# Hybrid Projection Algorithms for Asymptotically Strict Quasi-ϕ-Pseudocontractions

Authors: Xiaolong Qin; Tianze Wang; Sun Young Cho
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

A new nonlinear mapping is introduced. Hybrid projection algorithms are considered for the class of new nonlinear mappings. Strong convergence theorems are established in a real Banach space.


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## Body

## 1. Introduction

LetE be a real Banach space, C a nonempty subset of E, and T:C→C a nonlinear mapping. Denote by F(T) the set of fixed points of T. Recall that T is said to be nonexpansive if (1.1)‖Tx-Ty‖≤‖x-y‖,∀x,y∈C.
We remark that the mapping T is said to be quasinonexpansive if F(T)≠∅ and (1.1) holds for all x∈C and y∈F(T). T is said to be asymptotically nonexpansive if there exists a sequence {μn}⊂[0,∞) with μn→0 as n→∞ such that (1.2)‖Tnx-Tny‖≤(1+μn)‖x-y‖,∀x,y∈C.
We remark that the mapping T is said to be asymptotically quasinonexpansive if F(T)≠∅ and (1.2) holds for all x∈C and y∈F(T). The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] in 1972. They proved that if C is a nonempty bounded closed convex subset of a uniformly convex Banach space E, then every asymptotically nonexpansive selfmapping T has a fixed point in C. Further, the set F(T) of fixed points of T is closed and convex. Since 1972, many authors have studied the weak and strong convergence problems of iterative algorithms for the class of mappings.Recall thatT is said to be a strict pseudocontraction if there exists a constant κ∈[0,1) such that (1.3)‖Tx-Ty‖2≤‖x-y‖2+κ‖(I-T)x-(I-T)y‖2,∀x,y∈C.
We remark that the mapping T is said to be a strict quasipseudocontraction if F(T)≠∅ and (1.3) holds for all x∈C and y∈F(T).The class of strict pseudocontractions was introduced by Browder and Petryshyn [2]. In 2007, Marino and Xu [3] proved that the fixed point set of strict pseudocontractions is closed and convex. They also proved that I-T is demiclosed at the origin in real Hilbert spaces. A strong convergence theorem of hybrid projection algorithms for strict pseudocontractions was established; see [3] for more details.Recall thatT is said to be an asymptotically strict pseudocontraction if there exist a constant κ∈[0,1) and a sequence {μn}⊂[0,∞) with μn→0 as n→∞ such that (1.4)‖Tnx-Tny‖2≤(1+μn)‖x-y‖2+κ‖(I-Tn)x-(I-Tn)y‖2,∀x,y∈C.
We remark that the mapping T is said to be an asymptotically strict quasipseudocontraction if F(T)≠∅ and (1.4) holds for all x∈C and y∈F(T).The class of asymptotically strict pseudocontractions was introduced by Qihou [4] in 1996. Kim and Xu [5] proved that the fixed-point set of asymptotically strict pseudocontractions is closed and convex. They also obtained a strong convergence theorem for the class of asymptotically strict pseudocontractions by hybrid projection algorithms. To be more precise, they proved the following theorem.Theorem KX.
LetC be a closed convex subset of a Hilbert space  H, and let  T:C→C be an asymptotically  κ-strict pseudocontraction for some  0≤κ<1. Assume that the fixed-point set  F(T) of  T is nonempty and bounded. Let  {xn} be the sequence generated by the following (CQ) algorithm:
(1.5)x0∈Cchosen  arbitrarily,yn=αnxn+(1-αn)Tnxn,Cn={z∈C:‖yn-z‖2≤‖xn-z‖yn-z‖2‖2+[κ-αn(1-αn)]‖xn-Txn‖xn-z‖‖yn-z‖2Txn‖2+θn},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0,
where
(1.6)θn=Δn2(1-αn)μn⟶orn⟶∞,Δn=sup⁡{‖xn-z‖:p∈F(T)}.
Assume that the control sequence {αn} is chosen so that limsupn→∞αn<1-κ then {xn} converges strongly to PF(T)x0.It is well known that, in an infinite dimensional Hilbert space, the normal Mann iterative algorithm has only weak convergence, in general, even for nonexpansive mappings. Hybrid projection algorithms are popular tool to prove strong convergence of iterative sequences without compactness assumptions. Recently, hybrid projection algorithms have received rapid developments; see, for example, [3, 5–24]. In this paper, we will introduce a new mapping, asymptotically strict quasi-ϕ-pseudocontractions, and give a strong convergence theorem by a simple hybrid projection algorithm in a real Banach space.


## 2. Preliminaries

LetE be a Banach space with the dual space E*. We denote by J the normalized duality mapping from E to 2E* defined by (2.1)Jx={f*∈E*:〈x,f*〉=‖x‖2=‖f*‖2},∀x∈E,
where 〈·,·〉 denotes the generalized duality pairing of elements between E and E*; see [25]. It is well known that if E* is strictly convex, then J is single valued, and if E* is uniformly convex, then J is uniformly continuous on bounded subsets of E.It is also well known that ifC is a nonempty closed convex subset of a Hilbert space H and PC:H→C is the metric projection of H onto C, then PC is nonexpansive. This fact actually characterizes Hilbert spaces, and consequently, it is not available in more general Banach spaces. In this connection, Alber [26] recently introduced a generalized projection operator ΠC in a Banach space E which is an analogue of the metric projection in Hilbert spaces.Recall that a Banach spaceE 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. E is said to have Kadec-Klee property if a sequence {xn} of E satisfying that xn⇀x and ∥xn∥→∥x∥, then xn→x. It is known that if E is uniformly convex, then E enjoys Kadec-Klee property; see [25, 27] for more details. Let UE={x∈E:∥x∥=1} be the unit sphere of E then the Banach space E is said to be smooth provided (2.2)lim⁡t→0‖x+ty‖-‖x‖t
exists for each x,y∈UE. It is also said to be uniformly smooth if the limit is attained uniformly for x,y∈UE. It is well known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E.LetE be a smooth Banach space. Consider the functional defined by (2.3)ϕ(x,y)=‖x‖2-2〈x,Jy〉+‖y‖2,∀x,y∈E.
Observe that, in a Hilbert space H, (2.3) is reduced to ϕ(x,y)=∥x-y∥2 for all x,y∈H. The generalized projection ΠC:E→C is a mapping that assigns to an arbitrary point x∈E the minimum point of the functional ϕ(x,y), that is, ΠCx=x¯, where x¯ is the solution to the following minimization problem: (2.4)ϕ(x¯,x)=min⁡y∈Cϕ(y,x).
The existence and uniqueness of the operator ΠC follow from the properties of the functional ϕ(x,y) and the strict monotonicity of the mapping J; see, for example, [26–29]. In Hilbert spaces, ΠC=PC. It is obvious from the definition of the function ϕ that (2.5)(‖y‖-‖x‖)2≤ϕ(y,x)≤(‖y‖+‖x‖)2,∀x,y∈E,(2.6)ϕ(x,y)=ϕ(x,z)+ϕ(z,y)+2〈x-z,Jz-Jy〉,∀x,y,z∈E.Remark 2.1.
IfE is a reflexive, strictly convex, and smooth Banach space, then, for all x,y∈E, ϕ(x,y)=0 if and only if x=y. It is sufficient to show that if ϕ(x,y)=0, then x=y. From (2.5), we have ∥x∥=∥y∥. This implies that 〈x,Jy〉=∥x∥2=∥Jy∥2. From the definition of J, we see that Jx=Jy. It follows that x=y; see [25, 27] for more details.Now, we give some definitions for our main results in this paper.LetC be a closed convex subset of a real Banach space E and T:C→C a mapping.(1)
A pointp in C is said to be an asymptotic fixed point of T [30] if C contains a sequence {xn} which converges weakly to p such that limn→∞∥xn-Txn∥=0. The set of asymptotic fixed points of T will be denoted by F̃(T).(2)
T is said to be relatively nonexpansive [15, 31, 32] if
(2.7)F̃(T)=F(T)≠∅,ϕ(p,Tx)≤ϕ(p,x),∀x∈C,p∈F(T).
The asymptotic behavior of a relatively nonexpansive mapping was studied in [30–32].(3)
T is said to be relatively asymptotically nonexpansive [6, 11] if
(2.8)F̃(T)=F(T)≠∅,ϕ(p,Tnx)≤(1+μn)ϕ(p,x),∀x∈C,p∈F(T),
where {μn}⊂[0,∞) is a sequence such that μn→1 as n→∞.(4)
T is said to be ϕ-nonexpansive [14, 16, 17] if
(2.9)ϕ(Tx,Ty)≤ϕ(x,y),∀x,y∈C.(5)
T is said to be quasi-ϕ-nonexpansive [14, 16, 17] if
(2.10)F(T)≠∅,ϕ(p,Tx)≤ϕ(p,x),∀x∈C,p∈F(T).(6)
T is said to be asymptotically ϕ-nonexpansive [14] if there exists a real sequence {μn}⊂[0,∞) with μn→0 as n→∞ such that
(2.11)ϕ(Tnx,Tny)≤(1+μn)ϕ(x,y),∀x,y∈C.(7)
T is said to be asymptotically quasi-ϕ-nonexpansive [14] if there exists a real sequence {μn}⊂[0,∞) with μn→0 as n→∞ such that
(2.12)F(T)≠∅,ϕ(p,Tnx)≤(1+μn)ϕ(p,x),∀x∈C,p∈F(T).(8)
T is said to be a strict quasi-ϕ-pseudocontraction if F(T)≠∅, and there exists a constant κ∈[0,1) such that
(2.13)ϕ(p,Tx)≤ϕ(p,x)+κϕ(x,Tx),∀x∈C,p∈F(T).
We remark that T is said to be a quasistrict pseudocontraction in [13].(9)
T is said to be asymptotically regular on C if, for any bounded subset K of C,
(2.14)lim⁡n→∞sup⁡x∈K{‖Tn+1x-Tnx‖}=0.Remark 2.2.
The class of quasi-ϕ-nonexpansive mappings and the class of asymptotically quasi-ϕ-nonexpansive mappings are more general than the class of relatively nonexpansive mappings and the class of relatively asymptotically nonexpansive mappings. Quasi-ϕ-nonexpansive mappings and asymptotically quasi-ϕ-nonexpansive mappings do not require F(T)=F̃(T), where F̃(T) denotes the asymptotic fixed-point set of T.Remark 2.3.
In the framework of Hilbert spaces, quasi-ϕ-nonexpansive mappings and asymptotically quasi-ϕ-nonexpansive mappings are reduced to quasinonexpansive mappings and asymptotically quasinonexpansive mappings.In this paper, we introduce a new nonlinear mapping: asymptotically strict quasi-ϕ-pseudocontractions.Definition 2.4.
Recall that a mappingT:C→C is said to be an asymptotically strict quasi-ϕ-pseudocontraction if F(T)≠∅, and there exists a sequence {μn}⊂[0,∞) with μn→0 as n→∞ and a constant κ∈[0,1) such that
(2.15)ϕ(p,Tnx)≤(1+μn)ϕ(p,x)+κϕ(x,Tnx),∀x∈C,p∈F(T).Remark 2.5.
In the framework of Hilbert spaces, asymptotically strict quasi-ϕ-pseudocontractions are asymptotically strict quasipseudocontractions.Next, we give an example which is an asymptotically strict quasi-ϕ-pseudocontraction.LetE=l2:={x={x1,x2,…}:∑n=1∞|xn|2<∞}, and let BE be the closed unit ball in E. Define a mapping T:BE→BE by
(2.16)T(x1,x2,…)=(0,x12,a2x2,a3x3,…),
where {ai} is a sequence of real numbers such that a2>0, 0<aj<1, where i≠2, and Πi=2∞aj=1/2. Then
(2.17)ϕ(p,Tnx)=‖p-Tnx‖2≤2(Πi=2naj)‖p-x‖2+κ‖x-Tnx‖2=2(Πi=2naj)ϕ(p,x)+κϕ(x,Tnx),∀x∈BE,n≥2,
where p=(0,0,…) is a fixed point of T and κ∈[0,1) is a real number. In view of limn→∞2(Πi=2naj)=1, we see that T is an asymptotically strict quasi-ϕ-pseudocontraction.In order to prove our main results, we also need the following lemmas.Lemma 2.6 (see [29]).
LetE be a uniformly convex and smooth Banach space, and let  {xn},  {yn} be two sequences of  E. If  ϕ(xn,yn)→0 and either  {xn} or  {yn} is bounded, then  xn-yn→0.Lemma 2.7 (see [26]).
LetC be a nonempty closed convex subset of a smooth Banach space  E and  x∈E then  x0=ΠCx if and only if
(2.18)〈x0-y,Jx-Jx0〉≥0,∀y∈C.Lemma 2.8 (see [26]).
LetE be a reflexive, strictly convex, and smooth Banach space,  C a nonempty closed convex subset of  E, and  x∈E then
(2.19)ϕ(y,ΠCx)+ϕ(ΠCx,x)≤ϕ(y,x),∀y∈C.


## 3. Main Results

Theorem 3.1.
LetC be a nonempty closed and convex subset of a uniformly convex and smooth Banach space  E. Let  T:C→C be a closed and asymptotically strict quasi -ϕ-pseudocontraction with a sequence  {μn}⊂[0,∞) such that  μn→0 as  n→∞. Assume that  T is uniformly asymptotically regular on  C and  F(T) is nonempty and bounded. Let  {xn} be a sequence generated in the following manner:
(Υ)x0∈Echosen  arbitrarily,C1=C,x1=ΠC1x0,Cn+1={u∈Cn:ϕ(xn,Tnxn)≤21-κ〈xn-u,Jxn-JTnxn(‖y‖-‖x‖)2〉+μnMn1-κ},xn+1=ΠCn+1x0,∀n≥0,
where Mn=sup{ϕ(p,xn):p∈F(T)} then the sequence {xn} converges strongly to x¯=ΠF(T)x0.Proof.
The proof is split into five steps.
Step 1.
Show thatF(T) is closed and convex.
Let{pn} be a sequence in F(T) such that pn→p as n→∞. We see that p∈F(T). Indeed, we obtain from the definition of T that
(3.1)ϕ(pn,Tnp)≤(1+μn)ϕ(pn,p)+κϕ(p,Tnp).
In view of (2.6), we see that
(3.2)ϕ(pn,Tnp)=ϕ(pn,p)+ϕ(p,Tnp)+2〈pn-p,Jp-JTnp〉.
It follows that
(3.3)ϕ(pn,p)+ϕ(p,Tnp)+2〈pn-p,Jp-JTnp〉≤(1+μn)ϕ(pn,p)+κϕ(p,Tnp),
which implies that
(3.4)ϕ(p,Tnp)≤μn1-κϕ(pn,p)+21-κ〈p-pn,Jp-JTnp〉,
from which it follows that
(3.5)lim⁡n→∞ϕ(p,Tnp)=0.
From Lemma 2.6, we see that Tnp→p as n→∞. This implies that TTnp=Tn+1p→p as n→∞. From the closedness of T, we obtain that p∈F(T). This proves the closedness of F(T).
Next, we show the convexness ofF(T). Let p1,p2∈F(T) and pt=tp1+(1-t)p2, where t∈(0,1). We see that pt=Tpt. Indeed, we have from the definition of T that
(3.6)ϕ(p1,Tnpt)≤(1+μn)ϕ(p1,pt)+κϕ(pt,Tnpt),ϕ(p2,Tnpt)≤(1+μn)ϕ(p2,pt)+κϕ(pt,Tnpt).
By virtue of (2.6), we obtain that
(3.7)ϕ(pt,Tnpt)≤μn1-κϕ(p1,pt)+21-κ〈pt-p1,Jpt-JTnpt〉,(3.8)ϕ(pt,Tnpt)≤μn1-κϕ(p2,pt)+21-κ〈pt-p2,Jpt-JTnpt〉.
Multiplying t and (1-t) on both the sides of (3.7) and (3.8), respectively, yields that
(3.9)ϕ(pt,Tnpt)≤tμn1-κϕ(p1,pt)+(1-t)μn1-κϕ(p2,pt).
It follows that
(3.10)lim⁡n→∞ϕ(pt,Tnpt)=0.
In view of Lemma 2.6, we see that Tnpt→pt as n→∞. This implies that TTꍜpt=Tn+1pt→pt as n→∞. From the closedness of T, we obtain that pt∈F(T). This proves that F(T) is convex. This completes Step 1.Step 2.
Show thatCn is closed and convex for each n≥1.
It is not hard to see thatCn is closed for each n≥1. Therefore, we only show that Cn is convex for each n≥1. It is obvious that C1=C is convex. Suppose that Ch is convex for some h∈ℕ. Next, we show that Ch+1 is also convex for the same h. Let a,b∈Ch+1 and c=ta+(1-t)b, where t∈(0,1). It follows that
(3.11)ϕ(xh,Thxh)≤21-κ〈xh-a,Jxh-JThxh〉+μhMh1-κ,ϕ(xh,Thxh)≤21-κ〈xh-b,Jxh-JThxh〉+μhMh1-κ,
where a,b∈Ch. From the above two inequalities, we can get that
(3.12)ϕ(xh,Thxh)≤21-κ〈xh-c,Jxh-JThxh〉+μhMh1-κ,
where c∈Ch. It follows that Ch+1 is closed and convex. This completes Step 2.Step 3.
Show thatF(T)⊂Cn for each n≥1.
It is obvious thatF(T)⊂C=C1. Suppose that F(T)⊂Ch for some h∈ℕ. For any z∈F(T)⊂Ch, we see that
(3.13)ϕ(z,Thxh)≤(1+μh)ϕ(z,xh)+κϕ(xh,Thxh).
On the other hand, we obtain from (2.6) that
(3.14)ϕ(z,Thxh)=ϕ(z,xh)+ϕ(xh,Thxh)+2〈z-xh,Jxh-JThxh〉.
Combining (3.13) with (3.14), we arrive at
(3.15)ϕ(xh,Thxh)≤μh1-κϕ(z,xh)+21-κ〈xh-z,Jxh-JThxh〉≤μhMh1-κ+21-κ〈xh-z,Jxh-JThxh〉,
which implies that z∈Ch+1. This shows that F(T)⊂Ch+1. This completes Step 3.Step 4.
Show that the sequence{xn} is bounded.
In view ofxn=ΠCnx0, we see that
(3.16)〈xn-z,Jx0-Jxn〉≥0,∀z∈Cn.
In view of F(T)⊂Cn, we arrive at
(3.17)〈xn-w,Jx0-Jxn〉≥0,∀w∈F(T).
It follows from Lemma 2.8 that
(3.18)ϕ(xn,x0)=ϕ(ΠCnx0,x0)≤ϕ(ΠF(T)x0,x0)-ϕ(ΠF(T)x0,xn)≤ϕ(ΠF(T)x0,x0).
This implies that the sequence {ϕ(xn,x0)} is bounded. It follows from (2.5) that the sequence {xn} is also bounded. This completes Step 4.Step 5.
Show thatxn→x¯, where x¯=ΠF(T)x0, as n→∞.
Since{xn} is bounded and the space is reflexive, we may assume that xn→x¯ weakly. Since Cn is closed and convex, we see that x¯∈Cn. On the other hand, we see from the weakly lower semicontinuity of the norm that
(3.19)ϕ(x¯,x0)=‖x¯‖2-2〈x¯,Jx0〉+‖x0‖2≤liminf⁡n→∞(‖xn‖2-2〈xn,Jx0〉+‖x0‖2)=liminf⁡n→∞ϕ(xn,x0)≤limsup⁡n→∞ϕ(xn,x0)≤ϕ(x¯,x0),
which implies that ϕ(xn,x0)→ϕ(x¯,x0) as n→∞. Hence, ∥xn∥→∥x¯∥ as n→∞. In view of Kadec-Klee property of E, we see that xn→x¯ as n→∞.
Now, we are in a position to show thatx¯∈F(T). Notice that limn→∞∥xn+1-xn∥=0. On the other hand, we see from xn+1=ΠCn+1x0∈Cn+1⊂Cn that
(3.20)ϕ(xn,Tnxn)≤21-κ〈xn-xn+1,Jxn-JTnxn〉+μnMn1-κ,
from which it follows that ϕ(xn,Tnxn)→0 as n→∞. In view of Lemma 2.6, we arrive at
(3.21)lim⁡n→∞‖Tnxn-xn‖=0.
Note that xn→x¯ as n→∞ in view of
(3.22)‖Tnxn-x¯‖≤‖Tnxn-xn‖+‖xn-x¯‖.
It follows from (3.21) that
(3.23)Tnxn⟶x¯asn⟶∞.
On the other hand, we have
(3.24)‖Tn+1xn-x¯‖≤‖Tn+1xn-Tnxn‖+‖Tnxn-x¯‖Tn+1xn-Tnxn‖Tnxn‖.
It follows from the uniformly asymptotic regularity of T and (3.23) that
(3.25)Tn+1xn⟶x¯asn⟶∞
that is, TTnxn→x¯. From the closedness of T, we obtain that x¯=Tx¯.
Finally, we show thatx¯=ΠF(T)x0 which completes the proof. Indeed, we obtain from xn=ΠCnx0 that
(3.26)〈xn-w,Jx0-Jxn〉≥0,∀w∈Cn.
In particular, we have
(3.27)〈xn-w′,Jx0-Jxn〉≥0,∀w′∈F(T).
Taking the limit as n→∞ in (3.27), we obtain that
(3.28)〈x¯-w′,Jx0-Jx¯〉≥0,∀w′∈F(T).
Hence, we obtain from Lemma 2.7 that x¯=ΠF(T)x0. This completes the proof.As applications of Theorem3.1, we have the following.Corollary 3.2.
LetC be a nonempty closed and convex subset of a uniformly convex and smooth Banach space  E. Let  T:C→C be a closed and asymptotically quasi -ϕ-snonexpansive mapping with a sequence  {μn}⊂[0,∞) such that  μn→0 as  n→∞. Assume that  T is uniformly asymptotically regular on  C and  F(T) is nonempty and bounded. Let  {xn} be a sequence generated in the following manner:
(3.29)x0∈Echosen  arbitrarily,C1=C,x1=ΠC1x0,Cn+1={u∈Cn:ϕ(xn,Tnxn‖(I-T)x-(I-T)y‖2Tn∈)≤2〈xn-u,Jxn-JTnxn‖Tx-Ty‖2〉+μnMn},xn+1=ΠCn+1x0,∀n≥0,
where M={ϕ(p,xn):p∈F(T)} then the sequence {xn} converges strongly to x¯=ΠF(T)x0.Proof.
Puttingκ=0 in Theorem 3.1, we can conclude the desired conclusion easily.Next, we give two theorems in the framework of real Hilbert spaces.Theorem 3.3.
LetC a nonempty closed and convex subset of a real Hilbert space  H. Let  T:C→C be a closed and asymptotically strict quasipseudocontraction with a sequence  {μn}⊂[0,∞) such that  μn→0 as  n→∞. Assume that  T is uniformly asymptotically regular on  C and  F(T) is nonempty and bounded. Let  {xn} be a sequence generated by the following manner:
(3.30)x0∈Hchosen  arbitrarily,C1=C,x1=PC1x0,Cn+1={u∈Cn:‖xn-Tnxn‖2≤21-κ〈xn-u,xn-Tnxn‖Tx-Ty‖2‖Tx-Ty‖2〉+μnMn1-κ},xn+1=PCn+1x0,∀n≥0,
where Mn=sup{∥p-xn∥2:p∈F(T)} then the sequence {xn} converges strongly to x¯=PF(T)x0.Theorem 3.4.
LetC be a nonempty closed and convex subset of a real Hilbert space  H. Let  T:C→C be a closed and asymptotically quasinonexpansive mapping with a sequence  {μn}⊂[0,∞) such that  μn→0 as  n→∞. Assume that  T is uniformly asymptotically regular on  C and  F(T) is nonempty and bounded. Let  {xn} be a sequence generated by the following manner:
(3.31)x0∈Hchosen  arbitrarily,C1=C,x1=PC1x0,Cn+1={u∈Cn:‖xn-Tnxn‖2≤2〈xn-u,xn-Tnxn‖Tx-Ty‖2〉+μnMn},xn+1=PCn+1x0,∀n≥0,
where Mn={∥p-xn∥2:p∈F(T)} then the sequence {xn} converges strongly to x¯=PF(T)x0.



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*Source: 142626-2011-06-21.xml*