Composite Algorithms for Minimization over the Solutions of Equilibrium Problems and Fixed Point Problems

and Applied Analysis 3 Let C be a nonempty closed convex subset of a real Hilbert space H. Recall that a mapping A : C → H is called α-inverse strongly monotone if there exists a positive real number α such that 〈Ax − Ay, x − y〉 ≥ α‖Ax −Ay‖, for all x, y ∈ C. It is clear that any αinverse strongly monotone mapping is monotone and 1/α-Lipschitz continuous. Let f : C → H be a ρ-contraction; that is, there exists a constant ρ ∈ 0, 1 such that ‖f x −f y ‖ ≤ ρ‖x−y‖ for all x, y ∈ C. A mapping S : C → C is said to be nonexpansive if ‖Sx − Sy‖ ≤ ‖x − y‖, for all x, y ∈ C. Denote the set of fixed points of S by Fix S . Let A : C → H be a nonlinear mapping and F : C × C → R be a bifunction. The equilibrium problem is to find z ∈ C such that F ( z, y ) 〈 Az, y − z〉 ≥ 0, ∀y ∈ C. 1.9 The solution set of 1.9 is denoted by EP. If A 0, then 1.9 reduces to the following equilibrium problem of finding z ∈ C such that F ( z, y ) ≥ 0, ∀y ∈ C. 1.10 If F 0, then 1.9 reduces to the variational inequality problem of finding z ∈ C such that 〈Az, y − z〉 ≥ 0, ∀y ∈ C. 1.11 We note that the problem 1.9 is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games and others, see, for example, 1–4 . We next briefly review some historic approaches which relate to the fixed point problems and the equilibrium problems. In 2005, Combettes and Hirstoaga 5 introduced an iterative algorithm of finding the best approximation to the initial data and proved a strong convergence theorem. In 2007, by using the viscosity approximation method, S. Takahashi and W. Takahashi 6 introduced another iterative scheme for finding a common element of the set of solutions of the equilibrium problem and the set of fixed point points of a nonexpansive mapping. Subsequently, algorithms constructed for solving the equilibrium problems and fixed point problems have further developed by some authors. In particular, Ceng and Yao 7 introduced an iterative scheme for finding a common element of the set of solutions of the mixed equilibrium problem 1.9 and the set of common fixed points of finitely many nonexpansive mappings. Maingé and Moudafi 8 introduced an iterative algorithm for equilibrium problems and fixed point problems. Yao et al. 9 considered an iterative scheme for finding a common element of the set of solutions of the equilibrium problem and the set of common fixed points of an infinite nonexpansive mappings. Noor et al. 10 introduced an iterative method for solving fixed point problems and variational inequality problems. Their results extend and improve many results in the literature. Some works related to the equilibrium problem, fixed point problems, and the variational inequality problem in 1–45 and the references therein. However, we note that all constructed algorithms in 2, 4, 6–10, 14, 15, 21, 23–40 do not work to find the minimum-norm solution of the corresponding fixed point problems and the equilibrium problems. It is our main purpose in this paper that we devote to construct 4 Abstract and Applied Analysis some algorithms for finding the minimum-norm solution of the fixed point problems and the equilibrium problems. We first suggest two new composite algorithms one implicit and one explicit for solving the above minimization problem. Further, we prove that the proposed composite algorithms converge strongly to the minimum norm element x∗. 2. Preliminaries Let C be a nonempty closed convex subset of a real Hilbert space H. Throughout this paper, we assume that a bifunction F : C × C → R satisfies the following conditions: H1 F x, x 0, for all x ∈ C; H2 F is monotone, that is, F x, y F y, x ≤ 0 for all x, y ∈ C; H3 for each x, y, z ∈ C, limt↓0F tz 1 − t x, y ≤ F x, y ; H4 for each x ∈ C, y → F x, y is convex and lower semicontinuous. The metric or nearest point projection from H onto C is the mapping PC : H → C which assigns to each point x ∈ C the unique point PCx ∈ C satisfying the property ‖x − PCx‖ inf y∈C ∥ ∥x − y∥∥ : d x,C . 2.1 It is well known that PC is a nonexpansive mapping and satisfies 〈x − y, PCx − PCy〉 ≥ ∥ PCx − PCy ∥ ∥ 2 , ∀x, y ∈ H. 2.2 We need the following lemmas for proving our main results. Lemma 2.1 see 5 . Let C be a nonempty closed convex subset of a real Hilbert space H. Let F : C × C → R be a bifunction which satisfies conditions (H1)–(H4). Let r > 0 and x ∈ C. Then, there exists z ∈ C such that F ( z, y ) 1 r 〈 y − z, z − x〉 ≥ 0, ∀y ∈ C. 2.3 Further, if Tr x {z ∈ C : F z, y 1/r 〈y − z, z − x〉 ≥ 0, ∀y ∈ C}, then the following hold: i Tr is single-valued and Tr is firmly nonexpansive, that is, for any x, y ∈ H, ‖Trx − Try‖ ≤ 〈Trx − Try, x − y〉; ii EP is closed and convex and EP Fix Tr . Lemma 2.2 see 17 . Let C be a nonempty closed convex subset of a real Hilbert space H. Let the mapping A : C → H be α-inverse strongly monotone and r > 0 be a constant. Then, one has ∥ ∥ I − rA x − I − rA y∥∥2 ≤ ∥∥x − y∥∥2 r r − 2α ∥∥Ax −Ay∥∥2, ∀x, y ∈ C. 2.4 In particular, if 0 ≤ r ≤ 2α, then I − rA is nonexpansive. Abstract and Applied Analysis 5 Lemma 2.3 see 28 . Let C be a closed convex subset of a real Hilbert space H and let S : C → C be a nonexpansive mapping. Then, the mapping I − S is demiclosed, that is, if {xn} is a sequence in C such that xn → x∗ weakly and I − S xn → y strongly, then I − S x∗ y. Lemma 2.4 see 22 . Assume {an} is a sequence of nonnegative real numbers such that an 1 ≤ ( 1 − γn ) an δnγn, 2.5and Applied Analysis 5 Lemma 2.3 see 28 . Let C be a closed convex subset of a real Hilbert space H and let S : C → C be a nonexpansive mapping. Then, the mapping I − S is demiclosed, that is, if {xn} is a sequence in C such that xn → x∗ weakly and I − S xn → y strongly, then I − S x∗ y. Lemma 2.4 see 22 . Assume {an} is a sequence of nonnegative real numbers such that an 1 ≤ ( 1 − γn ) an δnγn, 2.5 where {γn} is a sequence in 0, 1 and {δn} is a sequence such that 1 ∑∞ n 1 γn ∞; 2 lim supn→∞δn ≤ 0 or ∑∞ n 1 |δnγn| < ∞. Then limn→∞an 0. 3. Main Results In this section we will introduce two algorithms for finding the minimum norm element x∗ of Γ : EP ∩ Fix S . Namely, we want to find the unique point x∗ which solves the following minimization problem: x∗ arg min x∈Γ ‖x‖. 3.1 Let S : C → C be a nonexpansive mapping and A : C → H be an α-inverse strongly monotone mapping. Let F : C×C → R be a bifunction which satisfies conditions H1 – H4 . Let r and μ be two constants such that r ∈ 0, 2α and μ ∈ 0, 1 . In order to find a solution of the minimization problem 3.1 , we construct the following implicit algorithm xt μPC 1 − t Sxt ( 1 − μ)Tr xt − rAxt , ∀t ∈ 0, 1 , 3.2 where Tr is defined as Lemma 2.1. We will show that the net {xt} defined by 3.2 converges to a solution of the minimization problem 3.1 . As matter of fact, in this paper, we will study the following general algorithm. Let f : C → H be a ρ-contraction. For each t ∈ 0, 1 , we consider the following mapping Wt given by Wtx μPC [ tf x 1 − t Sxt ] ( 1 − μ)Tr I − rA xt, ∀x ∈ C. 3.3 Since the mappings S, PC, Tr and I − rA are nonexpansive, then we can check easily that ‖Wtx−Wty‖ ≤ 1− 1−ρ μt ‖x−y‖which implies thatWt is a contraction. Using the Banach contraction principle, there exists a unique fixed point xt ofWt in C, that is, xt μPC [ tf xt 1 − t Sxt ] ( 1 − μ)Tr I − rA xt, t ∈ 0, 1 . 3.4 In this point, we would like to point out that algorithm 3.4 includes algorithm 3.2 as a special case due to the contraction f is a possible nonself-mapping. 6 Abstract and Applied Analysis In the sequel, we assume 1 C is a nonempty closed convex subset of a real Hilbert spaceH; 2 S : C → C is a nonexpansive mapping, A : C → H is an α-inverse strongly monotone mapping and f : C → H is a ρ-contraction; 3 F : C × C → R is a bifunction which satisfies conditions H1 – H4 ; 4 Γ/ ∅. In order to prove our first main result, we need the following lemmas. Lemma 3.1. The net {xt} generated by the implicit method 3.4 is bounded. Proof. Set ut Tr xt − rAxt and yt tf xt 1 − t Sxt for all t ∈ 0, 1 . Take z ∈ Γ. It is clear that Sz z Tr z − rAz . Since Tr is nonexpansive and A is α-inverse strongly monotone, we have from Lemma 2.2 that ‖ut − z‖ ≤ ‖xt − rAxt − z − rAz ‖ ≤ ‖xt − z‖ r r − 2α ‖Axt −Az‖ ≤ ‖xt − z‖. 3.5