Open Access

Approximating fixed points for a reversible semigroup of Lipschitzian mappings in a smooth Banach space

Journal of Inequalities and Applications20132013:555

https://doi.org/10.1186/1029-242X-2013-555

Received: 11 August 2013

Accepted: 10 October 2013

Published: 22 November 2013

Abstract

In this paper, we approximate a fixed point of the semigroup φ = { T s : s S } of Lipschitzian mappings from a nonempty compact convex subset C of a smooth Banach space E into C with a uniform Lipschitzian condition and with respect to a finite family of sequences { μ i , n } i = 1 , n = 1 m , of left strong regular invariant means defined on an appropriate invariant subspace of l ( S ) . Our result extends the main results announced by several others.

MSC:47H09, 47H10, 47J25.

Keywords

smooth Banach space asymptotically nonexpansive semigroup reversible semigroup invariant mean

1 Introduction

Let E be a real Banach space with the topological dual E , and let C be a nonempty closed and convex subset of E. Recall that a mapping T of C into itself is said to be
  1. (1)
    Lipschitzian with Lipschitz constant l > 0 if
    T x T y l x y , x , y C .
     
  2. (2)
    nonexpansive if
    T x T y x y , x , y C .
     
  3. (3)
    asymptotically nonexpansive if there exists a sequence { l n } of positive numbers such that lim n l n = 1 and
    T n x T n y l n x y , x , y C .
     

A semigroup S is called left reversible if any two closed right ideals of S have non-void intersection, i.e., a S b S for a , b S . In this case, ( S , ) is a directed set when the binary relation on S is defined by a b if and only if a S b S for a , b S .

Notation Throughout the rest of this paper, S will always denote a left reversible semigroup with an identity e.

In [1], Lau et al. studied iterative schemes for approximating a fixed point of the semigroup φ = { T ( s ) : s S } of nonexpansive mappings on a nonempty compact convex subset C of a smooth (and strictly convex) Banach space and introduced the following iteration process. Let x 1 = x C and
x n + 1 = α n x n + ( 1 α n ) T μ n x n , n 1 ,
(1)

where { μ n } n = 1 is a sequence of left strong regular invariant means defined on an appropriate invariant subspace of l ( S ) .

φ = { T ( s ) : s S } is called a representation of S as a Lipschitzian mapping on C with Lipschitz constant { l ( s ) : s S } if T ( s ) is a Lipschitzian with Lipschitz constant l ( s ) for each s S , T ( s t ) = T ( s ) T ( t ) for each t , s S and T ( e ) = I . φ is called an asymptotically nonexpansive semigroup on C if φ is a representation of S as a Lipschitzian mapping on C with Lipschitz constant { l ( s ) ; s s } and lim s l ( s ) 1 .

In 2008, Saeidi proved the following theorem.

Theorem 1.1 [2]

Let S be a left reversible semigroup and φ = { T s : s S } be a representation of S as a Lipschitzian mapping from a nonempty compact convex subset C of a smooth Banach space E into C, with uniform Lipschitzian constant lim s K ( s ) 1 , and let f be an α-contraction on C for some 0 < α < 1 . Let X be a left invariant φ-stable subspace of L ( φ ) containing 1, let { μ n } n = 1 be a sequence of left strong regular invariant means defined on X such that lim n μ n + 1 μ n = 0 , and let { c n } n = 1 be a sequence defined by
c n = sup x , y C ( T μ n x T μ n y x y ) , n 1 .

Let { α n } n = 1 , { β n } n = 1 and { γ n } n = 1 be sequences in ( 0 , 1 ) such that

(C1) α n + β n + γ n = 1 , n 1 ,

(C2) lim n α n = 0 ,

(C3) n = 1 α n = ,

(C4) 0 < lim inf n β n lim sup n β n < 1 ,

(C5) lim sup n c n α n 0 .

If { x n } n = 1 is a sequence generated by x 1 C and
x n + 1 = α n f ( x n ) + β n x n + γ n T μ n x n , n 1 ,
then the sequence { x n } n = 1 converges strongly to some z Fix ( φ ) , the set of common fixed points of φ, which is the unique solution of the variational inequality
( f I ) z , J ( y z ) 0 , y Fix ( φ ) .

Equivalently, one has z = P f z , where P is the unique sunny nonexpansive retraction of C onto F ( φ ) .

In 2007, Zhang et al. [3] introduced the following composite iteration scheme:
{ y n = β n x n + ( 1 β n ) T t n x n , x n + 1 = α n u + ( 1 α n ) y n ,
(2)

where { T ( t ) : t 0 } is a nonexpansive semigroup from C to C, u is an arbitrary (but fixed) element in C, { α n } ( 0 , 1 ) and { β n } [ 0 , 1 ] , { t n } R + , and proved some strong convergence theorems of an explicit composite iteration scheme for nonexpansive mappings in the framework of a reflexive Banach space with a uniformly Gâteaux differentiable norm, uniformly smooth Banach space and uniformly convex Banach space with a weakly continuous normalized duality mapping.

Motivated and inspired by Zhang et al. [3] and Saeidi [2], Katchang and Kumam proved the following theorem.

Theorem 1.2 [4]

Let S be a left reversible semigroup, and let φ = { T s : s S } be a representation of S as a Lipschitzian mapping from a nonempty compact convex subset C of a smooth Banach space E into C, with uniform Lipschitzian constant lim s K ( s ) 1 , and let f be an α-contraction on C for some 0 < α < 1 . Let X be a left invariant φ-stable subspace of L ( φ ) containing 1, let { μ n } n = 1 be a sequence of left strong regular invariant means defined on X such that lim n μ n + 1 μ n = 0 , and let { c n } n = 1 be a sequence defined by
c n = sup x , y C ( T μ n x T μ n y x y ) , n 1 .

Let { α n } n = 1 , { β n } n = 1 , { γ n } n = 1 and { δ n } n = 1 be sequences in ( 0 , 1 ) such that

(C1) α n + β n + γ n = 1 , n 1 ,

(C2) lim n α n = 0 ,

(C3) n = 1 α n = ,

(C4) 0 < lim inf n β n lim sup n β n < 1 ,

(C5) lim sup n c n α n 0 ,

(C6) lim n δ n = 0 .

If { x n } n = 1 is a sequence generated by x 1 C and
{ y n = δ n x n + ( 1 δ n ) T μ n x n , x n + 1 = α n f ( x n ) + β n x n + γ n y n , n 1 ,
(3)
then the sequence { x n } n = 1 converges strongly to some z Fix ( φ ) , which is the unique solution of the variational inequality
( f I ) z , J ( y z ) 0 , y F ( φ ) .

Equivalently, one has z = P f z , where P is the unique sunny nonexpansive retraction of C onto F ( φ ) .

Recently, many authors studied fixed point results for a nonlinear semigroup mapping, for example, [58].

In this paper, motivated and inspired by Qianglian et al. [9], Lau et al. [1], Zhang et al. [3], Saeidi [2], Katchang and Kumam [4], Sunthrayuth and Kumam [10, 11] and Wattanawitoon and Kumam [12], we introduce the composite explicit viscosity iterative schemes as follows:
x n + 1 = α n f ( y 1 , n ) + β n x n + γ n T μ 1 , n y 1 , n , y i , n = δ i , n y i + 1 , n + ( I δ i , n ) T μ i , n y i + 1 , n , i = 1 , 2 , , m , y m + 1 , n = x n
(4)
for an asymptotically nonexpansive semigroup φ = { T s : s S } on a compact convex subset C of a smooth Banach space E with respect to a finite family of left regular sequences { μ i , n } i = 1 , n = 1 m , of invariant means defined on an appropriate invariant subspace of l ( S ) . We prove, under certain appropriate assumptions on the sequences { α n } n = 1 , { β n } n = 1 , { γ n } n = 1 and { δ n } i = 1 , n = 1 m , , that { x n } n = 1 and { y n } i = 1 , n = 1 m , defined by (4) converge strongly to z Fix ( φ ) , which is the unique solution of the variational inequality
( f I ) z , J ( y z ) 0 , y Fix ( φ ) .

Our result improves and extends many previous results (e.g., [1, 2, 4, 1315] and many others).

2 Preliminaries

Let E be the topological dual of a real Banach space E. The value of j E at x E will be denoted by x , j or j ( x ) . With each x E , we associate the set
J ( x ) = { j E : x , j = x 2 = j 2 } .

Using the Hahn-Banach theorem, it is immediately clear that J ( x ) ϕ for each x E . The multi-valued mapping J from E into E is said to be the (normalized) duality mapping. A Banach space E is said to be smooth if the duality mapping J is single-valued. As is well known, the duality mapping is norm to weak-star continuous when E is smooth, see [16].

Let B ( S ) be the Banach space of all bounded real-valued functions defined on S with supremum norm. For each s S , we define the left and right translation operators l ( s ) f and r ( s ) f on B ( S ) by
( l ( s ) f ) ( t ) = f ( s t ) , and ( r ( s ) f ) ( t ) = f ( t s ) ,
for each s S and f B ( S ) , respectively. Let X be a subspace of B ( S ) containing 1 and let X be its topological dual. An element μ of X is said to be a mean on X if μ = μ ( 1 ) = 1 . Let μ X . Then we define r ( s ) μ , l ( s ) μ X by ( r ( s ) μ ) f = μ ( r ( s ) f ) , ( l ( s ) μ ) f = μ ( l ( s ) f ) for each f X and s S . It is easy to see that if μ is a mean on X, then r ( s ) μ and l ( s ) μ are also. We often write μ t ( f ( t ) ) instead of μ ( f ) for μ X and f X . Let X be left invariant (resp. right invariant), i.e., l ( s ) ( X ) X (resp. r ( s ) ( X ) X ) for each s S . A mean μ on X is said to be left invariant (resp. right invariant) if l ( s ) μ = μ (resp. r ( s ) μ = μ ) for each s S and f X . X is said to be left (resp. right) amenable if X has a left (resp. right) invariant mean. The semigroup S is amenable (i.e., S is both left and right amenable) when S is a commutative semigroup or a solvable group. However, the free group (or semigroup) on two generators is not left amenable. If a semigroup S is left amenable, then S is left reversible, but the converse is not true see [17], [[18], p.335]. A net { μ α } of means on X is said to be strongly left regular if
lim α l ( s ) μ α μ α = 0 ,

for each s S , where l ( s ) is the adjoint operator of l s .

Let φ = { T ( s ) : s S } be a representation of S as a Lipschitzian mapping on C with Lipschitz constant { l ( s ) : s S } . By Fix ( φ ) we denote the set of common fixed points of φ, i.e.,
Fix ( φ ) = s S { x C : T ( s ) x = x } .

We denote by C a the set of almost periodic elements in C, i.e., all x C such that { T ( s ) x : s S } is relatively compact in the norm topology of E. Let X be a subspace of B ( S ) such that the functions (i) s T ( s ) x , x and (ii) s T ( s ) x y on S are in X for all x , y C and x E . We will call a subspace X of B ( S ) satisfying (i) and (ii) φ-stable. We know that if X is a subspace of B ( S ) containing 1 and the function s T ( s ) x , x on S is in X for all x C and x E , then there exists a unique point x 0 E such that μ T ( ) x , x = x 0 , x for a mean μ on X, x C and x E . We denote such a point x 0 E by T μ x . See [19] for more details.

Lemma 2.1 [20]

Let S be a semigroup and C be a nonempty closed convex subset of a reflexive Banach space E. Let φ = { T t : t S } be a nonexpansive semigroup on H such that { T t x : t S } is bounded for some x C , let X be a subspace of B ( S ) such that 1 X and the mapping t T t x , y is an element of X for each x C and y E , and μ is a mean on X. If we write T μ x instead of T t x d μ ( t ) , then the following hold:
  1. (i)

    T μ is a nonexpansive mapping from C into C.

     
  2. (ii)

    T μ x = x for each x Fix ( φ ) .

     
  3. (iii)

    T μ x co ¯ { T t x : t S } for each x C .

     

Lemma 2.2 [21]

Let φ = { T ( s ) : s S } be a representation of S as a Lipschitzian mapping from a nonempty weakly compact convex subset C of a Banach space E into C, with uniform Lipschitzian constant lim s l ( s ) 1 on the Lipschitz constant of mappings. Let X be a left invariant and φ-stable subspace of B ( S ) , and let { μ n } n = 1 be an asymptotically left invariant sequence of means on X. If z C a and lim inf n T μ n z z = 0 , then z is a common fixed point of φ.

Lemma 2.3 [2]

Let φ = { T ( s ) : s S } be a representation of S as a Lipschitzian mapping from a nonempty weakly compact convex subset C of a Banach space E into C, with uniform Lipschitzian constant lim s l ( s ) 1 on the Lipschitz constant of mappings. Let X be a left invariant subspace of B ( S ) containing 1 such that the mapping s T ( s ) x , x on S is in X for all x C and x E , and { μ n } n = 1 is an asymptotically left invariant sequence of means on X. Then
lim sup n ( sup x , y C ( T μ n x T μ n y x y ) ) 0 .

Let D be a subset of B, where B is a subset of a Banach space E, and let P be a retraction of B onto D, that is, P x = x for each x D . Then P is said to be sunny [22] if for each x B and t 0 with P x + t ( x P x ) B , P ( P x + t ( x P x ) ) = P x . A subset D of B is said to be a sunny nonexpansive retract of B if there exists a sunny nonexpansive retraction P of B into D.

Lemma 2.4 [1]

Let φ = { T ( s ) : s S } be a representation of S as a Lipschitzian mapping from a nonempty compact convex subset C of a smooth Banach space E into C, with uniform Lipschitzian constant lim s l ( s ) 1 on the Lipschitz constant of mappings. Let X be a left invariant and φ-stable subspace of B ( S ) containing 1 and μ be a left invariant mean on X. Then Fix ( φ ) is a sunny nonexpansive retract of C, and the sunny nonexpansive retraction of C onto Fix ( φ ) is unique.

Lemma 2.5 [16]

Let C be a nonempty convex subset of a smooth Banach space E, let D be a nonempty subset of C, and let P : C D be a retraction. Then the following are equivalent:
  1. (a)

    P is sunny nonexpansive.

     
  2. (b)

    x P x , J ( y P x ) 0 for all x C and y D .

     
  3. (c)

    x y , J ( P x P y ) P x P y 2 for all x , y C .

     

Lemma 2.6 [3]

Let { x n } n = 1 and { y n } n = 1 be bounded sequences in a Banach space X, and let { α n } n = 1 be a sequence in [ 0 , 1 ] such that 0 < lim inf n α n lim sup n α n < 1 . Suppose x n + 1 = α n x n + ( 1 α n ) y n for all integers n 0 and
lim sup n ( y n + 1 y n x n + 1 x n ) 0 .

Then lim n y n x n = 0 .

Lemma 2.7 [19]

Let E be a real smooth Banach space and J be the duality mapping. Then
x + y 2 x 2 + 2 y , J ( x + y ) , x , y E .

Lemma 2.8 [23]

Let { a n } n = 1 be a sequence of nonnegative real numbers such that
a n + 1 ( 1 b n ) a n + b n c n , n 0 ,
where { b n } n = 1 and { c n } n = 1 are sequences of real numbers satisfying the following conditions:
  1. (i)

    { b n } n = 1 ( 0 , 1 ) , n = 0 b n = ,

     
  2. (ii)

    either lim sup n c n 0 or n = 0 | b n c n | < .

     

Then lim n a n = 0 .

Lemma 2.9 [16]

Let ( X , d ) be a metric space. A subset C of X is compact if and only if every sequence in C contains a convergent subsequence with limit in C.

3 The main result

In this section, we establish a strong convergence theorem for finding a common fixed point of an asymptotically nonexpansive semigroup in a smooth Banach space.

Theorem 3.1 Let φ = { T ( s ) : s S } be a representation of S as a Lipschitzian mapping from a nonempty compact convex subset C of a smooth Banach space E into C, with uniform Lipschitzian constant lim s l ( s ) 1 on the Lipschitz constant of mappings, such that Fix ( φ ) , and let f be a contraction of C into itself with constant α ( 0 , 1 ) . Let X be a left invariant and φ-stable subspace of B ( S ) containing 1 and the function t T t x , y is an element of X for each x C and y H , and let { μ i , n } i = 1 , n = 1 m , be a finite family of left regular sequences of invariant means on X such that for i = 1 , 2 , , m , lim n μ i , n + 1 μ i , n = 0 . Let { α n } n = 1 , { β n } n = 1 and { γ n } n = 1 be sequences in ( 0 , 1 ) satisfying conditions (C1)-(C4), and let { δ n } i = 1 , n = 1 m , be a sequence in ( 0 , 1 ) satisfying the condition

( C 5 ) lim n δ i , n = 1 , i = 1 , 2 , , m .

If { x n } n = 1 and { y i , n } i = 1 , n = 1 m , are sequences generated by x 1 C and
x n + 1 = α n f ( y 1 , n ) + β n x n + γ n T μ 1 , n y 1 , n , y i , n = δ i , n y i + 1 , n + ( I δ i , n ) T μ i , n y i + 1 , n , i = 1 , 2 , , m , y m + 1 , n = x n ,
(5)
then { x n } n = 1 and { y i , n } i = 1 , n = 1 m , converge strongly to z Fix ( φ ) , which is the unique solution of the variational inequality
( f I ) z , J ( y z ) 0 , y Fix ( φ ) .
(6)

Equivalently, z = P f ( z ) , where P denotes the unique sunny nonexpansive retraction of C onto Fix ( φ ) .

Proof From Lemma 2.1 and the definition of { y i , n } i = 1 , n = 1 m , , for every z Fix ( φ ) , we have
y i , n z = δ i , n y i + 1 , n + ( 1 δ i , n ) T μ i , n y i + 1 , n z δ i , n y i + 1 , n z + ( 1 δ i , n ) T μ i , n y i + 1 , n T μ i , n z = δ i , n y i + 1 , n z + ( 1 δ i , n ) y i + 1 , n z = y i + 1 , n z .
Therefore, we have
y 1 , n z y 2 , n z y m , n z x n z .
(7)

We shall divide the proof into several steps.

Step 1. Let { t n } n = 1 be a sequence in C. Then
lim n T μ i , n + 1 t n T μ i , n t n = 0 , i = 1 , 2 , , m .

Proof of Step 1. This assertion is proved in [24, 25].

Step 2. lim n x n + 1 x n = 0 .

Proof of Step 2. From the definition of { y i , n } i = 1 , n = 1 m , , we have
y i , n + 1 y i , n = δ i , n + 1 y i + 1 , n + 1 + ( 1 δ i , n + 1 ) T μ i , n + 1 y i + 1 , n + 1 δ i , n y i + 1 , n ( 1 δ i , n ) T μ i , n y i + 1 , n = δ i , n + 1 y i + 1 , n + 1 δ i , n + 1 y i + 1 , n + δ i , n + 1 y i + 1 , n + ( 1 δ i , n + 1 ) T μ i , n + 1 y i + 1 , n + 1 δ i , n y i + 1 , n ( 1 δ i , n ) T μ i , n y i + 1 , n δ i , n + 1 y i + 1 , n + 1 y i + 1 , n + | δ i , n + 1 δ i , n | y i + 1 , n + ( 1 δ i , n + 1 ) T μ i , n + 1 y i + 1 , n + 1 + ( 1 δ i , n ) T μ i , n y i + 1 , n y i + 1 , n + 1 y i + 1 , n + | δ i , n + 1 δ i + 1 , n | y i + 1 , n + ( 1 δ i , n + 1 ) T μ i , n + 1 y i + 1 , n + 1 + ( 1 δ i , n ) T μ i , n y i + 1 , n ,
which implies that
y i , n + 1 y i , n x n + 1 x n + j = i m ( | δ j , n + 1 δ j , n | y j + 1 , n + ( 1 δ i , n + 1 ) T μ i , n + 1 y i + 1 , n + 1 + ( 1 δ i , n ) T μ i , n y i + 1 , n ) .
(8)
Define
x n + 1 = ( 1 β n ) z n + β n x n , n 1 .
(9)
Observe that from the definition of z n , we obtain
z n + 1 z n = x n + 2 β n + 1 x n + 1 1 β n + 1 x n + 1 β n x n 1 β n = α n + 1 f ( y 1 , n + 1 ) + γ n + 1 T μ 1 , n + 1 y 1 , n + 1 1 β n + 1 α n f ( y 1 , n ) + γ n T μ 1 , n y 1 , n 1 β n = α n + 1 1 β n + 1 ( f ( y 1 , n + 1 ) T μ 1 , n + 1 y 1 , n + 1 ) α n 1 β n ( f ( y 1 , n ) T μ 1 , n y 1 , n ) + T μ 1 , n + 1 y 1 , n + 1 T μ 1 , n y 1 , n .
It follows that
z n + 1 z n α n + 1 1 β n + 1 f ( y 1 , n + 1 ) T μ 1 , n + 1 y 1 , n + 1 ) + α n 1 β n f ( y 1 , n ) T μ 1 , n y 1 , n + y 1 , n + 1 y 1 , n + T μ 1 , n + 1 y 1 , n T μ 1 , n y 1 , n .
(10)
Substituting (8) into (10), we obtain
z n + 1 z n x n + 1 x n α n + 1 1 β n + 1 f ( y 1 , n + 1 ) T μ 1 , n + 1 y 1 , n + 1 + α n 1 β n f ( y 1 , n ) T μ 1 , n y 1 , n + T μ 1 , n + 1 y 1 , n T μ 1 , n y 1 , n + j = 1 m ( | δ j , n + 1 δ j , n | y j + 1 , n + ( 1 δ i , n + 1 ) T μ i , n + 1 y i + 1 , n + 1 + ( 1 δ i , n ) T μ i , n y i + 1 , n ) .
(11)
It follows from Step 1, conditions (C2) and ( C 5 ) that
lim sup n ( z n + 1 z n x n + 1 x n ) 0 .
Applying Lemma 2.6 to (9), we get
lim n x n z n = 0 .
Consequently,
lim n x n + 1 x n = lim n ( 1 β n ) x n z n = 0 .
Step 3. We claim that ω ( { x n } n = 1 ) Fix ( φ ) , where
ω ( { x n } n = 1 ) : = { x C : { x n j } j = 1 { x n } n = 1 , lim j x n j x = 0 } .

Proof of Step 3. From Lemma 2.9, we get ω ( { x n } n = 1 ) .

Let x ω ( { x n } n = 1 ) . Then there exists a subsequence { x n j } j = 1 of { x n } n = 1 such that
lim j x n j x = 0 .
(12)
Observe that
x n T μ 1 , n x n x n x n + 1 + x n + 1 T μ 1 , n x n = x n x n + 1 + α n ( f ( y 1 , n ) T μ 1 , n x n ) + β n ( x n T μ 1 , n x n ) + γ n ( T μ 1 , n y 1 , n T μ 1 , n x n ) x n x n + 1 + α n f ( y 1 , n ) T μ 1 , n x n + β n x n T μ 1 , n x n + γ n T μ 1 , n y 1 , n T μ 1 , n x n x n x n + 1 + α n f ( y 1 , n ) T μ 1 , n x n + β n x n T μ 1 , n x n + γ n y 1 , n x n x n x n + 1 + α n f ( y 1 , n ) T μ 1 , n x n + β n x n T μ 1 , n x n + γ n i = 1 m y i , n y i + 1 , n = x n x n + 1 + α n f ( y 1 , n ) T μ 1 , n x n + β n x n T μ 1 , n x n + γ n i = 1 m ( 1 δ i , n ) y i + 1 , n T μ i , n y i + 1 , n .
Therefore, we have
( 1 β n ) x n T μ 1 , n x n x n x n + 1 + α n f ( y 1 , n ) T μ 1 , n x n + γ n i = 1 m ( 1 δ i , n ) y i + 1 , n T μ i , n y i + 1 , n .
(13)
From condition (C4), it follows that
lim inf n ( 1 β n ) > 0 .
(14)
By conditions (C2) and ( C 5 ), Step 2, (13) and (14), we have
lim n x n T μ 1 , n x n = 0 .
(15)
Indeed, observe that
lim sup j x T μ 1 , n j x lim sup j ( x x n j + x n j T μ 1 , n j x n j + T μ 1 , n j x n j T μ 1 , n j x ) .

Thus, due to (15), Lemma 2.2 and Lemma 2.3, we get x Fix ( φ ) .

Step 4. { x n } n = 1 converges strongly to z = P f ( z ) .

Proof of Step 4. From Lemma 2.4 there exists a unique sunny nonexpansive retraction P of C onto Fix ( φ ) . Since f is a contraction of C into itself, therefore Pf is a contraction. Then the Banach contraction guarantees that Pf has a unique fixed point z. By Lemma 2.5, z is the unique solution of the variational inequality
( f I ) z , J ( y z ) 0 , y Fix ( φ ) .
(16)
Let us show that
lim sup n ( f I ) z , J ( x n z ) 0 .
Indeed, we can choose a subsequence { x n k } of { x n } such that
lim sup n ( f I ) z , J ( x n z ) = lim k ( f I ) z , J ( x n k z ) .
(17)
Since C is compact, we may assume, with no loss of generality, that { x n k } converges strongly to some y C . By Step 3, y Fix ( φ ) . Because the duality mapping J is norm to weak-star continuous from (16) and (17), we have
lim sup n ( f I ) z , J ( x n z ) 0 .
(18)
Using Lemma 2.1, Lemma 2.7 and relation (7), we have
x n + 1 z 2 = α n ( f ( y 1 , n ) z ) + β n ( x n z ) + γ n ( T μ 1 , n y 1 , n z ) 2 β n ( x n z ) + γ n ( T μ 1 , n y 1 , n z ) 2 + 2 α n f ( y 1 , n ) z , J ( x n + 1 z ) [ β n x n z + γ n T μ 1 , n y 1 , n z ] 2 + 2 α n f ( y 1 , n ) f ( z ) , J ( x n + 1 z ) + 2 α n f ( z ) z , J ( x n + 1 z ) [ β n x n z + γ n y 1 , n z ] 2 + 2 α n α y 1 , n z J ( x n + 1 z ) + 2 α n f ( z ) z , J ( x n + 1 z ) [ β n x n z + γ n x n z ] 2 + 2 α n α x n z x n + 1 z + 2 α n f ( z ) z , J ( x n + 1 z ) ( 1 α n ) 2 x n z 2 + α n α ( x n z 2 + x n + 1 z 2 ) + 2 α n f ( z ) z , J ( x n + 1 z ) ,
and consequently,
x n + 1 z 2 ( 1 α n ) 2 + α n α 1 α n α x n z 2 + 2 α n 1 α n α f ( z ) z , J ( x n + 1 z ) ( 1 2 α n ( 1 α n ) 1 α n α ) x n z 2 + 2 α n ( 1 α n ) 1 α n α ( 1 1 α f ( z ) z , J ( x n + 1 z ) + α n 2 ( 1 α ) x n z 2 ) .
Then we have
x n + 1 z 2 ( 1 b n ) x n z 2 + b n c n ,
(19)
where b n = 2 α n ( 1 α n ) 1 α n α and
c n = 1 1 α f ( z ) z , J ( x n + 1 z ) + α n 2 ( 1 α ) x n z 2 .
It follows from conditions (C2), (C3) and (18) that
n = 1 b n = , lim sup n c n 0 .

Therefore, applying Lemma 2.8 to (19), we have that { x n } n = 1 converges strongly to z = P f ( z ) and since for i = 1 , 2 , , m , y i , n z x n z , therefore { y n } i = 1 , n = 1 m , converges strongly to z = P f ( z ) . This completes the proof. □

4 Applications

Let { g n , j } n , j = 0 be a family of real numbers. Then { g n , j } is said to be the strongly regular summation method [26, 27] if { g n , j } satisfies the following conditions:

(S1) g n , j 0 ,

(S2) j = 0 g n , j = 1 for every n,

(S3) lim n g n , j = 0 for every j,

(S4) lim n j = 0 | g n , j + 1 g n , j | = 0 .

Corollary 4.1 Let C be a compact convex subset of a smooth Banach space E, and let f be a contraction of C into itself with constant α ( 0 , 1 ) . Let T be an asymptotically nonexpansive mapping of C into itself with Lipschitz constants { k ( j ) } , and for i = 1 , 2 , , m , let { g ( i , n ) , j } n , j = 0 be a finite family of strongly regular summation methods such that
lim n j = 0 | g ( i , n ) , j + 1 g ( i , n ) , j | = 0 and n = 0 j = 0 g ( i , n ) , j ( k ( j ) 1 ) < .
Let { α n } n = 1 , { β n } n = 1 and { γ n } n = 1 be sequences in ( 0 , 1 ) satisfying conditions (C1)-(C4), and let { δ i , n } i = 1 , n = 1 m , be a sequence in ( 0 , 1 ) satisfying condition ( C 5 ). If { x n } n = 1 and { y i , n } i = 1 , n = 1 m , are sequences generated by x 1 C and
x n + 1 = α n f ( y 1 , n ) + β n x n + γ n j = 0 g ( 1 , n ) , j T ( j ) y 1 , n , y i , n = δ i , n y i + 1 , n + ( I δ i , n ) j = 0 g ( i , n ) , j T ( j ) y i + 1 , n , i = 1 , 2 , , m , y m + 1 , n = x n ,
then { x n } n = 1 and { y i , n } i = 1 , n = 1 m , converge strongly to z Fix ( T ) , which is the unique solution of the variational inequality
( f I ) z , J ( y z ) 0 , y Fix ( T ) .

Equivalently, z = P f ( z ) , where P denotes the unique sunny nonexpansive retraction of C onto Fix ( T ) .

Proof Denote by Z + = ( Z , + ) the semigroup of nonnegative integers. It is obvious that φ = { T j : j Z } is an asymptotically nonexpansive semigroup on C. For every n Z + and f l ( Z + ) , define
μ i , n f = j = 0 g ( i , n ) , j f ( j ) .
Hence { μ i , n } i = 1 , n = 1 m , is a strongly regular sequence of means on l ( Z + ) and lim n μ i , n + 1 μ i , n = 0 [28]. Further, for each y C , we have
T μ i , n = j = 0 g ( i , n ) , j T ( j ) y .

By Theorem 3.1, { x n } n = 1 and { y i , n } i = 1 , n = 1 m , converge strongly to z Fix ( T ) . This completes the proof. □

Example 4.2 Let C be a compact convex subset of a smooth Banach space E such that 0 C , and let f be a contraction of C into itself with constant α ( 0 , 1 ) . Let { α n } n = 1 , { β n } n = 1 and { γ n } n = 1 be sequences in ( 0 , 1 ) satisfying conditions (C1)-(C4), and let { δ n } n = 1 be a sequence in ( 0 , 1 ) satisfying condition ( C 5 ). Let { t n } n = 1 be sequences in ( 0 , 1 ) with t 1 > t 2 >  , lim n t n = 0 , lim n t n + 1 t n = 1 and n = 0 t n < . Let { x n } n = 1 and { y n } n = 1 be sequences generated by x 1 C and
x n + 1 = α n f ( y n ) + β n x n + γ n t n j = 0 ( 1 t n ) j ( 1 + 1 n 2 ) j n y n , y n = δ n x n + ( I δ n ) t n j = 0 ( 1 t n ) j ( 1 + 1 n 2 ) j n x n .

Then { x n } n = 1 and { y n } n = 1 converge strongly to z C .

Proof We define
T : C C , T x = ( 1 + 1 n 2 ) n x .
Obviously, T is an asymptotically nonexpansive mapping with Lipschitz constants l n = ( 1 + 1 n 2 ) . Define g n , j = t n ( 1 t n ) j . Then it follows that { q n , j } is a strongly regular summation method [28]. We also have
n = 0 j = 0 g n , j ( l n 1 ) = n = 0 j = 0 t n ( 1 t n ) j ( 1 + 1 n 2 1 ) = n = 0 j = 0 t n ( 1 t n ) j 1 n 2 n = 0 j = 0 t n 1 n 2 = n = 0 t n j = 0 1 n 2 < .

Therefore, by taking y 1 , n = y n and δ 1 , n = δ n in Corollary 4.1, we complete the proof. □

Let Q = { q n , m } n , m = 0 be a matrix satisfying the following conditions:
  1. (a)

    sup n 0 m = 0 | q n , m | < ,

     
  2. (b)

    m = 0 q n , m = 1 for every n N ,

     
  3. (c)

    lim n m = 0 | q n , m + 1 q n , m | = 0 .

     

Such a matrix Q is called strongly regular in the sense of Lorentz [29]. If Q is a strongly regular matrix, then for each m N , we have lim n q n , m = 0 , see [30]. Strongly regular matrices were used in the context of nonlinear ergodic theory in [31] and [32].

Corollary 4.3 Let C be a compact convex subset of a smooth Banach space E. Let T be an asymptotically nonexpansive mapping of C into itself, and let Q = { q n , m } n , m = 0 be a strongly regular matrix. Let { α n } n = 1 , { β n } n = 1 and { γ n } n = 1 be sequences in ( 0 , 1 ) satisfying conditions (C1)-(C4), and let { δ n } i = 1 , n = 1 m , be a sequence in ( 0 , 1 ) satisfying condition ( C 5 ). If { x n } n = 1 and { y i , n } i = 1 , n = 1 m , are sequences generated by x 1 C and
x n + 1 = α n f ( y 1 , n ) + β n x n + γ n m = 0 q ( 1 , n ) , m T m y 1 , n , y i , n = δ i , n y i + 1 , n + ( I δ i , n ) m = 0 q ( i , n ) , m T m y i + 1 , n , i = 1 , 2 , , m , y m + 1 , n = x n ,
then { x n } n = 1 and { y i , n } i = 1 , n = 1 m , converge strongly to z Fix ( T ) , which is the unique solution of the variational inequality
( f I ) z , J ( y z ) 0 , y Fix ( T ) .

Equivalently, z = P f ( z ) , where P denotes the unique sunny nonexpansive retraction of C onto Fix ( T ) .

Proof Let x C . For each n N , define
μ i , n f = m = 0 q ( i , n ) , m f ( m )
for each f L ( N ) . Hence { μ i , n } i = 1 , n = 1 m , is a strongly regular sequence of means on l ( Z + ) and lim n μ i , n + 1 μ i , n = 0 [33]. Further, for each y C , we have
T μ i , n = m = 0 q ( i , n ) , m T m y .

By Theorem 3.1, { x n } n = 1 and { y i , n } i = 1 , n = 1 m , converge strongly to z Fix ( T ) . This completes the proof. □

Example 4.4 Let C be a compact convex subset of a smooth Banach space E such that 0 C , and let f be a contraction of C into itself with constant α ( 0 , 1 ) . Let { α n } n = 1 , { β n } n = 1 and { γ n } n = 1 be sequences in ( 0 , 1 ) satisfying conditions (C1)-(C4), and let { δ n } n = 1 be a sequence in ( 0 , 1 ) satisfying condition ( C 5 ). Let { x n } n = 1 and { y n } n = 1 be sequences generated by x 1 C and
x n + 1 = α n f ( y n ) + β n x n + γ n m = 0 n 2 m n ( n 1 ) ( 1 1 n 2 ) m n y n , y n = δ n x n + ( I δ n ) m = 0 n 2 m n ( n 1 ) ( 1 1 n 2 ) m n x n .

Then { x n } n = 1 and { y n } n = 1 converge strongly to z C .

Proof We define
T : C C , T x = ( 1 1 n 2 ) n x .
Obviously, T is an asymptotically nonexpansive mapping with Lipschitz constants l n = ( 1 1 n 2 ) . Define
q n , m = { 2 m n ( n 1 ) , 0 m n , 0 , m > n .
Then it follows that { q n , m } is a strongly regular matrix. Further, we have
m = 0 q n , m = m = 0 n 2 m n ( n 1 ) = 2 n ( n 1 ) n ( n 1 ) 2 = 1 .
Therefore
sup n 0 m = 0 q n , m < .
On the other hand,
m = 0 | q n , m + 1 q n , m | = m = 0 n | q n , m + 1 q n , m | = m = 0 n 1 | q n , m + 1 q n , m | + | q n , n + 1 q n , n | = m = 0 n 1 | 2 ( m + 1 ) n ( n 1 ) 2 m n ( n 1 ) | + | 0 2 n n ( n 1 ) | = m = 0 n 1 | 2 ( m + 1 ) n ( n 1 ) 2 m n ( n 1 ) | + 2 n n ( n 1 ) = m = 0 n 1 2 n ( n 1 ) + 2 n n ( n 1 ) = 4 n n ( n 1 ) 0

as n . By taking y 1 , n = y n and δ 1 , n = δ n in Corollary 4.3, we complete the proof. □

Corollary 4.5 Let C be a compact convex subset of a smooth Banach space E such that 0 C , and let T be an asymptotically nonexpansive mapping of C into itself with Lipschitz constants { k ( j ) } satisfying j = 0 ( k ( j ) 1 ) < . Let f be a contraction of C into itself with constant α ( 0 , 1 ) , let { α n } n = 1 , { β n } n = 1 and { γ n } n = 1 be sequences in ( 0 , 1 ) satisfying conditions (C1)-(C4), and let { δ n } n = 1 be a sequence in ( 0 , 1 ) satisfying condition ( C 5 ). If { x n } n = 1 and { y n } n = 1 are sequences generated by x 1 C and
y n = δ n x n + 2 ( 1 δ n ) ( n + 1 ) 2 j = 0 ( n + 1 ) 2 T j x n , x n + 1 = α n f ( y n ) + β n x n + 2 γ n ( n + 1 ) 2 j = 0 ( n + 1 ) 2 T j y n ,
then { x n } n = 1 and { y n } n = 1 converge strongly to z Fix ( T ) , which is the unique solution of the variational inequality
( f I ) z , J ( y z ) 0 , y Fix ( T ) .
(20)

Equivalently, z = P f ( z ) , where P denotes the unique sunny nonexpansive retraction of C onto Fix ( T ) .

Proof Denote by Z + = ( Z , + ) the semigroup of nonnegative integers. It is obvious that φ = { T j : j Z } is an asymptotically nonexpansive semigroup on C. For every n Z + and f l ( Z + ) , define
μ n f = 1 ( n + 1 ) 2 j = 0 f ( j ) .
Hence { μ n } n = 1 is a strongly regular sequence of means on l ( Z + ) and lim n μ n + 1 μ n = 0 [28]. Further, for each y C , we have
T μ n = 1 ( n + 1 ) 2 j = 0 T j y .

By Theorem 3.1, { x n } n = 1 and { y n } n = 1 converge strongly to z Fix ( T ) . This completes the proof. □

Example 4.6 Let C be a compact convex subset of a smooth Banach space E such that 0 C . Let f be a contraction of C into itself with constant α ( 0 , 1 ) , let { α n } n = 1 , { β n } n = 1 and { γ n } n = 1 be sequences in ( 0 , 1 ) satisfying conditions (C1)-(C4), and let { δ n } n = 1 be a sequence in ( 0 , 1 ) satisfying condition ( C 5 ). If { x n } n = 1 and { y n } n = 1 are sequences generated by x 1 C and
y n = δ n x n + 2 ( 1 δ n ) ( n + 1 ) 2 j = 0 ( n + 1 ) 2 ( 1 + 1 n ln 2 n ) j n x n , x n + 1 = α n f ( y n ) + β n x n + 2 γ n ( n + 1 ) 2 j = 0 ( n + 1 ) 2 ( 1 + 1 n ln 2 n ) j n y n ,

then { x n } n = 1 and { y n } n = 1 converge strongly to z C .

Proof We define
T : C C , T ( x ) = ( 1 + 1 n ln 2 n ) n x .
Obviously, T is an asymptotically nonexpansive mapping with Lipschitz constants l n ( T ) = ( 1 + 1 n ln 2 n ) . Moreover,
n = 0 ( l n 1 ) = n = 0 ( 1 + 1 n ln 2 n 1 ) = n = 0 1 n ln 2 n < .

Therefore, applying Corollary 4.5, the result follows. □

Remark 4.7 For deducing some more applications, we refer to [13, 19, 24, 25, 28, 30].

Remark 4.8 Theorem 3.1 improves and extends [[4], Theorem 3.1] and [[2], Theorem 3.1] in the following aspects.
  1. (1)

    Theorem 3.1 extends [[4], Theorem 3.1] and [[2], Theorem 3.1] from one sequence of means to a finite family of sequences of means.

     
  2. (2)

    In Theorem 3.1, by taking T μ i , n = I for i = 1 , 2 , , m 1 , T μ m , n = T μ n and y m , n = y n , one can see that [[4], Theorem 3.1] is a special case of Theorem 3.1.

     
  3. (3)

    In Theorem 3.1, by taking T μ i , n = I for i = 1 , 2 , , m , one can see that [[2], Theorem 3.1] is a special case of Theorem 3.1.

     
  4. (4)

    Theorem 3.1 gives all consequences of [[4], Theorem 3.1] and [[2], Theorem 3.1] without assumption C 5 used in [[4], Theorem 3.1] and [[2], Theorem 3.1].

     

Declarations

Acknowledgements

The authors were supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (Grant No. NRU56000508).

Authors’ Affiliations

(1)
Department of Mathematics, University of Bonab
(2)
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT)

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© Piri and Kumam; licensee Springer. 2013

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