Invariant Means and Reversible Semigroup of Relatively Nonexpansive Mappings in Banach Spaces

and Applied Analysis 3 for x, y ∈ E. Observe that, in a Hilbert space H, (8) reduces to φ (x, y) = 󵄩 󵄩 󵄩 󵄩 x − y 󵄩 󵄩 󵄩 󵄩 2


Introduction
Let  be a real Banach space with the topological dual  * and let  be a closed and convex subset of .A mapping  of  into itself is called nonexpansive if ‖ − ‖ ≤ ‖ − ‖ for each ,  ∈ .
Three classical iteration processes are often used to approximate a fixed point of a nonexpansive mapping.The first one is introduced by Halpern [1] and is defined as follows: 0 =  ∈ , chosen arbitrarily, where {  } is a sequence in [0, 1].He pointed out that the conditions lim  → ∞   = 0 and ∑ ∞ =1   = ∞ are necessary in the sense that if the iteration (1) converges to a fixed point of , then these conditions must be satisfied.The second iteration process is known as Mann's iteration process [2] which is defined as follows: where the initial  1 is taken in  arbitrary and the sequence The third iteration process is referred to as Ishikawa's iteration process [3] which is defined as follows: where the initial  1 is taken in  arbitrary and {  } and {  } are sequences in [0, 1].
In 2007, Lau et al. [4] proposed the following modification of Halpern's iteration (1) for amenable semigroups of nonexpansive mappings in a Banach space.Theorem 1.Let  be a left reversible semigroup and let I = {() :  ∈ } be a representation of  as nonexpansive mappings from a compact convex subset  of a strictly convex and smooth Banach space  into , let  be an amenable and I-stable subspace of  ∞ (), and let {  } be a strongly left regular sequence of means on .Let {  } be a sequence in [0, 1] such that lim  → ∞   = 0 and ∑ ∞ =1   = ∞.Let  1 =  ∈  and let {  } be the sequence defined by  +1 =    + (1 −   )  (  )   ,  ≥ 2. ( Then {  } converges strongly to , where  denotes the unique sunny nonexpansive retraction of  onto (I).
Let  be a closed and convex subset of  and let  be a mapping from  into itself.We denote by () the set of fixed Theorem 2. Let  be a left reversible semigroup and let I = {() :  ∈ } be a representation of  as relatively nonexpansive mappings from a nonempty, closed, and convex subset  of a uniformly convex and uniformly smooth Banach space  into  with (I) ̸ = 0. Let  be a subspace of  ∞ () and let {  } be a asymptotically left invariant sequence of means on .Let {  } be a sequence in [0, 1] such that 0 <   < 1 and lim  → ∞   = 0. Let {  } be a sequence generated by the following algorithm: (5) Then {  } converges strongly to Π (I)  1 , where Π (I) is the generalized projection from  onto (I).
Let  be a semigroup.The purpose of this paper is to study modified Halpern type and Ishikawa type iterations for a semigroup of relatively nonexpansive mappings I = {() :  ∈ } on a nonempty closed convex subset  of a Banach space with respect to a sequence of asymptotically left invariant means {  } defined on an appropriate invariant subspace of  ∞ ().We prove that, given some mild conditions, we can generate iterative sequences which converge strongly to a common element of the set of fixed points (I), where (I) = ⋂{(()) :  ∈ }.
Let  be a real Banach space with norm ‖ ⋅ ‖ and let  * be the dual space of .Denote by ⟨⋅, ⋅⟩ the duality product.We denote by  the normalized duality mapping from  to 2  * defined by for  ∈ .A Banach space  is said to have the Kadec-Klee property if a sequence {  } of  satisfies that   ⇀  and ‖  ‖ → ‖‖ and then   → , where ⇀ and → denote the weak convergence and the strong convergence, respectively.We know the following: (1) the duality mapping  is monotone, that is, ⟨−,  * −  * ⟩ ≥ 0 whenever  * ∈  and  * ∈ ; (2) if  is strictly convex, then  is one-to-one; that is, if  ∩  is nonempty, then  = ; (3) if  is strictly convex, then  is strictly monotone; that is,  =  whenever ⟨ − ,  * −  * ⟩ = 0,  * ∈  and  * ∈ ; (4) if  is uniformly convex, then  has the Kadec-Klee property; (5 For more details, see [11]. As well known, if  is a nonempty, closed, and convex subset of a Hilbert space  and   :  →  is the metric projection of  onto , then   is nonexpansive (see, the reference therein).This fact actually characterizes Hilbert spaces.Consequently, it is not true to more general Banach spaces.In this connection, Alber [12] introduced a generalized projection operator Π  in a Banach space  which is an analogue of the metric projection in Hilbert spaces.Consider the function defined by for ,  ∈ .Observe that, in a Hilbert space , (8) reduces to for ,  ∈ .The generalized projection Π  :  →  is a mapping that assigns an arbitrary point  ∈  to the minimum point of the functional (, ); that is, Π   = , where  is the solution to the minimization problem: The existence and uniqueness of the operator Π  follow from the properties of the functional (, ) and strict monotonicity of the mapping  (see, e.g., [12,13]).In a Hilbert space, Π  =   .It is obvious from the definition of the function  that For more details see [14].
Let  be a semigroup.We denote by  ∞ () the Banach space of all bounded real-valued functionals on  with supremum norm.For each  ∈ , we define the left and right translation operators () and () on  ∞ () by for each  ∈  and  ∈  ∞ (), respectively.Let  be a subspace of  ∞ () containing 1.An element  in the dual space  * of  is said to be a mean on  if ‖‖ = (1) = 1.For  ∈ , we can define a point evaluation   by   () = () for each  ∈ .
It is well known that  is mean on  if and only if for each  ∈ .
Let  be a translation invariant subspace of  ∞ () (i.e., () ⊂  and () ⊂  for each  ∈ ) containing 1. Then a mean  on  is said to be left invariant (resp., right invariant) if for each  ∈  and  ∈ .A mean  on  is said to be invariant if  is both left and right invariant [15][16][17][18][19].  is said to be left (resp., right) amenable if  has a left (resp., right) invariant mean. is amenable if  is left and right amenable.We call a semigroup  amenable if  is amenable.Further, amenable semigroups include all commutative semigroups and solvable groups.However, the free group or semigroup of two generators is not left or right amenable (see [20][21][22]).
From now on  denotes a semigroup with an identity . is called left reversible if any two right ideals of  have nonvoid intersection; that is,  ∩  ̸ = 0 for ,  ∈ .In this case, (, ⪯) is a directed system when the binary relation "⪯" on  is defined by  ⪯  if and only if {} ∪  ⊇ {} ∪  for ,  ∈ .It is easy to see that  ⪯  for all ,  ∈ .Further, if  ⪯  then  ⪯  for all  ∈ .The class of left reversible semigroup includes all groups and commutative semigroups.If a semigroup  is left amenable, then  is left reversible.But the converse is not true [23][24][25][26][27][28].
Let  be a semigroup and let  be a closed and convex subset of .Let () denote the fixed point set of .Then I = {() :  ∈ } is called a representation of  as relatively nonexpansive mappings on  if () is relatively nonexpansive with () =  and () = ()() for each ,  ∈ .We denote by (I) the set of common fixed points of {() :  ∈ }; that is, We know that if  is a mean on  and if for each  * ∈  * the function   → ⟨(),  * ⟩ is contained in  and  is weakly compact, then there exists a unique point  0 of  such that ⟨(⋅),  * ⟩ = ⟨ 0 ,  * ⟩ for each  * ∈  * .We denote such a point  0 by   .Note that    is contained in the closure of the convex hull of {() :  ∈ } for each  ∈ .Note that    =  for each  ∈ (I); see [29][30][31].

Lemmas
We need the following lemmas for the proof of our main results.
Then (I) = (  ) ∩   , where   denotes the set of almost periodic elements in ; that is, all  ∈  such that {() :  ∈ } is relatively compact in the norm topology of .

Strong Convergence Theorems
In this section, we will establish two strong convergence theorems of various iterative sequences for finding common fixed point of relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach spaces (cf.[34][35][36]).
We begin with a strong convergence theorem of modified Halpern's type.Theorem 8. Let  be a left reversible semigroup and let I = {() :  ∈ } be a representation of  as relatively nonexpansive mappings from a nonempty, closed, and convex subset  of a uniformly convex and uniformly smooth Banach space  into itself.Let  be a subspace of  ∞ () and let {  } be an asymptotically left invariant sequence of means on .Let {  } be a sequence in (0, 1) such that lim  → ∞   = 0. Let {  } be a sequence generated by the following algorithm: If the interior of (I) is nonempty, then {  } converges strongly to some common fixed point (I).
We now establish a convergence theorem of modified Ishikawa type.Theorem 9. Let  be a left reversible semigroup and let I = {() :  ∈ } be a representation of  as relatively nonexpansive mappings from a nonempty, closed, and convex subset  of a uniformly convex and uniformly smooth Banach space  into itself.Let  be a subspace of  ∞ () and let {  } be an asymptotically left invariant sequence of means on .Let {  } and {  } be sequences of real numbers such that   ,   ∈ (0, 1) and lim  → ∞   = 0, lim  → ∞   = 1.Let {  } be a sequence generated by the following algorithm: If the interior of (I) is nonempty, then {  } converges strongly to some common fixed point (I).
Proof.Firstly, we show that {  } converges strongly in .
If we set   = 1, then the iteration (46) reduces modified Mann type.Hence we obtain the following corollary.
Corollary 10.Let  be a left reversible semigroup and let I = {() :  ∈ } be a representation of  as relatively nonexpansive mappings from a nonempty, closed, and convex subset  of a uniformly convex and uniformly smooth Banach space  into itself.Let  be a subspace of  ∞ () and let {  } be an asymptotically left invariant sequence of means on .Let {  } be a sequence of real number such that   ∈ (0, 1) and lim  → ∞   = 0. Let {  } be a sequence generated by the following algorithm:  0 ∈ , ℎ ,  +1 = Π   −1 (    + (1 −   )      ) , ∀ ≥ 0.

(69)
If the interior of (I) is nonempty, then {  } converges strongly to some common fixed point (I).