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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011, Article ID 840319, 25 pages doi:10.1155/2011/840319 Research Article The Shrinking Projection Method for Common Solutions of Generalized Mixed Equilibrium Problems and Fixed Point Problems for Strictly Pseudocontractive Mappings Thanyarat Jitpeera and Poom Kumam Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand Correspondence should be addressed to Poom Kumam, poom.kum@kmutt.ac.th Received 21 September 2010; Revised 14 December 2010; Accepted 20 January 2011 Academic Editor: Jewgeni Dshalalow Copyright q 2011 T. Jitpeera and P. Kumam. 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 the shrinking hybrid projection method for finding a common element of the set of fixed points of strictly pseudocontractive mappings, the set of common solutions of the variational inequalities with inverse-strongly monotone mappings, and the set of common solutions of generalized mixed equilibrium problems in Hilbert spaces. Furthermore, we prove strong convergence theorems for a new shrinking hybrid projection method under some mild conditions. Finally, we apply our results to Convex Feasibility Problems CFP. The results obtained in this paper improve and extend the corresponding results announced by Kim et al. 2010 and the previously known results. 1. Introduction Let H be a real Hilbert space with inner product ·, · and norm ·,andletE be a nonempty closed convex subset of H.LetT : E → E be a mapping. In the sequel, we will use FT to denote the set of fixed points of T,thatis,FT{x ∈ E : Tx  x}.Wedenoteweak convergence and strong convergence by notations  and → , respectively. Let S : E → E be a mapping. Then S is called 1 nonexpansive if   Sx − Sy   ≤   x − y   , ∀x, y ∈ E, 1.1 2 Journal of Inequalities and Applications 2 strictly pseudocontractive with the coefficient k ∈ 0, 1 if   Sx − Sy   2 ≤   x − y   2  k    I − S  x −  I − S  y   2 , ∀x, y ∈ E, 1.2 3 pseudocontractive if   Sx − Sy   2 ≤   x − y   2     I − S  x −  I − S  y   2 , ∀x, y ∈ E. 1.3 The class of strictly pseudocontractive mappings falls into the one between classes of nonexpansive mappings and pseudocontractive mappings. Within the past several decades, many authors have been devoted to the studies on the existence and convergence of fixed points for strictly pseudocontractive mappings. In 2008, Zhou 1 considered a convex combination method to study strictly pseudocontractive mappings. More precisely, take k ∈ 0, 1, and define a mapping S k by S k x  kx   1 − k  Sx, ∀x ∈ E, 1.4 where S is strictly pseudocontractive mappings. Under appropriate restrictions on k,itis proved that the mapping S k is nonexpansive. Therefore, the techniques of studying nonex- pansive mappings can be applied to study more general strictly pseudocontractive mappings. Recall that letting A : E → H be a mapping, then A is ca lled 1 monotone if  Ax − Ay, x − y  ≥ 0, ∀x, y ∈ E, 1.5 2 β-inverse-strongly monotone if there exists a constant β>0suchthat  Ax − Ay, x − y  ≥ β   Ax − Ay   2 , ∀x, y ∈ E. 1.6 The domain of the function ϕ : E → ∪{∞} is the set dom ϕ  {x ∈ E : ϕx < ∞}. Let ϕ : E → ∪{∞} be a proper extended real-valued function and let F be a bifunction of E × E into such that E ∩ dom ϕ /  ∅,where is the set of real numbers. There exists the generalized mixed equilibrium problem for finding x ∈ E such that F  x, y    Ax, y − x   ϕ  y  − ϕ  x  ≥ 0, ∀y ∈ E. 1.7 The set of solutions of 1.7 is denoted by GMEPF, ϕ, A,thatis, GMEP  F, ϕ, A    x ∈ E : F  x, y    Ax, y − x   ϕ  y  − ϕ  x  ≥ 0, ∀y ∈ E  . 1.8 Journal of Inequalities and Applications 3 We see that x is a solution of a problem 1.7 which implies that x ∈ dom ϕ  {x ∈ E : ϕx < ∞}. In particular, if A ≡ 0, then the problem 1.7 is reduced into the m ixed equilibrium problem 2 for finding x ∈ E such that F  x, y   ϕ  y  − ϕ  x  ≥ 0, ∀y ∈ E. 1.9 The set of solutions of 1.9 is denoted by MEPF, ϕ. If A ≡ 0andϕ ≡ 0, then the problem 1.7 is reduced into the equilibrium problem 3 for finding x ∈ E such that F  x, y  ≥ 0, ∀y ∈ E. 1.10 The set of solutions of 1.10 is denoted by EPF. This pro blem contains fixed point problems and includes as special cases numerous problems in physics, optimization, and economics. Some methods have been proposed to solve the equilibrium problem; please consult 4, 5. If F ≡ 0andϕ ≡ 0, then the problem 1.7 is reduced into the Hartmann-Stampacchia variational inequality 6 for finding x ∈ E such that  Ax, y − x  ≥ 0, ∀y ∈ E. 1.11 The set of solutions of 1.11 is denoted by VIE, A. The variational inequality has been extensively studied in the literature. See, for example, 7–10 and the references therein. Many authors solved the problems GMEPF, ϕ, A,MEPF, ϕ,andEPF based on iterative methods; see, for instance, 4, 5, 11–25 and reference therein. In 2007, Tada and Takahashi 26 intr oduced a hybrid method for finding the common element of the set of fixed point of nonexpansive mapping and the set of s olutions of equilibrium problems in Hilbert spaces. Let {x n } and {u n } be sequences generated by the following iterative algorithm: x 1  x ∈ H, F  u n ,y   1 r n  y − u n ,u n − x n  ≥ 0, ∀y ∈ E, w n   1 − α n  x n  α n Su n , E n  { z ∈ H :  w n − z  ≤  x n − z } , D n  { z ∈ H :  x n − z, x − x n  ≥ 0 } , x n1  P E n ∩D n x, ∀n ≥ 1. 1.12 Then, they proved that, under certain appropriate conditions imposed on {α n } and {r n },the sequence {x n } generated by 1.12 converges strongly to P FS∩EPF x. In 2009, Qin and Kang 27 introduced an explicit viscosity approximation method for finding a common element of the set of fixed point of strictly pseudocontractive mappings 4 Journal of Inequalities and Applications and the set of solutions of variational inequalities with inverse-strongly monotone mappings in Hilbert spaces: x 1 ∈ E, z n  P E  x n − μ n Cx n  , y n  P E  x n − λ n Bx n  , x n1   n f  x n   β n x n  γ n  α 1 n S k x n  α 2 n y n  α 3 n z n  , ∀n ≥ 1. 1.13 Then, they proved that, under certain appropriate conditions imposed on { n }, {β n }, {γ n }, {α 1 n }, {α 2 n },and{α 3 n },thesequence{x n } generated by 1.13 converges strongly to q ∈ FS ∩ VIE, B ∩ VIE, C,whereq  P FS∩VIE,B∩VIE,C fq. In 2010, Kumam and Jaiboon 28 introduced a new method for finding a common element of the set of fixed point of strictly pseudocontractive mappings, the set of common solutions of variational inequalities with inverse-strongly monotone mappings, and the set of common solutions of a system of generalized mixed equilibrium problems in Hilbert spaces. Then, they proved that, under certain a ppropriate conditions imposed on { n }, {β n },and {α i n },wherei  1, 2, 3, 4, 5. The sequence {x n } converges strongly to q ∈ Θ : FS∩VIE, B∩ VIE, C ∩ GMEPF 1 ,ϕ,A 1  ∩ GMEPF 2 ,ϕ,A 2 ,whereq  P Θ I − A  γfq. In this paper, motivate, by Tada and Takahashi 26, Qin and Kang 27,andKumam and Jaiboon 28, we introduce a new shrinking projection method for finding a common element of the set of fixed points of strictly pseudocontractive mappings, the set of common solutions of generalized mixed equilibrium problems, and the set of common solutions of the variational inequalities for inverse-strongly monotone mappings in Hilbert spaces. Finally, we apply our results to Convex Feasibility Problems CFP. The results obtained in this paper improve and extend the corresponding results announced by the previously known results. 2. Preliminaries Let H be a real Hilbert space, and let E be a nonempty closed convex subset of H.Inareal Hilbert space H,itiswellknownthat   λx   1 − λ  y   2  λ  x  2   1 − λ    y   2 − λ  1 − λ    x − y   2 , 2.1 for all x, y ∈ H and λ ∈ 0, 1. For any x ∈ H,thereexistsaunique nearest point in E, denoted by P E x,suchthat  x − P E x  ≤   x − y   , ∀y ∈ E. 2.2 The mapping P E is called the metric projection of H onto E. It is well known that P E is a firmly nonexpansive mapping of H onto E,thatis,  x − y, P E x − P E y  ≥   P E x − P E y   2 , ∀x, y ∈ H. 2.3 Journal of Inequalities and Applications 5 Moreover, P E x is characterized by the following properties: P E x ∈ E and  x − P E x, y − P E x  ≤ 0,   x − y   2 ≥  x − P E x  2    y − P E x   2 2.4 for all x ∈ H, y ∈ E. Lemma 2.1. Let E be a nonempty closed convex subset of a real Hilbert space H.Givenx ∈ H and z ∈ E,then, z  P E x ⇐⇒  x − z, y − z  ≤ 0, ∀y ∈ E. 2.5 Lemma 2.2. Let H be a Hilbert space, let E be a nonempty closed convex subset of H,andletB be a mapping of E into H.Letu ∈ E.Then,forλ>0, u ∈ VI  E, B  ⇐⇒ u  P E  u − λBu  , 2.6 where P E is the metric projection of H onto E. Lemma 2.3 see 1. Let E be a nonempty closed convex subset of a real H ilbert space H,andlet S : E → E be a k-strictly pseudocontractive mapping with a fixed point. Then FS is closed a nd convex. Define S k : E → E by S k  kx 1 − kSx for each x ∈ E.ThenS k is nonexpansive such that FS k FS. Lemma 2.4 see 29. Let E be a closed convex subset of a real Hilbert space H,andletS : E → E be a nonexpansive m apping. Then I − S is demiclosed at zero; that is, x n x, x n − Sx n −→ 0 2.7 implies x  Sx. Lemma 2.5 see 30. Each Hilbert space H satisfies the Kadec-Klee property, for any sequence {x n } with x n xand x n →x together implying x n − x→0. Lemma 2.6 see 31. Let E be a closed convex subset of H.Let{x n } be a bounded sequence in H. Assume that 1 the weak ω-limit set ω w x n  ⊂ E, 2 for each z ∈ E, lim n →∞ x n − z exists. Then {x n } is weakly convergent to a point in E. Lemma 2.7 see 32. Let E be a closed convex subset of H.Let{x n } be a sequence in H and u ∈ H. Let q  P E u.If{x n } is ω w x n  ⊂ E and satisfies the condition  x n − u  ≤   u − q   2.8 for all n,thenx n → q. 6 Journal of Inequalities and Applications Lemma 2.8 see 33. Let E be a nonempty closed convex subset of a strictly convex Banach space X.Let{T n : n ∈ } be a sequence of nonexpansive mappings on E. Suppose  ∞ n1 FT n  is nonempty. Let δ n be a sequence of positive number with  ∞ n1 δ n  1. Then a mapping S on E defined by Sx  ∞  n1 δ n T n x 2.9 for x ∈ E is well defined, nonexpansive, and FS  ∞ n1 FT n  holds. For solving the mixed equilibrium problem, let us give the following assumptions for the bifunction F, the function A,andthesetE: A1 Fx, x0forallx ∈ E A2 F is monotone, that is, Fx, yFy, x ≤ 0forallx, y ∈ E A3 for each x, y, z ∈ E, lim t →0 Ftz 1 − tx, y ≤ Fx, y A4 for each x ∈ E, y → Fx, y is convex and lower semicontinuous A5 for each y ∈ E, x → Fx, y is weakly upper semicontinuous B1 for each x ∈ H and r>0, there exists a bounded subset D x ⊆ E and y x ∈ E such that, for any z ∈ E \ D x , F  z, y x   ϕ  y x  − ϕ  z   1 r  y x − z, z − x  < 0, 2.10 B2 E is a bounded set. By similar argument as i n the proof of Lemma 2.9 in 34, we have the following lemma appearing. Lemma 2.9. Let E be a nonempty closed convex subset of H.LetF : E × E → be a bifunction that satisfies (A1)–(A5), and let ϕ: E → ∪{∞} 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 F r : H → E as follows: T F r  x    z ∈ E : F  z, y   ϕ  y  − ϕ  z   1 r  y − z, z − x  ≥ 0, ∀y ∈ E  , 2.11 for all z ∈ H. Then, the following hold: 1 for each x ∈ H, T F r x /  ∅, 2 T F r is single valued, 3 T F r is firmly nonexpansive, that is, for any x, y ∈ H,    T F r x − T F r y    2 ≤  T F r x − T F r y, x − y  , 2.12 4 FT F r MEPF, ϕ, 5 MEPF, ϕ is closed and convex. Journal of Inequalities and Applications 7 Lemma 2.10. Let H be a Hilbert space, let E be a nonempty closed convex subset of H,andlet A : E → H be ρ-inverse-strongly monotone. If 0 <r≤ 2ρ,thenI − ρA is a nonexpansive mapping in H. Proof. For all x, y ∈ E and 0 <r≤ 2ρ,wehave    I − rA  x −  I − rA  y   2     x − y  − r  Ax − Ay    2    x − y   2 − 2r  x − y, Ax − Ay   r 2   Ax − Ay   2 ≤   x − y   2 − 2rρ   Ax − Ay    r 2   Ax − Ay   2    x − y   2  r  r − 2ρ    Ax − Ay   2 ≤   x − y   2 . 2.13 So, I − ρA is a nonexpansive mapping of E into H. 3. Main Results In this section, we prove a strong convergence theorem of the new shrinking p rojection method for finding a common element of the set of fixed points of strictly pseudocontractive mappings, the set of common solutions of generalized mixed equilibrium problems and the set of common solutions of the variational inequalities with inverse-strongly monotone mappings in Hilbert spaces. Theorem 3.1. Let E be a nonempty closed convex subset of a real Hilbert space H.LetF 1 and F 2 be two bifunctions from E × E to satisfying (A1)–(A5), and let ϕ : E → ∪{∞} be a pr oper lower semicontinuous and convex function with either (B1) or (B2). Let A 1 , A 2 , B, C be four ρ, ω, β, ξ-inverse-strongly monotone mappings of E into H, respectively. Let S : E → E be a k-strictly pseudocontractive mapping with a fixed point. Define a mapping S k : E → E by S k x  kx1−kSx, for all x ∈ E. Suppose that Θ : F  S  ∩ GMEP  F 1 ,ϕ,A 1  ∩ GMEP  F 2 ,ϕ,A 2  ∩ VI  E, B  ∩ VI  E, C  /  ∅. 3.1 Let {x n } be a sequence generated by the following iterative algorithm: x 0 ∈ H, E 1  E, x 1  P E 1 x 0 ,u n ∈ E, v n ∈ E, F 1  u n ,u   ϕ  u  − ϕ  u n    A 1 x n ,u− u n   1 r n  u − u n ,u n − x n  ≥ 0, ∀u ∈ E, F 2  v n ,v   ϕ  v  − ϕ  v n    A 2 x n ,v− v n   1 s n  v − v n ,v n − x n  ≥ 0, ∀v ∈ E, y n  P E  x n − λ n Bx n  ,z n  P E  x n − μ n Cx n  , 8 Journal of Inequalities and Applications t n  α 1 n S k x n  α 2 n y n  α 3 n z n  α 4 n u n  α 5 n v n , E n1  { w ∈ E n :  t n − w  ≤  x n − w } , x n1  P E n1 x 0 , ∀n ≥ 0, 3.2 where {α i n } are sequences in 0, 1,wherei  1, 2, 3, 4, 5, r n ∈ 0, 2ρ, s n ∈ 0, 2ω,and{λ n }, {μ n } are positive sequences. Assume that the control sequences satisfy the following restrictions: C1  5 i1 α i n  1, C2 lim n →∞ α i n  α i ∈ 0, 1,wherei  1, 2, 3, 4, 5, C3 a ≤ r n ≤ 2ρ and b ≤ s n ≤ 2ω,wherea, b are two positive constants, C4 c ≤ λ n ≤ 2β and d ≤ μ n ≤ 2ξ,wherec, d are two positive constants, C5 lim n →∞ |λ n1 − λ n |  lim n →∞ |μ n1 − μ n |  0. Then, {x n } converges strongly to P Θ x 0 . Proof. Letting p ∈ Θ and by Lemma 2.9,weobtain p  P E  p − λ n Bp   P E  p − μ n Cp   T F 1 r n  I − r n A 1  p  T F 2 s n  I − s n A 2  p. 3.3 Note that u n  T F 1 r n I − r n A 1 x n ∈ dom ϕ and v n  T F 2 s n I − s n A 2 x n ∈ dom ϕ,thenwehave   u n − p    T F 1 r n  I − r n A 1  x n − T F 1 r n  I − r n A 1  p≤x n − p,   v n − p       T F 2 s n  I − s n A 2  x n − T F 2 s n  I − s n A 2  p    ≤   x n − p   . 3.4 Next, we will divide the proof into six steps. Step 1. We show that {x n } is well defined and E n is closed and convex for any n ≥ 1. From the assumption, we see that E 1  E is closed and convex. Suppose that E k is closed and convex for some k ≥ 1. Next, we show that E k1 is closed and convex for some k. For any p ∈ E k ,weobtain   t k − p   ≤   x k − p   3.5 is equivalent to   t k − p   2  2  t k − x k ,x k − p  ≤ 0. 3.6 Thus, E k1 is closed and convex. Then, E n is closed and convex for any n ≥ 1. This implies that {x n } is well defined. Journal of Inequalities and Applications 9 Step 2. We show that Θ ⊂ E n for each n ≥ 1. From the assumption, we see that Θ ⊂ E  E 1 . Suppose Θ ⊂ E k for some k ≥ 1. For any p ∈ Θ ⊂ E k ,sincey n  P E x n − λ n Bx n  and z n  P E x n − μ n Cx n ,foreachλ n ≤ 2β and μ n ≤ 2ξ by Lemma 2.10,wehaveI − λ n B and I − μ n C are nonexpansive. Thus, we obtain   y n − p      P E  x n − λ n Bx n  − P E  p − λ n Bp    ≤    x n − λ n Bx n  −  p − λ n Bp        I − λ n B  x n −  I − λ n B  p   ≤   x n − p   ,   z n − p      P E  x n − μ n Cx n  − P E  p − μ n Cp    ≤    x n − μ n Cx n  −  p − μ n Cp        I − μ n C  x n −  I − μ n C  p   ≤   x n − p   . 3.7 From Lemma 2.3,wehaveS k is nonexpansive with FS k FS. It follows that   t n − p       α 1 n S k x n  α 2 n y n  α 3 n z n  α 4 n u n  α 5 n v n − p    ≤ α 1 n   S k x n − p    α 2 n   y n − p    α 3 n   z n − p    α 4 n   u n − p    α 5 n   v n − p   ≤ α 1 n   x n − p    α 2 n   x n − p    α 3 n   x n − p    α 4 n   x n − p    α 5 n   x n − p      x n − p   . 3.8 It follows that p ∈ E k1 . This implies that Θ ⊂ E n for each n ≥ 1. Step 3. We claim that lim n →∞ x n1 − x n   0 and lim n →∞ x n − t n   0. From x n  P E n x 0 ,weget  x 0 − x n ,x n − y  ≥ 0 3.9 for each y ∈ E n .UsingΘ ⊂ E n ,wehave  x 0 − x n ,x n − p  ≥ 0foreachp ∈ Θ,n∈ . 3.10 10 Journal of Inequalities and Applications Hence, for p ∈ Θ,weobtain 0 ≤  x 0 − x n ,x n − p    x 0 − x n ,x n − x 0  x 0 − p   −  x 0 − x n ,x 0 − x n    x 0 − x n ,x 0 − p  ≤−  x 0 − x n  2   x 0 − x n    x 0 − p   . 3.11 It follows that  x 0 − x n  ≤   x 0 − p   , ∀p ∈ Θ,n∈ . 3.12 From x n  P E n x 0 and x n1  P E n1 x 0 ∈ E n1 ⊂ E n ,wehave  x 0 − x n ,x n − x n1  ≥ 0. 3.13 For n ∈ ,wecompute 0 ≤  x 0 − x n ,x n − x n1    x 0 − x n ,x n − x 0  x 0 − x n1   −  x 0 − x n ,x 0 − x n    x 0 − x n ,x 0 − x n1  ≤−  x 0 − x n  2   x 0 − x n ,x 0 − x n1  ≤−  x 0 − x n  2   x 0 − x n  x 0 − x n1  , 3.14 and then  x 0 − x n  ≤  x 0 − x n1  , ∀n ∈ . 3.15 Thus, the sequence {x n −x 0 } is a bounded and nondecreasing sequence, so lim n →∞ x n −x 0  exists; that is, there exists m such that m  lim n →∞  x n − x 0  . 3.16 [...]... 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Corporation Journal of Inequalities and Applications Volume 2011, Article ID 840319, 25 pages doi:10.1155/2011/840319 Research Article The Shrinking Projection Method for Common Solutions of Generalized Mixed Equilibrium Problems