A General Composite Algorithms for Solving General Equilibrium Problems and Fixed Point Problems in Hilbert Spaces

and Applied Analysis 3 literature. For some works related to the equilibrium problem, fixed point problems, and the variational inequality problem, please see 1–57 and the references therein. However, we note that all constructed algorithms in 7, 9–13, 16, 57 do not work to find the minimum-norm solution of the corresponding fixed point problems and the equilibrium problems. Very recently, Yao and Liou 46 purposed some algorithms for finding the minimum-norm solution of the fixed point problems and the equilibrium problems. They first suggested two new composite algorithms one implicit and one explicit for solving the above minimization problem. To be more precisely, let C be a nonempty, closed, convex subset of H , φ : C × C → a bifunction satisfying certain conditions, and S : C → C a nonexpansive mapping such that Ω : Fix S ∩ EP / ∅. Let f be a contraction on a Hilbert spaceH . For given x0 ∈ C arbitrarily, let the sequence {xn} be generated iteratively by φ ( un, y ) 〈 Axn, y − un 〉 1 r 〈 y − un, un − xn 〉 ≥ 0, ∀y ∈ C, xn 1 μnPC [ αnf xn 1 − αn Sxn ] ( 1 − μn ) un, n ≥ 0, 1.6 where A is an α-inverse strongly monotone mapping. They proved that if {αn} and {μn} are two sequences in 0,1 satisfying the following conditions: i limn→∞αn 0, ∑∞ n 0 αn ∞ and limn→∞ αn 1/αn 1, ii 0 < lim infn→∞μn ≤ lim supn→∞μn < 1 and limn→∞ μn 1 −μn /αn 1 0, then, the sequence {xn} generated by 1.6 converges strongly to x∗ ∈ Ω which is the unique solution of variational inequality 〈( I − f)x∗, x − x∗〉 ≥ 0, x ∈ Ω. 1.7 In particular, if we take f 0 in 1.6 , then the sequence {xn} generated by φ ( un, y ) 〈 Axn, y − un 〉 1 r 〈 y − un, un − xn 〉 ≥ 0, ∀y ∈ C, xn 1 μnPC 1 − αn Sxn ( 1 − μn ) un, n ≥ 0, 1.8 converges strongly to a solution of the minimization problemwhich is the problem of finding x∗ such that x∗ arg min x∈Ω ‖x‖, 1.9 where Ω stands for the intersection set of the solution set of the general equilibrium problem and the fixed points set of a nonexpansive mapping. On the other hand, iterative approximation methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, 25, 43, 44 and the references therein. Let B be a strongly positive bounded linear operator onH , that is, there is a constant γ > 0 with property 〈Bx, x〉 ≥ γ‖x‖ ∀ x ∈ H. 1.10 4 Abstract and Applied Analysis A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert spaceH min x∈Fix S 1 2 〈Bx, x〉 − 〈x, b〉, 1.11 where b is a given point in H . In 2003, Xu 43 proved that the sequence {xn} defined by the iterative method below, with the initial guess x0 ∈ H chosen arbitrarily: xn 1 I − αnB Txn αnu, n ≥ 0, 1.12 converges strongly to the unique solution of the minimization problem 1.11 provided the sequence {αn} satisfies certain conditions. Using the viscosity approximation method, Moudafi 29 introduced the following iterative process for nonexpansive mappings see 43 for further developments in both Hilbert and Banach spaces . Let f be a contraction on H . Starting with an arbitrary initial x0 ∈ H , define a sequence {xn} recursively by xn 1 1 − αn Txn αnf xn , n ≥ 0, 1.13 where {αn} is a sequence in 0, 1 . It is proved 29, 43 that under certain appropriate conditions imposed on {αn}, the sequence {xn} generated by 1.13 strongly converges to the unique solution x∗ in C of the variational inequality 〈( I − f)x∗, x − x∗〉 ≥ 0, x ∈ H. 1.14 Recently, Marino and Xu 28 mixed the iterative method 1.12 and the viscosity approximation method 1.13 introduced by Moudafi 29 and considered the following general iterative method: xn 1 I − αnB Txn αnγf xn , n ≥ 0, 1.15 where B is a strongly positive bounded linear operator onH . They proved that if the sequence {αn} of parameters satisfies the certain conditions, then the sequence {xn} generated by 1.15 converges strongly to the unique solution x∗ inH of the variational inequality 〈( B − γf)x∗, x − x∗〉 ≥ 0, x ∈ H 1.16 which is the optimality condition for the minimization problem: minx∈Fix S 1/2 〈Bx, x〉 − h x , where h is a potential function for γf i.e., h′ x γf x for x ∈ H . Recall that amapping F : H → H is called δ-strongly monotone if there exists a positive constant δ such that 〈 Fx − Fy, x − y〉 ≥ δ ∥∥x − y∥∥2, ∀x, y ∈ H. 1.17 Abstract and Applied Analysis 5 Recall also that a mapping F is called λ-strictly pseudocontractive if there exists a positive constant λ such thatand Applied Analysis 5 Recall also that a mapping F is called λ-strictly pseudocontractive if there exists a positive constant λ such that 〈 Fx − Fy, x − y〉 ≤ ∥∥x − y∥∥2 − λ∥∥(x − y) − (Fx − Fy)∥∥2, ∀x, y ∈ H. 1.18 It is easy to see that 1.18 can be rewritten as 〈 I − F x − I − F y, x − y〉 ≥ λ∥∥ I − F x − I − F y∥∥2. 1.19 Remark 1.1. If F is a strongly positive bounded linear operator on H with coefficient γ , then F is γ -strongly monotone and 12-strictly pseudocontractive. In fact, since F is a strongly positive, bounded, linear operator with coefficient γ , we have 〈 Fx − Fy, x − y〉 〈F(x − y), x − y〉 ≥ γ ∥∥x − y∥∥2. 1.20 Therefore, F is γ-strongly monotone. On the other hand, ∥ ∥ I − F x − I − F y∥∥2 〈(x − y) − (Fx − Fy), (x − y) − (Fx − Fy)〉 〈 x − y, x − y〉 − 2〈Fx − Fy, x − y〉 〈Fx − Fy, Fx − Fy〉 ∥ ∥x − y∥∥2 − 2〈Fx − Fy, x − y〉 ∥∥Fx − Fy∥∥2 ≤ ∥∥x − y∥∥2 − 2〈Fx − Fy, x − y〉 ‖F‖2∥∥x − y∥∥2. 1.21 Since F is strongly positive if and only if 1/‖F‖ F is strongly positive, we may assume, without loss of generality, that ‖F‖ 1. From 1.21 , we have 〈 Fx − Fy, x − y〉 ≤ ∥∥x − y∥∥2 − 1 2 ∥ ∥ I − F x − I − F y∥∥2 ∥ ∥x − y∥∥2 − 1 2 ∥ ∥ ( x − y) − (Fx − Fy)∥∥2. 1.22 Hence, F is 12-strictly pseudocontractive. In this paper, motivated by the above results, we introduce a general iterative scheme below in a real Hilbert spaceH , with the initial guess x0 ∈ C chosen arbitrary: φ ( un, y ) 〈 Axn, y − un 〉 1 r 〈 y − un, un − xn 〉 ≥ 0, ∀y ∈ C, yn αnγf xn I − αnF S n 1 i n 1 xn, xn 1 μnPC [ yn ] ( 1 − μn ) un, n ≥ 0, 1.23 where p n j 1 if jN < n ≤ j 1 N, j 1, 2, . . . and n jN i n , i n ∈ {1, 2, . . . ,N}, C is a nonempty, closed, convex subset of H , {αn} and {μn} are two sequences in 0,1 , 6 Abstract and Applied Analysis φ : C × C → is a bifunction satisfying certain conditions, S1, S2, . . . , SN : C → C is a finite family of asymptotically nonexpansive mappings with sequences {1 k n p n }, respectively, f : C → H is a contraction with coefficient 0 < ρ < 1, F is δ-strongly monotone and λstrictly pseudocontractive with δ λ > 1, γ is a positive real number such that γ < 1/ρ 1 − √ 1 − δ /λ , and A is an α-inverse strongly monotone mapping. We prove that the proposed algorithm converges strongly to x∗ ∈ Ω which is the unique solution of the following variational inequality: 〈( F − γf)x∗, x − x∗〉 ≥ 0, x ∈ Ω. 1.24 In particular, I if F is a strongly positive bounded linear operator on H , then x∗ is the unique solution of the variational inequality 1.16 , II if F I, the identity mapping on H and γ 1, then x∗ is the unique solution of the variational inequality 1.14 , III if F I, the identity mapping on H and f 0, then x∗ is the unique solution of minimization problem 1.9 . The results presented in this paper extend and improve the main results in Yao and Liou 46 , Marino and Xu 28 , and many others. 2. Preliminaries Let C be a nonempty, closed, convex subset of a real Hilbert spaceH . For every point x ∈ H , there exists a unique nearest point in C, denoted by PCx such that ‖x − PCx‖ ≤ ∥ ∥x − y∥∥, ∀y ∈ C. 2.1 PC is called the metric projection of H onto C. It is well known that PC is a nonexpansive mapping of H onto C and satisfies 〈 x − y, PCx − PCy 〉 ≥ ∥∥PCx − PCy ∥ ∥ 2 , 2.2 for every x, y ∈ H . Moreover, PCx is characterized by the following properties: PCx ∈ C and 〈 x − PCx, y − PCx 〉 ≤ 0, ∥ ∥x − y∥∥2 ≥ ‖x − PCx‖ ∥ ∥y − PCx ∥ ∥ 2 , 2.3 for all x ∈ H,y ∈ C. For more details, see 39 . We will make use of the following well-known result. Lemma 2.1. Let H be a Hilbert space. Then, the following inequality holds: ∥ ∥x y ∥ ∥≤ ‖x‖ 2〈y, x y〉, ∀x, y ∈ H. 2.4 Abstract and Applied Analysis 7 Throughout this paper, we assume that a bifunction φ : C × C → satisfies the following conditions: A1 φ x, x 0 for all x ∈ C, A2 φ is monotone, that is, φ x, y φ y, x ≤ 0 for all x, y ∈ C, A3 for each x, y, z ∈ C, limt↓0φ tz 1 − t x, y ≤ φ x, y , A4 for each x ∈ C, the mapping y → φ x, y is convex and lower semicontinuous. We need the following lemmas for proving our main results. Lemma 2.2 see 6 . Let C be a nonempty, closed, convex subset of a real Hilbert space H . Let φ : C × C → be a bifunction which satisfies conditions (A1)–(A4). Let r > 0 and x ∈ C. Then, there exists z ∈ C such that φ ( z, y ) 1 r 〈 y − z, z − x〉 ≥ 0, ∀y ∈ C. 2.5and Applied Analysis 7 Throughout this paper, we assume that a bifunction φ : C × C → satisfies the following conditions: A1 φ x, x 0 for all x ∈ C, A2 φ is monotone, that is, φ x, y φ y, x ≤ 0 for all x, y ∈ C, A3 for each x, y, z ∈ C, limt↓0φ tz 1 − t x, y ≤ φ x, y , A4 for each x ∈ C, the mapping y → φ x, y is convex and lower semicontinuous. We need the following lemmas for proving our main results. Lemma 2.2 see 6 . Let C be a nonempty, closed, convex subset of a real Hilbert space H . Let φ : C × C → be a bifunction which satisfies conditions (A1)–(A4). Let r > 0 and x ∈ C. Then, there exists z ∈ C such that φ ( z, y ) 1 r 〈 y − z, z − x〉 ≥ 0, ∀y ∈ C. 2.5 Further, if Tr x {z ∈ C : φ 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 ∥ ∥ 2 ≤ 〈Trx − Try, x − y 〉 , 2.6 ii EP is closed and convex and EP Fix Tr . Lemma 2.3 see 30 . Let C be a nonempty, closed, convex subset of a real Hilbert space H . Let the mappingA : C → H be α-inverse strongly monotone and r > 0 a constant. Then, one has ∥ ∥ I − rA x − I − rA y∥∥2 ≤ ∥∥x − y∥∥2 r r − 2α ∥∥Ax −Ay∥∥2, ∀x, y ∈ C. 2.7 In particular, if 0 ≤ r ≤ 2α, then I − rA is nonexpansive. Lemma 2.4 see 45 . Let S be an asymptotically nonexpansive mapping defined on a bounded, closed, convex subset C of a Hilbert space H . If {xn} is a sequence in C such that xn ⇀ x and ‖Sxn − xn‖ → 0 as n → ∞, then x ∈ Fix S . Lemma 2.5 see 44 . Assume {an} is a sequence of nonnegative real numbers such that an 1 ≤ 1 − αn an αnσn γn, n ≥ 0, 2.8 where {αn}, {σn}, and {γn} are nonnegative real sequences satisfying the following conditions: i {αn} ⊂ 0, 1 , ∑∞ n 1 αn ∞, ii lim supn→∞σn ≤ 0, iii ∑∞ n 1 γn < ∞. Then, limn→∞an 0. 8 Abstract and Applied Analysis Lemma 2.6 see 41 . Let E be a strictly convex Banach space and C a closed, convex subset of E. Let S1, S2, . . . , SN : C → C be a finite family of nonexpansive mappings of C into itself such that the set of common fixed points of S1, S2, . . . , SN is nonempty. Let T1, T2, . . . , TN : C → C be mappings given by Ti 1 − αi I αiSi, ∀i 1, 2, . . . ,N, 2.9 where I denotes the identity mapping on C. Then, the finite family {T1, T2, . . . , TN} satisfies the following: