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Synchronal and cyclic algorithms for fixed point problems and variational inequality problems in Banach spaces

Fixed Point Theory and Applications20132013:202

https://doi.org/10.1186/1687-1812-2013-202

  • Received: 17 May 2013
  • Accepted: 5 July 2013
  • Published:

Abstract

In this paper, we study synchronal and cyclic algorithms for finding a common fixed point x of a finite family of strictly pseudocontractive mappings, which solve the variational inequality

( γ f μ G ) x , j q ( x x ) 0 , x i = 1 N F ( T i ) ,

where f is a contraction mapping, G is an η-strongly accretive and L-Lipschitzian operator, N 1 is a positive integer, γ , μ > 0 are arbitrary fixed constants, and { T i } i = 1 N are N-strict pseudocontractions. Furthermore, we prove strong convergence theorems of such iterative algorithms in a real q-uniformly smooth Banach space. The results presented extend, generalize and improve the corresponding results recently announced by many authors.

MSC:47H06, 47H09, 47H10, 47J05, 47J20, 47J25.

Keywords

  • q-uniformly smooth Banach space
  • k-strict pseudocontractions
  • variational inequality
  • synchronal algorithm
  • cyclic algorithm
  • common fixed point

1 Introduction

Let E be a real Banach space, and let E be the dual of E. For some real number q ( 1 < q < ), the generalized duality mapping J q : E 2 E is defined by
J q ( x ) = { x E : x , x = x q , x = x q 1 } , x E ,
(1.1)

where , denotes the duality pairing between elements of E and those of E . In particular, J = J 2 is called the normalized duality mapping and J q ( x ) = x q 2 J 2 ( x ) for x 0 . If E is a real Hilbert space, then J = I , where I is the identity mapping. It is well known that if E is smooth, then J q is single-valued, which is denoted by j q .

Let C be a nonempty closed convex subset of E, and let G : E E be a nonlinear map. Then the variational inequality problem with respect to C and G is to find a point x C such that
G x , j q ( x x ) 0 , x C and j q ( x x ) J q ( x x ) .
(1.2)

We denote by VI ( G , C ) the set of solutions of this variational inequality problem.

If E = H , a real Hilbert space, the variational inequality problem reduces to the following: Find a point x C such that
G x , x x 0 , x C .
(1.3)
A mapping T : E E is said to be a contraction if, for some α [ 0 , 1 ) ,
T x T y α x y , x , y E .
(1.4)
The map T is said to be nonexpansive if
T x T y x y , x , y E .
(1.5)
The map T is said to be L-Lipschitzian if there exists L > 0 such that
T x T y L x y , x , y E .
(1.6)
A point x E is called a fixed point of the map T if T x = x . We denote by F ( T ) the set of all fixed points of the mapping T, that is,
F ( T ) = { x C : T x = x } .

We assume that F ( T ) in the sequel. It is well known that F ( T ) above is closed and convex (see, e.g., Goebel and Kirk [1]).

An operator F : E E is said to be accretive if x , y E , there exists j q ( x y ) J q ( x y ) such that
F x F y , j q ( x y ) 0 .
(1.7)
For some positive real numbers η, λ, the mapping F is said to be η-strongly accretive if for any x , y E , there exists j q ( x y ) J q ( x y ) such that
F x F y , j q ( x y ) η x y q ,
(1.8)
and it is called λ-strictly pseudocontractive if
F x F y , j q ( x y ) x y q λ x y ( F x F y ) q .
(1.9)
It is clear that (1.9) is equivalent to the following:
( I F ) x ( I F ) y , j q ( x y ) λ x y ( F x F y ) q ,
(1.10)

where I denotes the identity operator.

In Hilbert spaces, accretive operators are called monotone where inequality (1.7) holds with j q replaced by the identity map of H.

A bounded linear operator A on H is called strongly positive with coefficient γ if there is a constant γ > 0 with the property
A x , x γ x 2 , x H .

Let K be a nonempty closed convex and bounded subset of a Banach space E, and let the diameter of K be defined by d ( K ) : = sup { x y : x , y K } . For each x K , let r ( x , K ) : = sup { x y : y K } , and let r ( K ) : = inf { r ( x , K ) : x K } denote the Chebyshev radius of K relative to itself. The normal structure coefficient N ( E ) of E (see, e.g., [2]) is defined by N ( E ) : = inf { d ( K ) r ( K ) : d ( K ) > 0 } . A space E, such that N ( E ) > 1 , is said to have a uniform normal structure.

It is known that all uniformly convex and uniformly smooth Banach spaces have a uniform normal structure (see, e.g., [3, 4]).

Let μ be a continuous linear functional on l and ( a 0 , a 1 , ) l . We write μ n ( a n ) instead of μ ( ( a 0 , a 1 , ) ) . We call μ 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
lim inf n a n μ n ( a n ) lim sup n a n

for all { a n } l (see, e.g., [3, 5]).

Let S = { x E : x = 1 } denote the unit sphere of a real Banach space E.

The space E is said to have a Gâteaux differentiable norm if the limit
lim t 0 x + t y x t
(1.11)

exists for each x , y S . In this case, E is called smooth. E is said to be uniformly smooth if the limit (1.11) exists and is attained uniformly in x , y S . E is said to have a uniformly Gâteaux differentiable norm if, for any y S , the limit (1.11) exists uniformly for all x S .

The modulus of smoothness of E, with dim E 2 , is a function ρ E : [ 0 , ) [ 0 , ) defined by
ρ E ( τ ) = sup { x + y + x y 2 1 : x = 1 , y τ } .

A Banach space E is said to be uniformly smooth if lim t 0 + ρ E ( t ) t = 0 , and for q > 1 , E is said to be q-uniformly smooth if there exists a fixed constant c > 0 such that ρ E ( t ) c t q , t > 0 .

It is well known (see, e.g., [6]) that Hilbert spaces, L p (or l p ) spaces ( 1 < p < ) and Sobolev spaces, W m p ( 1 < p < ) are all uniformly smooth. More precisely, Hilbert spaces are 2-uniformly smooth, while
L p ( or  l p )  or  W m p  spaces are  { 2 -uniformly smooth if  2 p < , p -uniformly smooth if  1 < p 2 .

Also, it is well known (see, e.g., [7]) that q-uniformly smooth Banach spaces have a uniformly Gâteaux differentiable norm.

The variational inequality problem was initially introduced and studied by Stampacchia [8] in 1964. In the recent years, variational inequality problems have been extended to study a large variety of problems arising in structural analysis, economics and optimization. Thus, the problem of solving a variational inequality of the form (1.2) has been intensively studied by numerous authors. Iterative methods for approximating fixed points of nonexpansive mappings and their generalizations, which solve some variational inequality problems, have been studied by a number of authors (see, for example, [913] and the references therein).

Let H be a real Hilbert space. In 2001, Yamada [13] proposed a hybrid steepest descent method for solving variational inequality as follows: Let x 0 H be chosen arbitrarily and define a sequence { x n } by
x n + 1 = T x n μ λ n F ( T x n ) , n 0 ,
(1.12)
where T is a nonexpansive mapping on H, F is L-Lipschitzian and η-strongly monotone with L > 0 , η > 0 , 0 < μ < 2 η / L 2 . If { λ n } is a sequence in ( 0 , 1 ) satisfying the following conditions:
  1. (C1)

    lim n λ n = 0 ,

     
  2. (C2)

    n = 0 λ n = ,

     
  3. (C3)

    either n = 1 | λ n + 1 λ n | < or lim n λ n + 1 λ n = 1 ,

     
then he proved that the sequence { x n } converges strongly to the unique solution of the variational inequality
F x ˜ , x x ˜ 0 , x F ( T ) .
Besides, he also proposed the cyclic algorithm
x n + 1 = T λ n x n = ( I μ λ n F ) T [ n ] x n ,

where T [ n ] = T n ( mod N ) ; he also proved strong convergence theorems for the cyclic algorithm.

In 2006, Marino and Xu [10] considered the following general iterative method: Starting with an arbitrary initial point x 0 H , define a sequence { x n } by
x n + 1 = α n γ f ( x n ) + ( I α n A ) T x n , n 0 ,
(1.13)
where T is a nonexpansive mapping of H, f is a contraction, A is a linear bounded strongly positive operator, and { α n } is a sequence in ( 0 , 1 ) satisfying the following conditions:
  1. (M1)

    lim n α n = 0 ;

     
  2. (M2)

    n = 0 α n = ;

     
  3. (M3)

    n = 1 | α n + 1 α n | < or lim n α n + 1 α n = 1 .

     
They proved that the sequence { x n } converges strongly to a fixed point x ˜ of T, which solves the variational inequality
( γ f A ) x ˜ , x x ˜ 0 , x F ( T ) .
In 2010, Tian [11] combined the iterative method (1.13) with Yamada’s iterative method (1.12) and considered the following general iterative method:
x n + 1 = α n γ f ( x n ) + ( I μ α n F ) T x n , n 0 ,
(1.14)
where T is a nonexpansive mapping on H, f is a contraction, F is k-Lipschitzian and η-strongly monotone with k > 0 , η > 0 , 0 < μ < 2 η / k 2 . He proved that if the sequence { α n } of parameters satisfies conditions (M1)-(M3), then the sequence { x n } generated by (1.14) converges strongly to a fixed point x ˜ of T, which solves the variational inequality
( γ f μ F ) x ˜ , x x ˜ 0 , x F ( T ) .

Very recently, in 2011, Tian and Di [12] studied two algorithms, based on Tian’s [11] general iterative algorithm, and proved the following theorems.

Theorem 1.1 (Synchronal algorithm)

Let H be a real Hilbert space, and let T i : H H be a k i -strictly pseudocontraction for some k i ( 0 , 1 ) ( i = 1 , 2 , , N ) such that i = 1 N F ( T i ) , and f be a contraction with coefficient β ( 0 , 1 ) and λ i be a positive constant such that i = 1 N λ i = 1 . Let G : H H be an η-strongly monotone and L-Lipschitzian operator with L > 0 , η > 0 . Assume that 0 < μ < 2 η / L 2 , 0 < γ < μ ( η μ L 2 2 ) / β = τ / β . Let x 0 H be chosen arbitrarily, and let { α n } and { β n } be sequences in ( 0 , 1 ) satisfying the following conditions:
  1. (N1)

    lim n α n = 0 , n = 0 α n = ;

     
  2. (N2)

    n = 1 | α n + 1 α n | < , n = 1 | β n + 1 β n | < ;

     
  3. (N3)

    0 < max k i β n < a < 1 , n 0 .

     
Let { x n } be a sequence defined by the composite process
{ T β n = β n I + ( 1 β n ) i = 1 N λ i T i , x n + 1 = α n γ f ( x n ) + ( I α n μ G ) T β n x n , n 0 .
Then { x n } converges strongly to a common fixed point of { T i } i = 1 N , which solves the variational inequality
( γ f μ G ) x , x x 0 , x i = 1 N F ( T i ) .
(1.15)

Theorem 1.2 (Cyclic algorithm)

Let H be a real Hilbert space, and let T i : H H be a k i -strictly pseudocontraction for some k i ( 0 , 1 ) ( i = 1 , 2 , , N ) such that i = 1 N F ( T i ) , and f be a contraction with coefficient β ( 0 , 1 ) . Let G : H H be an η-strongly monotone and L-Lipschitzian operator with L > 0 , η > 0 . Assume that 0 < μ < 2 η / L 2 , 0 < γ < μ ( η μ L 2 2 ) / β = τ / β . Let x 0 H be chosen arbitrarily, and let { α n } and { β n } be sequences in ( 0 , 1 ) satisfying the following conditions:

  • (N1′) lim n α n = 0 , n = 0 α n = ;

  • (N2′) n = 1 | α n + 1 α n | < or lim n α n α n + N = 1 ;

  • (N3′) β [ n ] [ k , 1 ) , n 0 , where k = max { k i : 1 i N } .

Let { x n } be a sequence defined by the composite process
{ A [ n ] = β [ n ] I + ( 1 β [ n ] ) T [ n ] , x n + 1 = α n γ f ( x n ) + ( I α n μ G ) A [ n + 1 ] x n , n 0 ,

where T [ n ] = T i , with i = n ( mod N ) , 1 i N , namely T [ n ] is one of T 1 , T 2 , , T N cyclically. Then { x n } converges strongly to a common fixed point of { T i } i = 1 N , which solves the variational inequality (1.15).

In this paper, we study the synchronal and cyclic algorithms for finding a common fixed point x of finite strictly pseudocontractive mappings, which solves the variational inequality
( γ f μ G ) x , j q ( x x ) 0 , x i = 1 N F ( T i ) ,
(1.16)

where f is a contraction mapping, G is an η-strongly accretive and L-Lipschitzian operator, N 1 is a positive integer, γ , μ > 0 are arbitrary fixed constants, and { T i } i = 1 N are N-strict pseudocontractions defined on a closed convex subset C of a real q-uniformly smooth Banach space E whose norm is uniformly Gâteaux differentiable.

Let T be defined by
T : = i = 1 N λ i T i ,
where λ i > 0 such that i = 1 N λ i = 1 . We will show that a sequence { x n } generated by the following synchronal algorithm:
{ x 0 = x C chosen arbitrarily , T β n = β n I + ( 1 β n ) i = 1 N λ i T i , x n + 1 = α n γ f ( x n ) + ( I α n μ G ) T β n x n , n 0 ,
(1.17)

converges strongly to a solution of problem (1.16).

Another approach to problem (1.16) is the cyclic algorithm. For each i = 1 , , N , let A i = β i I + ( 1 β i ) T i , where the constant β i satisfies 0 < k i < β i < 1 . Beginning with x 0 C , define a sequence { x n } cyclically by
x 1 = α 0 γ f ( x 0 ) + ( I α 0 μ G ) ( A 1 x 0 ) , x 2 = α 1 γ f ( x 1 ) + ( I α 1 μ G ) ( A 2 x 1 ) , x N = α N 1 γ f ( x N 1 ) + ( I α N 1 μ G ) ( A N x N 1 ) , x N + 1 = α N γ f ( x N ) + ( I α N μ G ) ( A 1 x N ) ,
Indeed, the algorithm can be written in a compact form as follows:
{ x 0 = x C chosen arbitrarily , A [ n ] = β [ n ] I + ( 1 β [ n ] ) T [ n ] , x n + 1 = α n γ f ( x n ) + ( I α n μ G ) A [ n + 1 ] x n , n 0 ,
(1.18)

where T [ n ] = T i , with i = n ( mod N ) , 1 i N , namely T [ n ] is one of T 1 , T 2 , , T N cyclically. We will show that (1.18) is also strongly convergent to a solution of problem (1.16) if the sequences { α n } and { β n } of parameters are appropriately chosen.

Motivated by the results of Tian and Di [12], in this paper we aim to continue the study of fixed point problems and prove new theorems for the solution of variational inequality problems in the framework of a real Banach space, which is much more general than that of Hilbert.

Throughout this research work, we will use the following notations:
  1. 1.

    for weak convergence and → for strong convergence.

     
  2. 2.

    ω ω ( x n ) = { x : x n j x } denotes the weak ω-limit set of { x n } .

     

2 Preliminaries

In the sequel we shall make use of the following lemmas.

Lemma 2.1 (Zhang and Guo [14])

Let C be a nonempty closed convex subset of a real Banach space E. Given an integer N 1 , for each 1 i N , T i : C C is a λ i -strict pseudocontraction for some λ i [ 0 , 1 ) such that i = 1 N F ( T i ) . Assume that { γ i } i = 1 N is a sequence of positive numbers such that i = 1 N γ i = 1 . Then i = 1 N γ i T i is a λ-strict pseudocontraction with λ : = min { λ i : 1 i N } , and
F ( i = 1 N γ i T i ) = i = 1 N F ( T i ) .

Lemma 2.2 (Zhou [15])

Let E be a uniformly smooth real Banach space, and let C be a nonempty closed convex subset of E. Let T : C C be a k-strict pseudocontraction. Then ( I T ) is demiclosed at zero. That is, if { x n } C satisfies x n x and x n T x n 0 , as n , then T x = x .

Lemma 2.3 (Petryshyn [16])

Let E be a real Banach space, and let J q : E 2 E be the generalized duality mapping. Then, for any x , y E and j q ( x + y ) J q ( x + y ) ,
x + y q x q + q y , j q ( x + y ) .

Lemma 2.4 (Lim and Xu [4])

Suppose that E is a Banach space with a uniform normal structure, K is a nonempty bounded subset of E, and let T : K K be a uniformly k-Lipschitzian mapping with k < N ( E ) 1 2 . Suppose also that there exists a nonempty bounded closed convex subset C of K with the following property (P):
x C implies ω ω ( x ) C ,
(P)
where ω ω ( x ) is the ω-limit set of T at x, i.e., the set
{ y E : y = weak lim j T n j x for some n j } .

Then T has a fixed point in C.

Lemma 2.5 (Xu [17])

Let q > 1 , and let E be a real q-uniformly smooth Banach space, then there exists a constant d q > 0 such that for all x , y E and j q ( x ) J q ( x ) ,
x + y q x q + q y , j q ( x ) + d q y q .
Lemma 2.6 Let E be a real q-uniformly smooth Banach space with constant d q > 0 , q > 1 , and let C be a nonempty closed convex subset of E. Let F : C C be an η-strongly accretive and L-Lipschitzian operator with L > 0 , η > 0 . Assume that 0 < μ < ( q η d q L q ) 1 q 1 , τ = μ ( η d q μ q 1 L q q ) and t ( 0 , min { 1 , 1 τ } ) . Then, for any x , y C , the following inequality holds:
( I μ t F ) x ( I μ t F ) y ( 1 t τ ) x y .

That is, ( I μ t F ) is a contraction with coefficient ( 1 t τ ) .

Proof For any x , y C , we have, by Lemma 2.5, (1.6) and (1.8),
( I μ t F ) x ( I μ t F ) y q = ( x y ) μ t ( F x F y ) q x y q q μ t F x F y , j q ( x y ) + d q μ q t q F x F y q x y q q μ t η x y q + d q μ q t q L q x y q [ 1 t μ ( q η d q μ q 1 L q ) ] x y q = [ 1 q t μ ( η d q μ q 1 L q q ) ] x y q [ 1 t μ ( η d q μ q 1 L q q ) ] q x y q = ( 1 t τ ) q x y q .
From 0 < μ < ( q η d q L q ) 1 q 1 , q > 1 and t ( 0 , min { 1 , 1 τ } ) , we have ( 1 t τ ) ( 0 , 1 ) . It then follows that
( I μ t F ) x ( I μ t F ) y ( 1 t τ ) x y .

 □

Lemma 2.7 Let E be a real q-uniformly smooth Banach space with constant d q , q > 1 , and let C be a nonempty closed convex subset of E. Suppose that T : C C is a λ-strict pseudocontraction such that F ( T ) . For any α ( 0 , 1 ) , we define T α : C E by T α x = α x + ( 1 α ) T x for each x C . Then, as α [ μ , 1 ) , μ [ max { 0 , 1 ( λ q d q ) 1 q 1 } , 1 ) , T α is a nonexpansive mapping such that F ( T α ) = F ( T ) .

Proof For any x , y C , we have, by Lemma 2.5 and (1.10),
T α x T α y q = α x + ( 1 α ) T x α y ( 1 α ) T y q = x y ( 1 α ) [ x y ( T x T y ) ] q x y q q ( 1 α ) ( I T ) x ( I T ) y , j q ( x y ) + d q ( 1 α ) q x y ( T x T y ) q x y q λ q ( 1 α ) x y ( T x T y ) q + d q ( 1 α ) q x y ( T x T y ) q = x y q ( 1 α ) [ λ q d q ( 1 α ) q 1 ] x y ( T x T y ) q x y q ,

which shows that T α is a nonexpansive mapping.

It is clear that x = T α x x = T x . This proves our assertions. □

Lemma 2.8 (Xu [18])

Let { a n } be a sequence of nonnegative real numbers such that
a n + 1 ( 1 γ n ) a n + δ n , n 0 ,
where { γ n } is a sequence in ( 0 , 1 ) and { δ n } is a sequence in such that
  1. (i)

    lim n γ n = 0 and n = 0 γ n = ;

     
  2. (ii)

    lim sup n δ n γ n 0 or n = 1 | δ n | < .

     

Then lim n a n = 0 .

Lemma 2.9 Let E be a real q-uniformly smooth Banach space with constant d q , q > 1 , and let C be a nonempty closed convex subset of E. Suppose that T i : C C are k i -strict pseudocontractions for k i ( 0 , 1 ) ( i = 1 , 2 , , N ). Let T α i = α i I + ( 1 α i ) T i , k i < α i < 1 ( i = 1 , 2 , , N ). If i = 1 N F ( T i ) , then, as α i [ μ , 1 ) , μ [ max { 0 , 1 ( λ q d q ) 1 q 1 } , 1 ) , we have
F ( T α 1 T α 2 T α N ) = i = 1 N F ( T α i ) .
Proof We prove it by induction. For N = 2 , set T α 1 = α 1 I + ( 1 α 1 ) T 1 , T α 2 = α 2 I + ( 1 α 2 ) T 2 , k i < α i < 1 , i = 1 , 2 . Obviously,
F ( T α 1 ) F ( T α 2 ) F ( T α 1 T α 2 ) .
Now we prove
F ( T α 1 T α 2 ) F ( T α 1 ) F ( T α 2 ) .

For all y F ( T α 1 T α 2 ) , T α 1 T α 2 y = y , if T α 2 y = y , then T α 1 y = y , the conclusion holds. In fact, we can claim that T α 2 y = y . From Lemma 2.7, we know that T α 2 is nonexpansive and F ( T α 1 ) F ( T α 2 ) = F ( T 1 ) F ( T 2 ) .

Take x F ( T α 1 ) F ( T α 2 ) , then, by Lemma 2.5 and (1.10), we have
x y q = x T α 1 T α 2 y q = x [ α 1 ( T α 2 y ) + ( 1 α 1 ) T 1 T α 2 y ] q = x T α 2 y ( 1 α 1 ) [ x T α 2 y ( x T 1 T α 2 y ) ] q x T α 2 y q q ( 1 α 1 ) x T α 2 y ( x T 1 T α 2 y ) , j q ( x T α 2 y ) + d q ( 1 α 1 ) q x T α 2 y ( x T 1 T α 2 y ) q = T α 2 x T α 2 y q q ( 1 α 1 ) ( I T 1 ) x ( I T 1 ) T α 2 y , j q ( x T α 2 y ) + d q ( 1 α 1 ) q x T α 2 y ( x T 1 T α 2 y ) q T α 2 x T α 2 y q λ q ( 1 α 1 ) x T α 2 y ( x T 1 T α 2 y ) q + d q ( 1 α 1 ) q x T α 2 y ( x T 1 T α 2 y ) q x y q λ q ( 1 α 1 ) T 1 T α 2 y T α 2 y q + d q ( 1 α 1 ) q T 1 T α 2 y T α 2 y q = x y q ( 1 α 1 ) [ λ q d q ( 1 α 1 ) q 1 ] T 1 T α 2 y T α 2 y q .
So, we get
T 1 T α 2 y T α 2 y q 0 .
Namely T 1 T α 2 y = T α 2 y , that is,
T α 2 y F ( T 1 ) = F ( T α 1 ) , T α 2 y = T α 1 T α 2 y = y .
Suppose that the conclusion holds for N = k , we prove that
F ( T α 1 T α 2 T α k + 1 ) = i = 1 k + 1 F ( T α i ) .
It suffices to verify
F ( T α 1 T α 2 T α k + 1 ) i = 1 k + 1 F ( T α i ) .
For all y F ( T α 1 T α 2 T α k + 1 ) , T α 1 T α 2 T α k + 1 y = y . Using Lemma 2.5 and (1.10) again, take x i = 1 k + 1 F ( T α i ) , then
x y q = x T α 1 T α 2 T α k + 1 y q = x [ α 1 ( T α 2 T α k + 1 y ) + ( 1 α 1 ) T 1 T α 2 T α k + 1 y ] q = x T α 2 T α k + 1 y ( 1 α 1 ) [ x T α 2 T α k + 1 y ( x T 1 T α 2 T α k + 1 y ) ] q x T α 2 T α k + 1 y q q ( 1 α 1 ) x T α 2 T α k + 1 y ( x T 1 T α 2 T α k + 1 y ) , j q ( x T α 2 T α k + 1 y ) + d q ( 1 α 1 ) q x T α 2 T α k + 1 y ( x T 1 T α 2 T α k + 1 y ) q = x T α 2 T α k + 1 y q q ( 1 α 1 ) ( I T 1 ) x ( I T 1 ) T α 2 T α k + 1 y , j q ( x T α 2 T α k + 1 y ) + d q ( 1 α 1 ) q x T α 2 T α k + 1 y ( x T 1 T α 2 T α k + 1 y ) q x T α 2 T α k + 1 y q λ q ( 1 α 1 ) x T α 2 T α k + 1 y ( x T 1 T α 2 T α k + 1 y ) q + d q ( 1 α 1 ) q x T α 2 T α k + 1 y ( x T 1 T α 2 T α k + 1 y ) q x y q λ q ( 1 α 1 ) T 1 T α 2 T α k + 1 y T α 2 T α k + 1 y q + d q ( 1 α 1 ) q T 1 T α 2 T α k + 1 y T α 2 T α k + 1 y q = x y q ( 1 α 1 ) [ λ q d q ( 1 α 1 ) q 1 ] T 1 T α 2 T α k + 1 y T α 2 T α k + 1 y q .
So, we get
T 1 T α 2 T α k + 1 y T α 2 T α k + 1 y q 0 .
Thus, T 1 T α 2 T α k + 1 y = T α 2 T α k + 1 y , that is, T α 2 T α k + 1 y F ( T 1 ) = F ( T α 1 ) . Namely,
T α 2 T α k + 1 y = T α 1 T α 2 T α k + 1 y = y .
(2.1)
From (2.1) and inductive assumption, we get
y F ( T α 2 T α k + 1 ) = i = 2 k + 1 F ( T α i ) ,
that is,
T α i y = y , i = 2 , , k + 1 .
Substituting it into (2.1), we obtain T α 1 T α i y = y , i = 2 , , k + 1 , that is, T α 1 y = y , y F ( T α 1 ) , and hence
y i = 1 k + 1 F ( T α i ) .

 □

Lemma 2.10 (Ali et al. [9])

Let E be a real q-uniformly smooth Banach space with constant d q , q > 1 . Let f : E E be a contraction mapping with constant α ( 0 , 1 ) . Let T : E E be a nonexpansive mapping such that F ( T ) , and let A : E E be an η-strongly accretive mapping which is also k-Lipschitzian. Let μ ( 0 , min { 1 , ( q η d q k q ) 1 q 1 } ) and τ : = μ ( η μ q 1 d q k q q ) . For each t ( 0 , 1 ) and γ ( 0 , τ α ) , the path { x t } defined by
x t = t γ f ( x t ) + ( I t μ A ) T x t
converges strongly as t 0 to a fixed point x of T, which solves the variational inequality
( μ A γ f ) x , j q ( x z ) 0 , z F ( T ) .

Lemma 2.11 (Chang et al. [19])

Let E be a real Banach space with a uniformly Gâteaux differentiable norm. Then the generalized duality mapping J q : E 2 E is single-valued and uniformly continuous on each bounded subset of E from the norm topology of E to the w e a k topology of E .

Lemma 2.12 (Zhou et al. [20])

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

Lemma 2.13 (Mitrinović [21])

Suppose that q > 1 . Then, for any arbitrary positive real numbers x , y , the following inequality holds:
x y 1 q x q + ( q 1 q ) x q q 1 .

3 Synchronal algorithm

Theorem 3.1 Let E a real q-uniformly smooth Banach space, and let C be a nonempty closed convex subset of E. Let T i : C C be k i -strict pseudocontractions for k i ( 0 , 1 ) ( i = 1 , 2 , , N ) such that i = 1 N F ( T i ) . Let f be a contraction with coefficient β ( 0 , 1 ) , and let { λ i } i = 1 N be a sequence of positive numbers such that i = 1 N λ i = 1 . Let G : C C be an η-strongly accretive and L-Lipschitzian operator with L > 0 , η > 0 . Assume that 0 < μ < ( q η / d q L q ) 1 / q 1 , 0 < γ < μ ( η d q μ q 1 L q / q ) / β = τ / β . Let { α n } and { β n } be sequences in ( 0 , 1 ) satisfying the following conditions:
  1. (K1)

    lim n α n = 0 , n = 0 α n = ;

     
  2. (K2)

    n = 1 | α n + 1 α n | < , n = 1 | β n + 1 β n | < ;

     
  3. (K3)

    0 < k β n < a < 1 , where k = min { k i : 1 i N } ;

     
  4. (K4)

    α n , β n [ μ , 1 ) , where μ [ max { 0 , 1 ( λ q d q ) 1 q 1 } , 1 ) .

     

Let { x n } be a sequence defined by algorithm (1.17), then { x n } converges strongly to a common fixed point of { T i } i = 1 N , which solves the variational inequality (1.16).

Proof Let T : = i = 1 N λ i T i , then by Lemma 2.1 we conclude that T is a k-strict pseudocontraction and F ( T ) = i = 1 N F ( T i ) . We can rewrite algorithm (1.17) as follows:
{ x 0 = x C chosen arbitrarily , T β n = β n I + ( 1 β n ) T , x n + 1 = α n γ f ( x n ) + ( I α n μ G ) T β n x n , n 0 .
Furthermore, by Lemma 2.7 we have that T β n is a nonexpansive mapping and F ( T β n ) = F ( T ) . From condition (K1) we may assume, without loss of generality, that α n ( 0 , min { 1 , 1 τ } ) . Let p i = 1 N F ( T i ) , then the sequence { x n } satisfies
x n p max { x 0 p , γ f ( p ) μ G p τ γ β } , n 0 .

We prove this by mathematical induction as follows.

Obviously, it is true for n = 0 . Assume that it is true for n = k for some k N .

From (1.17) and Lemma 2.6, we have
x k + 1 p = α k γ f ( x k ) + ( I α k μ G ) T β k x k p = α k [ γ f ( x k ) μ G p ] + ( I α k μ G ) T β k x k ( I α k μ G ) p ( 1 α k τ ) x k p + α k γ [ f ( x k ) f ( p ) ] + γ f ( p ) μ G p ( 1 α k τ ) x k p + α k γ β x k p + α k γ f ( p ) μ G p = [ 1 α k ( τ γ β ) ] x k p + α k ( τ γ β ) γ f ( p ) μ G p τ γ β max { x k p , γ f ( p ) μ G p τ γ β } .

Hence the proof. Thus, the sequence { x n } is bounded and so are { T x n } , { G T β n x n } and { f ( x n ) } .

Observe that
x n + 2 x n + 1 = [ α n + 1 γ f ( x n + 1 ) + ( I α n + 1 μ G ) T β n + 1 x n + 1 ] [ α n γ f ( x n ) + ( I α n μ G ) T β n x n ] = [ α n + 1 γ f ( x n + 1 ) α n + 1 γ f ( x n ) ] + [ α n + 1 γ f ( x n ) α n γ f ( x n ) ] + [ ( I α n + 1 μ G ) T β n + 1 x n + 1 ( I α n + 1 μ G ) T β n x n ] + [ α n μ G T β n x n α n + 1 μ G T β n x n ] = α n + 1 γ [ f ( x n + 1 ) f ( x n ) ] + [ ( I α n + 1 μ G ) T β n + 1 x n + 1 ( I α n + 1 μ G ) T β n x n ] + ( α n + 1 α n ) γ f ( x n ) + ( α n α n + 1 ) μ G T β n x n ,
so that
x n + 2 x n + 1 α n + 1 γ β x n + 1 x n + ( 1 α n + 1 τ ) T β n + 1 x n + 1 T β n x n + | α n + 1 α n | ( γ f ( x n ) + μ G T β n x n ) α n + 1 γ β x n + 1 x n + ( 1 α n + 1 τ ) T β n + 1 x n + 1 T β n x n + | α n + 1 α n | M 1 ,
(3.1)

where M 1 is an appropriate constant such that M 1 sup n 1 { γ f ( x n ) + μ G T β n x n } .

On the other hand, we note that
T β n + 1 x n + 1 T β n x n T β n + 1 x n + 1 T β n + 1 x n + T β n + 1 x n T β n x n x n + 1 x n + [ β n + 1 x n + ( 1 β n + 1 ) T x n ] [ β n x n + ( 1 β n ) T x n ] = x n x n + β n + 1 ( x n T x n ) β n ( x n T x n ) x n + 1 x n + | β n + 1 β n | M 2 ,
(3.2)

where M 2 is an appropriate constant such that M 2 sup n 1 { x n T x n } .

Now, substituting (3.2) into (3.1) yields
x n + 2 x n + 1 α n + 1 γ β x n + 1 x n + ( 1 α n + 1 τ ) x n + 1 x n + | α n + 1 α n | M 1 + | β n + 1 β n | M 2 [ 1 α n + 1 ( τ γ β ) ] x n + 1 x n + ( | α n + 1 α n | + | β n + 1 β n | ) M 3 ,

where M 3 is an appropriate constant such that M 3 max { M 1 , M 2 } .

By Lemma 2.8 and conditions (K1), (K2), we have
x n + 1 x n 0 as  n .
(3.3)
From (1.17) and condition (K1), we have
x n + 1 T β n x n = α n γ f ( x n ) + ( I α n μ G ) T β n x n T β n x n α n γ f ( x n ) + μ G T β n x n 0 as  n .
(3.4)
On the other hand,
x n + 1 T β n x n = x n + 1 [ β n x n + ( 1 β n ) T x n ] = ( x n + 1 x n ) + ( 1 β n ) ( x n T x n ) ( 1 β n ) x n T x n x n + 1 x n ,
which implies, by condition (K3), that
x n T x n 1 1 β n ( x n + 1 x n + x n + 1 T β n x n ) 1 1 a ( x n + 1 x n + x n + 1 T β n x n ) .
Hence, from (3.3) and (3.4), we have
x n T x n 0 as  n .
(3.5)
From the boundedness of { x n } , without loss of generality, we may assume that x n p . Hence, by Lemma 2.2 and (3.5), we obtain T p = p . So, we have
ω ω ( x n ) F ( T ) .
(3.6)

We now prove that lim sup n ( γ f μ G ) x , j q ( x n + 1 x ) 0 , where x is obtained in Lemma 2.10. Put a n : = ( γ f μ G ) x , j q ( x n x ) , n 1 .

Define a map ϕ : E R by
ϕ ( x ) = μ n x n x q , x E .
Then ϕ is continuous, convex, and ϕ ( x ) as x . Since E is reflexive, there exists y C such that ϕ ( y ) = min z C ϕ ( z ) . Hence the set
K : = { y C : ϕ ( y ) = min z C ϕ ( z ) } .
Therefore, applying Lemma 2.4, we have K F ( T β n ) . Without loss of generality, assume x = y K F ( T β n ) . Let t ( 0 , 1 ) . Then it follows that ϕ ( x ) ϕ ( x + t ( γ f μ G ) x ) , and using Lemma 2.3, we obtain that
x n x t ( γ f μ G ) x q x n x q q t ( γ f μ G ) x , j q ( x n x t ( γ f μ G ) x ) .
This implies that
μ n ( γ f μ G ) x , j q ( x n x t ( γ f μ G ) x ) 0 .
By Lemma 2.11, j q is norm-to-weak uniformly continuous on a bounded subset of E, so we obtain, as t 0 , that
( γ f μ G ) x , j q ( x n x ) ( γ f μ G ) x , j q ( x n x t ( γ f μ G ) x ) 0 .
Hence, for all ϵ > 0 , there exists δ ϵ > 0 such that t ( 0 , δ ϵ ) and for all n 1 ,
( γ f μ G ) x , j q ( x n x ) ( γ f μ G ) x , j q ( x n x t ( γ f μ G ) x ) < ϵ .
Consequently,
μ n ( γ f μ G ) x , j q ( x n x ) μ n ( γ f μ G ) x , j q ( x n x t ( γ f μ G ) x ) + ϵ ϵ .
Since ϵ is arbitrary, we have
μ n ( γ f μ G ) x , j q ( x n x ) 0 .

Thus, μ n ( a n ) 0 for any Banach limit μ.

Furthermore, by (3.3), x n + 1 x n 0 as n . We therefore conclude that
lim sup n ( a n + 1 a n ) = lim sup n ( ( γ f μ G ) x , j q ( x n + 1 x ) ( γ f μ G ) x , j q ( x n x ) ) = lim sup n ( ( γ f μ G ) x , j q ( x n + 1 x ) j q ( x n x ) ) = 0 .
Hence, by Lemma 2.12 we obtain lim sup n a n 0 , that is,
lim sup n ( γ f μ G ) x , j q ( x n x ) 0 .
(3.7)
From (1.17), Lemmas 2.3, 2.6 and 2.13, we have
x n + 1 x q = x n + 1 x , j q ( x n + 1 x ) = α n [ γ f ( x n ) μ G x ] + ( I α n μ G ) ( T β n x n x ) , j q ( x n + 1 x ) = α n γ f ( x n ) μ G x , j q ( x n + 1 x ) + ( I α n μ G ) ( T β n x n x ) , j q ( x n + 1 x ) = α n γ f ( x n ) γ f ( x ) , j q ( x n + 1 x ) + α n γ f ( x ) μ G x , j q ( x n + 1 x ) + ( I α n μ G ) ( T β n x n x ) , j q ( x n + 1 x ) α n γ f ( x n ) f ( x ) x n + 1 x q 1 + α n ( γ f μ G ) x , j q ( x n + 1 x ) + ( I α n μ G ) T β n x n ( I α n μ G ) x x n + 1 x q 1 ( 1 α n τ ) x n x x n + 1 x q 1 + α n γ β x n x x n + 1 x q 1 + α n ( γ f μ G ) x , j q ( x n + 1 x ) = [ 1 α n ( τ γ β ) ] x n x x n + 1 x q 1 + α n ( γ f μ G ) x , j q ( x n + 1 x ) [ 1 α n ( τ γ β ) ] [ 1 q x n x q + ( q 1 q ) x n + 1 x q ] + α n ( γ f μ G ) x , j q ( x n + 1 x ) .
This implies that
x n + 1 x q 1 α n ( τ γ β ) 1 + α n ( q 1 ) ( τ γ β ) x n x q + q α n 1 + α n ( q 1 ) ( τ γ β ) ( γ f μ G ) x , j q ( x n + 1 x ) [ 1 α n ( τ γ β ) ] x n x q + q α n 1 + α n ( q 1 ) ( τ γ β ) ( γ f μ G ) x , j q ( x n + 1 x ) ( 1 γ n ) x n x q + δ n ,

where γ n : = α n ( τ γ β ) and δ n : = q α n 1 + α n ( q 1 ) ( τ γ β ) ( γ f μ G ) x , j q ( x n + 1 x ) . From (K1), lim n γ n = 0 , n = 0 γ n = . δ n γ n = q [ 1 + α n ( q 1 ) ( τ γ β ) ] ( τ γ β ) ( γ f μ G ) x , j q ( x n + 1 x ) . So, lim sup n δ n γ n 0 . Hence, by Lemma 2.8, we have that x n x as n . This completes the proof. □

4 Cyclic algorithm

Theorem 4.1 Let E be a real q-uniformly smooth Banach space, and let C be a nonempty closed convex subset of E. Let T i : C C be k i -strict pseudocontractions for k i ( 0 , 1 ) ( i = 1 , 2 , , N ) such that i = 1 N F ( T i ) , let f be a contraction with coefficient β ( 0 , 1 ) . Let G : C C be an η-strongly accretive and L-Lipschitzian operator with L > 0 , η > 0 . Assume that 0 < μ < ( q η / d q L q ) 1 / q 1 , 0 < γ < μ ( η d q μ q 1 L q / q ) / β = τ / β . Let { α n } and { β n } be sequences in ( 0 , 1 ) satisfying the following conditions:

  • (K1′) lim n α n = 0 , n = 0 α n = ;

  • (K2′) n = 1 | α n + 1 α n | < or lim n α n α n + N = 1 ;

  • (K3′) β [ n ] [ k , 1 ) , n 0 , where k = min { k i : 1 i N } ;

  • (K4′) α n , β n [ μ , 1 ) , where μ [ max { 0 , 1 ( λ q d q ) 1 q 1 } , 1 ) .

Let { x n } be a sequence defined by algorithm (1.18), then { x n } converges strongly to a common fixed point of { T i } i = 1 N , which solves the variational inequality (1.16).

Proof From condition (K1′) we may assume, without loss of generality, that α n ( 0 , min { 1 , 1 τ } ) . Let p i = 1 N F ( T i ) , then the sequence { x n } satisfies
x n p max { x 0 p , γ f ( p ) μ G p τ γ β } , n 0 .

We prove this by mathematical induction as follows.

Obviously, it is true for n = 0 . Assume it is true for n = k for some k N .

From (1.18) and Lemma 2.6, we have
x k + 1 p = α k γ f ( x k ) + ( I α k μ G ) A [ k + 1 ] x k p = α k [ γ f ( x k ) μ G p ] + ( I α k μ G ) A [ k + 1 ] x k ( I α k μ G ) p ( 1 α k τ ) x k p + α k γ [ f ( x k ) f ( p ) ] + γ f ( p ) μ G p ( 1 α k τ ) x k p + α k γ β x k p + α k γ f ( p ) μ G p = [ 1 α k ( τ γ β ) ] x k p + α k ( τ γ β ) γ f ( p ) μ G p τ γ β max { x k p , γ f ( p ) μ G p τ γ β } .

Hence the proof. Thus, the sequence { x n } is bounded and so are { T [ n ] x n } , { G A [ n ] x n } , { f ( x n ) } , and { A [ n ] x n } .

From (1.18) and Lemma 2.6, we have
x n + N + 1 x n + 1 = [ α n + N γ f ( x n + N ) + ( I α n + N μ G ) A [ n + 1 ] x n + N ] [ α n γ f ( x n ) ( I α n μ G ) A [ n + 1 ] x n ] = α n + N γ f ( x n + N ) α n + N γ f ( x n ) + α n + N γ f ( x n ) α n γ f ( x n ) + ( I α n + N μ G ) A [ n + 1 ] x n + N ( I α n + N μ G ) A [ n + 1 ] x n + ( I α n + N μ G ) A [ n + 1 ] x n ( I α n μ G ) A [ n + 1 ] x n = α n + N γ [ f ( x n + N ) f ( x n ) ] + ( α n + N α n ) γ f ( x n ) + ( I α n + N μ G ) A [ n + 1 ] x n + N ( I α n + N μ G ) A [ n + 1 ] x n + ( α n α n + N ) μ G A [ n + 1 ] x n α n + N γ β x n + N x n + | α n + N α n | γ f ( x n ) + ( 1 α n + N τ ) x n + N x n + | α n + N α n | μ G A [ n + 1 ] x n [ 1 α n + N ( τ γ β ) x n + N x n + | α n + N α n | M 4 ,
where M 4 is an appropriate constant such that M 4 sup n 1 { μ G A [ n + 1 ] x n + γ f ( x n ) } . By conditions (K1′), (K2′) and Lemma 2.8, we have
x n + N x n 0 as  n .
(4.1)
From (1.18) and condition (K1′), we have
x n + 1 A [ n + 1 ] x n = α n γ f ( x n ) + ( I α n μ G ) A [ n + 1 ] x n A [ n + 1 ] x n = α n γ f ( x n ) μ G A [ n + 1 ] x n 0 as  n .
(4.2)
Recursively,
x n + N A [ n + N ] x n + N 1 0 as  n , x n + N 1 A [ n + N 1 ] x n + N 2 0 as  n .
By condition (K3′) and Lemma 2.7, we know that A [ n + N ] is nonexpansive, so we get
A [ n + N ] x n + N 1 A [ n + N ] A [ n + N 1 ] x n + N 2 0 as  n .
Proceeding accordingly, we have
A [ n + N ] A [ n + N 1 ] x n + N 2 A [ n + N ] A [ n + N 1 ] A [ n + N 2 ] x n + N 3 0 as  n , A [ n + N ] A [ n + 2 ] x n + 1 A [ n + N ] A [ n + 1 ] x n 0 as  n .
Note that
x n + N A [ n + N ] A [ n + 1 ] x n x n + N A [ n + N ] x n + N 1 + A [ n + N ] x n + N 1 A [ n + N ] A [ n + N 1 ] x n + N 2 + + A [ n + N ] A [ n + 2 ] x n + 1 A [ n + N ] A [ n + 1 ] x n .
From the above inequality, we obtain
x n + N A [ n + N ] A [ n + 1 ] x n 0 as  n .
Since
x n A [ n + N ] A [ n + 1 ] x n x n x n + N + x n + N A [ n + N ] A [ n + 1 ] x n 0 as  n ,
we conclude that
x n A [ n + N ] A [ n + 1 ] x n 0 as  n .
(4.3)
Take a subsequence { x n j } { x n } , by (4.3) we get
x n j A [ n j + N ] A [ n j + 1 ] x n j 0 as  j .
Notice that for each n j , A [ n j + N ] A [ n j + N 1 ] A [ n j + 1 ] is some permutation of the mappings A 1 A 2 A N . Since A 1 , A 2 , , A N are finite, all the finite permutations are N!, there must be some permutation appearing infinitely many times. Without loss of generality, suppose this permutation is A 1 A 2 A N , we can take a subsequence { x n j k } { x n j } such that x n j k q ( k ) and
x n j k A 1 A 2 A N x n j k 0 as  k .
By Lemma 2.7, we conclude that A 1 , A 2 , , A N are all nonexpansive. It is clear that A [ n j + N ] A [ n j + N 1 ] A [ n j + 1 ] is nonexpansive, so is A 1 A 2 A N . By Lemma 2.2, we have A 1 A 2 A N q = q . From Lemmas 2.7 and 2.9, we obtain
q F ( A 1 A 2 A N ) = i = 1 N F ( A i ) = i = 1 N F ( T i ) ,
that is,
ω ω ( x n ) i = 1 N F ( T i ) .
(4.4)

We now prove that lim sup n ( γ f μ G ) x , j q ( x n + 1 x ) 0 , where x is obtained in Lemma 2.10. Put a n : = ( γ f μ G ) x , j q ( x n x ) , n 1 .

Define a map ϕ : E R by
ϕ ( x ) = μ n x n x q , x E .
Then ϕ is continuous, convex, and ϕ ( x ) as x . Since E is reflexive, there exists y C such that ϕ ( y ) = min z C ϕ ( z ) . Hence the set
K : = { y C : ϕ ( y ) = min z C ϕ ( z ) } .
Therefore, applying Lemma 2.4, we have K F ( T β n ) . Without loss of generality, assume x = y K F ( T β n ) . Let t ( 0 , 1 ) . Then it follows that ϕ ( x ) ϕ ( x + t ( γ f μ G ) x ) , and using Lemma 2.3, we obtain that
x n x t ( γ f μ G ) x q x n x q q t ( γ f μ G ) x , j q ( x n x t ( γ f μ G ) x ) .
This implies that
μ n ( γ f μ G ) x , j q ( x n x t ( γ f μ G ) x ) 0 .
By Lemma 2.11, j q is norm-to-weak uniformly continuous on a bounded subset of E, so we obtain, as t 0 , that
( γ f μ G ) x , j q ( x n x ) ( γ f μ G ) x , j q ( x n x t ( γ f μ G ) x ) 0 .
Hence, for all ϵ > 0 , there exists δ ϵ > 0 such that t ( 0 , δ ϵ ) , and for all n 1 ,
( γ f μ G ) x , j q ( x n x ) ( γ f μ G ) x , j q ( x n x t ( γ f μ G ) x ) < ϵ .
Consequently,
μ n ( γ f μ G ) x , j q ( x n x ) μ n ( γ f μ G ) x , j q ( x n x t ( γ f μ G ) x ) + ϵ ϵ .
Since ϵ is arbitrary, we have
μ n ( γ f μ G ) x , j q ( x n x ) 0 .

Thus, μ n ( a n ) 0 for any Banach limit μ.

Furthermore, by (4.1) x n + N x n 0 as n , we therefore conclude that
lim sup n ( a n + N a n ) = lim sup n ( ( γ f μ G ) x , j q ( x n + N x ) ( γ f μ G ) x , j q ( x n x ) ) = lim sup n ( ( γ f μ G ) x , j q ( x n + N x ) j q ( x n x ) ) = 0 .
Hence, by Lemma 2.12 we obtain lim sup n a n 0 , that is,
lim sup n ( γ f μ G ) x , j q ( x n x ) 0 .
(4.5)
From (1.18), Lemmas 2.3, 2.6 and 2.13, we have
x n + 1 x q = x n + 1 x , j q ( x n + 1 x ) = α n [ γ f ( x n ) μ G x ] + ( I α n μ G ) ( A [ n + 1 ] x n x ) , j q ( x n + 1 x ) = α n γ f ( x n ) μ G x , j q ( x n + 1 x ) + ( I α n μ G ) ( A [ n + 1 ] x n x ) , j q ( x n + 1 x ) = α n γ f ( x n ) γ f ( x ) , j q ( x n + 1 x ) + α n γ f ( x ) μ G x , j q ( x n + 1 x ) + ( I α n μ G ) ( A [ n + 1 ] x n x ) , j q ( x n + 1 x ) α n γ f ( x n ) f ( x ) x n + 1 x q 1 + α n ( γ f μ G ) x , j q ( x n + 1 x ) + ( I α n μ G ) A [ n + 1 ] x n ( I α n μ G ) x x n + 1 x q 1 ( 1 α n τ ) x n x x n + 1 x q 1 + α n γ β x n x x n + 1 x q 1 + α n ( γ f μ G ) x , j q ( x n + 1 x ) = [ 1 α n ( τ γ β ) ] x n x x n + 1 x q 1 + α n ( γ f μ G ) x , j q ( x n + 1 x ) [ 1 α n ( τ γ β ) ] [ 1 q x n x q + ( q 1 q ) x n + 1 x q ] + α n ( γ f μ G ) x , j q ( x n + 1 x ) .
This implies that
x n + 1 x q 1 α n ( τ γ β ) 1 + α n ( q 1 ) ( τ γ β ) x n x q + q α n 1 + α n ( q 1 ) ( τ γ β ) ( γ f μ G ) x , j q ( x n + 1 x ) [ 1 α n ( τ γ β ) ] x n x q + q α n 1 + α n ( q 1 ) ( τ γ β ) ( γ f μ G ) x , j q ( x n + 1 x ) ( 1 γ n ) x n x q + δ n ,

where γ n : = α n ( τ γ β ) and δ n : = q α n 1 + α n ( q 1 ) ( τ γ β ) ( γ f μ G ) x , j q ( x n + 1 x ) . From (K1′), lim n γ n = 0 , n = 0 γ n = . δ n γ n = q [ 1 + α n ( q 1 ) ( τ γ β ) ] ( τ γ β ) ( γ f μ G ) x , j q ( x n + 1 x ) . So, lim sup n δ n γ n 0 . Hence, by Lemma 2.8 we have that x n x as n . This completes the proof. □

5 Conclusion

Let E = H be a real Hilbert space, q = 2 , d q = 1 in Theorems 3.1 and 4.1, then we get the following result.

Corollary 5.1 (Tian and Di [12])

Let { x n } be a sequence generated by
{ T β n = β n I + ( 1 β n ) i = 1 N λ i T i , x n + 1 = α n γ f ( x n ) + ( I α n μ G ) T β n x n , n 0 .
Assume that { α n } and { β n } are sequences in ( 0 , 1 ) satisfying the conditions
  1. (K1)

    lim n α n = 0 , n = 1 α n = ,

     
  2. (K2)

    n = 1 | α n + 1 α n | < , n = 1 | β n + 1 β n | < ,

     
  3. (K3)

    0 < max k i β n < a < 1 , n 0 ,

     
then { x n } converges strongly to a common fixed point of { T i } i = 1 N , which solves the variational inequality
( γ f μ G ) x , x x 0 , x i = 1 N F ( T i ) .

Corollary 5.2 (Tian and Di [12])

Let { x n } be a sequence generated by
{ A [ n ] = β [ n ] I + ( 1 β [ n ] ) T [ n ] , x n + 1 = α n γ f ( x n ) + ( I α n μ G ) A [ n + 1 ] x n , n 0 ,

where T [ n ] = T i , with i = n ( mod N ) , 1 i N . Assume that { α n } and { β n } are sequences in ( 0 , 1 ) satisfying the conditions:

(K1′) lim n α n = 0 , n = 1 α n = ;

(K2′) n = 1 | α n + 1 α n | < or lim n α n α n + N = 1 ;

(K3′) β [ n ] [ k , 1 ) , n 0 , where k = max { k i : 1 i N } .

Then { x n } converges strongly to a common fixed point of { T i } i = 1 N , which solves the variational inequality
( γ f μ G ) x , x x 0 , x i = 1 N F ( T i ) .

Let E = H be a real Hilbert space; q = 2 , d q = 1 , n = 1 , β n = 0 , G = A , μ = 1 and T is a nonexpansive mapping in Theorems 3.1 and 4.1, then we get the following.

Corollary 5.3 (Tian [11])

Let { x n } be a sequence generated by x 0 H ,
x n