The purpose of this paper is to study modified Halpern type and Ishikawa type iteration for a semigroup of relatively nonexpansive mappings I={T(s):s∈S} on a nonempty closed convex subset C of a Banach space with respect to a sequence of asymptotically left invariant means {μn} defined on an appropriate invariant subspace of l∞(S), where S is a semigroup. 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 F(I), where F(I)=⋂{F(T(s)):s∈S}.
1. Introduction
Let E be a real Banach space with the topological dual E* and let C be a closed and convex subset of E. A mapping T of C into itself is called nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for each x,y∈C.
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:
(1)x0=u∈C,chosenarbitrarily,xn+1=αnu+(1-αn)Txn,∀n≥1,
where {αn} is a sequence in [0,1]. He pointed out that the conditions limn→∞αn=0 and ∑n=1∞αn=∞ are necessary in the sense that if the iteration (1) converges to a fixed point of T, then these conditions must be satisfied. The second iteration process is known as Mann’s iteration process [2] which is defined as follows:
(2)xn+1=αnxn+(1-αn)Txn,∀n≥1,
where the initial x1 is taken in C arbitrary and the sequence {αn} is in [0,1].
The third iteration process is referred to as Ishikawa’s iteration process [3] which is defined as follows:
(3)yn=βnxn+(1-βn)Txn,xn+1=αnxn+(1-αn)Tyn,∀n≥1,
where the initial x1 is taken in C arbitrary and {αn} and {βn} 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 S be a left reversible semigroup and let I={T(s):s∈S} be a representation of S as nonexpansive mappings from a compact convex subset C of a strictly convex and smooth Banach space E into C, let X be an amenable and I-stable subspace of l∞(S), and let {μn} be a strongly left regular sequence of means on X. Let {αn} be a sequence in [0,1] such that limn→∞αn=0 and ∑n=1∞αn=∞. Let x1=x∈C and let {xn} be the sequence defined by(4)xn+1=αnx+(1-αn)T(μn)xn,n≥2.
Then {xn} converges strongly to Px, where P denotes the unique sunny nonexpansive retraction of C onto F(I).
Let C be a closed and convex subset of E and let T be a mapping from C into itself. We denote by F(T) the set of fixed points of T. Point p in C is said to be an asymptotic fixed point of T [5] if C contains a sequence {xn} which converges weakly to p such that the strong limn→∞(Txn-xn)=0. The set of asymptotic fixed points of T will be denoted by F^(T). A mapping T from C into itself is called relatively nonexpansive [6–8], if F^(T)=F(T) and ϕ(p,Tx)≤ϕ(p,x) for all x∈C and p∈F(T). The asymptotic behavior of relatively nonexpansive mappings was studied in [6, 7, 9].
Recently, Kim [10] proved a strong convergence theorem for relatively nonexpansive mappings in a Banach space by using the shrinking method.
Theorem 2.
Let S be a left reversible semigroup and let I={T(s):s∈S} be a representation of S as relatively nonexpansive mappings from a nonempty, closed, and convex subset C of a uniformly convex and uniformly smooth Banach space E into C with F(I)≠∅. Let X be a subspace of l∞(S) and let {μn} be a asymptotically left invariant sequence of means on X. Let {αn} be a sequence in [0,1] such that 0<αn<1 and limn→∞αn=0. Let {xn} be a sequence generated by the following algorithm:
(5)x0∈C,chosenarbitrarily,C1=C,x1=ΠC1x0,yn=J-1(αnJx1+(1-αn)JTμnxn),Cn+1={z∈Cn:ϕ(z,yn)≤αnϕ(z,x1)+(1-αn)ϕ(z,xn)},xn+1=ΠCn+1x1,∀n≥1.
Then {xn} converges strongly to ΠF(I)x1, where ΠF(I) is the generalized projection from C onto F(I).
Let S 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={T(s):s∈S} on a nonempty closed convex subset C of a Banach space with respect to a sequence of asymptotically left invariant means {μn} defined on an appropriate invariant subspace of l∞(S). 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 F(I), where F(I)=⋂{F(T(s)):s∈S}.
2. Preliminaries
A real Banach space E is said to be strictly convex if ∥(x+y)/2∥<1 for all x,y∈E with ∥x∥=∥y∥=1 and x≠y. It is said to be uniformly convex if limn→∞∥xn-yn∥=0 for any two sequences {xn} and {yn} in E such that ∥xn∥=∥yn∥=1 and limn→∞∥(xn+yn)/2∥=1. Let U={x∈E:∥x∥=1} be the unit sphere of E. Then the Banach space E is said to be smooth if
(6)limt→0∥x+ty∥-∥x∥t
exists for each x,y∈U. It is said to be uniformly smooth if the limit is attained uniformly for x,y∈E.
Let E be a real Banach space with norm ∥·∥ and let E* be the dual space of E. Denote by 〈·,·〉 the duality product. We denote by J the normalized duality mapping from E to 2E* defined by
(7)Jx={f*∈E*:〈x,f*〉=∥x∥2=∥f*∥2},
for x∈E. A Banach space E is said to have the Kadec-Klee property if a sequence {xn} of E satisfies that xn⇀x and ∥xn∥→∥x∥ and then xn→x, where ⇀ and → denote the weak convergence and the strong convergence, respectively.
We know the following:
the duality mapping J is monotone, that is, 〈x-y,x*-y*〉≥0 whenever x*∈Jx and y*∈Jy;
if E is strictly convex, then J is one-to-one; that is, if Jx∩Jy is nonempty, then x=y;
if E is strictly convex, then J is strictly monotone; that is, x=y whenever 〈x-y,x*-y*〉=0, x*∈Jx and y*∈Jy;
if E is uniformly convex, then E has the Kadec-Klee property;
if E is uniformly convex, then E is reflexive and strictly convex;
if E is smooth, then J is single-valued and norm-to-weak* continuous;
if E is uniformly smooth, then J is uniformly norm-to-norm continuous on bounded subsets of E;
if E is reflexive, then J is onto;
if E is smooth and reflexive, then J is norm-to-weak continuous; that is, Jxn⇀Jx whenever xn→x;
if E is smooth, strictly convex, and reflexive, then J is single-valued, one-to-one and onto; in this case, the inverse mapping J-1 coincides with the duality mapping on E;
if E* is strictly convex, then J is single-valued;
the norm of E* is Fréchet differentiable if and only if E is strictly convex and reflexive Banach space which has the Kadec-Klee property.
For more details, see [11].
As well known, if C is a nonempty, closed, and convex subset of a Hilbert space H and PC:H→C is the metric projection of H onto C, then PC 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 ΠC in a Banach space E which is an analogue of the metric projection in Hilbert spaces. Consider the function defined by
(8)ϕ(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2,
for x,y∈E. Observe that, in a Hilbert space H, (8) reduces to
(9)ϕ(x,y)=∥x-y∥2,
for x,y∈H. The generalized projection ΠC:E→C is a mapping that assigns an arbitrary point x∈E to the minimum point of the functional ϕ(x,y); that is, ΠCx=x¯, where x¯ is the solution to the minimization problem:
(10)ϕ(x¯,x)=infy∈Cϕ(y,x).
The existence and uniqueness of the operator ΠC follow from the properties of the functional ϕ(x,y) and strict monotonicity of the mapping J (see, e.g., [12, 13]). In a Hilbert space, ΠC=PC. It is obvious from the definition of the function ϕ that
(∥x∥-∥y∥)2≤ϕ(x,y)≤(∥x∥+∥y∥)2 for all x,y∈E,
ϕ(x,y)=ϕ(x,z)+ϕ(z,y)+2〈x-z,Jz-Jy〉 for all x,y,z∈E,
ϕ(x,y)=〈x,Jx-Jy〉+〈y-x,Jy〉≤∥x∥∥Jx-Jy∥+∥y-x∥∥y∥ for all x,y∈E,
if E is a reflexive, strictly convex, and smooth Banach space, then, for all x,y∈E,
(11)ϕ(x,y)=0iffx=y.
For more details see [14].
Let S be a semigroup. We denote by l∞(S) the Banach space of all bounded real-valued functionals on S with supremum norm. For each s∈S, we define the left and right translation operators l(s) and r(s) on l∞(S) by
(12)(l(s)f)(t)=f(st),(r(s)f)(t)=f(ts),
for each t∈S and f∈l∞(S), respectively. Let X be a subspace of l∞(S) containing 1. An element μ in the dual space X* of X is said to be a mean on X if ∥μ∥=μ(1)=1. For s∈S, we can define a point evaluation δs by δs(f)=f(s) for each f∈X. It is well known that μ is mean on X if and only if
(13)infs∈Sf(s)≤μ(f)≤sups∈Sf(s),
for each f∈X.
Let X be a translation invariant subspace of l∞(S) (i.e., l(s)X⊂X and r(s)X⊂X for each s∈S) containing 1. Then a mean μ on X is said to be left invariant (resp., right invariant) if
(14)μ(l(s)f)=μ(f),(resp.,μ(r(s)f)=μ(f))
for each s∈S and f∈X. A mean μ on X is said to be invariant if μ is both left and right invariant [15–19]. X is said to be left (resp., right) amenable if X has a left (resp., right) invariant mean. X is amenable if X is left and right amenable. We call a semigroup S amenable if X 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 [20–22]).
A net {μα} of means on X is said to be asymptotically left (resp., right) invariant if
(15)limα(μα(l(s)f)-μα(f))=0,(resp.,limα(μα(r(s)f)-μα(f))=0),
for each f∈X and s∈S, and it is said to be left (resp., right) strongly asymptotically invariant (or strong regular) if(16)limα∥l*(s)μα-μα∥=0,(resp.,limα∥r*(s)μα-μα∥=0),
for each s∈S, where l*(s) and r*(s) are the adjoint operators of l(s) and r(s), respectively. Such nets were first studied by Day in [20] where they were called weak* invariant and norm invariant, respectively.
It is easy to see that if a semigroup S is left (resp., right) amenable, then the semigroup S′=S∪{e}, where es′=s′e=s′ for all s′∈S, is also left (resp., right) amenable and converse.
From now on S denotes a semigroup with an identity e. S is called left reversible if any two right ideals of S have nonvoid intersection; that is, aS∩bS≠∅ for a,b∈S. In this case, (S,⪯) is a directed system when the binary relation “⪯” on S is defined by a⪯b if and only if {a}∪aS⊇{b}∪bS for a,b∈S. It is easy to see that t⪯ts for all t,s∈S. Further, if t⪯s then pt⪯ps for all p∈S. The class of left reversible semigroup includes all groups and commutative semigroups. If a semigroup S is left amenable, then S is left reversible. But the converse is not true [23–28].
Let S be a semigroup and let C be a closed and convex subset of E. Let F(T) denote the fixed point set of T. Then I={T(s):s∈S} is called a representation of S as relatively nonexpansive mappings on C if T(s) is relatively nonexpansive with T(e)=I and T(st)=T(s)T(t) for each s,t∈S. We denote by F(I) the set of common fixed points of {T(s):s∈S}; that is,
(17)F(I)=⋂s∈SF(T(s))=⋂s∈S{x∈C:T(s)x=x}.
We know that if μ is a mean on X and if for each x*∈E* the function s↦〈T(s)x,x*〉 is contained in X and C is weakly compact, then there exists a unique point x0 of E such that μ〈T(·)x,x*〉=〈x0,x*〉 for each x*∈E*. We denote such a point x0 by Tμx. Note that Tμx is contained in the closure of the convex hull of {T(s)x:s∈S} for each x∈C. Note that Tμz=z for each z∈F(I); see [29–31].
3. Lemmas
We need the following lemmas for the proof of our main results.
Lemma 3 (see [9]).
Let E be a strictly convex and smooth Banach space, let C be a closed convex subset of E, and let T be a relatively nonexpansive mapping from C into itself. Then F(T) is closed and convex.
Lemma 4 (see [12, 32]).
Let E be a reflexive, strictly convex, and smooth Banach space and let C be a nonempty, closed, and convex subset of E and x∈E. Then
(18)ϕ(y,ΠCx)+ϕ(ΠCx,x)≤ϕ(y,x),
for all y∈C.
Lemma 5 (see [32]).
Let E be a uniformly convex and smooth Banach space and let {xn}, {yn} be two sequences of E. If limn→∞ϕ(xn,yn)=0 and either {xn} or {yn} is bounded, then limn→∞∥xn-yn∥=0.
Lemma 6 (see [4, 33]).
Let μ be a left invariant mean on X. Then F(I)=F(Tμ)∩Ca, where Ca denotes the set of almost periodic elements in C; that is, all x∈C such that {T(s)x:s∈S} is relatively compact in the norm topology of E.
Lemma 7 (cf. [4, 10]).
Let {μn} be an asymptotically left invariant sequence of means on X. If z∈Ca and liminfn→∞∥Tμnz-z∥=0, then z is a common fixed point of I.
4. 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. [34–36]).
We begin with a strong convergence theorem of modified Halpern’s type.
Theorem 8.
Let S be a left reversible semigroup and let I={T(s):s∈S} be a representation of S as relatively nonexpansive mappings from a nonempty, closed, and convex subset C of a uniformly convex and uniformly smooth Banach space E into itself. Let X be a subspace of l∞(S) and let {μn} be an asymptotically left invariant sequence of means on X. Let {αn} be a sequence in (0,1) such that limn→∞αn=0. Let {xn} be a sequence generated by the following algorithm:
(19)x0∈C,chosenarbitrarily,xn+1=ΠCJ-1(αnJx0+(1-αn)JTμnxn),∀n≥0.
If the interior of F(I) is nonempty, then {xn} converges strongly to some common fixed point F(I).
Proof.
We show first that the sequence {xn} converges strongly in C.
From Lemma 3, we know F(T) is closed and convex. So, we can define the generalized projection ΠC onto F(I). Most of all, we have
(20)∥Tμnxn∥=sup{|〈Tμnxn,x*〉|:x*∈E*,∥x*∥=1}=sup{|(μn)s〈T(s)xn,x*〉|:x*∈E*,∥x*∥=1}≤sup{(μn)s(∥T(s)xn∥∥x*∥):x*∈E*,∥x*∥=1}=(μn)s∥T(s)xn∥.
Then, from the definition of relatively nonexpansive, we have
(21)ϕ(u,Tμnxn)=∥u∥2-2〈u,JTμnxn〉+∥Tμnxn∥2=∥u∥2-2(μn)s〈u,JT(s)xn〉+(μn)s∥T(s)xn∥2=(μn)sϕ(u,T(s)xn)≤(μn)sϕ(u,xn)=ϕ(u,xn),
for all u∈F(I). From the convexity of ∥·∥2 and (21), we get
(22)ϕ(u,xn+1)=ϕ(u,ΠCJ-1(αnJx0+(1-αn)JTμnxn))≤ϕ(u,J-1(αnJx0+(1-αn)JTμnxn))=∥u∥2-2〈u,αnJx0+(1-αn)JTμnxn〉+∥αnJx0+(1-αn)JTμnxn∥2≤∥u∥2-2αn〈u,Jx0〉-2(1-αn)〈u,JTμnxn〉+αn∥x0∥2+(1-αn)∥Tμnxn∥2=αnϕ(u,x0)+(1-αn)ϕ(u,Tμnxn)≤αnϕ(u,x0)+(1-αn)ϕ(u,xn).
So, we have
(23)(1-αn){ϕ(u,xn+1)-ϕ(u,xn)}≤αn{ϕ(u,x0)-ϕ(u,xn+1)}≤αnϕ(u,x0).
Since limn→∞αn=0, we obtain
(24)limn→∞{ϕ(u,xn+1)-ϕ(u,xn)}≤0.
Therefore {ϕ(u,xn)} is bounded and limn→∞ϕ(u,xn) exists. Then {xn} is also bounded. This implies that {Tμnxn} is bounded. Since the interior of F(I) is nonempty, there exist p∈F(I) and r>0 such that
(25)p+rq∈F(I),
whenever ∥q∥≤1. By (ϕ2), we have
(26)ϕ(u,xn)=ϕ(u,xn+1)+ϕ(xn+1,xn)+2〈u-xn+1,Jxn+1-Jxn〉,
for any u∈F(I). This implies
(27)〈xn+1-u,Jxn-Jxn+1〉+12ϕ(xn+1,xn)=12(ϕ(u,xn)-ϕ(u,xn+1)).
Also, we have
(28)〈xn+1-p,Jxn-Jxn+1〉=〈xn+1-(p+rq)+rq,Jxn-Jxn+1〉=〈xn+1-(p+rq),Jxn-Jxn+1〉+r〈q,Jxn-Jxn+1〉.
On the other hand, by (24) and (25), we have that
(29)ϕ(p+rq,xn+1)≤ϕ(p+rq,xn).
From (27), we get
(30)0≤12(ϕ(p+rq,xn)-ϕ(p+rq,xn+1))=〈xn+1-(p+rq),Jxn-Jxn+1〉+12ϕ(xn+1,xn)=〈xn+1-p,Jxn-Jxn+1〉-r〈q,Jxn-Jxn+1〉+12ϕ(xn+1,xn).
Then, by (27), we have
(31)r〈q,Jxn-Jxn+1〉≤〈xn+1-p,Jxn-Jxn+1〉+12ϕ(xn+1,xn)=12(ϕ(p,xn)-ϕ(p,xn+1)),
for p∈F(I). Hence
(32)〈q,Jxn-Jxn+1〉≤12r(ϕ(p,xn)-ϕ(p,xn+1)).
Since q with ∥q∥≤1 is arbitrary, by (24), we have
(33)∥Jxn-Jxn+1∥≤12r(ϕ(p,xn)-ϕ(p,xn+1)).
So, we have
(34)∥Jxn+m-Jxn∥=∥Jxn+m-Jxn+m-1+Jxn+m-1-⋯-Jxn+1+Jxn+1-Jxn∥≤∑i=nn+m-1∥Jxi-Jxi+1∥≤12r∑i=nn+m-1(ϕ(p,xi)-ϕ(p,xi+1))=12r(ϕ(p,xn)-ϕ(p,xn+1)).
We know that {ϕ(p,xn)} converges. Hence, {Jxn} is a Cauchy sequence. Since E* is complete, {Jxn} converges strongly to some point in E*. Since E is uniformly convex, E* has a Fréchet differentiable norm. Then J-1 is continuous on E*. Hence {xn} converges strongly to some point v in C.
Now, we show that v∈F(I), where v=limn→∞ΠF(I)xn.
By (33) and the convergence of {ϕ(p,xn)}, we have
(35)limn→∞∥Jxn-Jxn+1∥=0.
Since J-1 is uniformly norm-to-norm continuous on bounded sets, it follows that
(36)limn→∞∥xn-xn+1∥=0.
Let zn=J-1(αnJx0+(1-αn)JTμnxn). Then, we have
(37)∥Jzn-JTμnxn∥=∥αnJx0+(1-αn)JTμnxn-JTμnxn∥=αn∥Jx0-JTμnxn∥.
Since limn→∞αn=0, we have
(38)limn→∞∥Jzn-JTμnxn∥=0.
Since J-1 is uniformly norm-to-norm continuous on bounded sets, we get
(39)limn→∞∥zn-Tμnxn∥=0.
From xn+1=ΠCzn and Lemma 4, we have
(40)ϕ(Tμnxn,xn+1)+ϕ(xn+1,zn)=ϕ(Tμnxn,ΠCzn)+ϕ(ΠCzn,zn)≤ϕ(Tμnxn,zn).
Since
(41)ϕ(Tμnxn,zn)=ϕ(Tμnxn,J-1(αnJx0+(1-αn)JTμnxn))=∥Tμnxn∥2-2〈Tμnxn,αnJx0+(1-αn)JTμnxn〉+∥αnJx0+(1-αn)JTμnxn∥2≤∥Tμnxn∥2-2αn〈Tμnxn,Jx0〉-2(1-αn)〈Tμnxn,JTμnxn〉+αn∥x0∥2+(1-αn)∥Tμnxn∥2=αnϕ(Tμnxn,x0)+(1-αn)ϕ(Tμnxn,Tμnxn)=αnϕ(Tμnxn,x0)
and limn→∞αn=0, we have
(42)limn→∞ϕ(Tμnxn,zn)=0.
From (40), we get
(43)Limn→∞ϕ(Tμnxn,xn+1)=limn→∞ϕ(xn+1,zn)=0.
By Lemma 5, we obtain(44)limn→∞∥Tμnxn-xn+1∥=limn→∞∥xn+1-zn∥=0.
Since ∥xn-Tμnxn∥≤∥xn-xn+1∥+∥xn+1-zn∥+∥zn-Tμnxn∥, from (36), (39), and (44), we have
(45)limn→∞∥xn-Tμnxn∥=0.
From Lemma 7, we have xn∈F(I). Since F(I) is closed and limn→∞xn=v, we have v∈F(I), where v=limn→∞ΠF(I)xn.
We now establish a convergence theorem of modified Ishikawa type.
Theorem 9.
Let S be a left reversible semigroup and let I={T(s):s∈S} be a representation of S as relatively nonexpansive mappings from a nonempty, closed, and convex subset C of a uniformly convex and uniformly smooth Banach space E into itself. Let X be a subspace of l∞(S) and let {μn} be an asymptotically left invariant sequence of means on X. Let {αn} and {βn} be sequences of real numbers such that αn,βn∈(0,1) and limn→∞αn=0, limn→∞βn=1. Let {xn} be a sequence generated by the following algorithm:
(46)x0∈C,chosenarbitrarily,yn=J-1(βnJxn+(1-βn)JTμnxn),xn+1=ΠCJ-1(αnJxn+(1-αn)JTμnyn),∀n≥0.
If the interior of F(I) is nonempty, then {xn} converges strongly to some common fixed point F(I).
Proof.
Firstly, we show that {xn} converges strongly in C.
From Lemma 3, we know F(T) is closed and convex. So, we can define the generalized projection ΠC onto F(I). Let u∈F(I). From the definition of relatively nonexpansive and the convexity of ∥·∥2, from (21), we have
(47)ϕ(u,yn)=ϕ(u,J-1(βnJxn+(1-βn)JTμnxn))≤βnϕ(u,xn)+(1-βn)ϕ(u,Tμnxn)≤ϕ(u,xn),
for all u∈F(I). From (47), we obtain
(48)ϕ(u,xn+1)=ϕ(u,ΠCJ-1(αnJxn+(1-αn)JTμnyn))≤ϕ(u,J-1(αnJxn+(1-αn)JTμnyn))≤αnϕ(u,xn)+(1-αn)ϕ(u,Tμnyn)≤αnϕ(u,xn)+(1-αn)ϕ(u,yn)≤ϕ(u,xn).
Hence, {ϕ(u,xn)} is bounded and limn→∞ϕ(u,xn) exists. This implies that {xn}, {Tμnxn}, and {yn} are bounded. Since the interior of F(I) is nonempty, similar to the proof of Theorem 8, we obtain that {xn} converges strongly to v in C.
Next, we show that v∈F(I), where v=limn→∞ΠF(I)xn.
Let
(49)zn=J-1(αnJxn+(1-αn)JTμnyn).
From Lemma 4, we have
(50)ϕ(xn,xn+1)+ϕ(xn+1,zn)=ϕ(xn,ΠCzn)+ϕ(ΠCzn,zn)≤ϕ(xn,zn).
Also,
(51)ϕ(xn,zn)=ϕ(xn,J-1(αnJxn+(1-αn)JTμnyn))≤αnϕ(xn,xn)+(1-αn)ϕ(xn,Tμnyn)≤αnϕ(xn,xn)+(1-αn)ϕ(xn,yn)≤ϕ(xn,yn),(52)∥Jxn-Jyn∥=∥Jxn-(βnJxn-(1-βn)JTμnxn)∥=(1-βn)∥Jxn-JTμnxn∥.
From limn→∞βn=1 and (52), we have
(53)limn→∞∥Jxn-Jyn∥=0.
Since J-1 is uniformly norm-to-norm continuous, we obtain
(54)limn→∞∥xn-yn∥=0.
Hence,
(55)ϕ(xn,yn)=∥xn∥2-2〈xn,Jyn〉+∥yn∥2=∥xn∥2-2〈xn,Jyn-Jxn〉-2〈xn,Jxn〉+∥yn∥2≤∥yn∥2-∥xn∥2+2∥xn∥∥Jyn-Jxn∥≤∥yn-xn∥(∥yn∥+∥xn∥)+2∥xn∥∥Jyn-Jxn∥.
By (53) and (54), we have
(56)limn→∞ϕ(xn,yn)=0.
From (50) and (51), we obtain
(57)limn→∞ϕ(xn,xn+1)=limn→∞ϕ(xn,zn)=0.
From Lemma 5, we get
(58)limn→∞∥xn-xn+1∥=limn→∞∥xn-zn∥=0.
Since
(59)∥Jzn-JTμnyn∥=∥αnJxn+(1-αn)JTμnyn-JTμnyn∥=αn∥Jxn-JTμnxn∥
and limn→∞αn=0, we have
(60)limn→∞∥Jzn-JTμnyn∥=0.
Since J-1 is uniformly norm-to-norm continuous, we obtain
(61)limn→∞∥zn-Tμnyn∥=0.
Since limn→∞∥xn-zn∥=0 and J is uniformly norm-to-norm continuous,
(62)limn→∞∥Jxn-Jzn∥=0.
By (46) and (49), we have
(63)JTμnxn=11-βn(Jyn-βnJxn),JTμnyn=11-αn(Jzn-αnJxn).
From (63), we obtain
(64)∥JTμnxn-JTμnyn∥=∥11-βn(Jyn-βnJxn)-11-αn(Jzn-αnJxn)∥=∥(Jzn+αn1-αn(Jzn-Jxn))Jyn+βn1-βn(Jyn-Jxn)-(Jzn+αn1-αn(Jzn-Jxn))βn1-βn∥≤∥Jyn-Jxn∥+∥Jxn-Jzn∥+βn1-βn∥Jyn-Jxn∥+αn1-αn∥Jzn-Jxn∥=11-αn∥Jzn-Jxn∥+11-βn∥Jyn-Jxn∥.
Combining (53), (62), and (64), we get
(65)limn→∞∥JTμnxn-JTμnyn∥=0.
Since J-1 is uniformly norm-to-norm continuous, we have
(66)limn→∞∥Tμnxn-Tμnyn∥=0.
Since
(67)∥xn-Tμnxn∥≤∥xn-zn∥+∥zn-Tμnyn∥+∥Tμnyn-Tμnxn∥,
therefore, by (58), (61), (66), and (67), we obtain
(68)limn→∞∥xn-Tμnxn∥=0.
From Lemma 7, we have xn∈F(I). Since F(I) is closed and limn→∞xn=v, we have v∈F(I), where v=limn→∞ΠF(I)xn.
If we set βn=1, then the iteration (46) reduces modified Mann type. Hence we obtain the following corollary.
Corollary 10.
Let S be a left reversible semigroup and let I={T(s):s∈S} be a representation of S as relatively nonexpansive mappings from a nonempty, closed, and convex subset C of a uniformly convex and uniformly smooth Banach space E into itself. Let X be a subspace of l∞(S) and let {μn} be an asymptotically left invariant sequence of means on X. Let {αn} be a sequence of real number such that αn∈(0,1) and limn→∞αn=0. Let {xn} be a sequence generated by the following algorithm:
(69)x0∈C,chosenarbitrarily,xn+1=ΠCJ-1(αnJxn+(1-αn)JTμnxn),∀n≥0.
If the interior of F(I) is nonempty, then {xn} converges strongly to some common fixed point F(I).
In a Hilbert space, J is the identity operator. Theorems 8 and 9 reduce to the following.
Corollary 11.
Let S be a left reversible semigroup and let I={T(s):s∈S} be a representation of S as relatively nonexpansive mappings from a nonempty, closed, and convex subset C of a Hilbert space H into itself. Let X be a subspace of l∞(S) and let {μn} be an asymptotically left invariant sequence of means on X. Let {αn} be a sequence in (0,1) such that limn→∞αn=0. Let {xn} be a sequence generated by the following algorithm:
(70)x0∈C,chosenarbitrarily,xn+1=PC(αnx0+(1-αn)Tμnxn),∀n≥0.
If the interior of F(I) is nonempty, then {xn} converges strongly to some common fixed point F(I), where PC is a metric projection.
Corollary 12.
Let S be a left reversible semigroup and let I={T(s):s∈S} be a representation of S as relatively nonexpansive mappings from a nonempty, closed, and convex subset C of a Hilbert space H into itself. Let X be a subspace of l∞(S) and let {μn} be an asymptotically left invariant sequence of means on X. Let {αn} and {βn} be sequences of real numbers such that αn,βn∈(0,1) and limn→∞αn=0, limn→∞βn=1. Let {xn} be a sequence generated by the following algorithm:
(71)x0∈C,chosenarbitrarily,yn=βnxn+(1-βn)Tμnxn,xn+1=PC(αnxn+(1-αn)Tμnyn),∀n≥0.
If the interior of F(I) is nonempty, then {xn} converges strongly to some common fixed point F(I), where PC is a metric projection.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author would like to thank Professor Anthony To-Ming Lau and Professor Jong Kyu Kim for their helpful suggestions. Also, special thanks are due to the referee for his/her deep insight which improved the paper. This work was supported by Kyungnam University Foundation Grant, 2013.
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