Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 274820, docx

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 274820, 19 pages doi:10.1155/2011/274820 Research Article Hybrid Algorithm for Finding Common Elements of the Set of Generalized Equilibrium Problems and the Set of Fixed Point Problems of Strictly Pseudocontractive Mapping Atid Kangtunyakarn Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand Correspondence should be addressed to Atid Kangtunyakarn, beawrock@hotmail.com Received 8 November 2010; Accepted 14 December 2010 Academic Editor: Qamrul Hasan Ansari Copyright q 2011 Atid Kangtunyakarn. 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. The purpose of this paper is to prove the strong convergence theorem for finding a common element of the set of fixed point problems of strictly pseudocontractive mapping in Hilbert spaces and two sets of generalized equilibrium problems by using the hybrid method. 1. Introduction Let C be a closed convex subset of a real Hilbert space H,andletF : C × C → R be a bifunction. Recall that the equilibrium problem for a bifunction F is to find x ∈ C such that F  x, y  ≥ 0, ∀y ∈ C. 1.1 The set of solutions of 1.1 is denoted by EPF. Given a mapping T : C → H,letFx, y Tx,y − x for all x, y ∈ C. Then, z ∈ EPF if and only if Tz,y − z≥0 for all y ∈ C; that is, z is a solution of the variational inequality. Let A : C → H be a nonlinear mapping. The variational inequality problem is to find a u ∈ C such that  v − u, Au  ≥ 0 1.2 2 Fixed Point Theory and Applications for all v ∈ C. The set of solutions of the variational inequality is denoted by VIC, A.Now, we consider the following generalized equilibrium problem: Find z ∈ C such that F  z, y    Az, y − z  ≥ 0, ∀y ∈ C. 1.3 The set of z ∈ C is denoted by EPF, A,thatis, EP  F, A    z ∈ C : F  z, y    Az, y − z  ≥ 0, ∀y ∈ C  . 1.4 In the case of A ≡ 0, EPF, A is denoted by EPF. In the case of F ≡ 0, EPF, A is also denoted by VIC, A. Numerous problems in physics, optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and economics are reduced to find a solution of 1.3; see, for instance, 1–3. A mapping A of C into H is called inverse strongly monotone mapping,see4, if there exists a positive real number α such that x − y, Ax − Ay≥α   Ax − Ay   2 1.5 for all x, y ∈ C. The following definition is well known. Definition 1.1. A mapping T : C → C is said to be a κ-strict pseudocontraction if there exists κ ∈ 0, 1 such that   Tx − Ty   2 ≤   x − y   2  κ    I − T  x −  I − T  y   2 , ∀x, y ∈ C. 1.6 A mapping T is called nonexpansive if   Tx − Ty   ≤   x − y   1.7 for all x, y ∈ C. We know that κ-strict pseudocontraction includes a class of nonexpansive mappings. If κ  1, T is said to be a pseudocontractive mapping. T is strong pseudocontraction if there exists a positive constant λ ∈ 0, 1 such that T  λI is pseudocontraction. In a real Hilbert space H, 1.6 is equivalent to  Tx − Ty,x − y  ≤   x − y   2 − 1 − κ 2   I − Tx − I − Ty   2 , ∀x, y ∈ D  T  . 1.8 T is pseudocontraction if and only if  Tx − Ty,x − y  ≤   x − y   2 , ∀x, y ∈ D  T  . 1.9 Then, T is strong pseudocontraction if there exists positive constant λ ∈ 0, 1  Tx − Ty,x − y  ≤  1 − λ    x − y   2 , ∀x, y ∈ D  T  . 1.10 Fixed Point Theory and Applications 3 The class of κ-strict pseudocontractions falls into the one between classes of nonex- pansive mappings, and the pseudocontraction mappings, and the class of strong pseudocon- traction mappings is independent of the class of κ-strict pseudocontraction. We denote by FT the set of fixed points of T.IfC ⊂ H is bounded, closed, and convex, and T is a nonexpansive mapping of C into itself, then FT is nonempty; for instance, see 5. Browder and Petryshyn 6 show that if a κ-strict pseudocontraction T has a fixed point in C, then starting with an initial x 0 ∈ C, the sequence {x n } generated by the recursive formula: x n1  αx n   1 − α  Tx n , 1.11 where α is a constant such that 0 <α<1, converges weakly to a fixed point of T.Marinoand Xu 7 have extended Browder and Petryshyns above-mentioned result by proving that the sequence {x n } generated by the following Manns algorithm 8: x n1  α n x n   1 − α n  Tx n 1.12 converges weakly to a fixed point of T provided the control sequence {α n } ∞ n0 satisfies the conditions that κ<α n < 1 for all n and  ∞ n0 α n − κ1 − α n ∞. In 1974, S. Ishikawa proved the following strong convergence theorem of pseudocontractive mapping. Theorem 1.2 see 9. Let C be a convex compact subset of a Hilbert space H, and let T : C → C be a Lipschitzian pseudocontractive mapping. For any x 1 ∈ C, suppose that the sequence {x n } is defined by y n   1 − β n  x n  β n Tx n , x n1   1 − α n  x n  α n Ty n , ∀n ∈ N, 1.13 where {α n }, {β n } are two real sequences in 0, 1 satisfying i α n ≤ β n , for all n ∈ N, ii lim n →∞ β n  0 , iii  ∞ n1 α n β n  ∞. Then {x n } converges strongly to a fixed point of T. In order to prove a strong convergence theorem of Mann algorithm 1.12 associated with strictly pseudocontractive mapping, in 2006, Marino and Xu 7 proved the following theorem for strict pseudocontractive mapping in Hilbert space by using CQ method. 4 Fixed Point Theory and Applications Theorem 1.3 see 7. Let C be a closed convex subset of a Hilbert space H.LetT : C → C be a κ-strict pseudocontraction for some 0 ≤ κ<1, and assume that the fixed point set FT of T is nonempty. Let {x n } ∞ n1 be the sequence generated by the following CQ algorithm: x 1 ∈ C, y n  α n x n   1 − α n  Tx n , C n   z ∈ C :   y n − z   2 ≤  1 − α n  κ − α n   x n − Tx n  2  , Q n  { z ∈ C :  x n − z, x 1 − x n } , x n1  P C n ∩Q n x 1 . 1.14 Assume that the control sequence {α n } ∞ n1 is chosen so that α n < 1 for all n ∈ N.Then{x n } converges strongly to P FT x 1 . Very recently, in 2010, [10] established the hybrid algorithm for Lipschitz pseudocontractive mapping as follows: For C 1  C, x 1  P C 1 x 1 , y n   1 − α n  x n  α n Tz n , z n   1 − β n  x n  β n Tx n , C n1   z ∈ C n :   α n  I − T  y n   2 ≤ 2α n  x n − z,  I − T  y n  2α n β n L  x n − Tx n    y n − x n  α n  I − T  y n    , x n1  P C n1 x 1 , ∀n ∈ N. 1.15 Under suitable conditions of {α n } and {β n }, they proved that the sequence {x n } defined by 1.15 converges strongly to P FT x 1 . Many authors study the problem for finding a common element of the set of fixed point problem and the set of equilibrium problem in Hilbert spaces, for instance, [2, 3, 11–15]. The motivation of 1.14, 1.15, and the research in this direction, we prove the strong convergence theorem for finding solution of the set of fixed points of strictly pseudocontractive mapping and two sets of generalized equilibrium problems by using the hybrid method. 2. Preliminaries In order to prove our main results, we need the following lemmas. Let C be closed convex subset of a real Hilbert space H,andletP C be the metric projection of H onto C;thatis,for x ∈ H, P C x satisfies the property  x − P C x   min y∈C   x − y   . 2.1 The following characterizes the projection P C . Fixed Point Theory and Applications 5 Lemma 2.1 see 5. Given that x ∈ H and y ∈ C,thenP C x  y if and only if the following inequality holds:  x − y, y − z  ≥ 0, ∀z ∈ C. 2.2 The following lemma is well known. Lemma 2.2. Let H be Hilbert space, and let C be a nonempty closed convex subset of H.LetT : C → C be κ-strictly pseudocontractive, then the fixed point set FT of T is closed and convex so that the projection P FT is well defined. Lemma 2.3 demiclosedness principlesee 16. If T is a κ-strict pseudocontraction on closed convex subset C of a real Hilbert space H,thenI − T is demiclosed at any point y ∈ H. To solve the equilibrium problem for a bifunction F : C × C → R, assume that F satisfies the following conditions: A1 Fx, x0 for all x ∈ C, A2 F is monotone, that is,Fx, yFy, x ≤ 0, for all x, y ∈ C, A3 for all x, y, z ∈ C, lim t → 0  F  tz   1 − t  x, y  ≤ F  x, y  , 2.3 A4 for all x ∈ C, y → Fx, y is convex and lower semicontinuous. The following lemma appears implicitly in 1. Lemma 2.4 see 1. Let C be a nonempty closed convex subset of H, and let F be a bifunction of C × C into R satisfying A1–A4.Letr>0, and x ∈ H. Then, there exists z ∈ C such that F  z, y   1 r  y − z, z − x  , 2.4 for all x ∈ C. Lemma 2.5 see 11. Assume that F : C × C → R satisfies A1–A4. For r>0 and x ∈ H, define a mapping T r : H → C as follows: T r  x    z ∈ C : F  z, y   1 r  y − z, z − x  ≥ 0, ∀y ∈ C  , 2.5 for all z ∈ H. Then, the following hold: 1 T r is single-valued; 6 Fixed Point Theory and Applications 2 T r is firmly nonexpansive, that is,   T r x − T r y   2 ≤  T r  x  − T r  y  ,x− y  , ∀x, y ∈ H, 2.6 3 FT r EPF; 4 EPF is closed and convex. Lemma 2.6 see 17. Let C be a closed convex subset of H.Let{x n } be a sequence in H and u ∈ H. Let q  P C u;if{x n } is such that ωx n  ⊂ C and satisfy the condition  x n − u  ≤   u − q   , ∀n ∈ N, 2.7 then x n → q, as n →∞. Lemma 2.7 see 7. For a real Hilbert space H, the following identities hold: if {x n } is a sequence in H weak convergence to z,then lim sup n →∞   x n − y   2  lim sup n →∞  x n − z  2    z − y   2 , 2.8 for all y ∈ H. 3. Main Result Theorem 3.1. Let C be a nonempty closed convex subset of a Hilbert space H.LetF and G be bifunctions from C × C into R satisfying A 1 –A 4 , respectively. Let A : C → H be an α-inverse strongly monotone mapping, and let B : C → H be a β-inverse strongly monotone mapping. Let T : C → C be a κ-strict pseudocontraction mapping with F  FT ∩ EPF, A ∩ EPG, B /  ∅.Let {x n } be a sequence generated by x 1 ∈ C  C 1 and F  u n ,u    Ax n ,u− u n   1 r n  u − u n ,u n − x n  ≥ 0, ∀u ∈ C, G  v n ,v    Bx n ,v− v n   1 s n  v − v n ,v n − x n  ≥ 0, ∀v ∈ C, z n  δ n u n   1 − δ n  v n , y n  α n z n   1 − α n  Tz n , C n1   z ∈ C n :   y n − z   ≤  x n − z   , x n1  P C n1 x 1 , ∀n ≥ 1, 3.1 where {α n } ∞ n0 is sequence in 0, 1, r n ∈ a, b ⊂ 0, 2α, and s n ⊂ c, d ⊂ 0, 2β satisfy the following conditions: Fixed Point Theory and Applications 7 i lim n →∞ δ n  δ ∈ 0, 1, ii 0 ≤ κ ≤ α n < 1, for all n ≥ 1. Then x n converges strongly to P F x 1 . Proof. First, we show that I −r n A is nonexpansive. Let x, y ∈ C. Since A is α-inverse strongly monotone mapping and r n < 2α, we have    I − r n A  x −  I − r n A  y   2    x − y − r n  Ax − Ay    2    x − y   2 − 2r n  x − y, Ax − Ay   r 2 n   Ax − Ay   2 ≤   x − y   2 − 2αr n   Ax − Ay   2  r 2 n   Ax − Ay   2    x − y   2  r n  r n − 2α    Ax − Ay   2 ≤   x − y   2 . 3.2 Thus I − r n A is nonexpansive, so are I − s n B, T r n I − r n A,andT s n I − s n B. Since F  u n ,u    Ax n ,u− u n   1 r n  u − u n ,u n − x n  ≥ 0, ∀u ∈ C, 3.3 then we have F  u n ,u   1 r n  u − u n ,u n −  I − r n A  x n  ≥ 0. 3.4 By Lemma 2.5, we have u n  T r n I − r n Ax n . By the same argument as above, we conclude that v n  T s n I − s n Bx n . Let z ∈ F. Then Fz, yy − z, Az≥0andGz, yy − z, Bz≥0. Hence F  z, y   1 r n  y − z, z − z  r n Az  ≥ 0, G  z, y   1 s n  y − z, z − z  s n Bz  ≥ 0. 3.5 Again by Lemma 2.5, we have z  T r n z − r n AzT s n z − s n Bz. By nonexpansiveness of T r n I − r n A and T s n I − s n B, we have  u n − z    T r n  I − r n A  x n − T r n  I − r n A  z  ≤  x n − z  ,  v n − z    T s n  I − s n A  x n − T s n  I − s n A  z  ≤  x n − z  . 3.6 8 Fixed Point Theory and Applications By 3.6, we h ave  z n − z  ≤  x n − z  . 3.7 Next, we show that C n is closed and convex for every n ∈ N. It is obvious that C n is closed. In fact, we know that, for z ∈ C n ,   y n − z   ≤  x n − z  is equivalent to   y n − x n   2  2  y n − x n ,x n − z  ≤ 0. 3.8 So, we have that for all z 1 ,z 2 ∈ C n and t ∈ 0, 1, it follows that   y n − x n   2  2  y n − x n ,x n −  tz 1   1 − t  z 2    t  2  y n − x n ,x n − z 1     y n − x n   2    1 − t   2  y n − x n ,x n − z 2     y n − x n   2  ≤ 0. 3.9 Then, we have that C n is convex. By Lemmas 2.5 and 2.2, we conclude that F is closed and convex. This implies that P F is well defined. Next, we show that F ⊂ C n for every n ∈ N. Taking p ∈ F, we have   y n − p   2    α n  z n − p    1 − α n   Tz n − p    2  α n   z n − p   2   1 − α n    Tz n − p   2 − α n  1 − α n   z n − Tz n  2 ≤ α n   z n − p   2   1 − α n     z n − p   2  κ    I − T  z n −  I − T  p   2  − α n  1 − α n  z n − Tz n  2  α n   z n − p   2   1 − α n    z n − p   2  κ  1 − α n   z n − Tz n  2 − α n  1 − α n   z n − Tz n  2    z n − p   2   κ − α n  1 − α n   z n − Tz n  2 ≤   z n − p   2 ≤   x n − p   2 . 3.10 It follows that p ∈ C n . Then, we have F ⊂ C n , for all n ∈ N. Since x n  P C n x 1 , for every w ∈ C n , we have  x n − x 1  ≤  w − x 1  , ∀n ∈ N. 3.11 Fixed Point Theory and Applications 9 In particular, we have  x n − x 1  ≤  P F x 1 − x 1  . 3.12 By 3.11, we have that {x n } is bounded, so are {u n }, {v n }, {z n }, {y n }. Since x n1  P C n1 x 1 ∈ C n1 ⊂ C n and x n  P C n x 1 , we have 0 ≤  x 1 − x n ,x n − x n1    x 1 − x n ,x n − x 1  x 1 − x n1  ≤−  x n − x 1  2   x n − x 1  x 1 − x n1  . 3.13 It is implied t hat  x n − x 1  ≤  x n1 − x 1  . 3.14 Hence, we have that lim n →∞ x n − x 1  exists. Since x n − x n1  2   x n − x 1  x 1 − x n1  2   x n − x 1  2  2  x n − x 1 ,x 1 − x n1    x 1 − x n1  2   x n − x 1  2  2  x n − x 1 ,x 1 − x n  x n − x n1    x 1 − x n1  2   x n − x 1  2 − 2  x n − x 1  2  2  x n − x 1 ,x n − x n1    x 1 − x n1  2 ≤  x 1 − x n1  2 −  x n − x 1  2 , 3.15 it is implied t hat lim n →∞  x n − x n1   0. 3.16 Since x n1  P C n1 x 1 ∈ C n1 , we have   y n − x n1   ≤  x n − x n1  , 3.17 And by 3.16, we have lim n →∞   y n − x n1    0. 3.18 Since   y n − x n   ≤   y n − x n1     x n1 − x n  , 3.19 10 Fixed Point Theory and Applications by 3.16 and 3.18, we have lim n →∞   y n − x n    0. 3.20 Next, we show that lim n →∞  u n − x n   0, lim n →∞  v n − x n   0. 3.21 Let p ∈ F,by3.10 and 3.7, we have   y n − p   2    α n  z n − p    1 − α n   Tz n − p    2  α n   z n − p   2   1 − α n    Tz n − p   2 − α n  1 − α n   z n − Tz n  2 ≤ α n   z n − p   2   1 − α n     z n − p   2  κ    I − T  z n −  I − T  p   2  − α n  1 − α n   z n − Tz n  2  α n   z n − p   2   1 − α n    z n − p   2  κ  1 − α n   z n − Tz n  2 − α n  1 − α n   z n − Tz n  2  α n   z n − p   2   1 − α n    z n − p   2   κ − α n  1 − α n   z n − Tz n  2 ≤ α n   x n − p   2   1 − α n    z n − p   2 ≤ α n   x n − p   2   1 − α n   δ n   u n − p   2   1 − δ n    v n − p   2  . 3.22 Since u n  T r n I − r n Ax n , p  T r n I − r n Ap, we have   u n − p   2    T r n  I − r n A  x n − T r n  I − r n A  p   2 ≤  I − r n A  x n −  I − r n A  p 2    x n − r n Ax n − p  r n Ap   2    x n − p − r n  Ax n − Ap    2    x n − p   2  r 2 n   Ax n − Ap   2 − 2r n  x n − p, Ax n − Ap  ≤   x n − p   2  r 2 n   Ax n − Ap   2 − 2r n α   Ax n − Ap   2    x n − p   2  r n  r n − 2α    Ax n − Ap   2 . 3.23 [...]... Nonlinear Analysis: Theory, Methods & Applications, vol 71, no 10, pp 4448–4460, 2009 13 A Kangtunyakarn and S Suantai, “Hybrid iterative scheme for generalized equilibrium problems and fixed point problems of finite family of nonexpansive mappings,” Nonlinear Analysis Hybrid Systems, vol 3, no 3, pp 296–309, 2009 Fixed Point Theory and Applications 19 14 S Takahashi and W Takahashi, “Viscosity approximation... 105–113, 2006 5 W Takahashi, Nonlinear Functional Analysis Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, Japan, 2000 6 F E Browder and W V Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol 20, pp 197–228, 1967 7 G Marino and H.-K Xu, “Weak and strong convergence theorems for strict pseudo-contractions... 3.51 Fixed Point Theory and Applications 17 By using the same method as 3.50 , we have q ∈ EP G, B Since xni q as i → ∞ and 3.38 , we have zni demiclosed at zero, and by 3.41 , we have 3.52 q as i → ∞ By Lemma 2.3, I − T is q∈F T 3.53 From 3.50 , 3.52 , and 3.53 , we have q ∈ F Hence ω xn ⊂ F Therefore, by 3.12 and Lemma 2.6, we have that {xn } converges strongly to PF x1 The proof is completed 4 Applications. .. equilibrium problems and fixed point problems in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol 331, no 1, pp 506–515, 2007 15 V Colao, G Marino, and H.-K Xu, “An iterative method for finding common solutions of equilibrium and fixed point problems,” Journal of Mathematical Analysis and Applications, vol 344, no 1, pp 340– 352, 2008 16 H Zhou, “Convergence theorems of fixed points for -strict... Bxn − Bp Axn − Ap Bxn − Bp , 3.34 Fixed Point Theory and Applications 15 and by 3.27 , 3.28 , 3.20 , and conditions i , ii , we have lim xn − un n→∞ 0 3.35 0 3.36 By using the same method as 3.35 , we have lim xn − vn n→∞ Since zn − xn ≤ δn un − xn 1 − δn vn − xn , 3.37 from 3.35 , 3.36 , and condition i , we have lim zn − xn 0 3.38 lim yn − zn 0 3.39 n→∞ By 3.20 and 3.38 , we have n→∞ Since yn − zn... S Takahashi and W Takahashi, “Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space,” Nonlinear Analysis: Theory, Methods & Applications, vol 69, no 3, pp 1025–1033, 2008 4 H Iiduka and W Takahashi, “Weak convergence theorems by Ces´ ro means for nonexpansive a mappings and inverse-strongly-monotone mappings,” Journal of Nonlinear and Convex Analysis,... Analysis: Theory, Methods & Applications, vol 74, pp 380–385, 2011 11 P L Combettes and S A Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol 6, no 1, pp 117–136, 2005 12 A Kangtunyakarn and S Suantai, “A new mapping for finding common solutions of equilibrium problems and fixed point problems of finite family of nonexpansive mappings,” Nonlinear Analysis: Theory, ... − δn sn − 2β sn 1 − δn sn − 2β 2 2 3.25 12 Fixed Point Theory and Applications It is implied that 1 − αn δn rn 2α − rn Axn − Ap 2 2 ≤ xn − p 2 − yn − p sn 1 − αn 1 − δn sn − 2β ≤ xn − p yn − p Bxn − Bp 2 3.26 xn − yn By 3.20 and condition i , we have lim Axn − Ap 0 3.27 0 n→∞ 3.28 By using the same method as 3.27 , we have lim Bxn − Bp n→∞ By Lemma 2.5 and firm nonexpansiveness of Trn , we have un... desired References 1 E Blum and W Oettli, “From optimization and variational inequalities to equilibrium problems,” Mathematics Student, vol 63, no 1–4, pp 123–145, 1994 2 A Moudafi and M Th´ ra, “Proximal and dynamical approaches to equilibrium problems,” in e Ill-Posed Variational Problems and Regularization Techniques (Trier, 1998), vol 477 of Lecture Notes in Economics and Mathematical Systems, pp... By 3.31 , it is implied that vn − p 2 ≤ xn − p 2 − xn − vn 2 − s2 Bxn − Bp n 2 2sn xn − vn , Bxn − Bp ≤ xn − p 2 − xn − vn 2 − s2 Bxn − Bp n 2 2sn xn − vn Bxn − Bp 3.32 14 Fixed Point Theory and Applications Substituting 3.30 and 3.32 into 3.22 , we have 2 ≤ αn xn − p 2 1 − αn ≤ αn xn − p yn − p 2 1 − αn xn − p × δn 1 − δn 2 δn un − p − xn − un 2 xn − p 2 2 1 − δn 2 − rn Axn − Ap − xn − vn 2sn xn . Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 274820, 19 pages doi:10.1155/2011/274820 Research Article Hybrid Algorithm for Finding. Hybrid Systems, vol. 3, no. 3, pp. 296–309, 2009. Fixed Point Theory and Applications 19 14 S. Takahashi and W. Takahashi, “Viscosity approximation methods for equilibrium problems and fixed point. 3.17 And by 3.16, we have lim n →∞   y n − x n1    0. 3.18 Since   y n − x n   ≤   y n − x n1     x n1 − x n  , 3.19 10 Fixed Point Theory and Applications by 3.16 and