n    w n1 v n − w n v n   α n 1 − β n W n v n − α n1 1 − β n1 W n1 v n . 3.5 Now, we estimate W n1 v n − W n v n  and W n1 v n1 − W n1 v n . From the definition of {W n }, 1.6, and since T i , U n,i are nonexpansive, we deduce that, for each n ≥ 1, W n1 v n − W n v n   t 1 T 1 U n1,2 v n − t 1 T 1 U n,2 v n  ≤ t 1 U n1,2 v n − U n,2 v n   t 1 t 2 T 2 U n1,3 v n − t 2 T 2 U n,3 v n  ≤ t 1 t 2 U n1,3 v n − U n,3 v n  . . . ≤  n  i1 t i  U n1,n1 v n − U n,n1 v n  ≤ M n  i1 t i , 3.6 for some constant M>0 such that sup{U n1,n1 v n − U n,n1 v n ,n ≥ 1}≤M. And 10 Fixed Point Theory and Applications we note that W n1 v n1 − W n1 v n ≤  v n1 − v n    P C  y n1 − λ n1 Ay n1  − P C  y n − λ n Ay n   ≤  y n1 − λ n1 Ay n1  −  y n − λ n Ay n   ≤  I − λ n1 A  y n1 −  I − λ n1 A  y n   | λ n − λ n1 | Ay n  ≤y n1 − y n   | λ n − λ n1 | Ay n , 3.7 y n1 − y n    s n1 P C  u n1 − λ n1 Au n1    1 − s n1  x n1 −s n P C  u n − λ n Au n  −  1 − s n  x n    s n1 P C  u n1 − λ n1 Au n1  − s n1 P C  u n − λ n Au n    s n1 − s n  P C  u n − λ n Au n    1 − s n1  x n1 −  1 − s n1  s n1 − s n  x n  ≤ s n1   u n1 − λ n1 Au n1  −  u n − λ n Au n    | s n1 − s n | u n − λ n Au n    1 − s n1  x n1 − x n   | s n1 − s n | x n  ≤ s n1 {   u n1 − λ n1 Au n1  −  u n − λ n Au n    | λ n − λ n1 | Au n  }  | s n1 − s n |  u n   λ n Au n   x n     1 − s n1  x n1 − x n  ≤ s n1 u n1 − u n   s n1 | λ n − λ n1 | Au n   | s n1 − s n | Q   1 − s n1   x n1 − x n  , 3.8 where Q  sup{u n ,λ n Au n , x n  : n ≥ 1}. Combining 3.7 and 3.8,weobtain v n1 − v n ≤s n1 u n1 − u n   s n1 | λ n − λ n1 | Au n   | s n1 − s n | Q   1 − s n1  x n1 − x n   | λ n − λ n1 | Ay n . 3.9 On the other hand, from u n  T r n x n and u n1  T r n1 x n1 ,wenotethat Θ  u n ,y   ϕ  y  − ϕ  u n   1 r n  y − u n ,u n − x n  ≥ 0, ∀y ∈ C, 3.10 Θ  u n1 ,y   ϕ  y  − ϕ  u n1   1 r n1  y − u n1 ,u n1 − x n1  ≥ 0, ∀y ∈ C. 3.11 [...]... 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