A Viscosity Hybrid Steepest Descent Method for Generalized Mixed Equilibrium Problems and Variational Inequalities for Relaxed Cocoercive Mapping in Hilbert Spaces

and Applied Analysis 3 The generalized mixed equilibrium problem is very general in the sense that it includes, as special cases, fixed point problems, variational inequality problems, optimization problems, Nash equilibrium problems in noncooperative games, the equilibrium problem, and Numerous problems in physics, economics, and others. Some methods have been proposed to solve problem 1.2 ; see, for instance, 3, 4 and the references therein. Let B : C → H be a nonlinear mapping. Now, we recall the following definitions. d1 B is said to be monotone if for each x, y ∈ C 〈 Bx − By, x − y ≥ 0. 1.8 d2 B is said to be ρ-strongly monotone if there exists a positive real number ρ such that 〈 Bx − By, x − y ≥ ρ∥x − y∥2, ∀x, y ∈ C. 1.9 d3 B is said to be ω-Lipschitz continuous if there exists a positive real number ω such that ∥Bx − By∥ ≤ ω∥x − y∥, ∀x, y ∈ C. 1.10 d4 B is said to be ξ-inverse-strongly monotone if there exists a constant ξ > 0 such that 〈 Bx − By, x − y ≥ ξ∥Bx − By∥2, ∀x, y ∈ C. 1.11 d5 B is said to be relaxed u, v -cocoercive if there exist positive real numbers u, v such that 〈 Bx − By, x − y ≥ −u ∥Bx − By∥2 v∥x − y∥2, ∀x, y ∈ C. 1.12 d6 A set-valued mapping Q : H → 2 is called monotone if for all x, y ∈ H, f ∈ Qx and g ∈ Qy imply 〈x − y, f − g〉 ≥ 0. d7 A monotone mapping Q : H → 2 is called maximal if the graph G Q of Q is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mapping Q is maximal if and only if for x, f ∈ H × H, 〈x − y, f − g〉 ≥ 0 for every y, g ∈ G Q implies f ∈ Qx. For finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of variational inequalities for a ξ-inverse-stronglymonotonemapping, Takahashi and Toyoda 5 introduced the following iterative scheme: x0 ∈ C chosen arbitrary, xn 1 γnxn ( 1 − γn ) SPC xn − αnBxn , ∀n ≥ 0, 1.13 4 Abstract and Applied Analysis where B is a ξ-inverse-strongly monotone mapping, {γn} is a sequence in 0, 1 , and {αn} is a sequence in 0, 2ξ . They showed that if F S ∩ VI C,B is nonempty, then the sequence {xn} generated by 1.13 converges weakly to some z ∈ F S ∩ VI C,B . For finding an element of VI C,B , Iiduka et al. 6 introduced the following iterative scheme: x0 ∈ C chosen arbitrary, xn 1 PC ( γnxn ( 1 − γn ) PC xn − αnBxn ) , ∀n ≥ 0, 1.14 where B is a ξ-inverse-strongly monotone mapping, {γn} is a sequence in −1, 1 , and {αn} is a sequence in 0, 2ξ . They showed that if VI C,B is nonempty, then the sequence {xn} generated by 1.14 converges weakly to some z ∈ VI C,B . For finding a common element of F S ∩ VI C,B , let S : H → H be a nonexpansive mapping. Yamada 7 introduced the following iterative scheme called the hybrid steepest descent method: xn 1 Sxn − αnμBSxn, ∀n ≥ 1, 1.15 where x1 x ∈ H, {αn} ⊂ 0, 1 , B : H → H is a strongly monotone and Lipschitz continuous mapping, and μ is a positive real number. He proved that the sequence {xn} generated by 1.15 converges strongly to the unique solution of the F S ∩ VI C,B . The hybrid steepest descent method is constructed by blending important ideas in the steepest descent method and in the fixed point theory. The remarkable applicability of this method to the convexly constrained generalized pseudoinverse problem as well as to the convex feasibility problem is demonstrated by constructing nonexpansive mappings whose fixed point sets are the feasible sets of the problems. On the other hand, Shang et al. 8 introduced a new iterative process for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequalities for relaxed u, v -cocoercive mappings in a real Hilbert space by using viscosity approximation method. Let S : C → C be a nonexpansive mapping and let f : C → C be a contraction mapping. Starting with arbitrary initial x1 ∈ C and define sequences {xn} recursively by xn 1 nf xn βnxn γnSPC xn − αnBxn , ∀n ≥ 1. 1.16 They proved that under certain appropriate conditions imposed on { n}, {βn}, {γn}, and {αn}, the sequence {xn} converges strongly to z ∈ F S ∩VI C,B , where z PF S ∩VI C,B f z . For finding a common element of F S ∩GEF Θ,Ψ , letC be a nonempty closed convex subset of a real Hilbert space H. Let Ψ be a ξ-inverse-strongly monotone mapping of C into Abstract and Applied Analysis 5 H and let S be a nonexpansive mapping of C into itself. S. Takahashi and W. Takahashi 9 introduced the following iterative scheme:and Applied Analysis 5 H and let S be a nonexpansive mapping of C into itself. S. Takahashi and W. Takahashi 9 introduced the following iterative scheme: Θ ( un, y ) 〈 Ψxn, y − un 〉 1 rn 〈 y − un, un − xn 〉 ≥ 0, ∀y ∈ C, yn αnx 1 − αn un, xn 1 γnxn ( 1 − γn ) Syn, 1.17 where {αn} ⊂ 0, 1 , {γn} ⊂ 0, 1 , and {rn} ⊂ 0, 2ξ satisfy some parameters controlling conditions. They proved that the sequence {xn} defined by 1.17 converges strongly to a common element of F S ∩GEF Θ,Ψ . Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, 7, 10–12 and the references therein. Convex minimization problems have a great impact and influence in the development of almost all branches of pure and applied sciences. A typical problem is to minimize a quadratic function over the set of fixed points of a nonexpansive mapping defined on a real Hilbert spaceH: min x∈F 1 2 〈Ax, x〉 − 〈x, b〉, 1.18 where F is the fixed point set of a nonexpansive mapping S defined on H and b is a given point in H. A linear bounded operator A is strongly positive if there exists a constant γ > 0 with the property 〈Ax, x〉 ≥ γ‖x‖, ∀x ∈ H. 1.19 Recently, Marino and Xu 13 introduced a new iterative scheme by the viscosity approximation method: xn 1 nγf xn 1 − nA Sxn. 1.20 They proved that the sequence {xn} generated by 1.20 converges strongly to the unique solution of the variational inequality: 〈 γfz −Az, x − z ≤ 0, ∀x ∈ F S , 1.21 which is the optimality condition for the minimization problem: min x∈F S 1 2 〈Ax, x〉 − h x , 1.22 where h is a potential function for γf . 6 Abstract and Applied Analysis In 2008, Qin et al. 14 proposed the following iterative algorithm: Θ ( un, y ) 1 rn 〈 y − un, un − xn 〉 ≥ 0, ∀y ∈ H, xn 1 nγf xn I − nA SPC un − αnBun , 1.23 whereA is a strongly positive linear bounded operator and B is a relaxed cocoercive mapping of C into H. They proved that if the sequences { n}, {αn}, and {rn} of parameters satisfy appropriate condition, then the sequence {xn} defined by 1.23 converges strongly to the unique solution z of the variational inequality: 〈 γfz −Az, x − z ≤ 0, ∀x ∈ F S ∩ VI C,B ∩ EP Θ , 1.24 which is the optimality condition for the minimization problem: min x∈F S ∩VI C,B ∩EP Θ 1 2 〈Ax, x〉 − h x , 1.25 where h is a potential function for γf . In this paper, we introduce an iterative scheme by using a viscosity hybrid steepest descent method for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of fixed points of a nonexpansive mapping, and the set of solutions of variational inequality problem for a relaxed cocoercive mapping in a real Hilbert space. The results shown in this paper improve and extend the recent ones announced by many others. 2. Preliminaries Throughout this paper, we always assume thatH is a real Hilbert space and C is a nonempty closed convex subset of H. For a sequence {xn}, the notation of xn ⇀ x and xn → x means that the sequence {xn} converges weakly and strongly to x, respectively. The following lemmata give some characterizations and useful properties of themetric projection PC in a real Hilbert space. The metric or nearest point projection from H onto C is the mapping PC : H → C which assigns to each point x ∈ H the unique point PCx ∈ C satisfying the following property: ‖x − PCx‖ inf y∈C ∥x − y∥. 2.1 Lemma 2.1. It is well known that the metric projection PC has the following properties: m1 for each x ∈ H and z ∈ C, z PCx ⇐⇒ 〈x − z, y − z〉 ≤ 0, ∀y ∈ C; 2.2 Abstract and Applied Analysis 7 m2 PC : H → C is nonexpansive, that is, ∥ ∥PCx − PCy ∥ ∥ ≤ ∥x − y∥, ∀x, y ∈ H; 2.3and Applied Analysis 7 m2 PC : H → C is nonexpansive, that is, ∥ ∥PCx − PCy ∥ ∥ ≤ ∥x − y∥, ∀x, y ∈ H; 2.3 m3 PC is firmly nonexpansive, that is, ∥ ∥PCx − PCy ∥ ∥2 ≤ PCx − PCy, x − y 〉 ∀x, y ∈ H. 2.4 In order to prove our main results, we also need the following lemmata. Lemma 2.2 see 2 . Let H be a Hilbert space, let C be a nonempty closed convex subset of H, and let B be a mapping of C intoH. Let x∗ ∈ C. Then, for λ > 0, x∗ ∈ VI C,B ⇐⇒ x∗ PC x∗ − λBx∗ , 2.5