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Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 915629, 17 pages doi:10.1155/2011/915629 Research Article Iterative Algorithms for Finding Common Solutions to Variational Inclusion Equilibrium and Fixed Point Problems J. F. Tan and S. S. Chang Department of Mathematics, Yibin University, Yibin, Sichuan 644007, China Correspondence should be addressed to S. S. Chang, changss@yahoo.cn Received 30 October 2010; Accepted 9 November 2010 Academic Editor: Qamrul Hasan Ansari Copyright q 2011 J. F. Tan and S. S. Chang. 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 main purpose of this paper is to introduce an explicit iterative algorithm to study the existence problem and the approximation problem of solution to the quadratic minimization problem. Under suitable conditions, some strong convergence theorems for a family of nonexpansive mappings are proved. The results presented in the paper improve and extend the corresponding results announced by some authors. 1. Introduction Throughout this paper, we assume that H is a real Hilbert space with inner product ·, · and norm ·, C is a nonempty closed convex subset of H,andFT{x ∈ H : Tx x} is the setoffixedpointsofmappingT. A mapping S : C → C is called nonexpansive if Sx − Sy ≤ x − y , ∀x, y ∈ C. 1.1 Let A : H → H be a single-valued nonlenear mapping and M : H → 2 H be a multivalued mapping. The so-called quasivariational inclusion problem see 1–3 is to find u ∈ H such that θ ∈ A u M u . 1.2 The set of solutions to quasivariational inclusion problem 1.2 is denoted by VIH, A, M. 2 Fixed Point Theory and Applications Special Cases I If M ∂φ : H → 2 H ,whereφ : H → ∪{∞} is a proper convex lower semi-continuous function and ∂φ is the subdifferential of φ, then the quasivariational inclusion problem 1.2 is equivalent to finding u ∈ H such that A u ,y− u φ y − φ u ≥ 0, ∀y ∈ H, 1.3 which is called the mixed quasivariational inequality see 4. II If M ∂δ C ,whereC is a nonempty closed convex subset of H and δ C : H → 0, ∞ is the indicator function of C,thatis, δ C x ⎧ ⎨ ⎩ 0,x∈ C, ∞,x / ∈ C, 1.4 then the quasivariational inclusion problem 1.2 is equivalent to finding u ∈ C such that A u ,v− u ≥ 0, ∀v ∈ C. 1.5 This problem is called the Hartman-Stampacchia variational inequality see 5.Thesetof solutions to variational inequality 1.5 is denoted by VIA, C. Let B : C → H be a nonlinear mapping and F : C × C → be a bifunction. The so-called generalized equilibrium problem is to find a point u ∈ C such that F u, y B u ,y− u ≥ 0, ∀y ∈ C. 1.6 The set of solutions to 1.6 is denoted by GEP see 5, 6.IfB 0, then 1.6 reduces to the following equilibrium problem: to find u ∈ C such that F u, y ≥ 0, ∀y ∈ C. 1.7 The set of solutions to 1.7 is denoted by EP. Iterative methods for nonexpansive mappings and equilibrium problems have been applied to solve convex minimization problems see 7–9. 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 1 2 x 2 , 1.8 where F is the fixed point set of a nonexpansive mapping T on H. In 2010, Zhang et al. see 10 proposed the following iteration method for variational inclusion problem 1.5 and equilibrium problem 1.6 in a Hilbert space H: x t SP C 1 − t J M,λ I − λA T μ I − μB x t ,t∈ 0, 1 . 1.9 Fixed Point Theory and Applications 3 Under suitable conditions, they proved the sequence {x n } generated by 1.9 converges strongly to the fixed point x ∗ , which solves the quadratic minimization problem 1.8. Motivated and inspired by the researches going on in this direction, especially inspired by Zhang et al. 10, the purpose of this paper is to introduce an explicit iterative algorithm to studying the existence problem and the approximation problem of the solution to the quadratic minimization problem 1.8 and prove some strong convergence theorems for a family of nonexpansive mappings in the setting of Hilbert spaces. 2. Preliminaries Let H be a real Hilbert space, and C be a nonempty closed convex subset of H.Foranyx ∈ H, there exists a unique nearest point in C, denoted by P C x,suchthat x − P C x ≤ x − y , ∀y ∈ C. 2.1 Such a mapping P C from H onto C is called the metric projection. It is well-known that the metric projection P C : H → C is nonexpansive. In the sequel, we use x n xand x n → x to denote the weak convergence and the strong convergence of the sequence {x n }, respectively. Definition 2 .1. A mapping A : H → H is called α-inverse strongly monotone if there exists an α>0suchthat Ax − Ay, x − y ≥ α Ax − Ay 2 , ∀x, y ∈ H. 2.2 A multivalued mapping M : H → 2 H is called monotone if ∀x, y ∈ H, u ∈ Mx, v ∈ My, u − v, x − y ≥ 0. 2.3 A multivalued mapping M : H → 2 H is called maximal monotone if it is monotone and for any x, u ∈ H × H,when u − v, x − y ≥ 0 for every y, v ∈ Graph M , 2.4 then u ∈ Mx. Proposition 2.2 see 11. Let A : H → H be an α-inverse strongly monotone mapping. Then, the following statements hold: i A is an 1/α-Lipschitz continuous and monotone mapping; ii if λ is any constant in 0, 2α, then the mapping I − λA is nonexpansive, wher e I is the identity mapping on H. 4 Fixed Point Theory and Applications Lemma 2.3 see 12. Let X be a strictly convex Banach space, C be a closed convex subset of X, and {T n : C → C} be a sequence of nonexpansive mappings. Suppose ∞ n1 FT n / ∅.Let{λ n } be a sequence of positive numbers with Σ ∞ n1 λ n 1. Then the mapping S : C → C defined by Sx Σ ∞ n1 λ n T n x, x ∈ C 2.5 is well defined. And it is nonexpansive and F S ∞ n1 F T n . 2.6 Definition 2.4. Let H be a Hilbert space and M : H → 2 H be a multivalued maximal monotone mapping. Then, the single-valued mapping J M,λ : H → H defined by J M,λ u I λM −1 u ,u∈ H 2.7 is called the resolvent operator associated with M,whereλ is any positive number and I is the identity mapping. Proposition 2.5 see 11. i The resolvent operator J M,λ associated with M is single-valued and nonexpansive for all λ>0,thatis, J M,λ x − J M,λ y ≤ x − y , ∀x, y ∈ H, ∀λ>0. 2.8 ii The resolvent operator J M,λ is 1-inverse strongly monotone, that is, J M,λ x − J M,λ y 2 ≤ x − y, J M,λ x − J M,λ y , ∀x, y ∈ H. 2.9 Definition 2.6. A single-valued mapping A : H → H is said to be hemicontinuous if for any x, y, z ∈ H,functiont →Ax ty,z is continuous at 0. It is well-known that every continuous mapping must be hemicontinuous. Lemma 2.7 see 13. Let {x n } and {y n } be bounded sequences in a Banach space X.Let{β n } be a sequence in 0, 1 with 0 < lim inf n →∞ β n ≤ lim sup n →∞ β n < 1. 2.10 Suppose that x n1 1 − β n y n β n x n , ∀n ≥ 0, lim sup n →∞ y n1 − y n − x n1 − x n ≤ 0. 2.11 Fixed Point Theory and Applications 5 Then, lim n →∞ y n − x n 0. 2.12 Lemma 2.8 see 14. Let X be a real Banach space, X ∗ be the dual space of X, T : X → 2 X ∗ be a maximal monotone mapping, and P : X → X ∗ be a hemicontinuous bound monotone mapping with DP X. Then, the mapping S T P : X → 2 X ∗ is a maximal monotone mapping. Lemma 2.9 see 15. Let X be a uniformly convex Banach space, let C be a nonempty closed convex subset of X,andT : C → C be a nonexpansive mapping with a fixed point. Then, I − T is demiclosed in the sense that if {x n } is a sequence in C satisfying x n x, I − T n −→ 0, 2.13 then I − T x 0 . 2.14 Throughout this paper, we assume that the bifunction F : C × C → satisfies the following conditions: H 1 Fx, x0forallx ∈ C; H 2 F is monotone, that is, F x, y F y, x ≤ 0, ∀x, y ∈ C, 2.15 H 3 for each x, y, z ∈ C, lim t↓0 F tz 1 − t x, y ≤ F x, y , 2.16 H 4 for each x ∈ C, y → Fx, y is convex and lower semi-continuous. Lemma 2.10 see 16. Let H be a real Hilbert space, C be a nonempty closed convex subset of H, and F : C × C → be a bifunction satisfying the conditions (H 1 )–(H 4 ). Let μ>0 and x ∈ H.Then, there exists a point z ∈ C such that F z, y 1 μ y − z, z − x ≥ 0, ∀y ∈ C. 2.17 Moreover, if T μ : H → C is a mapping defined by T μ x z ∈ C : F z, y 1 μ y − z, z − x ≥ 0, ∀y ∈ C ,x∈ H, 2.18 then the following results hold: 6 Fixed Point Theory and Applications i T μ is single-valued and firmly nonexpansive, that is, for any x, y ∈ H, T μ x − T μ y 2 ≤ T μ x − T μ y, x − y , 2.19 ii EP is closed and convex, and EP FT μ . Lemma 2.11. isee 11 u ∈ H is a solution of variational inclusion 1.2 ifandonlyif u J M,λ u − λAu , ∀λ>0, 2.20 that is, VI H, A, M F J M,λ u − λAu , ∀λ>0. 2.21 iisee 10 u ∈ C is a solution of generalized equilibrium problem 1.6 if and only if u T μ u − μBu , ∀μ>0, 2.22 that is, GEP F T μ u − μBu , ∀μ>0. 2.23 iiisee 10 Let A : H → H be an α-inverse strongly monotone mapping and B : C → H be a β-inverse strongly monotone mapping. If λ ∈ 0, 2α and μ ∈ 0, 2β,thenVIH, A, M is a closed convex subset in H and GEP is a closed convex subset in C. Lemma 2.12 see 17. Assume that {a n } is a sequence of nonnegative real numbers such that a n1 ≤ 1 − γ n a n δ n , ∀n ≥ 1, 2.24 where {γ n } is a sequence in 0, 1 and {δ n } is a sequence such that: i ∞ n1 γ n ∞; ii lim sup n →∞ δ n /γ n ≤ 0 or ∞ n1 |δ n | < ∞. Then, lim n →∞ a n 0. 3. Main Results Theorem 3.1. Let H be a real Hilbert space, C be a nonempty closed convex subset of H, A : H → H be an α-inverse strongly monotone mapping and B : C → H be a β-inverse strongly monotone Fixed Point Theory and Applications 7 mapping. Let M : H → 2 H be a maximal monotone mapping, {T n : C → C} be a sequence of nonexpansive mappings with ∞ n1 FT n / ∅, S : C → C be the nonexpansive mapping defined by 2.5,andF : C × C → be a bifunction satisfying conditions (H 1 )–(H 4 ). Let {x n } be the sequence defined by x n1 α n x n 1 − α n SP C 1 − t n J M,λ I − λA T μ I − μB x n , 3.1 where the mapping T μ : H → C is defined by 2.18,andλ, μ are two constants with λ ∈ 0, 2α,μ∈ 0, 2β,and t n ∈ 0, 1 ,t n −→ 0 n −→ ∞ , ∞ n1 t n ∞, 0 <a<α n <b<1. 3.2 If Ω : F S ∩ VI H, A, M ∩ GEP / ∅, 3.3 where VI H, A, M and GEP is the set of solutions of variational inclusion 1.2 and generalized equilibrium problem 1.6, respectively, then the sequence {x n } defined by 3.1 converges strongly to x ∗ ∈ Ω, which is the unique solution of the following quadratic minimization problem: x ∗ 2 min x∈Ω x 2 . 3.4 Proof. We divide the proof of Theorem 3 .1 into four steps. Step 1 The sequence {x n } is bounded.Set u n T μ I − μB x n ,y n J M,λ I − λA u n ,z n SP C 1 − t n y n . 3.5 Taking z ∈ Ω, then it follows from Lemma 2.11 that z T μ z − μBz J M,λ z − λAz SP C z. 3.6 Since both T μ and J M,λ are nonexpansive, A and B are α-inverse strongly monotone and β- inverse strongly monotone, respectively, from Proposition 2.2,wehave 8 Fixed Point Theory and Applications u n − z 2 T μ I − μBx n − T μ z − μBz 2 ≤ I − μBx n − z − μBz 2 ≤ x n − z 2 μ μ − 2β Bx n − Bz 2 , 3.7 y n − z 2 J M,λ I − λAu n − J M,λ z − λAz 2 ≤ I − λAu n − z − λAz 2 ≤ u n − z 2 λ λ − 2α Au n − Az 2 ≤ x n − z 2 λ λ − 2α Au n − Az 2 μ μ − 2β Bx n − Bz 2 . 3.8 This implies that y n − z ≤ u n − z ≤ x n − z . 3.9 It follows from 3.1 and 3.9 that x n1 − z α n x n 1 − α n SP C 1 − t n y n − z α n x n − z 1 − α n SP C 1 − t n y n − SP C z ≤ α n x n − z 1 − α n SP C 1 − t n y n − SP C z ≤ α n x n − z 1 − α n 1 − t n y n − z ≤ α n x n − z 1 − α n 1 − t n y n − z t n z ≤ α n x n − z 1 − α n 1 − t n x n − z t n z ≤ 1 − t n 1 − α n x n − z t n 1 − α n z ≤ max { x n − z , z } ≤ max { x n−1 − z , z } ≤···≤max { x 1 − z , z } M, 3.10 where M max{x 1 − z, z}. This shows that {x n } is bounded. Hence, it follows from 3.9 that the sequence {u n } and {y n } are also bounded. It follows from 3.5, 3.6,and3.9 that z n − z SP C 1 − t n y n − SP C z ≤ 1 − t n y n − z ≤ 1 − t n y n − z t n z ≤ 1 − t n x n − z t n z ≤ M. 3.11 This shows that {z n } is bounded. Fixed Point Theory and Applications 9 Step 2. Now, we prove that lim n →∞ x n − u n lim n →∞ u n − y n lim n →∞ x n − y n 0, lim n →∞ x n − Sx n 0. 3.12 Since SP C is nonexpansive, from 3.5 and 3.9,wehavethat y n1 − y n ≤ u n1 − u n ≤ x n1 − x n , 3.13 z n1 − z n SP C 1 − t n1 y n1 − SP C 1 − t n y n ≤ 1 − t n1 y n1 − 1 − t n y n 1 − t n1 y n1 − y n 1 − t n1 − 1 − t n y n ≤ 1 − t n1 y n1 − y n | t n1 − t n | y n ≤ y n1 − y n | t n1 − t n | y n ≤ u n1 − u n | t n1 − t n | y n ≤ x n1 − x n | t n1 − t n | y n . 3.14 Let n →∞in 3.14, in view of condition t n → 0n →∞,wehave lim n →∞ z n1 − z n − x n1 − x n 0. 3.15 By virtue of Lemma 2.7,wehave lim n →∞ x n − z n 0. 3.16 This implies that lim n →∞ x n1 − x n lim n →∞ 1 − α n z n − x n 0. 3.17 10 Fixed Point Theory and Applications We derive from 3.17 that lim n →∞ x n − z 2 − x n1 − z 2 lim n →∞ x n − x n1 2 2 x n − x n1 ,x n1 − z ≤ lim n →∞ x n − x n1 2 2 x n − x n1 · x n1 − z 0. 3.18 From 3.1 and 3.8,wehave x n1 − z 2 ≤ α n x n − z 1 − α n 1 − t n y n − z 2 ≤ α n x n − z 2 1 − α n 1 − t n y n − z − t n z 2 α n x n − z 2 1 − α n 1 − t n 2 y n − z 2 − 2t n 1 − t n z, y n − z t 2 n z 2 ≤ α n x n − z 2 1 − α n y n − z 2 t n M 1 ≤ α n x n − z 2 1 − α n × x n − z 2 λ λ − 2α Au n − Az 2 μ μ − 2β Bx n − Bz 2 t n M 1 x n − z 2 1 − α n λ λ − 2α Au n − Az 2 μ μ − 2β Bx n − Bz 2 t n M 1 , 3.19 where M 1 sup n z 2 2 λ u n − y n μ x n − y n < ∞, 3.20 that is, 1 − α n λ 2α − λ Au n − Az 2 μ 2β − μ Bx n − Bz 2 ≤ x n − z 2 − x n1 − z 2 1 − α n t n M 1 . 3.21 Let n →∞, noting the assumptions that λ ∈ 0, 2α, μ ∈ 0, 2β,from3.2 and 3.18,we have lim n →∞ Au n − Az lim n →∞ Bx n − Bz 0. 3.22 [...]... “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 S.-S Zhang, H.-W Lee, and C.-K Chan, “Quadratic minimization for equilibrium problem variational inclusion and fixed point problem,” Applied Mathematics and Mechanics, vol 31, no 7, pp 917–928, 2010 11 S.-S Zhang, J H W Lee, and. .. 3.56 Fixed Point Theory and Applications 17 where VI A, C and GEP are the sets of solutions of variational inclusion 1.5 and generalized equilibrium problem 1.6 , then the sequence {xn } defined by 3.54 converges strongly to x∗ ∈ Ω2 , which is the unique solution of the following quadratic minimization problem: x∗ 2 min x 2 x∈Ω2 3.57 References 1 M A Noor and K I Noor, “Sensitivity analysis for quasi -variational. .. mappings in general Banach spaces,” Fixed Point Theory and Applications, no 1, pp 103–123, 2005 14 D Pascali, Nonlinear Mappings of Monotone Type, Sijthoff and Noordhoff International Publishers, The Netherlands, 1978 15 K Goebel and W A Kirk, Topics in Metric Fixed Point Theory, vol 28, Cambridge University Press, Cambridge, UK, 1990 16 P L Combettes and S A Hirstoaga, Equilibrium programming in Hilbert... “Generalized set-valued variational inclusions and resolvent equations,” Journal of Mathematical Analysis and Applications, vol 228, no 1, pp 206–220, 1998 5 E Blum and W Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol 63, no 1–4, pp 123–145, 1994 6 F Tang, “Strong convergence theorem for a generalized equilibrium problems and a family of infinitely... 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Bun Let n → ∞ in 3.34 , in view of condition H4 and un 2 ≥ 0 x∗ , we have yt − x∗ , Byt ≥ F yt , x∗ 3.36 It follows from conditions H1 , H4 and 3.36 that 0 F yt , yt ≤ tF yt , y ≤ tF yt , y tF yt , y 1 − t F yt , x∗ 1 − t yt − x∗ , Byt 1 − t t y − x∗ , Byt , 3.37 14 Fixed Point Theory and Applications that is, 1 − t y − x∗ , Byt 0 ≤ F yt , y 3.38 Let t to 0 in 3.38 , we have F x∗ , y y − x∗ , Bx∗ ≥ . Corporation Fixed Point Theory and Applications Volume 2011, Article ID 915629, 17 pages doi:10.1155/2011/915629 Research Article Iterative Algorithms for Finding Common Solutions to Variational Inclusion. 917–928, 2010. 11 S S. Z hang, J. H. W. Lee, and C. K. Chan, Algorithms of common solutions to quasi variational inclusion and fixed point problems,” Applied Mathematics and Mechanics, vol. 29, no. 5, pp 2009. 9 V. Colao, G. Marino, and H K. Xu, “An iterative method for finding common solutions of equilibrium and xedpointproblems,”Journal of Mathematical Analysis and Applications, vol. 344, no.
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