Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009, Article ID 362191, 21 pages doi:10.1155/2009/362191 Research Article Strong Convergence of an Iterative Method for Equilibrium Problems and Variational Inequality Problems HongYu Li 1 and HongZhi Li 2 1 Department of Mathematics, TianJin Polytechnic University, TianJin 300160, China 2 Department of Mathematics, Agricultural University of Hebei, BaoDing 071001, China Correspondence should be addressed to HongYu Li, lhy x1976@eyou.com Received 26 August 2008; Revised 11 November 2008; Accepted 9 January 2009 Recommended by Massimo Furi We introduce an iterative method for finding a common element of the set of solutions of equilibrium problems, the set of solutions of variational inequality problems, and the set of fixed points of finite many nonexpansive mappings. We prove strong convergence of the iterative sequence generated by the proposed iterative algorithm to the unique solution of a variational inequality, which is the optimality condition for the minimization problem. Copyright q 2009 H. Li and H. Li. 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. 1. Introduction Let H be a real Hilbert space with inner product ·, ·and norm ·, respectively. Suppose that C is nonempty, closed convex subset of H and F is a bifunction from C × C to R, where R is the set of real number. The equilibrium problem is to find a x ∈ C such that Fx, y ≥ 0, ∀y ∈ C. 1.1 The set of such solutions is denoted by EPf. Numerous problems in physics, optimization, and economics reduce to find a solution of equilibrium problem. Some methods have been proposed to solve the equilibrium problems in Hilbert space, see, for instance, Blum and Oettli 1, Combettes and Hirstoaga 2, and Moudafi 3. A mapping A : C → H is called monotone if Au − Av, u − v≥0. A is called relaxed u, v-cocoercive, if there exist constants u>0andv>0 such that Ax − Ay, x − y≥−uAx − Ay 2  vx − y 2 , ∀x, y ∈ C, 1.2 2 Fixed Point Theory and Applications when u  0, A is called v-strong monotone; when v  0, A is called relaxed u-cocoercive. Let A : C → H be a monotone operator, the variational inequality problem is to find a point u ∈ C, such that Au, v − u≥0, ∀v ∈ C. 1.3 The set of solutions of variational inequality problem is denoted by VIC, A. The variational inequality problem has been extensively studied in literatures, see, for example, 4, 5 and references therein. Let B be a strong positive bounded linear operator on H with coefficient γ,thatis, there exists a constant γ>0 such that Bx, x≥γx 2 , for all x ∈ H. A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space H: min x∈FT Ax, x−x, b, 1.4 where T is a nonexpansive mapping on H and b is a point on H. A mapping T from C into itself is called nonexpansive, if Tx−Ty≤x−y, ∀x, y ∈ C. The set of fixed points of T is denoted by FT.Let{T i } N i1 be a finite family of nonexpansive mappings and F   N i1 FT i  /  ∅, define the mappings U n,1  λ n,1 T 1   1 − λ n,1  I, U n,2  λ n,2 T 2 U n,1   1 − λ n,2  I, . . . U n,N−1  λ n,N−1 T N−1 U n,N−2   1 − λ n,N−1  I, W n  U n,N  λ n,N T N U n,N−1   1 − λ n,N  I, 1.5 where {λ n,i } N i1 ⊂ 0, 1 for all n ≥ 1. Such a mapping W n is called W-mapping generated by T 1 ,T 2 , ,T N and {λ n,i } N i1 . We know that W n is nonexpansive and FW n   N i1 FT i ,see6. Let S : C → C be a nonexpansive mapping and f : C → C is a contractive with coefficient α ∈ 0, 1.MarinoandXu7 considered the following general iterative scheme: x n1  α n γf  x n    1 − α n B  Sx n . 1.6 They proved that {x n } converges strongly to z  P FS I − B γfz, where P FS is the metric projection from H onto FS. Fixed Point Theory and Applications 3 By combining equilibrium problems and 1.6, Plutbieng and Pumpaeng 8 proposed the following algorithm: F  u n ,y   1 r n  y − u n ,u n − x n  ≥ 0, ∀y ∈ H, x n1  α n γf  x n    I − α n B  Su n . 1.7 They proved that if the sequences {α n } and {r n } satisfy some appropriate conditions, then sequence {x n } convergence to the unique solution z of the variational inequality  B − γfz, x − z  ≥ 0, ∀x ∈ FS ∩ EPF. 1.8 Motivated by 8, Colao et al. 9 introduced an iterative method for equilibrium problem and finite family of nonexpansive mappings F  u n ,y   1 r n  y − u n ,u n − x n  ≥ 0, ∀y ∈ H, x n1  α n γf  x n   βx n   1 − βI − α n B  W n u n , 1.9 and proved that {x n } converges strongly to a point x ∗ ∈ F ∩ EPF and x ∗ also solves the variational inequality 1.8. For equilibrium problems, also see 10, 11. On the other hand, let A : C → C be a α-cocoercive mapping, for finding common element of the solution of variational inequality problems and the set of fixed point of nonexpansive mappings, Takahashi and Toyoda 12 introduced iterative scheme x n1  α n x n   1 − α n  SP C  I − λ n A  x n . 1.10 They proved that {x n } converges weakly to z ∈ FS ∩ VIC, A. Inspired by 1.10 and 13, Y. Yao and J C. Yao 14 given the following iterative process: y n  P C  I − λ n A  x n , x n1  α n u  β n x n  γ n SP C  I − λ n A  y n , 1.11 and proved that {x n } converges strongly to z ∈ FS ∩ VIC, A. By combining viscosity approximation method and 1.10, Chen et al. 15 introduced the process x n1  α n f  x n   β n SP C  I − λ n A  x n , 1.12 4 Fixed Point Theory and Applications and studied the strong convergence of sequence {x n } generated by 1.12. Motivated by 1.6, 1.11,and1.12, Qin et al. 16 introduced the following general iterative process y n  P C  I − s n A  x n , x n1  α n γf  W n x n    I − α n B  W n P C  I − t n A  y n , 1.13 and established a strong convergence theorem of {x n } to an element of  N i1 FT i  ∩ VIC, A. The purpose of this paper is to introduce the iterative process: x 1 ∈ H and Fu n ,y 1 r n  y − u n ,u n − x n  ≥ 0, ∀y ∈ C, y n  b n u n   1 − b n  W n P C  I − s n A  u n , x n1  α n γf  W n x n   βx n   1 − βI − α n B  W n P C  I − t n A  y n , 1.14 where W n is defined by 1.5, A is u, v-cocoercive, and B is a bounded linear operator. We should show that the sequences {x n } converge strongly to an element of  N i1 FT i ∩VIC, A∩ EPF. Our result extends the corresponding results of Qin et al. 16 and Colao et al. 9,and many others. 2. Preliminaries Let H be a real Hilbert space and C a nonempty, closed convex subset of H. We denote strong convergence of {x n } to x by x n → x and weak convergence by x n x.LetP C : C → H is a mapping such that for every point x ∈ H, there exists a unique P C x ∈ C satisfying x − P C x≤x − y, for all y ∈ C. P C is called the metric projection of H onto C. It is known that P C is a nonexpansive mapping from H onto C. It is also known that P C x ∈ C and  x − P C x, P C x − y  ≥ 0, ∀x ∈ H, y ∈ C, 2.1  x − y, P c x − P C y  ≥   P C x − P C y   2 , ∀x, y ∈ H. 2.2 Let A : C → H be a monotone mapping of C into H, then u ∈ VIC, A if and only if u  P C u − λAu, for all λ>0. The following result is useful in the rest of this paper. Lemma 2.1 see 17. Assume {a n } is a sequence of nonegative real number such that a n1 ≤  1 − α n  a n  δ n , ∀n ≥ 0, 2.3 where {α n } is a sequence in 0, 1, and {δ n } is a sequence in R such that 1  ∞ n0 α n  ∞, 2 lim sup n →∞ δ n /α n  ≤ 0 or  ∞ n0 |δ n | < ∞. Then, lim n →∞ a n  0. Fixed Point Theory and Applications 5 Lemma 2.2 see 18. Let {x n }, {u n } be bounded sequences in Banach space E satisfying x n1  τ n x n 1 − τ n u n for all n ≥ 0 and lim inf n →∞ u n1 − u n −x n1 − x n  ≤ 0.Letτ n be a sequence in 0, 1 with 0 < lim inf n →∞ τ n ≤ lim sup n →∞ τ n < 1. Then, lim n →∞ x n − u n   0. Lemma 2.3. For all x, y ∈ H, there holds the inequality x  y≤x 2  2y, x  y. 2.4 Lemma 2.4 see 7. Assume that A is a strong positive linear bounded operator on a Hilbert space H with coefficient γ>0 and 0 <ρ≤A −1 .ThenI − ρA≤1 − ργ. For solving the equilibrium problem for a bifunction F : C × C → R, we assume that F satisfies the following conditions: A1 Fx, x0 for all x ∈ C; A2 F is monotone: Fx, yFy, x ≤ 0 for all x, y ∈ C; A3 for all x, y, z ∈ C, lim sup t↓0 Ftz 1 − tx, y ≤ Fx, y; A4 for all x ∈ C, Fx, ·  is convex and lower semicontinuous. The following result is in Blum and Oettli 1. Lemma 2.5 see 1. Let C be a nonempty closed convex subset of a Hilbert space E,letF be a bifunction from C × C into R satisfying (A1)–(A4), let r>0, and let x ∈ H. Then there exists z ∈ C such that Fz, y 1 r y − z, z − x≥0, ∀y ∈ C. 2.5 We also know the following lemmas. Lemma 2.6 see 19. Let C be a nonempty closed convex subset of Hilbert space H,letF be a bifunction from C × C to R satisfying (A1)–(A4), let r>0, and let 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  , 2.6 for all x ∈ H. Then, the following holds: 1 T r is single-valued; 2 T r is firmly nonexpansive-type mapping, that is, for all x, y ∈ H,   T r x − T r y   2 ≤  T r x − T r y, x − y  ; 2.7 3 FT r EPF; 4 EPF is closed and convex. 6 Fixed Point Theory and Applications A monotone operator T : H → 2 H is said to be maximal monotone if its graph GT {u, v : v ∈ Tu} is not properly contained in the graph of any other monotone operators. Let A be a monotone mapping of C into H and let N C v be the normal cone for C at a point v ∈ C,thatis N C v  x ∈ H : v − y, x≥0, ∀y ∈ C  . 2.8 Define Tv  ⎧ ⎨ ⎩ Av  N C v,v∈ C, ∅,v / ∈ C. 2.9 It is known that in this case T is maximal monotone, and 0 ∈ Tv if and only if v ∈ VIC, A. 3. Strong Convergence Theorem Theorem 3.1. Let H be a real Hilbert space and C be a nonempty closed convex subset of H. {T i } N i1 a finite family of nonexpansive mappings from C into itself and F : C × C → R a bifunction satisfying (A1)–(A4). Let A : C → H be relaxed u, v-cocoercive and μ-Lipschitzian. Let f : C → C be an α-contraction with 0 ≤ α<1 and B a strong positive linear bounded operator with coefficient γ>0, γ is a constant with 0 <γ< γ/α. Let sequences {α n }, {b n } be in 0, 1 and {r n } be in 0, ∞, β is a constant in 0, 1. Assume C 0   N i1 FT i  ∩ VIC, A ∩ EPF /  ∅ and i lim n →∞ α n  0,  ∞ n1 α n  ∞; ii lim n →∞ |r n1 − r n |  0, lim inf n →∞ r n > 0; iii {s n }, {t n }∈a, b for some a, b with 0 ≤ a ≤ b ≤ 2v − uμ 2 /μ 2 and lim n →∞ |s n1 −s n |  lim n →∞ |t n1 − t n |  0; iv lim n →∞ |λ n1 − λ n |  lim n →∞ |b n1 − b n |  0. Then the sequence {x n } generated by 1.14 converges strongly to x ∗ ∈ C 0 and x ∗ solves the variational inequality x ∗  P C 0 I − B − γfx ∗ , that is,  γfx ∗ − Bx ∗ ,x− x ∗  ≤ 0, ∀x ∈ C 0 . 3.1 Proof. Without loss of generality, we can assume α n ≤ 1 − βB −1 . Then from Lemma 2.4 we know   1 − βI − α n B   1 − β     I − α n 1 − β B     ≤ 1 − β  1 − α n 1 − β γ   1 − β − α n γ. 3.2 Since A is relaxed u, v-cocoercive and μ-Lipschitzian and iii holds, we know from 14 that for all x, y ∈ C and n ≥ 1, the following holds:    I − s n A  x −  I − s n A  y   ≤x − y,    I − t n A  x −  I − t n A  y   ≤x − y. 3.3 Fixed Point Theory and Applications 7 We divide the proof into several steps. Step 1. {x n } is bounded. Take p ∈ C 0 ,noticethatu n  T r n x n and form Lemma 2.62 that T r n is nonexpansive, we have   u n − p      T r n x n − T r n p   ≤   x n − p   . 3.4 Since p  W n P C p − s n Ap, we have   y n − p      b n u n   1 − b n  W n P C  u n − s n Au n  − p   ≤ b n   u n − p     1 − b n    W n P C  u n − s n Au n  − p   ≤ b n   u n − p     1 − b n    u n − s n Au n −  p − s n Ap    ≤ b n   u n − p     1 − b n    u n − p   ≤   x n − p   . 3.5 Then we have   x n1 − p      α n γf  W n x n   βx n   1 − βI − α n B  W n P C  I − t n A  y n − p      α n  γf  W n x n  − Bp   β  x n − p    1 − βI − α n B  W n  P C  I − t n A  y n − p     α n   γf  W n x n  − Bp    β   x n − p     1 − β − α n γ    y n − p   . 3.6 Thus From 3.5 we have   x n1 − p   ≤ α n γ   f  W n x n  − f  p     α n   γfp − Bp    β   x n − p     1 − β − α n γ    x n − p   ≤ α n γα   x n − p    α n   γfp − Bp   1 − α n γ   x n − p     1 − α n γ − αγ    x n − p    α n   γfp − Bp     1 − α n γ − αγ    x n − p    α n γ − αγ ·   γf  p  − Bp   γ − αγ ≤ max    x n − p   ,   γf  p  − Bp   γ − αγ  ≤ max    x 1 − p   ,   γf  p  − Bp   γ − αγ  , 3.7 hence {x n } is bounded, so is {u n }, {y n }. 8 Fixed Point Theory and Applications Step 2. lim n →∞ x n1 − x n   0. Let x n1  βx n 1 − βz n , for all n ≥ 0, where z n  1 1 − β  α n γf  W n x n    1 − βI − α n B  W n P C  y n − t n Ay n  . 3.8 Then we have   z n1 − z n    1 1 − β   γ  α n1 f  W n1 x n1  − α n f  W n x n    1 − βI − α n1 B  W n1 P C  y n1 − t n1 Ay n1  −  1 − βI − α n B  W n P C  y n − t n Ay n         γ 1 − β  α n1 f  W n1 x n1  − α n f  W n x n    W n1 P C  y n1 − t n1 Ay n1  − W n P C  y n − t n Ay n  − 1 1 − β  α n1 BW n1 P C  y n1 − t n1 Ay n1  − α n BW n P C  y n − t n Ay n      ≤   W n1 P C  y n1 − t n1 Ay n1  − W n1 P C  y n − t n1 Ay n       W n1 P C  y n − t n1 Ay n  − W n1 P C  y n − t n Ay n       W n1 P C  y n − t n Ay n  − W n P C  y n − t n Ay n     K 1 , 3.9 where K 1  α n1 1 − β  γ   f  W n1 x n1       BW n1 P C  y n1 − t n1 Ay n1      α n 1 − β  γ   f  W n x n       BW n P C  y n − t n Ay n     . 3.10 Next we estimate W n1 P C y n1 −t n1 Ay n1 −W n1 P C y n −t n1 Ay n , W n1 P C y n −t n1 Ay n − W n1 P C y n − t n Ay n  and W n1 P C y n − t n Ay n  − W n P C y n − t n Ay n .Atfirst   W n1 P C  y n − t n1 Ay n  − W n1 P C  y n − t n Ay n    ≤   t n Ay n − t n1 Ay n      t n1 − t n   ·   Ay n   . 3.11 Fixed Point Theory and Applications 9 Put v n  P C y n − t n Ay n , we have   W n1 P C  y n − t n Ay n  − W n P C  y n − t n Ay n       W n1 v n − W n v n      U n1,N v n − U n,N v n      λ n1,N T N U n1,N−1 v n   1 − λ n1,N  v n − λ n,N T N U n,N−1 v n −  1 − λ n,N  v n   ≤   λ n1,N T N U n1,N−1 v n − λ n,N T N U n,N−1 v n      λ n1,N − λ n,N   ·   v n   ≤ λ n1,N   U n1,N−1 v n − U n,N−1 v n      λ n1,N − λ n,N   ·   T N U n,N−1 v n      λ n1,N − λ n,N   ·   v n   ≤   U n1,N−1 v n − U n,N−1 v n      λ n1,N − λ n,N      T N U n,N−1 v n      v n    . 3.12 By recursion we get   W n1 P C  y n − t n Ay n  − W n P C  y n − t n Ay n    ≤ M · N  i1   λ n1,i − λ n,i   , 3.13 for some M>0. Similarly, we also get   W n1 P C  u n − s n Au n  − W n P C  u n − s n Au n    ≤ M · N  i1   λ n1,i − λ n,i   . 3.14 Since F  u n ,y   1 r n  y − u n ,u n − x n  ≥ 0, ∀y ∈ C, F  u n1 ,y   1 r n1  y − u n1 ,u n1 − x n1  ≥ 0, ∀y ∈ C. 3.15 Put y  u n1 in the first inequality and y  u n in the second one, we have F  u n ,u n1   1 r n  u n1 − u n ,u n − x n  ≥ 0, F  u n1 ,u n   1 r n1  u n − u n1 ,u n1 − x n1  ≥ 0. 3.16 Adding both inequality, by A2 we have  u n1 − u n , 1 r n  u n − x n  − 1 r n1  u n1 − x n1   ≥ 0, 3.17 10 Fixed Point Theory and Applications therefore, we have  u n1 − u n ,u n − u n1   x n1 − x n   r n1 − r n r n1  u n1 − x n1   ≥ 0, 3.18 which implies that   u n1 − u n   2 ≤  u n1 − u n ,  x n1 − x n   r n1 − r n r n1  u n1 − x n1   ≤   u n1 − u n   ·    x n1 − x n    |r n1 − r n | r n1   u n1 − x n1    . 3.19 Hence we have   u n1 − u n   ≤   x n1 − x n    |r n1 − r n | r n1   u n1 − x n1   , 3.20 so, by 3.20 and the property I − t n Ax − I − t n Ay≤x − y, we arrive at   W n1 P C  y n1 − t n1 Ay n1  − W n1 P C  y n − t n1 Ay n    ≤   y n1 − y n      b n1 u n1   1 − b n1  W n1 P C  I − s n1 A  u n1 − b n u n −  1 − b n  W n P C  I − s n A  u n      b n1  u n1 − u n    b n1 − b n  u n   1 − b n1  W n1 P C  u n1 − s n1 Au n1  − W n1 P C  u n − s n1 Au n    1 − b n1  W n1 P C  u n − s n1 Au n  − W n1 P C  u n − s n Au n  ×  1 − b n1  W n1 P C  u n − s n Au n  − W n P C  u n − s n Au n    b n − b n1  W n P C  u n − s n Au n    ≤ b n1   u n1 − u n    |b n1 − b n |·   u n     1 − b n1    u n1 − u n     1 − b n1  |s n − s n1 |·   Au n     1 − b n1    W n1 P C  u n − s n Au n  − W n P C  u n − s n Au n       b n − b n1   ·   W n P C  u n − s n Au n       u n1 − u n     1 − b n1    W n1 P C  u n − s n Au n  − W n P C  u n − s n Au n     K 2 , 3.21 [...]... equlibrium problems and fixed point problems in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol 336, no 1, pp 455– 469, 2007 9 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 10 Y Yao, Y.-C Liou, and J.-C Yao, Convergence. .. 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