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Fixed point theorems for left amenable semigroups of non-Lipschitzian mappings in Banach spaces

Abstract

In this paper, we provide the existence and convergence theorems of fixed points for left amenable semigroups of asymptotically nonexpansive type mappings in general Banach spaces, which extend and improve many recent results in this area.

MSC:47H09, 47H10, 47H20.

1 Introduction

Let E be a Banach space and C a nonempty bounded closed convex subset of E. A mapping T on C is said to be nonexpansive if TxTyxy for all x,yC. A well-known result of Browder [1] asserts that if E is uniformly convex, then every nonexpansive mapping on C has a fixed point. Kirk [2], Belluce and Kirk [3] extended this result to the case that X has a normal structure or Opial’s property. Goebel and Kirk [4] proved that if E is a uniformly convex Banach space, then every asymptotically nonexpansive mapping on C has a fixed point.

As is well known, not every semigroup of nonexpansive mappings on a subset of a Banach space has a fixed point [5]. The existence and convergence of fixed points for semigroups of various mappings have been studied extensively [610]. Recently, Suzuki and Takahashi [8], Takahashi and Zembayashi [9], Zhu and Li [10] proved the existence theorems of fixed points for semigroups ={T(t):t0} of nonexpansive, asymptotically nonexpansive and asymptotically nonexpansive type mappings, respectively. For instance, in [9], Takahashi and Zembayashi proved the following theorem:

Theorem 1.1 [9]

Let C be a nonempty compact convex subset of a Banach space E and ={T(t):t0} be a semigroup of asymptotically nonexpansive mappings on C, then the set of common fixed points F() of is nonempty.

Many results are known in the case that the semigroup G is commutative, amenable or reversible [1124]. In the case of an amenable semigroup, the first result was established by Takahashi [21] where he proved:

Theorem 1.2 [21]

Let C be a nonempty compact convex subset of a Banach space E. Let ={T(t):tG} be an amenable semigroup of nonexpansive mappings on C. Then C contains a common fixed point for .

Theorem 1.2 was proved for a commutative semigroup by DeMarr [11]. Later in [13], Lau showed that the fixed point property is equivalent to the existence of a left invariant mean on AP(), the space of almost periodic functions on the semigroup . It should be pointed out that if is left reversible, then AP() always has a left invariant mean [13], but the converse is false [14]. And in [16], Lau, Miyake and Takahashi gave the following existence theorem:

Theorem 1.3 [16]

Let C be a nonempty weakly compact convex subset of a Banach space E. Let G be a left reversible semigroup (with identity) and ={T(t):tG} be a semigroup of nonexpansive mappings on C. Let X be a left invariant -stable subspace of l (G) containing 1, and μ be a left invariant mean on X. Then F()=F( T μ ) C a , where C a denotes the set of almost periodic elements in C, i.e., all xC such that {T(s)x:sG} is relatively compact in the norm topology of E. Further, if C is compact, then the set F() is nonempty.

In [20], Saeidi extended Theorem 1.3 to the case for left reversible semigroups of asymptotically nonexpansive mappings. Inspired and motivated by [810, 16, 20, 21, 23], we investigate the existence and convergence of fixed points for left amenable semigroups of asymptotically nonexpansive type mappings in Banach spaces. We first provide the existence theorem of fixed points for left amenable semigroups of asymptotically nonexpansive type mappings in Banach spaces. Utilizing this result, we obtain a strong convergence theorem of iterative sequences for left amenable semigroups of asymptotically nonexpansive type mappings. The results obtained in this paper extend and improve many recent results in [810, 16, 20, 23].

2 Preliminaries

Let C be a nonempty bounded subset of a Banach space E. Let G be a semitopological semigroup, i.e., G is a semigroup with a Hausdorff topology such that for sG the mappings sst and sts from G to G are continuous. Let ={T(t):tG} be a continuous representation of G on C, i.e., T(ts)x=T(t)T(s)x, t,sG, xC and the mapping (t,x)T(t)x from G×C into C is continuous when G×C has the product topology. Recall that is said to be

  1. (1)

    nonexpansive if for all x,yC and tG,

    T ( t ) x T ( t ) y xy;
  2. (2)

    asymptotically nonexpansive [2527] if there exists a function k:G[0,+) with inf s G sup t G k(ts)1 such that for all x,yC and tG,

    T ( t ) x T ( t ) y k(t)xy;
  3. (3)

    asymptotically nonexpansive type [2527] if for each xC, there exists a function r(,x):G[0,+) with inf s G sup t G r(ts,x)=0 such that for all x,yC and tG,

    T ( t ) x T ( t ) y xy+r(t,x).

It is easily seen that (1)(2)(3) and that both inclusions are proper [2527].

Let l (G) be the Banach space of all bounded real valued functions on G with the supremum norm. Then, for each sG and f l (G), we can define l s f in l (G) by ( l s f)(t)=f(st) for all tG. Let X be a subspace of l (G) containing 1 and X be its dual space. An element μ X is called a mean on X if μ=μ(1)=1. We always denote the value of μ at fX by μ t f(t)=μ(f). Let X be left invariant, i.e., l s (X)X for all sG. A mean μ on X is said to be left invariant if μ( l s f)=μ(f) for all sG and fX. Further, X is called left amenable if X has a left invariant mean. In this case, we also say that G is a left amenable semigroup. Recall that a semigroup G is called left reversible if any two closed right ideals of G have nonvoid intersection. In this case, (G,) is a directed system when the binary relation ≤ on G is defined by st if and only if {s} s G ¯ {t} t G ¯ , s,tG. As is well known, the class of left reversible semigroups includes all commutative semigroups and if a semigroup G is left amenable, then G is left reversible. But the converse is false [28].

Let ={T(t):tG} be an asymptotically nonexpansive type semigroup on C. Let F() denote the set of all fixed points of , i.e., F()={xC:T(s)x=x,sG}. A subspace X of l (G) is called -stable if functions sT(s)x, x and sT(s)xy on G are in X for all x,yC and x E . We know that if μ is a mean on X and if for each x E the function sT(s)x, x is contained in X and C is weakly compact, then there exists a unique point x 0 of E such that μ s T(s)x, x = x 0 , x for all x E . Such a point x 0 is always denoted by T μ x. Obviously, T μ x=x for each xF().

3 Main results

Lemma 3.1 Let C be a nonempty weakly compact convex subset of a Banach space E. Let G be a left reversible semigroup and ={T(t):tG} be a continuous representation of G as asymptotically nonexpansive type mappings on C, with the condition lim sup s G r(s,x)=0 for all xC. Let X be a left invariant -stable subspace of l (G) containing 1, and μ be a left invariant mean on X. Then F()=F( T μ ) C a .

Proof If F( T μ ) C a is empty, then so is F() as F()F( T μ ) C a . Let zF( T μ ) C a and define d= μ s T(s)zz, then for all tG, we have

T ( t ) z z = T ( t ) z T μ z = sup { | T ( t ) z T μ z , x | : x E , x = 1 } = sup { | μ s T ( t ) z T ( s ) z , x | : x E , x = 1 } sup { μ s T ( t ) z T ( s ) z x : x E , x = 1 } = μ s T ( t ) z T ( s ) z = μ s T ( t ) z T ( t s ) z (by  μ -left invariant) μ s T ( s ) z z + r ( t , z ) = d + r ( t , z ) ,

i.e., for all tG,

T ( t ) z z d+r(t,z).
(3.1)

Next, we shall show d=0. In fact, if d>0, then for each tG,

d= μ s T ( s ) z z = μ s T ( t s ) z z sup s G T ( t s ) z z ,

i.e.,

sup s G T ( t s ) z z d,tG.
(3.2)

By lim sup s G r(s,z)=0, then for any nN, there exists s n G such that

sup t s n r(t,z)< 1 4 n .
(3.3)

It follows from (3.2) that we can choose a cluster point u 1 of the net {T(s)z:sG} in the set C with u 1 zd and there exists t n ( 1 ) G satisfying t n ( 1 ) s n and T( t n ( 1 ) )z u 1 < 1 4 n . Combining it with (3.1) and (3.3), we get

u 1 z u 1 T ( t n ( 1 ) ) z + T ( t n ( 1 ) ) z z 1 4 n + d + r ( t n ( 1 ) , z ) d + 1 2 n (by  t n ( 1 ) s n ) .

Hence u 1 zd and so u 1 z=d. It follows from (3.1) and (3.3) that

T ( t n ( 1 ) s n s ) z u 1 T ( t n ( 1 ) s n s ) z T ( t n ( 1 ) ) z + T ( t n ( 1 ) ) z u 1 T ( s n s ) z z + r ( t n ( 1 ) , z ) + T ( t n ( 1 ) ) z u 1 d + r ( s n s , z ) + 1 2 n d + 3 4 n
(3.4)

for all sG. Noting

d = u 1 z = u 1 T μ z = sup { | u 1 T μ z , x | : x E , x = 1 } = sup { | μ s u 1 T ( s ) z , x | : x E , x = 1 } μ s u 1 T ( s ) z ,
(3.5)

we obtain

μ s ( T ( t n ( 1 ) s n s ) z z + T ( t n ( 1 ) s n s ) z u 1 ) = μ s T ( t n ( 1 ) s n s ) z z + μ s T ( t n ( 1 ) s n s ) z u 1 = μ s T ( s ) z z + μ s T ( s ) z u 1 2 d .

This implies that

sup s G [ T ( t n ( 1 ) s n s ) z z + T ( t n ( 1 ) s n s ) z u 1 ] 2d.

Thus there exists s n ( 1 ) G such that

T ( t n ( 1 ) s n s n ( 1 ) ) z z + T ( t n ( 1 ) s n s n ( 1 ) ) z u 1 2d 1 4 n .
(3.6)

Since {T(s)z:sG} is a relatively compact set, {T( t n ( 1 ) s n s n ( 1 ) )z}, as a subset of {T(s)z:sG}, has a strong convergent subsequence. Without loss of generality, we can assume that T( t n ( 1 ) s n s n ( 1 ) )z u 2 C. Setting t n ( 2 ) = t n ( 1 ) s n s n ( 1 ) , then t n ( 2 ) t n ( 1 ) s n , T( t n ( 2 ) )z u 2 and by (3.6),

u 2 z+ u 2 u 1 2d.
(3.7)

On the other hand,

u 2 z u 2 T ( t n ( 2 ) ) z + T ( t n ( 2 ) ) z z u 2 T ( t n ( 2 ) ) z + d + r ( t n ( 2 ) , z ) (by (3.1)) u 2 T ( t n ( 2 ) ) z + d + 1 4 n (by (3.3))

and

u 2 u 1 u 2 T ( t n ( 2 ) ) z + T ( t n ( 2 ) ) z u 1 = u 2 T ( t n ( 2 ) ) z + T ( t n ( 1 ) s n s n ( 1 ) ) z u 1 u 2 T ( t n ( 2 ) ) z + d + 3 4 n (by (3.4)) .

Thus we can conclude u 2 zd and u 2 u 1 d. So by (3.7),

u 2 z= u 2 u 1 =d.

Similar to the proof of (3.5), we can prove μ s u 2 T(s)zd and

μ s ( T ( t n ( 2 ) s n s ) z z + T ( t n ( 2 ) s n s ) z u 1 + T ( t n ( 2 ) s n s ) z u 2 ) = μ s T ( t n ( 2 ) s n s ) z z + μ s T ( t n ( 2 ) s n s ) z u 1 + μ s T ( t n ( 2 ) s n s ) z u 2 = μ s T ( s ) z z + μ s T ( s ) z u 1 + μ s T ( s ) z u 2 3 d .

This means

sup s G ( T ( t n ( 2 ) s n s ) z z + T ( t n ( 2 ) s n s ) z u 1 + T ( t n ( 2 ) s n s ) z u 2 ) 3d.

Thus there exists s n ( 2 ) G such that

T ( t n ( 2 ) s n s n ( 2 ) ) z z + T ( t n ( 2 ) s n s n ( 2 ) ) z u 1 + T ( t n ( 2 ) s n s n ( 2 ) ) z u 2 3d 1 n .

Therefore,

T ( t n ( 2 ) s n s n ( 2 ) ) z z d + r ( t n ( 2 ) s n s n ( 2 ) , z ) d + 1 4 n , T ( t n ( 2 ) s n s n ( 2 ) ) z u 2 T ( t n ( 2 ) s n s n ( 2 ) ) z T ( t n ( 2 ) ) z + T ( t n ( 2 ) ) z u 2 T ( t n ( 2 ) s n s n ( 2 ) ) z u 2 T ( s n s n ( 2 ) ) z z + r ( t n ( 2 ) , z ) + T ( t n ( 2 ) ) z u 2 T ( t n ( 2 ) s n s n ( 2 ) ) z u 2 d + r ( s n s n ( 2 ) , z ) + r ( t n ( 2 ) , z ) + T ( t n ( 2 ) ) z u 2 T ( t n ( 2 ) s n s n ( 2 ) ) z u 2 d + 1 2 n + T ( t n ( 2 ) ) z u 2

and

T ( t n ( 2 ) s n s n ( 2 ) ) z u 1 T ( t n ( 2 ) s n s n ( 2 ) ) z T ( t n ( 1 ) ) z + T ( t n ( 1 ) ) z u 1 T ( t n ( 1 ) s n s n ( 1 ) s n s n ( 2 ) ) z T ( t n ( 1 ) ) z + T ( t n ( 1 ) ) z u 1 T ( s n s n ( 1 ) s n s n ( 2 ) ) z z + r ( t n ( 1 ) , z ) + T ( t n ( 1 ) ) z u 1 d + r ( s n s n ( 1 ) s n s n ( 2 ) , z ) + r ( t n ( 1 ) , z ) + T ( t n ( 1 ) ) z u 1 d + 1 2 n + T ( t n ( 1 ) ) z u 1 .

Since {T( t n ( 2 ) s n s n ( 2 ) )z} has a strong convergent subsequence, without loss of generality, we can assume that T( t n ( 2 ) s n s n ( 2 ) )z u 3 C. Setting t n ( 3 ) = t n ( 2 ) s n s n ( 2 ) , then t n ( 3 ) t n ( 2 ) , T( t n ( 3 ) )z u 3 ,

u 3 zd, u 3 u 2 d, u 3 u 1 d

and

u 3 z+ u 3 u 1 + u 3 u 2 3d.

Thus we have found u 3 C such that

u 3 u 1 = u 3 u 2 = u 3 z=d.

Now, by mathematical induction, we can find a sequence { u i }C satisfying

u i z=d, u i u j =d(i,jN,ij).

Since T( t n ( i ) )z u i , we can seek out t n i ( i ) G with T( t n i ( i ) )z u i d 4 . Thus

T ( t n i ( i ) ) z T ( t n j ( j ) ) z d 2 (i,jN,ij),

which is a contradiction with the relative compactness of {T( t n i ( i ) )z:iN}. Therefore, we can conclude d=0.

In the following, we shall show zF(). Indeed, for any hG, T(h):CC is continuous at z, then for all ε>0, there exists a δ>0 (δ<ε) such that for all xC with xz<δ,

T ( h ) x T ( h ) z <ε.

By (3.1) and the definition of r(,z), we can get

inf s G sup t G T ( t s ) z z d+ inf s G sup t G r(ts,z)=0

and so we can find a s δ G such that sup t G T(t s δ )zz<δ, i.e., for all tG,

T ( t s δ ) z z <δ.

Hence

T ( h ) z z T ( h ) z T ( h ) T ( t s δ ) z + T ( h ) T ( t s δ ) z z = T ( h ) z T ( h ) T ( t s δ ) z + T ( h t s δ ) z z < ε + δ < 2 ε .

Since ε>0 is arbitrary, we get zF(). This completes the proof. □

Now we can give the existence theorem of fixed points for left amenable semigroups of non-Lipschitzian mappings in Banach spaces.

Theorem 3.1 Let C be a nonempty compact convex subset of a Banach space E. Let G be a left reversible semigroup and ={T(t):tG} be a continuous representation of G as asymptotically nonexpansive type mappings on C, with the condition lim sup s G r(s,x)=0 for all xC. Let X be a left invariant -stable subspace of l (G) containing 1, and μ be a left invariant mean on X. Then the set F() is nonempty.

Proof For all x,yC and tG, we have

T μ x T μ y = sup { | T μ x T μ y , x | : x E , x = 1 } = sup { | μ s T ( s ) x T ( s ) y , x | : x E , x = 1 } sup { μ s T ( s ) x T ( s ) y x : x E , x = 1 } = μ s T ( s ) x T ( s ) y = μ s T ( t s ) x T ( t s ) y sup s G T ( t s ) x T ( t s ) y x y + sup s G r ( t s , z ) ,

and so by lim sup s G r(s,z)=0, we get T μ x T μ yxy, i.e., T μ is a nonexpansive mapping from C into itself. Since a nonexpansive mapping of a compact convex subset of a Banach space into itself has a fixed point [29], T μ has a fixed point z. By Lemma 3.1, zF(). This completes the proof. □

Remark 3.1 Theorem 3.1 is an extension of the main results in [810, 16, 20, 23] to the case for left amenable semigroups of asymptotically nonexpansive type mappings in Banach spaces.

Recall that for each sG, we define a point evaluation δ s on X by δ s (f)=f(s) for every fX. A convex combination of point evaluation is called a finite mean on G. If λ is a finite mean on G, say λ= Σ i = 1 n a i δ s i , where s i G, a i 0, i=1,2,,n, and Σ i = 1 n a i =1, then λ(t)T(t)x, x = Σ i = 1 n a i T( s i )x, x = Σ i = 1 n a i T( s i )x, x for all x E . For convenience, we denote it by λ(t)T(t)x= Σ i = 1 n a i T( s i )x. A net { λ α :αI} of finite means on G is said to be strongly left regular if

lim α I λ α l s λ α =0

for all sG, where A is a directed system and l s is the conjugate operator of l s .

Corollary 3.1 Let C be a nonempty compact convex subset of a Banach space E. Let G be a left reversible semigroup and ={T(t):tG} be a continuous representation of G as asymptotically nonexpansive type mappings on C, with the condition lim sup s G r(s,x)=0 for all xC. Let X be a left invariant -stable subspace of l (G) containing 1 and { λ α :αI} be a net of strongly left regular finite means on G. If zC satisfies

lim inf α I λ α ( t ) T ( t ) z z =0,

then zF().

Proof Since lim inf α I λ α (t)T(t)zz=0 and { λ α :αI} D , we can find a subnet { λ α β :βA} of { λ α :αI} such that lim β A λ α β (t)T(t)z=z and ω lim β A λ α β =μ, where A is a directed system. Hence μ is a left invariant mean on X (see [30]) and T μ z=z, which implies zF(). This completes the proof. □

Remark 3.2 Corollary 3.1 is an extension of the main results in [810, 23].

Next we shall prove the strong convergence theorem for the iterative sequences of left reversible semigroups of asymptotically nonexpansive type mappings. We need a lemma which plays a crucial role in the proof of Theorem 3.2.

Lemma 3.2 [30]

Let z n and w n be bounded sequences in a Banach space X and let α n be a sequence in (0,1) with 0< lim inf n α n lim sup n α n <1. Suppose that z n + 1 = α n w n +(1 α n ) z n for all nN and

lim sup n ( w n w n + k z n z n + k ) 0

for all kN. Then lim inf n w n z n =0.

Theorem 3.2 Let C be a nonempty compact convex subset of a Banach space X and G be a left reversible semigroup. Let ={T(t):tG} be a continuous representation of G as asymptotically nonexpansive type mappings on C, with the condition lim sup s G r(s,x)=0 for all xC. Let X be a left invariant -stable subspace of l (G) containing 1, and μ be a left invariant mean on X. Let x 1 C and define a sequence { x n } in C by

x n + 1 = α n T μ x n +(1 α n ) x n ,

for all nN, where α n [0,1] satisfies 0< lim inf n α n lim sup n α n <1. Then x n converges strongly to a fixed point zF().

Proof It follows from

T μ x n + 1 x n + 1 T μ x n + 1 T μ x n + T μ x n x n + 1 = T μ x n + 1 T μ x n + ( 1 α n ) T μ x n x n = T μ x n x n + T μ x n + 1 T μ x n x n + 1 x n T μ x n x n

that lim n T μ x n x n exists. By Lemma 3.1, we get lim inf n T μ x n x n =0 and so lim n T μ x n x n =0. Since C is compact, there exists a subsequence { x n k }{ x n } such that x n k zC. Hence, z is a fixed point of T μ . By Lemma 3.1, we have zF() and

x n + 1 z = α n T μ x n + ( 1 α n ) x n z α n T μ x n z + ( 1 α n ) x n z x n z .

Hence lim n x n z exists. Thus lim n x n z= lim k x n k z=0, which implies that x n converges strongly to zF(). This completes the proof. □

In the following, we shall give an example of a semigroup which is asymptotically nonexpansive type but not asymptotically nonexpansive on a compact set.

Example 3.1 [27]

Let Δ be the Cantor ternary set. Define the Cantor ternary function

τ(x)={ n = 1 + b n 2 n , x = n = 1 + 2 b n 3 n Δ ( b n = 0 , 1 ) , sup { τ ( y ) , y x , y Δ } , x [ 0 , 1 ] Δ

then τ:[0,1][0,1] is a continuous and increasing but not absolutely continuous function with τ(0)=0, τ( 1 2 )= 1 2 (see [31]). Since a Lipschitzian function is absolutely continuous, τ is non-Lipschitzian. For all t>0, we define T(t):[0,1][0,1] by

T(t)x={ x 2 t , 0 x 1 2 , τ ( 1 x ) 2 t , 1 2 < x 1 .

Then T(t) is continuous but not Lipschitzian continuous (since τ is non-Lipschitzian) and for all x,y[0,1], |T(t)x| 1 2 t + 1 ,

| T ( t ) x T ( t ) y | 1 2 t |xy|+ 1 2 t .

Therefore, we can conclude that the semigroup ={T(t):t>0} is asymptotically nonexpansive type but not an asymptotically nonexpansive on [0,1]. Also, 0 is a fixed point of .

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Acknowledgement

This manuscript has benefited greatly from the constructive comments and helpful suggestions of the anonymous referees, the authors would like to express their deep gratitude to them. This research is supported by the Natural Science Foundation of China (10971182, 11201410), the Natural Science Foundation of Jiangsu Province (BK2009179, BK2010309 and BK2012260), the Jiangsu Government Scholarship for Overseas Studies, the Natural Science Foundation of Jiangsu Education Committee (10KJB110012) and the Natural Science Foundation of Yangzhou University.

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Correspondence to Lanping Zhu.

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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, 116 (2012). https://doi.org/10.1186/1687-1812-2012-116

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Keywords

  • asymptotically nonexpansive type mapping
  • left amenable semigroup
  • reversible semigroup
  • fixed point