AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2014/678147 678147 Research Article Strong Convergence of an Iterative Algorithm for Hierarchical Problems Kumam Poom 1 Jitpeera Thanyarat 2 Du Wei-Shih 1 Department of Mathematics, Faculty of Science King Mongkut’s University of Technology Thonburi, Bang Mod, Thung Khru, Bangkok 10140 Thailand kmutt.ac.th 2 Department of Mathematics Faculty of Science and Agriculture Rajamangala University of Technology Lanna Phan, Chiangrai 57120 Thailand rmutl.ac.th 2014 2072014 2014 26 04 2014 17 06 2014 27 06 2014 20 7 2014 2014 Copyright © 2014 Poom Kumam and Thanyarat Jitpeera. 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.

We introduce the triple hierarchical problem over the solution set of the variational inequality problem and the fixed point set of a nonexpansive mapping. The strong convergence of the algorithm is proved under some mild conditions. Our results extend those of Yao et al., Iiduka, Ceng et al., and other authors.

1. Introduction

Let C be a closed convex subset of a real Hilbert space H with inner product · , · and norm · . We denote weak convergence and strong convergence by notations and , respectively. Let A be a nonlinear mapping. The Hartman-Stampacchia variational inequality  is to find x C such that A x , y - x 0 , y C . The set of solutions is denoted by VI ( C , A ) . f : C C is said to be a ρ -contraction if there exists a constant ρ [ 0,1 ) such that f ( x ) - f ( y ) ρ x - y , x , y C . A mapping A : H H is said to be monotone if A x - A y , x - y 0 , x , y H . A mapping A : H H is said to be α - strongly monotone if there exists a positive real number α such that A x - A y , x - y α x - y 2 , x , y H . A mapping A : H H is said to be β -inverse-strongly monotone if there exists a positive real number β such that A x - A y , x - y β A x - A y 2 , x , y H . A mapping A : H H is said to be L -Lipschitz continuous if there exists a positive real number L such that A x - A y L x - y , x , y H . A linear bounded operator A is said to be strongly positive on H if there exists a constant γ ¯ > 0 with the property A x , x γ ¯ x 2 , x H . A mapping T : C C is said to be nonexpansive if T x - T y x - y , x , y C .

A point x C is a fixed point of T provided T x = x . Denote by F ( T ) the set of fixed points of T ; that is, F ( T ) = { x C : T x = x } . If C is bounded closed convex and T is a nonexpansive mapping of C into itself, then F ( T ) is nonempty (see ).

We discuss the following variational inequality problem over the fixed point set of a nonexpansive mapping (see ), which is said to be the hierarchical problem. Let a monotone, continuous mapping A : H H and a nonexpansive mapping T : H H . Find x VI ( F ( T ) , A ) = { x F ( T ) : A x , y - x 0 , y F ( T ) } , where F ( T ) . This solution set is denoted by Ξ .

We introduce the following variational inequality problem over the solution set of variational inequality problem and the fixed point set of a nonexpansive mapping (see [17, 18]), which is said to be the triple hierarchical problem. Let an inverse-strongly monotone A : H H , a strongly monotone and Lipschitz continuous B : H H , and a nonexpansive mapping T : H H . Find x VI ( Ξ , B ) = { x Ξ : B x , y - x 0 , y Ξ } , where Ξ : = VI ( F ( T ) , A ) .

In 2009, Yao et al.  considered the following two-step iterative algorithm with the initial guess x 0 C which is chosen arbitrarily: (1) x n + 1 = α n f ( x n ) + ( 1 - α n ) T y n , y n = β n S x n + ( 1 - β n ) x n , n 0 , where α n , β n ( 0,1 ) satisfies certain assumptions. Let S , T be two nonexpansive mappings and let f : C C be a contraction mapping. Then, they proved that the above iterative sequence { x n } converges strongly to fixed point.

Next, Iiduka  introduced a monotone variational inequality with variational inequality constraint over the fixed point set of a nonexpansive mapping; the sequence { x n } defined by the iterative method below, with the initial guess x 1 H , is chosen arbitrarily: (2) y n = T ( x n - λ n A 1 x n ) , x n + 1 = y n - μ α n A 2 y n , n 0 , where α n ( 0,1 ] and λ n ( 0,2 α ] satisfy certain conditions, A 1 : H H is an inverse-strongly monotone, A 2 : H H is a strongly monotone and Lipschitz continuous, and T : H H is a nonexpansive mapping; then the strongly convergence analysis of the sequence generated by (2) is proved under some appropriate conditions.

In 2011, Yao et al.  studied the hierarchical problem over the fixed point set. Let the sequences { x n } be generated by these two following algorithms:

implicit algorithm x t = T P C [ I - t ( A - γ f ) ] x t , t ( 0,1 )

explicit algorithm x n + 1 = β n x n + ( 1 - β n ) T P C [ I - α n ( A - γ f ) ] x n , n 0 .

They illustrated that these two algorithms converge strongly to the unique solution of the variational inequality which is to find x * F ( T ) such that (3) ( A - γ f ) x * , x - x * 0 , x F ( T ) , where A : C H is a strongly positive linear bounded operator, f : C H is a ρ -contraction, and T : C C is a nonexpansive mapping satisfying some conditions.

Very recently, Ceng et al.  studied the following new algorithms. For x 0 C is chosen arbitrarily, they defined a sequence { x n } by (4) x n + 1 = P C [ λ n γ ( α n f ( x n ) + ( 1 - α n ) S x n ) + ( I - λ n μ F ) T x n ] , w w w w w w w w w w w w w w w w w w w w w w w w w w w i n 0 , where the mappings S , T are nonexpansive mappings with F ( T ) . Let F : C H be a Lipschitzian and strongly monotone operator and let f : C H be a contraction mapping satisfying some appropriate conditions. They proved that the proposed algorithms strongly converge to the minimum norm fixed point of T .

In this paper, we consider a new iterative algorithm for solving the triple hierarchical problem over the solution set of the variational inequality problem and the fixed point set of a nonexpansive mapping which contain algorithms (1) and (4) as follows: (5) y n = P C [ β n S x n + ( 1 - β n ) x n ] , x n + 1 = γ λ n ϕ ( x n ) + ( I - λ n μ F ) T y n , n 0 , where the mappings S , T are nonexpansive mappings with F ( T ) . Let F : C H be a Lipschitzian and strongly monotone operator, and let ϕ : H H be a contraction mapping satisfying some mild conditions. Find a point x * F ( T ) such that (6) ( I - S ) x * , x - x * 0 , x F ( T ) . This solution set of (6) is denoted by Ω : = V I ( F ( T ) , S ) . The strong convergence for the proposed algorithms to the solution is solved under some appropriate assumptions. Our results improve the results of Ceng et al. , Iiduka , Yao et al. , Yao et al. , and some authors.

2. Preliminaries

Let C be a nonempty closed convex subset of H . There holds the following inequality in an inner product space x + y 2 x 2 + 2 y , x + y , x , y H . For every point x H , there exists a unique nearest point in C , denoted by P C x , such that (7) x - P C x x - y , y C . P C is called the metric projection of H onto C . It is well known that P C is a nonexpansive mapping of H onto C and satisfies (8) x - y , P C x - P C y P C x - P C y 2 , for every x , y H . Moreover, P C x is characterized by the following properties: P C x C and (9) x - P C x , y - P C x 0 , (10) x - y 2 x - P C x 2 + y - P C x 2 , for all x H , y C . Let B be a monotone mapping of C into H . In the context of the variational inequality problem the characterization of projection (9) implies the following: (11) u V I ( C , B ) u = P C ( u - λ B u ) , λ > 0 . It is also known that H satisfies the Opial’s condition ; that is, for any sequence { x n } H with x n x , the inequality liminf n x n - x < liminf n x n - y holds for every y H with x y .

Lemma 1 (see [<xref ref-type="bibr" rid="B1">23</xref>]).

Let C be a closed convex subset of a real Hilbert space H and let T : C C be a nonexpansive mapping. Then I - T is demiclosed at zero; that is, x n x   and   x n - T x n 0 imply x = T x .

Lemma 2 (see [<xref ref-type="bibr" rid="B13">24</xref>]).

Let { x n } and { y n } be bounded sequences in a Banach space X and let { β n } be a sequence in [ 0,1 ] with 0 < liminf n β n limsup n β n < 1 . Suppose x n + 1 = ( 1 - β n ) y n + β n x n for all integers n 0 and limsup n ( y n + 1 - y n - x n + 1 - x n ) 0 . Then, lim n y n - x n = 0 .

Lemma 3 (see [<xref ref-type="bibr" rid="B16">10</xref>]).

Let B : H H be β -strongly monotone and L -Lipschitz continuous and let μ ( 0 , 2 β / L 2 ) . For λ [ 0,1 ] , define T λ : H H by T λ ( x ) : = x - λ μ B ( x ) for all x H . Then, for all x , y H , T λ ( x ) - T λ ( y ) ( 1 - λ τ ) x - y hold, where τ : = 1 - 1 - μ ( 2 β - μ L 2 ) ( 0,1 ] .

Lemma 4 (see [<xref ref-type="bibr" rid="B25">25</xref>]).

Assume that { a n } is a sequence of nonnegative real numbers such that (12) a n + 1 ( 1 - γ n ) a n + δ n , n 0 , where { γ n } ( 0,1 ) and { δ n } is a sequence in R such that

n = 1 γ n = ;

limsup n ( δ n / γ n ) 0 or n = 1 | δ n | < .

Then lim n a n = 0 .

3. Strong Convergence Theorem

In this section, we introduce an iterative algorithm of triple hierarchical for solving monotone variational inequality problems for κ -Lipschitzian and η -strongly monotone operators over the solution set of variational inequality problems and the fixed point set of a nonexpansive mapping.

Theorem 5.

Let C be a nonempty closed and convex subset of a real Hilbert space H . Let F : C C be κ -Lipschitzian and η -strongly monotone operators with constant κ and η > 0 , respectively, and let ϕ : C C be a ρ -contraction with coefficient ρ [ 0,1 ) . Let T : C C be a nonexpansive mapping with F ( T ) , and let S : H H be a nonexpansive mapping. Let 0 < μ < 2 η / κ 2 and 0 < γ < τ , where τ = 1 - 1 - μ ( 2 η - μ κ 2 ) . Suppose that { x n } is a sequence generated by the following algorithm where x 0 C is chosen arbitrarily: (13) y n = P C [ β n S x n + ( 1 - β n ) x n ] , x n + 1 = γ λ n ϕ ( x n ) + ( I - λ n μ F ) T y n , n 0 , where { β n } , { λ n } , ( 0,1 ) satisfy the following conditions:

β n k λ n ;

lim n λ n = 0 , lim n ( ( λ n - λ n - 1 ) / λ n ) = 0 , n = 0 λ n = ;

lim n ( ( β n - β n - 1 ) / β n ) = 0 .

Then { x n } converges strongly to x * Ω , which is the unique solution of another variational inequality: (14) ( μ F - γ ϕ ) x * , x - x * 0 , x Ω , where Ω : = V I ( F ( T ) , S ) .

Proof.

We will divide the proof into four steps.

Step  1. We will show that { x n } is bounded. Indeed, for any x * F ( T ) , we have (15) y n - x * = P C [ β n S x n + ( 1 - β n ) x n ] - P C x * β n S x n + ( 1 - β n ) x n - x * = β n ( S x n - S x * ) + ( 1 - β n ) ( x n - x * ) + β n ( S x * - x * ) β n x n - x * + ( 1 - β n ) x n - x * + β n S x * - x * x n - x * + β n S x * - x * . From (13), we deduce that (16) x n + 1 - x * = γ λ n ϕ ( x n ) + ( I - λ n μ F ) T y n - x * = γ λ n ( ϕ ( x n ) - ϕ ( x * ) ) + ( I - λ n μ F ) ( T y n - x * ) + λ n ( γ ϕ ( x * ) - μ F x * ) ( ϕ ( x n ) - ϕ ( x * ) ) γ λ n ϕ ( x n ) - ϕ ( x * ) + ( I - λ n μ F ) T y n - x * + λ n γ ϕ ( x * ) - μ F x * γ ρ λ n x n - x * + ( 1 - λ n τ ) y n - x * + λ n γ ϕ ( x * ) - μ F x * . Substituting (15) into (16), we obtain (17) x n + 1 - x * γ ρ λ n x n - x * + ( 1 - λ n τ ) { x n - x * + β n S x * - x * } + λ n γ ϕ ( x * ) - μ F x * γ ρ λ n x n - x * + ( 1 - λ n τ ) x n - x * + β n S x * - x * + λ n γ ϕ ( x * ) - μ F x * [ 1 - λ n ( τ - γ ρ ) ] x n - x * + k λ n S x * - x * + λ n γ ϕ ( x * ) - μ F x * [ 1 - λ n ( τ - γ ρ ) ] x n - x * + λ n ( k S x * - x * + γ ϕ ( x * ) - μ F x * ) max { x n - x * + 1 τ - γ ρ × ( k S x * - x * + γ ϕ ( x * ) - μ F x * ) 1 τ - γ ρ } . By induction, it follows that (18) x n - x * max { x 0 - x * + 1 τ - γ ρ w w w w w w × ( k S x * - x * + γ ϕ ( x * ) - μ F x * ) 1 τ - γ ρ } , w w w w w w w w w w w w w w w w w w w w w w w w w n 0 . Therefore, { x n } is bounded and so are { y n } , { T y n } , { S x n } , { ϕ ( x n ) } , and { F T ( y n ) } .

Step  2. We will show that lim n x n - T x n = 0 . Setting v n : = β n S x n + ( 1 - β n ) x n , we obtain (19) v n - v n - 1 = β n S x n + ( 1 - β n ) x n - β n - 1 S x n - 1 - ( 1 - β n - 1 ) x n - 1 = β n ( S x n - S x n - 1 ) + ( β n - β n - 1 ) S x n - 1 + ( 1 - β n ) ( x n - x n - 1 ) + ( β n - 1 - β n ) x n - 1 β n x n - x n - 1 + | β n - β n - 1 | ( S x n - 1 + x n - 1 ) + ( 1 - β n ) x n - x n - 1 x n - x n - 1 + | β n - β n - 1 | ( S x n - 1 + x n - 1 ) , which implies that (20) y n - y n - 1 = P C v n - P C v n - 1 v n - v n - 1 x n - x n - 1 + | β n - β n - 1 | ( S x n - 1 + x n - 1 ) . It follows from (13) that (21) x n + 1 - x n = γ λ n ϕ ( x n ) + ( I - λ n μ F ) T y n - γ λ n - 1 ϕ ( x n - 1 ) - ( I - λ n - 1 μ F ) T y n - 1 = γ λ n ( ϕ ( x n ) - ϕ ( x n - 1 ) ) + ( λ n - λ n - 1 ) γ ϕ ( x n - 1 ) + ( I - λ n μ F ) T y n - ( I - λ n - 1 μ F ) T y n - 1 γ ρ λ n x n - x n - 1 + | λ n - λ n - 1 | γ ϕ ( x n - 1 ) + ( I - λ n μ F ) T y n - ( I - λ n μ F ) T y n - 1 + ( I - λ n μ F ) T y n - 1 - ( I - λ n - 1 μ F ) T y n - 1 γ ρ λ n x n - x n - 1 + | λ n - λ n - 1 | γ ϕ ( x n - 1 ) + ( 1 - λ n τ ) y n - y n - 1 + | λ n - λ n - 1 | μ F T y n - 1 γ ρ λ n x n - x n - 1 + | λ n - λ n - 1 | × ( γ ϕ ( x n - 1 ) + μ F T y n - 1 ) + ( 1 - λ n τ ) { x n - x n - 1 + | β n - β n - 1 | × ( S x n - 1 + x n - 1 ) } [ 1 - λ n ( τ - γ ρ ) ] x n - x n - 1 + | λ n - λ n - 1 | ( γ ϕ ( x n - 1 ) + μ F T y n - 1 ) + | β n - β n - 1 | ( S x n - 1 + x n - 1 ) = [ 1 - λ n ( τ - γ ρ ) ] x n - x n - 1 + ( | λ n - λ n - 1 | λ n + | β n - β n - 1 | λ n ) λ n M 1 [ 1 - λ n ( τ - γ ρ ) ] x n - x n - 1 + ( | λ n - λ n - 1 | λ n + k | β n - β n - 1 | β n ) λ n M 1 , where M 1 is a constant such that (22) sup n 0 { ( γ ϕ ( x n ) + μ F T y n ) , ( S x n + x n ) } M 1 . Hence, conditions (C2) and (C3) allow us to apply Lemma 4; then we get (23) lim n x n + 1 - x n = 0 . By (21), we get (24) x n + 1 - x n λ n [ 1 - λ n ( τ - γ ρ ) ] x n - x n - 1 λ n + | λ n - λ n - 1 | + | β n - β n - 1 | λ n M 1 = [ 1 - λ n ( τ - γ ρ ) ] x n - x n - 1 λ n - 1 + [ 1 - λ n ( τ - γ ρ ) ] ( x n - x n - 1 λ n - x n - x n - 1 λ n - 1 ) + | λ n - λ n - 1 | + | β n - β n - 1 | λ n M 1 [ 1 - λ n ( τ - γ ρ ) ] x n - x n - 1 λ n - 1 + λ n x n - x n - 1 1 λ n | 1 λ n - 1 λ n - 1 | + M 1 λ n | λ n - λ n - 1 | + | β n - β n - 1 | λ n 2 . Using the conditions (C2) and (C3), we can apply Lemma 4 to conclude that (25) lim n x n + 1 - x n λ n = 0 . By (13), we compute (26) x n + 1 - T y n = γ λ n ϕ ( x n ) + ( I - λ n μ F ) T y n - T y n = γ λ n ϕ ( x n ) + T y n - λ n μ F T y n - T y n λ n γ ϕ ( x n ) - μ F T y n . From the condition (C2), we note that lim n x n + 1 - T y n = 0 . At the same time, from (13), we also have (27) y n - x n = P C [ β n S x n + ( 1 - β n ) x n ] - P C x n β n S x n + ( 1 - β n ) x n - x n β n S x n - x n . By the conditions (C1) and (C2), we note that lim n y n - x n = 0 . Consider (28) y n - T y n y n - x n + x n - x n + 1 + x n + 1 - T y n 0 . From (23), (26), and (27), we obtain (29) lim n y n - T y n = 0 . We set v n = β n S x n + ( 1 - β n ) x n ; then we get (30) y n - v n = P C v n - v n v n - v n 0 , as    n . From (13), we have (31) T y n - T x n = T P C [ β n S x n + ( 1 - β n ) x n ] - T P C x n β n S x n + ( 1 - β n ) x n - x n β n S x n - x n . By the conditions (C1) and (C2) again, we note that lim n T y n - T x n = 0 . Consider (32) x n - T x n x n - y n + y n - T y n + T y n - T x n 0 . From (29), lim n x n - y n = 0 , and lim n T y n - T x n = 0 , we obtain (33) lim n x n - T x n = 0 .

Step  3. We will show that limsup n μ F x * - γ ϕ ( x * ) , x n - x * 0 . Rewrite (13) as (34) x n + 1 = γ λ n ϕ ( x n ) + ( I - μ λ n F ) T y n - v n + β n S x n + ( 1 - β n ) x n . We observe that (35) x n - x n + 1 = x n - γ λ n ϕ ( x n ) - ( I - μ λ n F ) T y n + v n - β n S x n - x n + β n x n = λ n ( μ F - γ ϕ ) x n - λ n μ F x n - ( I - μ λ n F ) T y n + ( I - μ λ n F ) y n - ( I - μ λ n F ) y n + v n + β n ( I - S ) x n = λ n ( μ F - γ ϕ ) x n + λ n μ ( F y n - F x n ) + ( y n - T y n ) - μ λ n F ( y n - T y n ) + ( v n - y n ) + β n ( I - S ) x n = λ n ( μ F - γ ϕ ) x n + λ n μ ( F y n - F x n ) + ( y n - T y n ) - μ λ n F ( y n - T y n ) + λ n ( y n - T y n ) - λ n ( y n - T y n ) + ( v n - y n ) + β n ( I - S ) x n = λ n ( μ F - γ ϕ ) x n + λ n μ ( F y n - F x n ) + λ n ( I - μ F ) ( y n - T y n ) + ( 1 - λ n ) ( y n - T y n ) + ( v n - y n ) + β n ( I - S ) x n . Set (36) z n = x n - x n + 1 λ n , n 0 . We note from (35) that (37) z n = ( μ F - γ ϕ ) x n + μ ( F y n - F x n ) + ( I - μ F ) ( y n - T y n ) + 1 - λ n λ n ( y n - T y n ) + 1 λ n ( v n - y n ) + β n λ n ( I - S ) x n . This yields that, for each x * F ( T ) , (38) z n , x n - x * = ( μ F - γ ϕ ) x n , x n - x * + μ ( F y n - F x n ) , x n - x * + ( I - μ F ) y n - ( I - μ F ) T y n , x n - x * + 1 - λ n λ n y n - T y n , x n - x * + 1 λ n v n - y n , x n - x * + β n λ n ( I - S ) x n , x n - x * = ( μ F - γ ϕ ) x * , x n - x * + ( μ F - γ ϕ ) x n - ( μ F - γ ϕ ) x * , x n - x * + μ ( F y n - F x n ) , x n - x * + ( I - μ F ) y n - ( I - μ F ) T y n , x n - x * + 1 - λ n λ n y n - T y n , x n - x * + 1 λ n v n - y n , x n - x * + β n λ n ( I - S ) x n , x n - x * . In view of (38), ( μ F - γ ϕ ) x n - ( μ F - γ ϕ ) x * , x n - x * is nonnegative due to the monotonicity of μ F - γ ϕ . From (38), we derive that (39) z n , x n - x * ( μ F - γ ϕ ) x * , x n - x * + μ ( F y n - F x n ) , x n - x * + ( I - μ F ) y n - ( I - μ F ) T y n , x n - x * + 1 - λ n λ n y n - T y n , x n - x * + 1 λ n v n - y n , x n - x * + β n λ n ( I - S ) x n , x n - x * . Since (29) implies ( I - μ F ) y n - ( I - μ F ) T y n 0 , as n , from (25), then we get z n 0 . Using (C1) and (30), y n - x n 0 , as n and { x n } is bounded. We obtain from (39) that (40) limsup n ( μ F - γ ϕ ) x * , x n - x * 0 , x * F ( T ) . Since the sequence { x n } is bounded, we can take a subsequence { x n j } of { x n } such that (41) limsup n ( μ F - γ ϕ ) x * , x n - x * = limsup j ( μ F - γ ϕ ) x * , x n j - x * and x n j x ~ . From (33), by the demiclosed principle of the nonexpansive mapping, it follows that x ~ F ( T ) . Then (42) limsup j ( μ F - γ ϕ ) x * , x n j - x * = ( μ F - γ ϕ ) x * , x ~ - x * 0 .

Step  4. Finally, we will prove x n + 1 x * . From (13), we note that (43) y n - x * 2 = P C [ β n S x n + ( 1 - β n ) x n ] - P C x * 2 [ β n S x n + ( 1 - β n ) x n ] - x * 2 β n ( S x n - S x * ) + ( 1 - β n ) ( x n - x * ) + β n ( S x * - x * ) 2 β n ( S x n - S x * ) + ( 1 - β n ) ( x n - x * ) 2 + 2 β n S x * - x * , y n - x * β n x n - x * 2 + ( 1 - β n ) x n - x * 2 + 2 β n S x * - x * , y n - x * x n - x * 2 + 2 β n S x * - x * y n - x * . Using (43), we compute (44) x n + 1 - x * 2 = γ λ n ϕ ( x n ) + ( I - λ n μ F ) T y n - x * 2 = γ λ n ( ϕ ( x n ) - ϕ ( x * ) ) + ( I - λ n μ F ) T y n - ( I - λ n μ F ) x * + ( I - λ n μ F ) x * - x * + γ λ n ϕ ( x * ) 2 = γ λ n ( ϕ ( x n ) - ϕ ( x * ) ) + ( I - λ n μ F ) ( T y n - x * ) + λ n ( γ ϕ ( x * ) - μ F x * ) 2 γ λ n ( ϕ ( x n ) - ϕ ( x * ) ) + ( I - λ n μ F ) ( T y n - x * ) 2 + 2 λ n γ ϕ ( x * ) - μ F x * , x n + 1 - x * γ 2 λ n 2 ϕ ( x n ) - ϕ ( x * ) 2 + ( 1 - λ n τ ) 2 T y n - x * 2 + 2 λ n γ ϕ ( x * ) - μ F x * , x n + 1 - x * + 2 γ λ n ( ϕ ( x n ) - ϕ ( x * ) ) , ( I - μ λ n F ) ( T y n - x * ) γ 2 ρ 2 λ n 2 x n - x * 2 + ( 1 - 2 λ n τ + λ n 2 τ 2 ) y n - x * 2 + 2 λ n γ ϕ ( x * ) - μ F x * , x n + 1 - x * + 2 γ λ n ϕ ( x n ) - ϕ ( x * ) , ( I - μ λ n F ) T y n - ( I - μ λ n F ) x * = γ 2 ρ 2 λ n 2 x n - x * 2 + ( 1 - 2 λ n τ + λ n 2 τ 2 ) y n - x * 2 + 2 λ n γ ϕ ( x * ) - μ F x * , x n + 1 - x * + 2 γ λ n ϕ ( x n ) - ϕ ( x * ) , ( T y n - x * ) - μ λ n F ( T y n - x * ) = γ 2 ρ 2 λ n 2 x n - x * 2 + ( 1 - 2 λ n τ + λ n 2 τ 2 ) y n - x * 2 + 2 λ n γ ϕ ( x * ) - μ F x * , x n + 1 - x * + 2 γ λ n ϕ ( x n ) - ϕ ( x * ) , T y n - x * - 2 γ λ n ϕ ( x n ) - ϕ ( x * ) , μ λ n F ( T y n - x * ) γ 2 ρ 2 λ n 2 x n - x * 2 + ( 1 - 2 λ n τ + λ n 2 τ 2 ) × { x n - x * 2 + 2 β n S x * - x * y n - x * } + 2 λ n γ ϕ ( x * ) - μ F x * , x n + 1 - x * + 2 γ ρ λ n x n - x * T y n - x * - 2 γ ρ μ λ n 2 x n - x * F ( T y n - x * ) [ 1 - λ n ( 2 τ - λ n τ 2 - λ n γ 2 ρ 2 ) ] x n - x * 2 + 2 ε n λ n S x * - x * y n - x * + 2 λ n γ ϕ ( x * ) - μ F x * , x n + 1 - x * + 2 γ ρ λ n x n - x * T y n - x * - 2 γ ρ μ λ n 2 x n - x * F ( T y n - x * ) . Since { x n } , { T y n } , and { F T y n } are all bounded, we can choose a constant M 2 > 0 such that (45) sup n 0 1 2 τ - λ n τ 2 - λ n γ 2 ρ 2 × { 2 γ ρ μ x n - x * F ( T y n - x * ) } M 2 . It follows that (46) x n + 1 - x * 2 [ 1 - λ n ( 2 τ - λ n τ 2 - λ n γ 2 ρ 2 ) ] x n - x * 2 + λ n ( 2 τ - λ n τ 2 - λ n γ 2 ρ 2 ) δ n , where (47) δ n = 2 ε n 2 τ - λ n τ 2 - λ n γ 2 ρ 2 S x * - x * y n - x * + 2 2 τ - λ n τ 2 - λ n γ 2 ρ 2 γ ϕ ( x * ) - μ F x * , x n + 1 - x * + 2 2 τ - λ n τ 2 - λ n γ 2 ρ 2 γ ρ x n - x * T y n - x * - λ n M 2 . Now, applying Lemma 4 and (35), we conclude that x n x * . This completes the proof.

Corollary 6.

Let C be a nonempty closed and convex subset of a real Hilbert space H . Let F : C C be κ -Lipschitzian and η -strongly monotone operators with constant κ and η > 0 , respectively. Let T : C C be a nonexpansive mapping with F ( T ) , and let S : H H be a nonexpansive mapping. Let 0 < μ < 2 η / κ 2 and 0 < γ < τ , where τ = 1 - 1 - μ ( 2 η - μ κ 2 ) . Suppose { x n } is a sequence generated by the following algorithm x 0 C arbitrarily: (48) x n + 1 = ( I - λ n μ F ) T P C [ β n S x n + ( 1 - β n ) x n ] , n 0 , where { β n } , { λ n } ( 0,1 ) satisfy the following conditions (C1)–(C3). Then { x n } converges strongly to x * Ω , which is the unique solution of variational inequality: (49) ( I - μ F ) x * , x - x * 0 , x Ω , where Ω : = V I ( F ( T ) , S ) .

Proof.

Putting ϕ 0 in Theorem 5, we can obtain the desired conclusion immediately.

Corollary 7.

Let C be a nonempty closed and convex subset of a real Hilbert space H . Let ϕ : H H be a ρ -contraction with coefficient ρ [ 0,1 ) , and let T : C C be a nonexpansive mapping with F ( T ) and S : H H a nonexpansive mapping. Suppose { x n } is a sequence generated by the following algorithm, x 0 C , arbitrarily: (50) y n = P C [ β n S x n + ( 1 - β n ) x n ] , x n + 1 = λ n ϕ ( x n ) + ( 1 - λ n ) T y n , n 0 , where { β n } , { λ n } ( 0,1 ) satisfy the following conditions (C1)–(C3). Then { x n } converges strongly to x * Ω , which is the unique solution of variational inequality: (51) ( I - ϕ ) x * , x - x * 0 , x Ω , where Ω : = V I ( F ( T ) , S ) .

Proof.

Putting γ = 1 , μ = 2 , and F I / 2 in Theorem 5, we can obtain the desired conclusion immediately.

Corollary 8.

Let C be a nonempty closed and convex subset of a real Hilbert space H . Let T : C C be a nonexpansive mapping with F ( T ) and let S : H H be a nonexpansive mapping. Suppose { x n } is a sequence generated by the following algorithm, x 0 C , arbitrarily: (52) x n + 1 = ( 1 - λ n ) T P C [ β n S x n + ( 1 - β n ) x n ] , n 0 , where { β n } , { λ n } ( 0,1 ) satisfy the following conditions (C1)–(C3). Then { x n } converges strongly to x * F ( T ) , which is the unique solution of variational inequality: (53) ( I - S ) x * , x - x * 0 , x F ( T ) .

Proof.

Putting ϕ 0 in Corollary 7, we can obtain the desired conclusion immediately.

Corollary 9.

Let C be a nonempty closed and convex subset of a real Hilbert space H . Let ϕ : H H be a ρ -contraction with coefficient ρ [ 0,1 ) , and let T : C C be a nonexpansive mapping with F ( T ) and S : C C a nonexpansive mapping. Suppose { x n } is a sequence generated by the following algorithm, x 0 C , arbitrarily: (54) x n + 1 = λ n x n + ( 1 - λ n ) T [ β n S x n + ( 1 - β n ) x n ] , w w w w w w w w w w w w w w w w w w w w w w i i n 0 , where { β n } , { λ n } ( 0,1 ) satisfy the following conditions (C1)–(C3). Then { x n } converges strongly to x * F ( T ) , which is the unique solution of variational inequality: (55) ( I - S ) x * , x - x * 0 , x F ( T ) .

Proof.

Putting P C I in Corollary 7, we can obtain the desired conclusion immediately.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The first author was supported by the Thailand Research Fund and the King Mongkut’s University of Technology Thonburi (Grant no. RSA5780059). The second author was supported by the Commission on Higher Education, the Thailand Research Fund, and Rajamangala University of Technology Lanna Chiangrai under Grant no. MRG5680157 during the preparation of this paper.

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