Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 852789, potx

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 852789, 22 pages doi:10.1155/2011/852789 Research Article Iterative Methods for Family of Strictly Pseudocontractive Mappings and System of Generalized Mixed Equilibrium Problems and Variational Inequality Problems Yekini Shehu Mathematics Institute, African University of Science and Technology , Abuja, Nigeria Correspondence should be addressed to Yekini Shehu, deltanougt2006@yahoo.com Received 6 September 2010; Accepted 25 November 2010 Academic Editor: S. Al-Homidan Copyright q 2011 Yekini Shehu. 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 by hybrid method for finding a common element of the set of common fixed points of infinite family of k-strictly pseudocontractive mappings and the set of common solutions to a system of generalized mixed equilibrium problems and the set of solutions to a variational inequality problem in a real Hilbert space. We then prove strong convergence of the scheme to a common element of the three above described sets. We give an application of our results. Our results extend important recent results from the current literature. 1. Introduction Let K be a nonempty closed and convex subset of a real Hilbert space H. A mapping A : K → H is called monotone if  Ax − Ay, x − y  ≥ 0, ∀x, y ∈ K. 1.1 A mapping A : K → H is called inverse-strongly monotone see, e.g., 1, 2 if there exists a positive real number λ such that Ax − Ay, x − y≥λAx − Ay 2 ,forallx, y ∈ K.Forsuch acase,A is called λ-inverse-strongly monotone. A λ-inverse-strongly monotone is sometime called λ-cocoercive. A mapping A is said to be relaxed λ-cocoercive if there exists λ>0suchthat  Ax − Ay, x − y  ≥−λ   Ax − Ay   2 , ∀x, y ∈ K. 1.2 2 Fixed Point Theory and Applications A is said to be relaxed λ, γ-cocoercive if there exist λ, γ > 0suchthat Ax − Ay, x − y≥−λ   Ax − Ay   2  γ   x − y   2 , ∀x, y ∈ K. 1.3 A mapping A : H → H is said to be μ-Lipschitzian if there exists μ ≥ 0suchthat   Ax − Ay   ≤ μ   x − y   ,x,y∈ H. 1.4 Let A : K → H be a nonlinear mapping. The variational inequality problem is to find an x ∗ ∈ K such that  Ax ∗ ,y− x ∗  ≥ 0, ∀y ∈ K. 1.5 See, e.g., 3, 4. We will denote the set of solutions of the variational inequality problem 1.5 by VIK, A. A monotone mapping A is said to be maximal if the graph GA is not properly contained in the graph of any other monotone map, where GA : {x, y ∈ H × H : y ∈ Ax} for a multivalued mapping A.ItisalsoknownthatA is maximal if and only if for x, f ∈ H × H, x − y, f − g≥0 for every y, g ∈ GA implies f ∈ Ax.LetA be a monotone mapping defined from K into H and N K q a normal cone to K at q ∈ K,thatis, N K q  {p ∈ H : q − u, p≥0, for all u ∈ K}. Define a mapping M by Mq  ⎧ ⎨ ⎩ Aq  N K q, q ∈ K, ∅,q / ∈ K. 1.6 Then, M is maximal monotone and x ∗ ∈ M −1 0 ⇔ x ∗ ∈ VIK, Asee, e.g., 5. A mapping T : K → K is said to be k-strictly pseudocontractive if there exists a constant k ∈ 0, 1 such that   Tx − Ty   2 ≤   x − y   2  k   I − Tx − I − Ty   2 , 1.7 for all x, y ∈ K.Ifk  0, then the mapping T is nonexpansive. Apointx ∈ K is called afixed point of T if Tx  x.ThefixedpointssetofT is the set FT : {x ∈ K : Tx  x}. Iterative approximation of fixed points of k-strictly pseudocontractive mappings have been studied extensivelybymanyauthorssee, e.g., 1, 6–9 and the references contained therein. Let ϕ : K → be a real-valued function and A : K → H a nonlinear mapping. Suppose F : K × K into is an equilibrium bifunction. That is, Fu, u0, forall u ∈ K.The generalized mixed equilibrium problem is to find x ∈ K see, e.g., 10–12 such that F  x, y   ϕ  y  − ϕ  x    Ax, y − x  ≥ 0, 1.8 for all y ∈ K. We shall denote the set of solutions of this generalized mixed e quilibrium problem by GMEPF, A, ϕ.Thus, GMEP  F, A, ϕ  :  x ∗ ∈ K : F  x ∗ ,y   ϕ  y  − ϕ  x ∗    Ax ∗ ,y− x ∗  ≥ 0, ∀y ∈ K  . 1.9 Fixed Point Theory and Applications 3 If ϕ  0,A  0, then problem 1.8 reduces to equilibrium problem studied by many authors see, e.g., 8, 13–17,whichistofindx ∗ ∈ K such that F  x ∗ ,y  ≥ 0, 1.10 for all y ∈ K. The set of solutions of 1.10 is denoted by EPF. If ϕ  0, then problem 1.8 reduces to generalized equilibrium problem studied by many authors see, e.g., 18–20,whichistofindx ∗ ∈ K such that F  x ∗ ,y    Ax ∗ ,y− x ∗  ≥ 0, 1.11 for all y ∈ K. The set of solutions of 1.11 is denoted by EP. If A  0, then problem 1.8 reduces to mixed equilibrium problem considered by many authors see, e.g., 21–23,whichistofindx ∗ ∈ K such that F  x ∗ ,y   ϕ  y  − ϕ  x ∗  ≥ 0, 1.12 for all y ∈ K. The set of solutions of 1.12 is denoted by MEP. The generalized mixed equilibrium problems include fixed-point problems, optimiza- tion problems, variational inequality problems, Nash equilibrium problems, and equilibrium problems as special cases see, e.g., 24. Numerous problems in Physics, optimization, and economics reduce to find a solution of problem 1.8. S everal methods have been proposed to solve the fixed-point problems, variational inequality problems and equilibrium problems in the literature see, e.g., 5, 11, 12, 20, 25–30. Recently, Ceng and Yao 25 introduced a new iterative scheme of approximating a common element of the set of solutions to mixed equilibrium problem and set of common fixed points of finite family of nonexpansive mappings in a real Hilbert space H.Intheir results, they imposed the following condition on a nonempty closed and convex subset K of H: E A : K → is η-strongly convex and its derivative A  is sequentially continuous from weak topology to the strong topology. We remark here that this condition E has been used by many authors for approximation of solution to mixed equilibrium problem in a real Hilbert space see, e.g., 31, 32. However, it is observed that the condition E does not include the case Axx 2 /2andηx, yx − y.Furthermore,PengandYao21, R. Wangkeeree and R. Wangkeeree 30, and many other authors replaced condition E with the following conditions: B1 for each x ∈ H and r>0, there exists a bounded subset D x ⊆ K and y x ∈ K such that for any z ∈ K \ D x , F  z, y x   ϕ  y x  − ϕ  z   1 r  y x − z, z − x  < 0, 1.13 or B2 K is a bounded set. 4 Fixed Point Theory and Applications Consequently, conditions B1 and B2 have been used by many authors in approximating solution to generalized mixed equilibrium mixed equilibrium problems in a real Hilbert space see, e.g., 21, 30. Recently, Takahashi et al. 33 proved the following convergence theorem using hybrid method. Theorem 1.1 Takahashi et al. 33. Let K be a nonempty closed and convex subset of a real Hilbert space H.LetT be a nonexpansive m apping of K into itself such that FT /  ∅.ForC 1  K, x 1  P C 1 x 0 , define sequences {x n } ∞ n0 and {y n } ∞ n1 of K as follows: y n  α n x n   1 − α n  Tx n ,n≥ 1, C n1   z ∈ C n :   y n − z   ≤  x n − z   ,n≥ 1, x n1  P C n1 x 0 ,n≥ 1. 1.14 Assume that {α n } ∞ n1 ⊂ 0, 1 satisfies 0 ≤ α n <α<1.Then,{x n } ∞ n0 converges strongly to P FT x 0 . Motivated by the results of Takahashi et al. 33,Kumam28 studied the problem of approximating a common element of set of solutions to an equilibrium problem, set of solutions to variational inequality problem and the set of fixed points of a nonexpansive mapping in a real Hilbert space. In particular, he proved the following theorem. Theorem 1.2 Kumam, 28. Let K be a nonempty closed convex subset of a real Hilbert space H.LetF be a bifunction from K × K satisfying (A1)–(A4) and let B be a β-inverse-strongly monotone mapping of K into H.LetT be a nonexpansive m apping of K into H such that FT  EPF  VIK, B /  ∅.ForC 1  K, x 1  P C 1 x 0 , define sequences {x n } ∞ n0 and {z n } ∞ n1 of K as follows: F  z n ,y   1 r n  y − z n ,z n − x n  ≥ 0, ∀y ∈ K, y n  α n x n   1 − α n  TP K  z n − λ n Bz n  ,n≥ 1, C n1   z ∈ C n :   y n − z   ≤  x n − z   ,n≥ 1, x n1  P C n1 x 0 ,n≥ 1. 1.15 Assume that {α n } ∞ n1 ⊂ 0, 1, {r n } ∞ n1 ⊂ 0, ∞ and {λ n } ∞ n1 ⊂ 0, 2β satisfy lim inf n →∞ r n > 0, 0 <c≤ λ n ≤ f<2β, lim n →∞ α n  0. 1.16 Then, {x n } ∞ n0 converges strongly to P FT  EPF  VIK,B x 0 . Motivated by the ongoing research and the above-mentioned results, we introduce a new iterative scheme for finding a common element of the set of fixed points of an infinite family of k-strictly pseudocontractive mappings, the set of common solutions to a system of generalized mixed equilibrium pr oblems and the set of solutions to a variational inequality problem in a real Hilbert space. Furthermore, w e show that our new iterative scheme converges strongly to a common Fixed Point Theory and Applications 5 element of the three afore mentioned sets. In our results, we use conditions B1 and B2 mentioned above. Our result extends many important recent results. Finally, we give some applications of our results. 2. Preliminaries Let H be a real Hilbert space with inner product ·, · and norm ·and let K be a nonempty closed and convex s ubset of H. The strong convergence of {x n } ∞ n0 to x is denoted by x n → x as n →∞. For any point u ∈ H, there exists a unique point P K u ∈ K such that  u − P K u  ≤   u − y   , ∀y ∈ K. 2.1 P K is called the metric projection of H onto K. We know that P K is a nonexpansive mapping of H onto K.ItisalsoknownthatP K satisfies  x − y, P K x − P K y  ≥   P K x − P K y   2 , 2.2 for all x, y ∈ H.Furthermore,P K x is characterized by the properties P K x ∈ K and x − P K x, P K x − y≥0, 2.3 for all y ∈ K and  x − P K x  2 ≤   x − y   2 −   y − P K x   2 , ∀x ∈ H, y ∈ K. 2.4 In the context of the variational inequality problem, 2.3 implies that x ∗ ∈ VI  A, K  ⇐⇒ x ∗  P K  x ∗ − λAx ∗  , ∀λ>0. 2.5 If A is α-inverse-strongly monotone mapping of K into H,thenitisobviousthatA is 1/α- Lipschitz continuous. We also have that for all x, y ∈ K and r>0,   I − rAx − I − rAy   2    x − y − rAx − Ay   2    x − y   2 − 2r  Ax − Ay, x − y   r 2   Ax − Ay   2 ≤   x − y   2  r  r − 2α    Ax − Ay   2 . 2.6 So, if r ≤ 2α,thenI − rA is a nonexpansive mapping of K into H. For solving the generalized mixed equilibrium problem for a bifunction F : K × K → , let us assume that F satisfies the following conditions: A1 Fx, x0forallx ∈ K, 6 Fixed Point Theory and Applications A2 F is monotone, that is, Fx, yFy, x ≤ 0forallx, y ∈ K, A3 for each x, y, z ∈ K, lim t → 0 Ftz 1 − tx, y ≤ Fx, y, A4 for each x ∈ K, y → Fx, y is convex and lower semicontinuous. We need the following technical result. Lemma 2.1 R. Wangkeeree and R. Wangkeeree 30. Assume that F : K × K → satisfies (A1)–(A4) and let ϕ : K → 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,ϕ r : H → K as follows: T F,ϕ r  x    z ∈ K : F  z, y   ϕ  y  − ϕ  z   1 r  y − z, z − x  ≥ 0, ∀y ∈ K  2.7 for all z ∈ H. Then, the following hold: 1 for each x ∈ H, T F,ϕ r /  ∅, 2 T F,ϕ r is single-valued, 3 T F,ϕ r is firmly nonexpansive, that is, for any x, y ∈ H,    T F,ϕ r x − T F,ϕ r y    ≤  T F,ϕ r x − T F,ϕ r y, x − y  , 2.8 4 FT F,ϕ r MEPF, 5 MEPF is closed and convex. 3. Main Results Theorem 3.1. Let K be a nonempty closed and convex subset of a real Hilbert space H.Foreach m  1, 2,letF m be a bifunction from K × K satisfying (A1)–(A4), ϕ m : K → ∪{∞} aproper lower semicontinuous and convex function with assumption (B1)or(B2), A an α-inverse-strongly monotone mapping of K into H, B a β-inverse-strongly monotone mapping of K into H and for each i  1, 2, ,letT i : K → K be a k i -strictly pseudocontractive mapping for some 0 ≤ k i < 1 such that ∩ ∞ i1 FT i  /  ∅.LetD be a μ-Lipschitzian, relaxed λ, γ-cocoercive mapping of K into H. Suppose Ω : ∩ ∞ i1 FT i ∩GMEPF 1 ,A,ϕ 1 ∩GMEPF 2 ,B,ϕ 2 ∩VIK, D /  ∅.Let{z n } ∞ n1 , {u n } ∞ n1 , Fixed Point Theory and Applications 7 {w n } ∞ n1 , {y n,i } ∞ n1 i  1, 2,  and {x n } ∞ n0 be generated by x 0 ∈ K, C 1,i  K, C 1  ∩ ∞ i1 C 1,i , x 1  P C 1 x 0 z n  T F 1 ,ϕ 1  r n  x n − r n Ax n  , u n  T F 2 ,ϕ 2  λ n  z n − λ n Bz n  , w n  P K  u n − s n Du n  , y n,i  α n,i w n   1 − α n,i  T i w n , C n1,i   z ∈ C n,i :   y n,i − z   ≤  x n − z   , C n1  ∩ ∞ i1 C n1,i , x n1  P C n1 x 0 ,n≥ 1. 3.1 Assume that {α n,i } ∞ n1 ⊂ 0, 1i  1, 2, , {r n } ∞ n1 ⊂ 0, 2α and {λ n } ∞ n1 ⊂ 0, 2β satisfy i 0 <a≤ r n ≤ b<2α, ii 0 <c≤ λ n ≤ f<2β, iii 0 ≤ k i ≤ α n,i ≤ d i < 1, iv 0 <h≤ s n ≤ j<2γ − λμ 2 /μ 2 . Then, {x n } ∞ n0 converges strongly to P Ω x 0 . Proof. For all x, y ∈ K and s n ∈ 0, 2γ − λμ 2 /μ 2 ,weobtain   I − s n Dx − I − s n Dy   2    x − y − s n Dx − Dy   2    x − y   2 − 2s n  x − y, Dx − Dy   s 2 n   Dx − Dy   2 ≤   x − y   2 − 2s n  −λ   Dx − Dy   2  γ   x − y   2   s 2 n   Dx − Dy   2 ≤   x − y   2  2s n μ 2 λ   x − y   2 − 2s n γ   x − y   2  μ 2 s 2 n   x − y   2   1  2s n μ 2 λ − 2s n γ  μ 2 s 2 n    x − y   2 ≤   x − y   2 . 3.2 This shows that I − s n D is nonexpansive for each n ≥ 1. Let x ∗ ∈ Ω.Then  w n − x ∗  2   P K u n − s n Du n  − P K x ∗ − s n Dx ∗   2 ≤  u n − s n Du n  − x ∗ − s n Dx ∗   2 ≤  u n − x ∗  2 . 3.3 8 Fixed Point Theory and Applications Since both I−r n A and I−λ n B are nonexpansive for each n ≥ 1andx ∗  T F 1 ,ϕ 1  r n x ∗ −r n Ax ∗ ,x ∗  T F 2 ,ϕ 2  λ n x ∗ − λ n Bx ∗ ,from2.6,wehave  u n − x ∗  2     T F 2 ,ϕ 2  λ n z n − λ n Bz n  − x ∗    2     T F 2 ,ϕ 2  λ n z n − λ n Bz n  − T F 2 ,ϕ 2  λ n x ∗ − λ n Bx ∗     2 ≤  I − λ n Bz n −  I − λ n B  x ∗  2 ≤  z n − x ∗  2  λ n  λ n − 2β   Bz n − Bx ∗  2 ≤  z n − x ∗  2  since λ n < 2β, ∀n ≥ 1  ,  z n − x ∗  2     T F 1 ,ϕ 1  r n x n − r n Ax n  − x ∗    2     T F 1 ,ϕ 1  r n x n − r n Ax n  − T F 1 ,ϕ 1  r n x ∗ − r n Ax ∗     2 ≤  I − r n Ax n − I − r n Ax ∗  2 ≤  x n − x ∗  2  r n  r n − 2α  Ax n − Ax ∗  2 ≤  x n − x ∗  2 . 3.4 Therefore,  u n − x ∗  ≤  x n − x ∗  . 3.5 Let n  1, then C 1,i  K is closed convex for each i  1, 2, Now assume that C n,i is closed convex for some n>1. Then, from definition of C n1,i , we know that C n1,i is closed convex for the same n>1. Hence, C n,i is closed convex for n ≥ 1andforeachi  1, 2, This implies that C n is closed convex for n ≥ 1. Furthermore, we show that Ω ⊂ C n .Forn  1, Ω ⊂ K  C 1,i . For n ≥ 2, let x ∗ ∈ Ω. Then,   y n,i − x ∗   2  α n,i  w n − x ∗  2   1 − α n,i  T i w n − x ∗  2 − α n,i  1 − α n,i  T i w n − w n  2 ≤ α n,i  w n − x ∗  2   1 − α n,i    w n − x ∗  2  k i  T i w n − w n  2  − α n,i  1 − α n,i  T i w n − w n  2   w n − x ∗  2   1 − α n,i  k i − α n,i  T i w n − w n  2 ≤  u n − x ∗  2 ≤  x n − x ∗  2 , 3.6 Fixed Point Theory and Applications 9 which shows that x ∗ ∈ C n,i ,foralln ≥ 2, for all i  1, 2, Thus,Ω ⊂ C n,i ,foralln ≥ 1, for all i  1, 2, Hence, it follows that ∅ / Ω ⊂ C n ,foralln ≥ 1. Therefore, {x n } ∞ n0 is well defined. Since x n  P C n x 0 ,foralln ≥ 1andx n1 ∈ C n1 ⊂ C n ,foralln ≥ 1, we have  x n − x 0  ≤  x n1 − x 0  , ∀n ≥ 1. 3.7 Also, as Ω ⊂ C n ,by2.1 it follows that  x n − x 0  ≤  v − x 0  ,v∈ Ω, ∀n ≥ 1. 3.8 From 3.7 and 3.8, we have that lim n →∞ x n − x 0  exists. Hence, {x n } ∞ n0 is bounded and so are {z n } ∞ n1 , {Ax n } ∞ n1 , {u n } ∞ n1 , {Du n } ∞ n1 , {Bz n } ∞ n1 , {w n } ∞ n1 , {T i w n } ∞ n1 and {y n,i } ∞ n1 , i  1, 2, Form>n≥ 1, we have that x m  P C m x 0 ∈ C m ⊂ C n .By2.4,weobtain  x m − x n  2 ≤  x m − x 0  2 −  x n − x 0  2 . 3.9 Letting m, n →∞and taking the limit in 3.9,wehavex m − x n → 0, m, n →∞,which shows that {x n } ∞ n0 is Cauchy. In particular, lim n →∞ x n1 − x n   0. Since, {x n } ∞ n0 is Cauchy and K is closed, there exists z ∈ K such that x n → z, n →∞.Sincex n1  P C n1 x 0 ∈ C n1 , therefore   y n,i − x n1   ≤  x n − x n1  , 3.10 and it follows that   y n,i − x n   ≤   y n,i − x n1     x n − x n1  ≤ 2  x n − x n1  . 3.11 Thus, lim n →∞   y n,i − x n    0,i 1, 2, 3.12 10 Fixed Point Theory and Applications Furthermore,   y n,i − x ∗   2  α n,i  w n − x ∗  2   1 − α n,i  T i w n − x ∗  2 − α n,i  1 − α n,i  T i w n − w n  2 ≤ α n,i  w n − x ∗  2   1 − α n,i    w n − x ∗  2  k i  T i w n − w n  2  − α n,i  1 − α n,i  T i w n − w n  2  α n,i  w n − x ∗  2   1 − α n,i  w n − x ∗  2 −  1 − α n,i  α n,i − k i  T i w n − w n  2 ≤ α n,i  u n − x ∗  2   1 − α n,i  u n − x ∗  2 ≤ α n,i  u n − x ∗  2   1 − α n,i     T F 2 ,ϕ 2  λ n z n − λ n Bz n  − T F 2 ,ϕ 2  λ n x ∗ − λ n Bx ∗     2 ≤ α n,i  u n − x ∗  2   1 − α n,i  z n − λ n Bz n  − x ∗ − λ n Bx ∗   2 ≤ α n,i  u n − x ∗  2   1 − α n,i    z n − x ∗  2  λ n  λ n − 2β   Bz n − Bx ∗  2  ≤  u n − x ∗  2   1 − α n,i  λ n  λ n − 2β   Bz n − Bx ∗  2 ≤  x n − x ∗  2   1 − α n,i  λ n  λ n − 2β   Bz n − Bx ∗  2 . 3.13 Since 0 <c≤ λ n ≤ f<2 β,0≤ k i ≤ α n,i ≤ d i < 1, we have  1 − d i  c  2β − f   Bz n − Bx ∗  2 ≤  x n − x ∗  2 −   y n,i − x ∗   2 ≤   y n,i − x n     x n − x ∗     y n,i − x ∗    . 3.14 Hence, lim n →∞ Bz n − Bx ∗   0. 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Kumam, “A hybrid projection method for generalized. x ∗ − s n Dx ∗   2 ≤  u n − x ∗  2 . 3.3 8 Fixed Point Theory and Applications Since both I−r n A and I−λ n B are nonexpansive for each n ≥ 1andx ∗  T F 1 ,ϕ 1  r n x ∗ −r n Ax ∗ ,x ∗  T F 2 ,ϕ 2  λ n x ∗ −