Open Access

Relaxed and composite viscosity methods for variational inequalities, fixed points of nonexpansive mappings and zeros of accretive operators

  • Lu-Chuan Ceng1,
  • Abdullah Al-Otaibi2,
  • Qamrul Hasan Ansari3, 4 and
  • Abdul Latif2Email author
Fixed Point Theory and Applications20142014:29

https://doi.org/10.1186/1687-1812-2014-29

Received: 15 July 2013

Accepted: 7 January 2014

Published: 4 February 2014

Abstract

In this paper, we present relaxed and composite viscosity methods for computing a common solution of a general systems of variational inequalities, common fixed points of infinitely many nonexpansive mappings and zeros of accretive operators in real smooth and uniformly convex Banach spaces. The relaxed and composite viscosity methods are based on Korpelevich’s extragradient method, the viscosity approximation method and the Mann iteration method. Under suitable assumptions, we derive some strong convergence theorems for relaxed and composite viscosity algorithms not only in the setting of a uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gâteaux differentiable norm. The results presented in this paper improve, extend, supplement, and develop the corresponding results given in the literature.

1 Introduction

The theory of variational inequalities is well established and a tool to solve many problems arising from science, engineering, social sciences, etc., see, for example, [14] and the references therein. One of the interesting directions, from the research view point, in the theory of variational inequalities is to develop some new iterative methods for computing the approximate solutions of different kinds of variational inequalities. In 1976, Korpelevich [5] proposed an iterative algorithm for solving variational inequalities (VI) in the finite dimensional space setting, It is now known as the extragradient method. Korpelevich’s extragradient method has received great attention by many authors, who improved it in various ways and in different directions, see, for example [616] and the references therein. In the recent past, several iterative methods for solving VI were proposed and analyzed in [1724] in the setting of Banach spaces. In the last three decades, the system of variational inequalities is used as a tool to study the Nash equilibrium problem for a finite or infinite number of players, see, for example, [2, 3, 25, 26] and the references therein. Cai and Bu [20] considered a system of two variational inequalities (SVI) in the setting of real smooth Banach spaces. They proposed and analyzed an iterative method for computing the approximate solutions of system of variational inequalities. Such a solution is also a common fixed point of a family of nonexpansive mappings.

One of the most interesting problems in nonlinear analysis is to find a zero of an accretive operator. In 2007, Aoyama et al. [27] suggested a Halpern type iterative method for finding a common fixed point of a countable family of nonexpansive mappings and a zero of an accretive operator. They studied the strong convergence of the sequence generated by the proposed method in the setting of a uniformly convex Banach space having a uniformly Gâreaux differentiable norm. Ceng et al. [28] introduced and analyzed the composite iterative scheme to compute a zero of m-accretive operator A defined on a uniformly smooth Banach space or a reflexive Banach space having a weakly sequentially continuous duality mapping. It is shown that the iterative process in each case converges strongly to a zero of A. Subsequently, Jung [29] studied a viscosity approximation method, which generalizes the composite method in [28], to investigate the zero of an accretive operator.

During the last decade, several iterative methods have been proposed and analyzed to find a common solution of two different fixed point problems, a fixed point problem and a variational inequality problem, a fixed point problem for a family of nonexpansive mappings and a variational inequality problem or a fixed point problem and a system of variational inequalities, etc. See, for example, [8, 16, 20, 30, 31] and the references therein.

In the present paper, we mainly propose two different methods, namely, relaxed viscosity method and composite viscosity method, to find a common fixed point of an infinite family of nonexpansive mappings, a system of variational inequalities and zero of an accretive operator in the setting of a uniformly convex and 2-uniformly smooth Banach spaces. These methods are based on Korpelevich’s extragradient method, viscosity approximation method and Mann iteration method. Under suitable assumptions, we derive some strong convergence theorems for relaxed and composite viscosity algorithms not only in the setting of a uniformly convex and 2-uniformly smooth Banach space but also in the setting of uniformly convex Banach spaces having a uniformly Gâteaux differentiable norm. The results presented in this paper improve, extend, supplement, and develop the corresponding results in [10, 20, 24, 29, 30].

2 Preliminaries

Throughout the paper, unless otherwise specified, we adopt the following assumptions and notations.

Let X be a real Banach space whose dual space is denoted by X . Let C be a nonempty closed convex subset of X. We denote by Ξ C the set of all contractive mappings from C into itself.

The normalized duality mapping J : X 2 X is defined by
J ( x ) = { x X : x , x = x 2 = x 2 } , x X ,

where , denotes the generalized duality pairing. It is an immediate consequence of the Hahn-Banach Theorem that J ( x ) is nonempty for each x X .

Let U = { x X : x = 1 } denote the unite sphere in X. A Banach space X is said to be uniformly convex if for each ϵ ( 0 , 2 ] , there exists δ > 0 such that for all x , y U ,
x y ϵ x + y 2 1 δ .
It is well known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space X is said to be smooth if the limit
lim t 0 x + t y x t ,
exists for all x , y U ; in this case, X is also said to have a Gâteaux differentiable norm. X is said to have a uniformly Gâteaux differentiable norm if for each y U , the limit is attained uniformly for all x U . Moreover, it is said to be uniformly smooth if this limit is attained uniformly for all x , y U . The norm of X is said to be Fréchet differentiable if, for each x U , this limit is attained uniformly for all y U . A function ρ : [ 0 , ) [ 0 , ) defined by
ρ ( τ ) = sup { 1 2 ( x + y + x y ) 1 : x , y X , x = 1 , y = τ }

is called the modulus of smoothness of X. It is well known that X is uniformly smooth if and only if lim τ 0 ρ ( τ ) / τ = 0 . Let q be a fixed real number with 1 < q 2 . Then a Banach space X is said to be q-uniformly smooth if there exists a constant c > 0 such that ρ ( τ ) c τ q for all τ > 0 . As pointed out in [32], no Banach space is q-uniformly smooth for q > 2 . In addition, it is also known that J is single-valued if and only if X is smooth, whereas if X is uniformly smooth, then the mapping J is norm-to-norm uniformly continuous on bounded subsets of X. If X has a uniformly Gâteaux differentiable norm then the duality mapping J is norm-to-weak uniformly continuous on bounded subsets of X. For further details of the geometry of Banach spaces, we refer to [3335].

Now, we present some lemmas which will be used in the sequel.

Lemma 2.1 [36]

Let X be a 2-uniformly smooth Banach space. Then
x + y 2 x 2 + 2 y , J ( x ) + 2 κ y 2 , x , y X ,

where κ is the 2-uniformly smooth constant of X.

The following lemma is an immediate consequence of the subdifferential inequality of the function 1 2 2 .

Lemma 2.2 [37]

Let X be a real Banach space X. Then, for all x , y X ,
  1. (a)

    x + y 2 x 2 + 2 y , j ( x + y ) , j ( x + y ) J ( x + y ) ;

     
  2. (b)

    x + y 2 x 2 + 2 y , j ( x ) , j ( x ) J ( x ) .

     

Lemma 2.3 [36]

Given a number r > 0 . A real Banach space X is uniformly convex if and only if there exists a continuous strictly increasing function g : [ 0 , ) [ 0 , ) , g ( 0 ) = 0 , such that
λ x + ( 1 λ ) y 2 λ x 2 + ( 1 λ ) y 2 λ ( 1 λ ) g ( x y )

for all λ [ 0 , 1 ] and x , y X such that x r and y r .

Lemma 2.4 [38]

Let X be a uniformly convex Banach space and B r = { x X : x r } , r > 0 . Then there exists a continuous, strictly increasing, and convex function g : [ 0 , ] [ 0 , ] , g ( 0 ) = 0 such that
α x + β y + γ z 2 α x 2 + β y 2 + γ z 2 α β g ( x y )

for all x , y , z B r and all α , β , γ [ 0 , 1 ] with α + β + γ = 1 .

Proposition 2.1 [22]

Let X be a real smooth and uniform convex Banach space and r > 0 . Then there exists a strictly increasing, continuous, and convex function g : [ 0 , 2 r ] R , g ( 0 ) = 0 such that
g ( x y ) x 2 2 x , J ( y ) + y 2 , x , y B r ,

where B r = { x X : x r } .

Lemma 2.5 [39]

Let C be a nonempty closed convex subset of a strictly convex Banach space X. Let { T n } n = 0 be a sequence of nonexpansive mappings from C into itself such that n = 0 Fix ( T n ) is nonempty. Let { λ n } be a sequence of positive numbers with n = 0 λ n = 1 . Then a mapping S : C C defined by S x = n = 0 λ n T n x , for all x C , is well defined and nonexpansive, and Fix ( S ) = n = 0 Fix ( T n ) .

Lemma 2.6 [40]

Let { x n } and { z n } be bounded sequences in a Banach space X and { β n } be a sequence of nonnegative numbers in [ 0 , 1 ] with 0 < lim inf n β n lim sup n β n < 1 . Suppose that x n + 1 = β n x n + ( 1 β n ) z n for all integers n 0 and lim sup n ( z n + 1 z n x n + 1 x n ) 0 . Then lim n x n z n = 0 .

Lemma 2.7 [41]

Let { s n } be a sequence of nonnegative real numbers satisfying
s n + 1 ( 1 α n ) s n + α n β n + γ n , n 0 ,
where { α n } , { β n } , and { γ n } satisfy the conditions:
  1. (i)

    { α n } [ 0 , 1 ] and n = 0 α n = ;

     
  2. (ii)

    lim sup n β n 0 ;

     
  3. (iii)

    γ n 0 , n 0 , and n = 0 γ n < .

     

Then lim sup n s n = 0 .

A mapping T : C C is called nonexpansive if T x T y x y for every x , y C . The set of fixed points of T is denoted by Fix ( T ) . A mapping A : C X is said to be
  1. (a)
    accretive if for each x , y C , there exists j ( x y ) J ( x y ) such that
    A x A y , j ( x y ) 0 ;
     
  2. (b)
    α-strongly accretive if for each x , y C , there exists j ( x y ) J ( x y ) such that
    A x A y , j ( x y ) α x y 2 , for some  α ( 0 , 1 ) ;
     
  3. (c)
    β-inverse strongly accretive if for each x , y C , there exists j ( x y ) J ( x y ) such that
    A x A y , j ( x y ) β A x A y 2 , for some  β > 0 ;
     
  4. (d)
    λ-strictly pseudocontractive [18, 42] if for each x , y C , there exists j ( x y ) J ( x y ) such that
    A x A y , j ( x y ) x y 2 λ x y ( A x A y ) 2 , for some  λ ( 0 , 1 ) .
     

It is worth to emphasize that the definition of the inverse strongly accretive mapping is based on that of the inverse strongly monotone mapping [43].

Lemma 2.8 [[20], Lemma 2.8]

Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space X and for each i = 1 , 2 , B i : C X be an α i -inverse strongly accretive mapping. Then, for each i = 1 , 2 ,
( I μ i B i ) x ( I μ i B i ) y 2 x y 2 + 2 μ i ( μ i κ 2 α i ) B i x B i y 2 , x , y C ,

where μ i > 0 . In particular, if 0 < μ i α i κ 2 , then I μ i B i is nonexpansive for each i = 1 , 2 .

Let C be a nonempty closed convex subset of a Banach space X and T : C C be a nonexpansive mapping with Fix ( T ) . For all t ( 0 , 1 ) and f Ξ C , let x t C be a unique fixed point of the contraction x t f ( x ) + ( 1 t ) T x on C, that is,
x t = t f ( x t ) + ( 1 t ) T x t .

Lemma 2.9 [44, 45]

Let X be an uniformly smooth Banach space, or a reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. Let C be a nonempty closed convex subset of X, T : C C be a nonexpansive mapping with Fix ( T ) , and f Ξ C . Then the net { x t } defined by x t = t f ( x t ) + ( 1 t ) T x t converges strongly to a point in Fix ( T ) . If we define a mapping Q : Ξ C Fix ( T ) by Q ( f ) : = s lim t 0 x t , f Ξ C , then Q ( f ) solves the VIP
( I f ) Q ( f ) , J ( Q ( f ) p ) 0 , f Ξ C , p Fix ( T ) .
Recall that a (possibly set-valued mapping) operator A X × X with domain D ( A ) and range R ( A ) in X is accretive if, for each x i D ( A ) and y i A x i ( i = 1 , 2 ), there exists a j ( x 1 x 2 ) J ( x 1 x 2 ) such that y 1 y 2 , j ( x 1 x 2 ) 0 . An accretive operator A is said to satisfy the range condition if D ( A ) ¯ R ( I + r A ) for all r > 0 . An accretive operator A is m-accretive if R ( I + r A ) = X for each r > 0 . If A is an accretive operator which satisfies the range condition, then we define a mapping J r : R ( I + r A ) D ( A ) by J r = ( I + r A ) 1 for each r > 0 , which is called the resolvent of A. It is well known that J r is nonexpansive and Fix ( J r ) = A 1 0 for all r > 0 . Therefore,
Fix ( J r ) = A 1 0 = { z D ( A ) : 0 A z } .

If A 1 0 , then the inclusion 0 A z is solvable.

Proposition 2.2 (Resolvent Identity [46])

For λ > 0 , μ > 0 and x X ,
J λ x = J μ ( μ λ x + ( 1 μ λ ) J λ x ) .
Let D be a subset of C. A mapping Π : C D is said to be sunny if
Π [ Π ( x ) + t ( x Π ( x ) ) ] = Π ( x ) ,

whenever Π ( x ) + t ( x Π ( x ) ) C for all x C and t 0 . A mapping Π : C C is called a retraction if Π 2 = Π . If a mapping Π : C C is a retraction, then Π ( z ) = z for every z R ( Π ) where R ( Π ) is the range of Π. A subset D of C is called a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction from C onto D.

Lemma 2.10 [23]

Let C be a nonempty closed convex subset of a real smooth Banach space X, D be a nonempty subset of C and Π be a retraction of C onto D. Then the following statements are equivalent:
  1. (a)

    Π is sunny and nonexpansive;

     
  2. (b)

    Π ( x ) Π ( y ) 2 x y , J ( Π ( x ) Π ( y ) ) , x , y C ;

     
  3. (c)

    x Π ( x ) , J ( y Π ( x ) ) 0 , x C , y D .

     

It is well known that if X = H a Hilbert space, then a sunny nonexpansive retraction Π C is coincident with the metric projection from X onto C, that is, Π C = P C . If C is a nonempty closed convex subset of a strictly convex and uniformly smooth Banach space X and if T : C C is a nonexpansive mapping with the fixed point set Fix ( T ) , then the set Fix ( T ) is a sunny nonexpansive retract of C.

Lemma 2.11 [[20], Lemma 2.9]

Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space X and Π C be a sunny nonexpansive retraction from X onto C. For each i = 1 , 2 , let B i : C X be an α i -inverse strongly accretive mapping and G : C C be defined by
G x = Π C [ Π C ( x μ 2 B 2 x ) μ 1 B 1 Π C ( x μ 2 B 2 x ) ] , x C .

If 0 < μ i α i κ 2 for each i = 1 , 2 , then G : C C is nonexpansive.

Let f Ξ C with a contractive coefficient ρ ( 0 , 1 ) , { T n } n = 0 be a sequence of nonexpansive self-mappings on C and { λ n } n = 0 be a sequence of nonnegative numbers in [ 0 , 1 ] . For any n 0 , a self-mapping W n on C defined by
{ U n , n + 1 = I , U n , n = λ n T n U n , n + 1 + ( 1 λ n ) I , U n , n 1 = λ n 1 T n 1 U n , n + ( 1 λ n 1 ) I , U n , k = λ k T k U n , k + 1 + ( 1 λ k ) I , U n , k 1 = λ k 1 T k 1 U n , k + ( 1 λ k 1 ) I , U n , 1 = λ 1 T 1 U n , 2 + ( 1 λ 1 ) I , W n = U n , 0 = λ 0 T 0 U n , 1 + ( 1 λ 0 ) I
(2.1)

is called W-mapping [47] generated by T n , T n 1 , , T 0 and λ n , λ n 1 , , λ 0 .

Lemma 2.12 [[37], Lemma 3.2]

Let C be a nonempty closed convex subset of a strictly convex Banach space X. Let { T n } n = 0 be a sequence of nonexpansive self-mappings on C such that n = 0 Fix ( T n ) and { λ n } n = 0 be a sequence of positive numbers in ( 0 , b ] for some b ( 0 , 1 ) . Then, for every x C and k 0 , the limit lim n U n , k x exists.

B using Lemma 2.12, we define a W-mapping W : C C generated by the sequences { T n } n = 0 and { λ n } n = 0 by
W x = lim n W n x = lim n U n , 0 x , for every  x C .

Throughout this paper, we assume that { λ n } n = 0 is a sequence of positive numbers in ( 0 , b ] for some b ( 0 , 1 ) .

Lemma 2.13 [[37], Lemma 3.3]

Let C be a nonempty closed convex subset of a strictly convex Banach space X. Let { T n } n = 0 be a sequence of nonexpansive self-mappings on C such that n = 0 Fix ( T n ) and let { λ n } n = 0 be a sequence of positive numbers in ( 0 , b ] for some b ( 0 , 1 ) . Then Fix ( W ) = n = 0 Fix ( T n ) .

Let μ be a continuous linear functional on l and s = ( a 0 , a 1 , ) l . We write μ n ( a n ) instead of μ ( s ) . μ is called a Banach limit if μ satisfies μ = μ n ( 1 ) = 1 and μ n ( a n + 1 ) = μ n ( a n ) for all ( a 0 , a 1 , ) l . If μ is a Banach limit, then the following implications hold:
  1. (a)

    for all n 0 , a n c n implies μ n ( a n ) μ n ( c n ) ;

     
  2. (b)

    μ n ( a n + r ) = μ n ( a n ) for any fixed positive integer r;

     
  3. (c)

    lim inf n a n μ n ( a n ) lim sup n a n for all ( a 0 , a 1 , ) l .

     

Lemma 2.14 [48]

Let a R be a real number and a sequence { a n } l satisfy the condition μ n ( a n ) a for all Banach limits μ. If lim sup n ( a n + r a n ) 0 , then lim sup n a n a .

In particular, if r = 1 in Lemma 2.14, then we obtain the following corollary.

Corollary 2.1 [49]

Let a R be a real number and a sequence { a n } l satisfy the condition μ n ( a n ) a for all Banach limits μ. If lim sup n ( a n + 1 a n ) 0 , then lim sup n a n a .

3 Formulations

Let C be a nonempty closed convex subset of a smooth Banach space X, B 1 , B 2 : C X be nonlinear mappings and μ 1 and μ 2 be two positive constants. The problem of system of variational inequalities (SVI) in the setting of a real smooth Banach space X is to find ( x , y ) C × C such that
{ μ 1 B 1 y + x y , J ( x x ) 0 , x C , μ 2 B 2 x + y x , J ( x y ) 0 , x C .
(3.1)

The set of solutions of SVI (3.1) is denoted by SVI ( C , B 1 , B 2 ) . Very recently, Cai and Bu [20] constructed an iterative algorithm for solving SVI (3.1) and a common fixed point problem of an infinite family of nonexpansive mappings in a uniformly convex and 2-uniformly smooth Banach space. They studied the strong convergence of the proposed algorithm.

In particular, if X = H , a real Hilbert space, then SVI (3.1) reduces to the following problem of SVI of finding ( x , y ) C × C such that
{ μ 1 B 1 y + x y , x x 0 , x C , μ 2 B 2 x + y x , x y 0 , x C .
(3.2)
Further, if B 1 = B 2 = A , where A : C X is an operator, and x = y , then the SVI (3.2) reduces to the classical variational inequality problem (VIP) of finding x C such that
A x , x x 0 , x C .
(3.3)

The solution set of the VIP (3.3) is denoted by VI ( C , A ) . For details and applications of theory of variational inequalities, we refer to [14] and the references therein.

Recently, Ceng et al. [10] transformed problem (3.2) into a fixed point problem in the following way.

Lemma 3.1 [10]

For given x ¯ , y ¯ C , ( x ¯ , y ¯ ) is a solution of problem (3.2) if and only if x ¯ is a fixed point of the mapping G : C C defined by
G ( x ) = P C [ P C ( x μ 2 B 2 x ) μ 1 B 1 P C ( x μ 2 B 2 x ) ] , x C ,
(3.4)

where y ¯ = P C ( x ¯ μ 2 B 2 x ¯ ) and P C is the projection of H onto C.

In particular, if for each i = 1 , 2 , B i : C H is a β i -inverse strongly monotone mapping, then G is a nonexpansive mapping provided μ i ( 0 , 2 β i ) for each i = 1 , 2 .

In particular, whenever X is a real smooth Banach space, B 1 B 2 A and x = y , then SVI (3.1) reduces to the variational inequality problem (VIP) of finding x C such that
A x , J ( x x ) 0 , x C ,
(3.5)

which was considered by Aoyama et al. [17]. Note that VIP (3.5) is connected with the fixed point problem for nonlinear mapping [44], the problem of finding a zero point of a nonlinear operator [50] and so on. It is clear that VIP (3.5) extends VIP (3.3) from Hilbert spaces to Banach spaces. For further study on VIP in the setting of Banach spaces, we refer to [17, 21] and the references therein.

Define a mapping G : C C by
G ( x ) : = Π C ( I μ 1 B 1 ) Π C ( I μ 2 B 2 ) x , x C .
(3.6)

The fixed point set of G is denoted by Ω.

Lemma 3.2 Let C be a nonempty closed convex subset of a smooth Banach space X. Let Π C be a sunny nonexpansive retraction from X onto C and B 1 , B 2 : C X be nonlinear mappings. Then ( x , y ) C × C is a solution of SVI (3.1) if and only if x = Π C ( y μ 1 B 1 y ) , where y = Π C ( x μ 2 B 2 x ) .

Proof We rewrite SVI (3.1) as
{ x ( y μ 1 B 1 y ) , J ( x x ) 0 , x C , y ( x μ 2 B 2 x ) , J ( x y ) 0 , x C ,
which is obviously equivalent to
{ x = Π C ( y μ 1 B 1 y ) , y = Π C ( x μ 2 B 2 x ) ,

because of Lemma 2.10. This completes the proof. □

In terms of Lemma 3.2, we observe that
x = Π C [ Π C ( x μ 2 B 2 x ) μ 1 B 1 Π C ( x μ 2 B 2 x ) ] ,

which implies that x is a fixed point of the mapping G.

Motivated and inspired by the research going on in this area, we introduce some relaxed and composite viscosity methods for finding a zero of an accretive operator A X × X such that D ( A ) ¯ C r > 0 R ( I + r A ) , solving SVI (3.1) and the common fixed point problem of an infinite family { T n } of nonexpansive self-mappings on C. Our methods are based on Korpelevich’s extragradient method, the viscosity approximation method, and Mann’s iteration method. Under suitable assumptions, we derive some strong convergence theorems for relaxed and composite viscosity algorithms not only in the setting of uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gâteaux differentiable norm. The results presented in this paper improve, extend, supplement, and develop the corresponding results given in [10, 20, 24, 29, 48].

4 Relaxed viscosity algorithms and convergence criteria

In this section, we introduce relaxed viscosity algorithms in the setting of real smooth uniformly convex Banach spaces and study the strong convergence of the sequences generated by the proposed algorithms.

Throughout this paper, we denote by Ω the fixed point set of the mapping G = Π C ( I μ 1 B 1 ) Π C ( I μ 2 B 2 ) .

Assumption 4.1 Let { α n } , { β n } , { γ n } , { δ n } , { σ n } be the sequences in ( 0 , 1 ) such that α n + β n + γ n + δ n = 1 for all n 0 . Suppose that the following conditions hold:
  1. (i)

    lim n α n = 0 and n = 0 α n = ;

     
  2. (ii)

    { γ n } , { δ n } [ c , d ] for some c , d ( 0 , 1 ) ;

     
  3. (iii)

    lim n ( | σ n σ n 1 | + | β n β n 1 | + | γ n γ n 1 | + | δ n δ n 1 | ) = 0 ;

     
  4. (iv)

    n = 1 | r n r n 1 | < and r n ε > 0 for all n 0 ;

     
  5. (v)

    0 < lim inf n β n lim sup n β n < 1 and 0 < lim inf n σ n lim sup n σ n < 1 .

     
Theorem 4.1 Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space X. Let Π C be a sunny nonexpansive retraction from X onto C and A X × X be an accretive operator such that D ( A ) ¯ C r > 0 R ( I + r A ) . For each i = 1 , 2 , let B i : C X be α i -inverse strongly accretive mapping and f : C C be a contraction with coefficient ρ ( 0 , 1 ) . Let { T i } i = 0 be an infinite family of nonexpansive mappings from C into itself such that F : = i = 0 Fix ( T i ) Ω A 1 0 with 0 < μ i < α i κ 2 for i = 1 , 2 . Assume that Assumption  4.1 holds. For arbitrarily given x 0 C , let { x n } be a sequence generated by
{ y n = σ n x n + ( 1 σ n ) J r n G x n , x n + 1 = α n f ( y n ) + β n x n + γ n W n y n + δ n J r n G y n , n 0 ,
(4.1)
where W n is the W-mapping generated by (2.1). Then
  1. (a)

    lim n x n + 1 x n = 0 ;

     
  2. (b)
    the sequence { x n } n = 0 converges strongly to some q F which is a unique solution of the following variational inequality problem (VIP):
    ( I f ) q , J ( q p ) 0 , p F ,
     

provided β n β for some fixed β ( 0 , 1 ) .

Proof We first claim that the sequence { x n } is bounded. Indeed, take a fixed p F arbitrarily. Then we get p = G p , p = W n p , and p = J r n p for all n 0 . By Lemma 2.11, G is nonexpansive. Then, from (4.1), we have
y n p σ n x n p + ( 1 σ n ) J r n G x n p σ n x n p + ( 1 σ n ) G x n p σ n x n p + ( 1 σ n ) x n p = x n p
(4.2)
and
x n + 1 p α n f ( y n ) p + β n x n p + γ n W n y n p + δ n J r n G y n p α n ( f ( y n ) f ( p ) + f ( p ) p ) + β n x n p + γ n y n p + δ n G y n p α n ( ρ y n p + f ( p ) p ) + β n x n p + γ n y n p + δ n y n p α n ( ρ x n p + f ( p ) p ) + β n x n p + γ n x n p + δ n x n p = ( 1 α n ( 1 ρ ) ) x n p + α n ( 1 ρ ) f ( p ) p 1 ρ max { x n p , f ( p ) p 1 ρ } .
By induction, we obtain
x n p max { x 0 p , f ( p ) p 1 ρ } , n 0 .
(4.3)

Hence, { x n } is bounded, and so are the sequences { y n } , { G x n } , { G y n } , and { f ( y n ) } .

Next we show that
lim n x n + 1 x n = 0 .
(4.4)
We note that x n + 1 can be rewritten as follows:
x n + 1 = β n x n + ( 1 β n ) z n ,
where z n = α n f ( y n ) + γ n W n y n + δ n J r n G y n 1 β n . Observe that
z n z n 1 = α n f ( y n ) + γ n W n y n + δ n J r n G y n 1 β n α n 1 f ( y n 1 ) + γ n 1 W n 1 y n 1 + δ n 1 J r n 1 G y n 1 1 β n 1 = x n + 1 β n x n 1 β n x n β n 1 x n 1 1 β n 1 = x n + 1 β n x n 1 β n x n β n 1 x n 1 1 β n + x n β n 1 x n 1 1 β n x n β n 1 x n 1 1 β n 1 x n + 1 β n x n 1 β n x n β n 1 x n 1 1 β n + x n β n 1 x n 1 1 β n x n β n 1 x n 1 1 β n 1 = 1 1 β n x n + 1 β n x n ( x n β n 1 x n 1 ) + | 1 1 β n 1 1 β n 1 | x n β n 1 x n 1 = 1 1 β n x n + 1 β n x n ( x n β n 1 x n 1 ) + | β n β n 1 | ( 1 β n 1 ) ( 1 β n ) x n β n 1 x n 1 = 1 1 β n × α n f ( y n ) + γ n W n y n + δ n J r n G y n α n 1 f ( y n 1 ) γ n 1 W n 1 y n 1 δ n 1 J r n 1 G y n 1 + | β n β n 1 | ( 1 β n 1 ) ( 1 β n ) x n β n 1 x n 1 1 1 β n [ α n f ( y n ) f ( y n 1 ) + γ n W n y n W n 1 y n 1 + δ n J r n G y n J r n 1 G y n 1 + | α n α n 1 | f ( y n 1 ) + | γ n γ n 1 | W n 1 y n 1 + | δ n δ n 1 | J r n 1 G y n 1 ] + | β n β n 1 | ( 1 β n 1 ) ( 1 β n ) x n β n 1 x n 1 .
(4.5)
On the other hand, if r n 1 r n , using the resolvent identity in Proposition 2.2,
J r n x n = J r n 1 ( r n 1 r n x n + ( 1 r n 1 r n ) J r n x n ) ,
we get
J r n G x n J r n 1 G x n 1 = J r n 1 ( r n 1 r n G x n + ( 1 r n 1 r n ) J r n G x n ) J r n 1 G x n 1 r n 1 r n G x n G x n 1 + ( 1 r n 1 r n ) J r n G x n G x n 1 x n x n 1 + r n r n 1 r n J r n G x n G x n 1 x n x n 1 + 1 ε | r n r n 1 | J r n G x n G x n 1 .
If r n r n 1 , then it is easy to see that
J r n G x n J r n 1 G x n 1 x n 1 x n + 1 ε | r n 1 r n | J r n 1 G x n 1 G x n .
By combining the above cases, we obtain
J r n G x n J r n 1 G x n 1 x n 1 x n + | r n 1 r n | ε sup n 1 { J r n G x n G x n 1 + J r n 1 G x n 1 G x n } , n 1 .
Similarly, we have
J r n G y n J r n 1 G y n 1 y n 1 y n + | r n 1 r n | ε sup n 1 { J r n G y n G y n 1 + J r n 1 G y n 1 G y n } , n 1 .
Therefore, we obtain
{ J r n G x n J r n 1 G x n 1 x n 1 x n + | r n 1 r n | M 0 , J r n G y n J r n 1 G y n 1 y n 1 y n + | r n 1 r n | M 0 , n 1 ,
(4.6)
where
sup n 1 { 1 ε ( J r n G x n G x n 1 + J r n 1 G x n 1 G x n ) } M 0 ,
and
sup n 1 { 1 ε ( J r n G y n G y n 1 + J r n 1 G y n 1 G y n ) } M 0 ,
for some M 0 > 0 . Since T i and U n , i are nonexpansive, from (2.1), we deduce that for each n 1
W n y n 1 W n 1 y n 1 = λ 0 T 0 U n , 1 y n 1 λ 0 T 0 U n 1 , 1 y n 1 W n y n 1 W n 1 y n 1 λ 0 U n , 1 y n 1 U n 1 , 1 y n 1 W n y n 1 W n 1 y n 1 = λ 0 λ 1 T 1 U n , 2 y n 1 λ 1 T 1 U n 1 , 2 y n 1 W n y n 1 W n 1 y n 1 λ 0 λ 1 U n , 2 y n 1 U n 1 , 2 y n 1 W n y n 1 W n 1 y n 1 W n y n 1 W n 1 y n 1 ( i = 0 n 1 λ i ) U n , n y n 1 U n 1 , n y n 1 W n y n 1 W n 1 y n 1 M i = 0 n 1 λ i , for some constant  M > 0 .
(4.7)
By simple computations, we obtain
y n y n 1 = σ n ( x n x n 1 ) + ( σ n σ n 1 ) ( x n 1 J r n 1 G x n 1 ) + ( 1 σ n ) ( J r n G x n J r n 1 G x n 1 ) .
It follows from (4.6) that
y n y n 1 σ n x n x n 1 + | σ n σ n 1 | x n 1 J r n 1 G x n 1 + ( 1 σ n ) J r n G x n J r n 1 G x n 1 σ n x n x n 1 + | σ n σ n 1 | x n 1 J r n 1 G x n 1 + ( 1 σ n ) [ x n 1 x n + | r n 1 r n | M 0 ] x n x n 1 + | σ n σ n 1 | x n 1 J r n 1 G x n 1 + | r n 1 r n | M 0 .
(4.8)
Taking into account that 0 < lim inf n β n lim sup n β n < 1 , without loss of generality, we may assume that { β n } [ c ˆ , d ˆ ] . Utilizing (4.5)-(4.8), we have
z n z n 1 1 1 β n [ α n f ( y n ) f ( y n 1 ) + γ n W n y n W n 1 y n 1 + δ n J r n G y n J r n 1 G y n 1 + | α n α n 1 | f ( y n 1 ) + | γ n γ n 1 | W n 1 y n 1 + | δ n δ n 1 | J r n 1 G y n 1 ] + | β n β n 1 | ( 1 β n 1 ) ( 1 β n ) x n β n 1 x n 1 1 1 β n [ α n f ( y n ) f ( y n 1 ) + γ n W n y n W n y n 1 + δ n J r n G y n J r n 1 G y n 1 + | α n α n 1 | f ( y n 1 ) + | γ n γ n 1 | W n 1 y n 1 + | δ n δ n 1 | J r n 1 G y n 1 + γ n W n y n 1 W n 1 y n 1 ] + | β n β n 1 | ( 1 β n 1 ) ( 1 β n ) x n β n 1 x n 1 1 1 β n [ α n ρ y n y n 1 + γ n y n y n 1 + δ n ( y n 1 y n + | r n 1 r n | M 0 ) + | α n α n 1 | f ( y n 1 ) + | γ n γ n 1 | W n 1 y n 1 + | δ n δ n 1 | J r n 1 G y n 1 + γ n M i = 0 n 1 λ i ] + | β n β n 1 | ( 1 β n 1 ) ( 1 β n ) x n β n 1 x n 1 = 1 1 β n [ ( 1 β n α n ( 1 ρ ) ) y n y n 1 + 1 1 β n [ δ n | r n 1 r n | M 0 + | α n α n 1 | f ( y n 1 ) ] + | γ n γ n 1 | W n 1 y n 1 + | δ n δ n 1 | J r n 1 G y n 1 + γ n M i = 0 n 1 λ i ] + | β n β n 1 | ( 1 β n 1 ) ( 1 β n ) α n 1 f ( y n 1 ) + γ n 1 W n 1 y n 1 + J r n 1 G y n 1 = ( 1 α n ( 1 ρ ) 1 β n ) y n y n 1 + 1 1 β n [ δ n | r n 1 r n | M 0 + | α n α n 1 | f ( y n 1 ) + | γ n γ n 1 | W n 1 y n 1 + | δ n δ n 1 | J r n 1 G y n 1 + γ n M i = 0 n 1 λ i ] + | β n β n 1 | ( 1 β n 1 ) ( 1 β n ) α n 1 f ( y n 1 ) + γ n 1 W n 1 y n 1 + J r n 1 G y n 1 y n y n 1 + 1 1 β n [ δ n | r n 1 r n | M 0 + | α n α n 1 | f ( y n 1 ) + | γ n γ n 1 | W n 1 y n 1 + | δ n δ n 1 | J r n 1 G y n 1 + γ n M i = 0 n 1 λ i ] + | β n β n 1 | ( 1 β n 1 ) ( 1 β n ) α n 1 f ( y n 1 ) + γ n 1 W n 1 y n 1 + J r n 1 G y n 1 x n x n 1 + | σ n σ n 1 | x n 1 J r n 1 G x n 1 + | r n 1 r n | M 0 + 1 1 β n [ δ n | r n 1 r n | M 0 + | α n α n 1 | f ( y n 1 ) + | γ n γ n 1 | W n 1 y n 1 + | δ n δ n 1 | J r n 1 G y n 1 + γ n M i = 0 n 1 λ i ] + | β n β n 1 | ( 1 β n 1 ) ( 1 β n ) α n 1 f ( y n 1 ) + γ n 1 W n 1 y n 1 + δ n 1 J r n 1 G y n 1 x n x n 1 + [ | σ n σ n 1 | + | α n α n 1 | + | β n β n 1 | + | γ n γ n 1 | + | δ n δ n 1 | + i = 0 n 1 λ i ] M 1 ,
(4.9)
where sup n 0 { 1 ( 1 d ˆ ) 2 ( f ( y n ) + W n y n + J r n G y n + x n J r n G x n + M + 2 M 0 ) } M 1 for some M 1 > 0 . Thus, it follows from (4.9) and conditions (i), (iii), (iv) that
lim n ( z n z n 1 x n x n 1 ) 0 .
Since 0 < lim inf n β n lim sup n β n < 1 , by Lemma 2.6, we get
lim n x n z n = 0 .
Consequently,
lim n x n + 1 x n = lim n ( 1 β n ) z n x n = 0 .
Now we show that x n G x n 0 as n . Indeed, by Lemma 2.3 and (4.1), we get
y n p 2 = σ n ( x n p ) + ( 1 σ n ) ( J r n G x n p ) 2 σ n x n p 2 + ( 1 σ n ) J r n G x n p 2 σ n ( 1 σ n ) g ( x n J r n G x n ) σ n x n p 2 + ( 1 σ n ) x n p 2 σ n ( 1 σ n ) g ( x n J r n G x n ) = x n p 2 σ n ( 1 σ n ) g ( x n J r n G x n ) .
(4.10)
By Lemma 2.2(a), (4.1), and (4.10), we obtain
x n + 1 p 2 = α n ( f ( y n ) f ( p ) ) + β n ( x n p ) + γ n ( W n y n p ) + δ n ( J r n G y n p ) + α n ( f ( p ) p ) 2 α n ( f ( y n ) f ( p ) ) + β n ( x n p ) + γ n ( W n y n p ) + δ n ( J r n G y n p ) 2 + 2 α n f ( p ) p , J ( x n + 1 p ) α n f ( y n ) f ( p ) 2 + β n x n p 2 + γ n W n y n p 2 + δ n J r n G y n p 2 + 2 α n f ( p ) p , J ( x n + 1 p ) α n ρ 2 y n p 2 + β n x n p 2 + γ n y n p 2 + δ n G y n p 2 + 2 α n f ( p ) p , J ( x n + 1 p ) α n ρ y n p 2 + β n x n p 2 + γ n y n p 2 + δ n y n p 2 + 2 α n f ( p ) p , J ( x n + 1 p ) = ( 1 β n α n ( 1 ρ ) ) y n p 2 + β n x n p 2 + 2 α n f ( p ) p , J ( x n + 1 p ) ( 1 β n α n ( 1 ρ ) ) [ x n p 2 σ n ( 1 σ n ) g ( x n J r n G x n ) ] + β n x n p 2 + 2 α n f ( p ) p , J ( x n + 1 p ) = ( 1 α n ( 1 ρ ) ) x n p 2 ( 1 β n α n ( 1 ρ ) ) σ n ( 1 σ n ) g ( x n J r n G x n ) + 2 α n f ( p ) p , J ( x n + 1 p ) x n p 2 ( 1 β n α n ( 1 ρ ) ) σ n ( 1 σ n ) g ( x n J r n G x n ) + 2 α n f ( p ) p x n + 1 p ,
and thus
( 1 β n α n ( 1 ρ ) ) σ n ( 1 σ n ) g ( x n J r n G x n ) x n p 2 x n + 1 p 2 + 2 α n f ( p ) p x n + 1 p ( x n p + x n + 1 p ) x n x n + 1 + 2 α n f ( p ) p x n + 1 p .
Since α n 0 and x n + 1 x n 0 , from condition (v) and the boundedness of { x n } , it follows that
lim n g ( x n J r n G x n ) = 0 .
Utilizing the properties of g, we have
lim n x n J r n G x n = 0 ,
(4.11)
and thus,
lim n y n x n = lim n ( 1 σ n ) J r n G x n x n = 0 .
(4.12)
For simplicity, we put q = Π C ( p μ 2 B 2 p ) , u n = Π C ( x n μ 2 B 2 x n ) and v n = Π C ( u n μ 1 B 1 u n ) . Then v n = G x n for all n 0 . From Lemma 2.8, we have
u n q 2 = Π C ( x n μ 2 B 2 x n ) Π C ( p μ 2 B 2 p ) 2 x n p μ 2 ( B 2 x n B 2 p ) 2 x n p 2 2 μ 2 ( α 2 κ 2 μ 2 ) B 2 x n B 2 p 2 ,
(4.13)
and
v n p 2 = Π C ( u n μ 1 B 1 u n ) Π C ( q μ 1 B 1 q ) 2 u n q μ 1 ( B 1 u n B 1 q ) 2 u n q 2 2 μ 1 ( α 1 κ 2 μ 1 ) B 1 u n B 1 q 2 .
(4.14)
By combining (4.13) and (4.14), we obtain
v n p 2 x n p 2 2 μ 2 ( α 2 κ 2 μ 2 ) B 2 x n B 2 p 2 2 μ 1 ( α 1 κ 2 μ 1 ) B 1 u n B 1 q 2 .
(4.15)
By the convexity of 2 , we have, from (4.1) and (4.15),
y n p 2 σ n x n p 2 + ( 1 σ n ) J r n G x n p 2 σ n x n p 2 + ( 1 σ n ) v n p 2 σ n x n p 2 + ( 1 σ n ) [ x n p 2 2 μ 2 ( α 2 κ 2 μ 2 ) B 2 x n B 2 p 2 2 μ 1 ( α 1 κ 2 μ 1 ) B 1 u n B 1 q 2 ] = x n p 2 2 ( 1 σ n ) [ μ 2 ( α 2 κ 2 μ 2 ) B 2 x n B 2 p 2 + μ 1 ( α 1 κ 2 μ 1 ) B 1 u n B 1 q 2 ] ,
and thus
2 ( 1 σ n ) [ μ 2 ( α 2 κ 2 μ 2 ) B 2 x n B 2 p 2 + μ 1 ( α 1 κ 2 μ 1 ) B 1 u n B 1 q 2 ] x n p 2 y n p 2 ( x n p + y n p ) x n y n .
Since x n y n 0 and 0 < μ i < α i κ 2 for i = 1 , 2 , and { x n } and { y n } are bounded, we obtain from condition (v) that
lim n B 2 x n B 2 p = 0 and lim n B 1 u n B 1 q = 0 .
(4.16)
Utilizing Proposition 2.2 and Lemma 2.10, we have
u n q 2 = Π C ( x n μ 2 B 2 x n ) Π C ( p μ 2 B 2 p ) 2 x n μ 2 B 2 x n ( p μ 2 B 2 p ) , J ( u n q ) = x n p , J ( u n q ) + μ 2 B 2 p B 2 x n , J ( u n q ) 1 2 [ x n p 2 + u n q 2 g 1 ( x n u n ( p q ) ) ] + μ 2 B 2 p B 2 x n u n q ,
which implies that
u n q 2 x n p 2 g 1 ( x n u n ( p q ) ) + 2 μ 2 B 2 p B 2 x n u n q .
(4.17)
In the same way, we derive
v n p 2 = Π C ( u n μ 1 B 1 u n ) Π C ( q μ 1 B 1 q ) 2 u n μ 1 B 1 u n ( q μ 1 B 1 q ) , J ( v n p ) = u n q , J ( v n p ) + μ 1 B 1 q B 1 u n , J ( v n p ) 1 2 [ u n q 2 + v n p 2 g 2 ( u n v n + ( p q ) ) ] + μ 1 B 1 q B 1 u n v n p ,
and we get
v n p 2 u n q 2 g 2 ( u n v n + ( p q ) ) + 2 μ 1 B 1 q B 1 u n v n p .
(4.18)
Combining (4.17) and (4.18), we get
v n p 2 x n p 2 g 1 ( x n u n ( p q ) ) g 2 ( u n v n + ( p q ) ) + 2 μ 2 B 2 p B 2 x n u n q + 2 μ 1 B 1 q B 1 u n v n p .
(4.19)
By the convexity of 2 , we have, from (4.1) and (4.19),
y n p 2 σ n x n p 2 + ( 1 σ n ) J r n G x n p 2 σ n x n p 2 + ( 1 σ n ) v n p 2 σ n x n p 2 + ( 1 σ n ) [ x n p 2 g 1 ( x n u n ( p q ) ) g 2 ( u n v n + ( p q ) ) + 2 μ 2 B 2 p B 2 x n u n q + 2 μ 1 B 1 q B 1 u n v n p ] x n p 2 ( 1 σ n ) [ g 1 ( x n u n ( p q ) ) + g 2 ( u n v n + ( p q ) ) ] + 2 μ 2 B 2 p B 2 x n u n q + 2 μ 1 B 1 q B 1 u n v n p ,
and hence
( 1 σ n ) [ g 1 ( x n u n ( p q ) ) + g 2 ( u n v n + ( p q ) ) ] x n p 2 y n p 2 + 2 μ 2 B 2 p B 2 x n u n q + 2 μ 1 B 1 q B 1 u n v n p ( x n p + y n p ) x n y n + 2 μ 2 B 2 p B 2 x n u n q + 2 μ 1 B 1 q B 1 u n v n p .
From (4.12), (4.16), condition (v), and the boundedness of { x n } , { y n } , { u n } , and { v n } , we deduce
lim n g 1 ( x n u n ( p q ) ) = 0 and lim n g 2 ( u n v n + ( p q ) ) = 0 .
Utilizing the properties of g 1 and g 2 , we obtain
lim n x n u n ( p q ) = 0 and lim n u n v n + ( p q ) = 0 .
(4.20)
Hence,
x n v n x n u n ( p q ) + u n v n + ( p q ) 0 as  n ,
that is,
lim n x n G x n = 0 .
(4.21)
Next, we show that
lim n J r n x n x n = 0 and lim n W n x n x n = 0 .
Indeed, observe that x n + 1 can be rewritten as
x n + 1 = α n f ( y n ) + β n x n + γ n W n y n + δ n J r n G y n = α n f ( y n ) + β n x n + ( γ n + δ n ) γ n W n y n + δ n J r n G y n γ n + δ n = α n f ( y n ) + β n x n + e n z ˆ n ,
(4.22)
where e n = γ n + δ n and z ˆ n = γ n W n y n + δ n J r n G y n γ n + δ n . Utilizing Lemma 2.4 and (4.22), we have
x n + 1 p 2 = α n ( f ( y n ) p ) + β n ( x n p ) + e n ( z ˆ n p ) 2 α n f ( y n ) p 2 + β n x n p 2 + e n z ˆ n p 2 β n e n g 3 ( z ˆ n x n ) = α n f ( y n ) p 2 + β n x n p 2 β n e n g 3 ( z ˆ n x n ) + e n γ n W n y n + δ n J r n G y n γ n + δ n p 2 = α n f ( y n ) p 2 + β n x n p 2 β n e n g 3 ( z ˆ n x n ) + e n γ n γ n + δ n ( W n y n p ) + δ n γ n + δ n ( J r n G y n p ) 2 α n f ( y n ) p 2 + β n x n p 2 β n e n g 3 ( z ˆ n x n ) + e n [ γ n γ n + δ n W n y n p 2 + δ n γ n + δ n J r n G y n p 2 ] α n f ( y n ) p 2 + β n x n p 2 β n e n g 3 ( z ˆ n x n ) + e n [ γ n γ n + δ n y n p + δ n γ n + δ n y n p 2 ] α n f ( y n ) p 2 + β n x n p 2 β n e n g 3 ( z ˆ n x n ) + e n [ γ n γ n + δ n x n p + δ n γ n + δ n x n p 2 ] = α n f ( y n ) p 2 + ( 1 α n ) x n p 2 β n e n g 3 ( z ˆ n x n ) α n f ( y n ) p 2 + x n p 2 β n e n g 3 ( z ˆ n x n ) ,
which implies that
β n e n g 3 ( z ˆ n x n ) α n f ( y n ) p 2 + x n p 2 x n + 1 p 2 α n f ( y n ) p 2 + ( x n p + x n + 1 p ) x n x n + 1 .
Utilizing (4.4), conditions (i), (ii), (v), and the boundedness of { x n } and { f ( y n ) } , we obtain
lim n g 3 ( z ˆ n x n ) = 0 .
From the properties of g 3 , we have
lim n z ˆ n x n = 0 .
Utilizing Lemma 2.3 and the definition of z ˆ n , we have
z ˆ n p 2 = γ n W n y n + δ n J r n G y n γ n + δ n p 2 = γ n γ n + δ n ( W n y n p ) + δ n γ n + δ n ( J r n G y n p ) 2 γ n γ n + δ n W n y n p 2 + δ n γ n + δ n J r n G y n p 2 γ n δ n ( γ n + δ n ) 2 g 4 ( J r n G y n W n y n ) y n p 2 γ n δ n ( γ n + δ n ) 2 g 4 ( J r n G y n W n y n ) x n p 2 γ n δ n ( γ n + δ n ) 2 g 4 ( J r n G y n W n y n ) ,
and thus
γ n δ n ( γ n + δ n ) 2 g 4 ( J r n G y n W n y n ) x n p 2 z ˆ n p 2 ( x n p + z ˆ n p ) x n z ˆ n .
Since { x n } and { z ˆ n } are bounded and z ˆ n x n 0 as n , we deduce from condition (ii) that
lim n g 4 ( W n y n J r n G y n ) = 0 .
From the properties of g 4 , we have
lim n W n y n J r n G y n = 0 .
(4.23)
On the other hand, x n + 1 can also be rewritten as
x n + 1 = α n f ( y n ) + β n x n + γ n W n y n + δ n J r n G y n = β n x n + γ n W n y n + ( α n + δ n ) α n f ( y n ) + δ n J r n G y n α n + δ n = β n x n + γ n W n y n + d n z ˜ n ,
where d n = α n + δ n and z ˜ n = α n f ( y n ) + δ n J r n G y n α n + δ n . Utilizing Lemma 2.4 and the convexity of 2 , we have
x n + 1 p 2 = β n ( x n p ) + γ n ( W n y n p ) + d n ( z ˜ n p ) 2 β n x n p 2 + γ n W n y n p 2 + d n z ˜ n p 2 β n γ n g 5 ( x n W n y n ) = β n x n p 2 + γ n W n y n p 2 + d n α n f ( y n ) + δ n J r n G y n α n + δ n p 2 β n γ n g 5 ( x n W n y n ) = β n x n p 2 + γ n W n y n p 2 + d n α n α n + δ n ( f ( y n ) p ) + δ n α n + δ n ( J r n G y n p ) 2 β n γ n g 5 ( x n W n y n ) β n x n p 2 + γ n y n p 2 + d n [ α n α n + δ n f ( y n ) p 2 + δ n α n + δ n J r n G y n p 2 ] β n γ n g 5 ( x n W n y n ) β n x n p 2 + γ n y n p 2 + d n [ α n α n + δ n f ( y n ) p 2 + δ n α n + δ n y n p 2 ] β n γ n g 5 ( x n W n y n ) α n f ( y n ) p 2 + ( β n + γ n ) x n p 2 + δ n x n p 2 β n γ n g 5 ( x n W n y n ) = α n f ( y n ) p 2 + ( 1 α n ) x n p 2 β n γ n g 5 ( x n W n y n ) α n f ( y n ) p 2 + x n p 2 β n γ n g 5 ( x n W n y n ) ,
which implies that
β n γ n g 5 ( x n W n y n ) α n f ( y n ) p 2 + x n p 2