Skip to main content

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

Abstract

In this paper, we approximate a fixed point of the semigroup φ={ T s :sS} 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.

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

    TxTylxy,x,yC.
  2. (2)

    nonexpansive if

    TxTyxy,x,yC.
  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 xy,x,yC.

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

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):sS} 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 =xC and

x n + 1 = α n x n +(1 α n ) T μ n x n ,n1,
(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):sS} is called a representation of S as a Lipschitzian mapping on C with Lipschitz constant {l(s):sS} if T(s) is a Lipschitzian with Lipschitz constant l(s) for each sS, T(st)=T(s)T(t) for each t,sS 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);ss} 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 :sS} 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 ) ,n1.

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, n1,

(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 ,n1,

then the sequence { x n } n = 1 converges strongly to some zFix(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,yFix(φ).

Equivalently, one has z=Pfz, 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):t0} 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 :sS} 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 ) ,n1.

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, n1,

(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 zFix(φ), which is the unique solution of the variational inequality

( f I ) z , J ( y z ) 0,yF(φ).

Equivalently, one has z=Pfz, 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 :sS} 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 zFix(φ), which is the unique solution of the variational inequality

( f I ) z , J ( y z ) 0,yFix(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 xE will be denoted by x,j or j(x). With each xE, 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 xE. 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 sS, we define the left and right translation operators l(s)f and r(s)f on B(S) by

( l ( s ) f ) (t)=f(st),and ( r ( s ) f ) (t)=f(ts),

for each sS and fB(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 fX and sS. 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 fX. Let X be left invariant (resp. right invariant), i.e., l(s)(X)X (resp. r(s)(X)X) for each sS. 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 sS and fX. 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 sS, where l ( s ) is the adjoint operator of l s .

Let φ={T(s):sS} be a representation of S as a Lipschitzian mapping on C with Lipschitz constant {l(s):sS}. 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 xC such that {T(s)x:sS} is relatively compact in the norm topology of E. Let X be a subspace of B(S) such that the functions (i) sT(s)x, x and (ii) sT(s)xy on S are in X for all x,yC 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 sT(s)x, x on S is in X for all xC and x E , then there exists a unique point x 0 E such that μT()x, x = x 0 , x for a mean μ on X, xC 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 :tS} be a nonexpansive semigroup on H such that { T t x:tS} is bounded for some xC, let X be a subspace of B(S) such that 1X and the mapping t T t x, y is an element of X for each xC and y E , and μ is a mean on X. If we write T μ x instead of T t xdμ(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 xFix(φ).

  3. (iii)

    T μ x co ¯ { T t x:tS} for each xC.

Lemma 2.2 [21]

Let φ={T(s):sS} 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 zz=0, then z is a common fixed point of φ.

Lemma 2.3 [2]

Let φ={T(s):sS} 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 sT(s)x, x on S is in X for all xC 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, Px=x for each xD. Then P is said to be sunny [22] if for each xB and t0 with Px+t(xPx)B, P(Px+t(xPx))=Px. 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):sS} 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:CD be a retraction. Then the following are equivalent:

  1. (a)

    P is sunny nonexpansive.

  2. (b)

    xPx,J(yPx)0 for all xC and yD.

  3. (c)

    xy,J(PxPy) P x P y 2 for all x,yC.

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 n0 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,yE.

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 ,n0,

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):sS} 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 xC and yH, 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 zFix(φ), which is the unique solution of the variational inequality

( f I ) z , J ( y z ) 0,yFix(φ).
(6)

Equivalently, z=Pf(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 zFix(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 ,n1.
(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 xω( { 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 xFix(φ).

Step 4. { x n } n = 1 converges strongly to z=Pf(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,yFix(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 yC. By Step 3, yFix(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=Pf(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=Pf(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 |=0and 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 zFix(T), which is the unique solution of the variational inequality

( f I ) z , J ( y z ) 0,yFix(T).

Equivalently, z=Pf(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 :jZ} 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 yC, 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 zFix(T). This completes the proof. □

Example 4.2 Let C be a compact convex subset of a smooth Banach space E such that 0C, 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 zC.

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 nN,

  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 mN, 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 zFix(T), which is the unique solution of the variational inequality

( f I ) z , J ( y z ) 0,yFix(T).

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

Proof Let xC. For each nN, 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 yC, 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 zFix(T). This completes the proof. □

Example 4.4 Let C be a compact convex subset of a smooth Banach space E such that 0C, 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 zC.

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 0C, 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 zFix(T), which is the unique solution of the variational inequality

( f I ) z , J ( y z ) 0,yFix(T).
(20)

Equivalently, z=Pf(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 :jZ} 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 yC, 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 zFix(T). This completes the proof. □

Example 4.6 Let C be a compact convex subset of a smooth Banach space E such that 0C. 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 zC.

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,,m1, 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].

References

  1. Lau AT, Miyake H, Takahashi W: Approximation of fixed points for amenable semigroups of nonexpansive mappings in Banach spaces. Nonlinear Anal. 2007, 67(4):1211–1225. 10.1016/j.na.2006.07.008

    MATH  MathSciNet  Article  Google Scholar 

  2. Saeidi S: Approximating common fixed points of Lipschitzian semigroup in smooth Banach spaces. Fixed Point Theory Appl. 2008., 2008: Article ID 363257

    Google Scholar 

  3. Zhang SS, Yang L, Liu JA: Strong convergence theorems for nonexpansive mappings in Banach spaces. Appl. Math. Mech. 2007, 28: 1287–1297. 10.1007/s10483-007-1002-x

    MATH  MathSciNet  Article  Google Scholar 

  4. Katchang P, Kumam P: A composite explicit iterative process with a viscosity method for Lipschitzian semigroup in smooth Banach space. Bull. Iran. Math. Soc. 2011, 37: 143–159.

    MATH  MathSciNet  Google Scholar 

  5. Kumam P, Wattanawitoon K: A general composite explicit iterative scheme of fixed point solutions of variational inequalities for nonexpansive semigroups. Math. Comput. Model. 2011, 53: 998–1006. 10.1016/j.mcm.2010.11.057

    MATH  MathSciNet  Article  Google Scholar 

  6. Kumam P, Plubtieng S, Katchang P: Viscosity approximation to a common solution of variational inequality problems and fixed point problems for Lipschitzian semigroup in Banach spaces. Math. Sci. 2013., 7: Article ID 28

    Google Scholar 

  7. Saewan S, Kumam P: Explicit iterations for Lipschitzian semigroups with Meir-Keeler type contraction in Banach spaces. J. Inequal. Appl. 2012., 2012: Article ID 279

    Google Scholar 

  8. Sunthrayuth P, Wattanawitoon K, Kumam P: Convergence theorems of a general composite iterative method for nonexpansive semigroups in Banach spaces. ISRN Math. Anal. 2011., 2011: Article ID 576135

    Google Scholar 

  9. Huang Q, Zhu L, Li G: Fixed point theorems for left amenable semigroups of non-Lipschitzian mappings in Banach spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 116 10.1186/1687-1812-2012-116

    Google Scholar 

  10. Sunthrayuth P, Kumam P: A general iterative algorithm for the solution of variational inequalities for a nonexpansive semigroup in Banach spaces. J. Nonlinear Anal. Optim. 2010, 1(1):139–150.

    MathSciNet  Google Scholar 

  11. Sunthrayuth P, Kumam P: A new composite general iterative scheme for nonexpansive semigroups in Banach spaces. Int. J. Math. Math. Sci. 2011., 2011: Article ID 560671

    Google Scholar 

  12. Wattanawitoon K, Kumam P: Strong convergence to common fixed points for countable families of asymptotically nonexpansive mappings and semigroups. Fixed Point Theory Appl. 2010., 2010: Article ID 301868 10.1155/2010/301868

    Google Scholar 

  13. Atsushiba S, Takahashi W: Strong convergence theorems for one-parameter nonexpansive semigroups with compact domains. 3. In Fixed Point Theory and Applications. Edited by: Cho YJ, Kim JK, Kang SM. Nova Science Publishers, New York; 2002:15–31.

    Google Scholar 

  14. Lau AT: Invariant means on almost periodic functions and fixed point properties. Rocky Mt. J. Math. 1973, 3: 69–76. 10.1216/RMJ-1973-3-1-69

    MATH  Article  Google Scholar 

  15. Lau AT: Semigroup of nonexpansive mappings on a Hilbert space. J. Math. Anal. Appl. 1985, 105(2):514–522. 10.1016/0022-247X(85)90066-6

    MATH  MathSciNet  Article  Google Scholar 

  16. Agarwal RP, O’Regan D, Sahu DR Topological Fixed Point Theory and Its Applications 6. In Fixed Point Theory for Lipschitzian-Type Mappings with Applications. Springer, New York; 2009.

    Google Scholar 

  17. Kim KS: Nonlinear ergodic theorems of nonexpansive type mappings. J. Math. Anal. Appl. 2009, 358: 261–272. 10.1016/j.jmaa.2009.04.045

    MATH  MathSciNet  Article  Google Scholar 

  18. Holmes RD, Lau AT: Non-expansive actions of topological semigroups and fixed points. J. Lond. Math. Soc. 1972, 5: 330–336.

    MATH  MathSciNet  Article  Google Scholar 

  19. Takahashi W: Nonlinear Functional Analysis: Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama; 2000.

    Google Scholar 

  20. Lau AT, Shioji N, Takahashi W: Existence of nonexpansive retractions for amenable semigroups of nonexpansive mappings and nonlinear ergodic theorems in Banach spaces. J. Funct. Anal. 1999, 161(1):62–75. 10.1006/jfan.1998.3352

    MATH  MathSciNet  Article  Google Scholar 

  21. Kim KS: Convergence of a hybrid algorithm for a reversible semigroup of nonlinear operators in Banach spaces. Nonlinear Anal. 2010, 73(3):3413–3419.

    MATH  MathSciNet  Article  Google Scholar 

  22. Reich S: Asymptotic behavior of contraction in Banach spaces. J. Math. Anal. Appl. 1973, 44: 57–70. 10.1016/0022-247X(73)90024-3

    MATH  MathSciNet  Article  Google Scholar 

  23. Xu HK: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 2003, 116(3):659–678. 10.1023/A:1023073621589

    MATH  MathSciNet  Article  Google Scholar 

  24. Piri H: Hybrid pseudo-viscosity approximation schemes for systems of equilibrium problems and fixed point problems of infinite family and semigroup of non-expansive mappings. Nonlinear Anal. 2011, 74: 6788–6804. 10.1016/j.na.2011.06.056

    MATH  MathSciNet  Article  Google Scholar 

  25. Piri H, Badali AH: Strong convergence theorem for amenable semigroups of nonexpansive mappings and variational inequalities. Fixed Point Theory Appl. 2011., 2011: Article ID 55

    Google Scholar 

  26. Brézis H, Browder FE: Nonlinear ergodic theorem. Bull. Am. Math. Soc. 1976, 82: 959–961. 10.1090/S0002-9904-1976-14233-4

    MATH  Article  Google Scholar 

  27. Brézis H, Browder FE: Remark on nonlinear ergodic theory. Adv. Math. 1977, 25: 165–177. 10.1016/0001-8708(77)90003-2

    MATH  Article  Google Scholar 

  28. Eshita K, Miyake H, Takahashi W: Strong convergence theorems for asymptotically nonexpansive mappings in general Banach spaces. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 2006, 13: 621–640.

    MATH  MathSciNet  Google Scholar 

  29. Lorentz GG: A contribution to the theory of divergent series. Acta Math. 1948, 80: 167–190. 10.1007/BF02393648

    MATH  MathSciNet  Article  Google Scholar 

  30. Hirano N, Kido K, Takahashi W: Nonexpansive retractions and nonlinear ergodic theorems in Banach spaces. Nonlinear Anal. 1988, 28: 1269–1281.

    MathSciNet  Article  Google Scholar 

  31. Reich S: Nonlinear evolution equations and nonlinear ergodic theorems. Nonlinear Anal. 1977, 1: 319–330. 10.1016/S0362-546X(97)90001-8

    MATH  Article  Google Scholar 

  32. Reich S: Almost convergence and nonlinear ergodic theorems. J. Approx. Theory 1978, 24: 269–272. 10.1016/0021-9045(78)90012-6

    MATH  MathSciNet  Article  Google Scholar 

  33. Miyake H, Takahashi W: Nonlinear mean ergodic theorems for nonexpansive semigroup in Banach spaces. J. Fixed Point Theory Appl. 2007, 2: 360–382.

    MathSciNet  Article  Google Scholar 

Download references

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).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Poom Kumam.

Additional information

Competing interests

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

Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Piri, H., Kumam, P. Approximating fixed points for a reversible semigroup of Lipschitzian mappings in a smooth Banach space. J Inequal Appl 2013, 555 (2013). https://doi.org/10.1186/1029-242X-2013-555

Download citation

  • Received:

  • Accepted:

  • Published:

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

Keywords

  • smooth Banach space
  • asymptotically nonexpansive semigroup
  • reversible semigroup
  • invariant mean