# Cyclic Iterative Method for Strictly Pseudononspreading in Hilbert Space

Authors: Bin-Chao Deng; Tong Chen; Zhi-Fang Li
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

Let{Ti}i=1N be N strictly pseudononspreading mappings defined on closed convex subset C of a real Hilbert space H. Consider the problem of finding a common fixed point of these mappings and introduce cyclic algorithms based on general viscosity iteration method for solving this problem. We will prove the strong convergence of these cyclic algorithm. Moreover, the common fixed point is the solution of the variational inequality 〈(γf-μB)x*,v-x*〉≤0,∀v∈⋂i=1NFix(Ti).


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## Body

## 1. Introduction

Throughout this paper, we always assume thatC is a nonempty, closed, and convex subset of a real Hilbert space H. Let B:C→H be a nonlinear mapping. Recall the following definitions.Definition 1.1.
B is said to be(i)
monotone if(1.1)〈Bx-By,x-y〉≥0,∀x,y∈C,(ii)
strongly monotone if there exists a constantα>0 such that
(1.2)〈Bx-By,x-y〉≥α‖x-y‖2,∀x,y∈C,
for such a case, B is said to be α-strongly-monotone,(iii)
inverse-strongly monotone if there exists a constantα>0 such that
(1.3)〈Bx-By,x-y〉≥α‖Bx-By‖2,∀x,y∈C,
for such a case, B is said to be α-inverse-strongly monotone,(iv)
k-Lipschitz continuous if there exists a constant k≥0 such that
(1.4)‖Bx-By‖≤k‖x-y‖∀x,y∈C.Remark 1.2.
LetF=μB-γf,where B is a k-Lipschitz and η-strongly monotone operator on H with  k>0  and f is a Lipschitz mapping on H with coefficient L>0,0<γ≤μη/L.It is a simple matter to see that the operator  F  is (μη-γL)-strongly monotone over  H,  that is:
(1.5)〈Fx-Fy,x-y〉≥(μη-γL)‖x-y‖2,∀(x,y)∈H×H.
Following the terminology of Browder-Petryshyn [1], we say that a mapping T:D(T)⊆H→H is(1)
k-strict pseudocontraction if there exists k∈[0,1) such that
(1.6)‖Tx-Ty‖2≤‖x-y‖2+k‖x-Tx-(y-Ty)‖2,∀x,y∈D(T),(2)
k-strictly pseudononspreading if there exists k∈[0,1) such that
(1.7)‖Tx-Ty‖2≤‖x-y‖2+k‖x-Tx-(y-Ty)‖2+2〈x-Tx,y-Ty〉,
for all x,y∈D(T),(3)
nonspreading in [2] if
(1.8)‖Tx-Ty‖2≤‖Tx-y‖2+‖Ty-x‖2,∀x,y∈C.
It is shown in [3] that (1.8) is equivalent to
(1.9)‖Tx-Ty‖2≤‖x-y‖2+2〈x-Tx,y-Ty〉,∀x,y∈C.Clearly every nonspreading mapping is0-strictly pseudononspreading. Iterative methods for strictly pseudononspreading mapping have been extensively investigated; see [2, 4–6].LetC be a closed convex subset of H, and let {Ti}i=1N be n  ki-strictly pseudocontractive mappings on C such that ⋂i=1N‍Fix(Ti)≠∅. Let x1∈C and {αn}n=1∞ be a sequence in (0,1). In [7], Acedo and Xu introduced an explicit iteration scheme called the followting cyclic algorithm for iterative approximation of common fixed points of {Ti}i=1N in Hilbert spaces. They define the sequence {xn} cyclically by
(1.10)x1=α0x0+(1-α0)T0x0;x2=α1x1+(1-α1)T1x1;⋮xN=αN-1xN-1+(1-αN-1)TN-1xN-1;xN+1=αNxN+(1-αN)T0xN;(1.11)⋮.
In a more compact form, they rewrite xn+1 as
(1.12)xN+1=αNxN+(1-αN)TNxn,
where TN=Ti, with i=n (mod N), 0≤i≤N-1. Using the cyclic algorithm (1.12), Acedo and Xu [7] show that this cyclic algorithm (1.12) is weakly convergent if the sequence {αn} of parameters is appropriately chosen.Motivated and inspired by Acedo and Xu [7], we consider the following cyclic algorithm for finding a common element of the set of solutions of ki-strictly pseudononspreading mappings {Ti}i=1N. The sequence {xn}i=1∞ generated from an arbitrary x1∈H as follows:
(1.13)x1=α0γf(x0)+(I-μα0B)Tω0x0;x2=α1γf(x1)+(I-μα1B)Tω1x1;⋮xN=αN-1γf(xN-1)+(I-μαN-1B)TωN-1xN-1;xN+1=αNγf(xN)+(I-μαNB)Tω0xN;(1.14)⋮.
Indeed, the algorithm above can be rewritten as
(1.15)xn+1=αnγf(xn)+(I-μαnB)Tω[n]xn,
where Tω[n]=(I-ω[n])I+ω[n]T[n], T[n]=TnmodN; namely, T[N] is one of T1,T2,…,TN circularly.


## 2.   Preliminaries

Throughout this paper, we writexn⇀x to indicate that the sequence {xn} converges weakly to x. xn→x implies that {xn} converges strongly to x. The following definitions and lemmas are useful for main results.Definition 2.1.
A mappingT is said to be demiclosed, if for any sequence {xn}which weakly converges to y, and if the sequence {Txn} strongly converges to z, then T(y)=z.Definition 2.2.
T
:
H
→
H is called demicontractive on H, if there exists a constant α<1 such that
(2.1)‖Tx-q‖2≤‖x-q‖2+α‖x-Tx‖2,∀(x,q)∈H×Fix(T).Remark 2.3.
Everyk-strictly pseudononspreading mapping with a nonempty fixed point set Fix(T) is demicontractive (see [8, 9]).Remark 2.4 (see [10]).
LetT be a α-demicontractive mapping on H with Fix(T)≠∅ and Tω=(1-ω)I+ωT for ω∈(0,∞):  (A1)
Tα-demicontractive is equivalent to
(2.2)〈x-Tx,x-q〉≥12(1-α)‖x-Tx‖2,∀(x,q)∈H×Fix(T),(A2)
Fix(T)=Fix(Tω) if ω≠0.Remark 2.5.
According toI-Tω=ω(I-T) with T being a k-strictly pseudononspreading mapping, we obtain
(2.3)〈x-Tωx,x-q〉≥ω(1-k)2‖x-Tx‖2,∀(x,q)∈H×Fix(T).Proposition 2.6 (see [2]).
LetC be a nonempty closed convex subset of a real Hilbert space H, and let T:C→C be a k-strictly pseudononspreading mapping. If Fix(T)≠∅, then it is closed and convex.Proposition 2.7 (see [2]).
LetC be a nonempty closed convex subset of a real Hilbert space H, and let T:C→C be a k-strictly pseudononspreading mapping. Then (I-T) is demiclosed at 0.Lemma 2.8 (see [11]).
Let{𝒯n} be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence {𝒯nj}j≥0 of {𝒯n} which satisfies 𝒯nj<𝒯nj+1 for all j≥0. Also consider the sequence of integers {δ(n)}n≥n0 defined by
(2.4)δ(n)=max⁡{k≤n∣Tk<Tk+1}.
Then {δ(n)}n≥n0 is a nondecreasing sequence verifying limn→∞δ(n)=∞,∀n≥n0; it holds that 𝒯δ(n)<𝒯δ(n)+1 and we have
(2.5)Tn<Tδ(n)+1.Lemma 2.9.
LetH be a real Hilbert space. The following expressions hold: (i)
∥tx+(1-t)y∥2=t∥x∥2+(1-t)∥y∥2-t(1-t)∥x-y∥2,∀x,y∈H,∀t∈[0,1],(ii)
∥x+y∥2≤∥x∥2+2〈y,x+y〉,∀x,y∈H.Lemma 2.10 (see [6]).
LetC be a closed convex subset of a Hilbert space H, and let T:H→H be a k-strictly pseudononspreading mapping with a nonempty fixed point set. Let k≤ω<1 be fixed and define TωC→C by
(2.6)Tω(x)=(1-ω)(x)+ωT(x),∀x∈C.
Then Fix(Tω)=Fix(T).Lemma 2.11.
AssumeC is a closed convex subset of a Hilbert space H. (a)
Given an integerN≥1, assume, Ti:H→H is a ki-strictly pseudononspreading mapping for some ki∈[0,1), (i=1,2,…,N). Let {λi}i=1N be a positive sequence such that ∑i=1N‍λi=1. Suppose that {Ti}i=1N has a common fixed point and ⋂i=1N‍Fix(Ti)≠∅. Then,
(2.7)Fix(∑i=1N‍λiTi)=⋂i=1N‍Fix(Ti).(b)
AssumingTi:H→H is a ki-strictly pseudononspreading mapping for some ki∈[0,1), (i=1,2,…,N), let Tωi=(1-ωi)I+ωiTi, 1≤i≤N. If ⋂i=1N‍Fix(Ti)≠∅, then
(2.8)Fix(Tω1Tω2⋯TωN)=⋂i=1N‍Fix(Tωi).Proof.
To prove (a), we can assumeN=2. It suffices to prove that Fix(F)⊂Fix(T1)∩Fix(T2), where F=(1-λ)T1+λT2 with 0<λ<1. Let x∈Fix(F) and write V1=I-T1 and V2=I-T2.
From Lemma2.9 and taking z∈Fix(T1)∩Fix(T2) to deduce that
(2.9)‖z-x‖2=‖(1-λ)(z-T1x)+λ(z-T2x)‖2=(1-λ)‖z-T1x‖2+λ‖z-T2x‖2-λ(1-λ)‖T1x-T2x‖2≤(1-λ)(‖z-x‖2+k‖x-T1x‖2)+λ(‖z-x‖2+k‖x-T2x‖2)-λ(1-λ)‖T1x-T2x‖2=‖z-x‖2+k[(1-λ)‖V1x‖2+λ‖V2x‖2]-λ(1-λ)‖V1x-V2x‖2,
it follows that
(2.10)λ(1-λ)‖V1x-V2x‖2≤k[(1-λ)‖V1x‖2+λ‖V2x‖2].
Since (1-λ)V1x+λV2x=0, we obtain
(2.11)(1-λ)‖V1x‖2+λ‖V2x‖2=λ(1-λ)‖V1x-V2x‖2.
This together with (2.10) implies that
(2.12)(1-k)λ(1-λ)‖V1x-V2x‖2≤0.
Since 0<λ<1 and k<1, we get ∥V1x-V2x∥=0 which implies T1x=T2x which in turn implies that T1x=T2x=x since (1-λ)T1x+λT2x=x. Thus, x∈Fix(T1)∩Fix(T2).
By induction, we also claim thatFix(∑i=1N‍λiTi)=⋂i=1N‍Fix(Ti) with {λi}i=1N is a positive sequence such that ∑i=1N‍λi=1, (i=1,2,…,N).
To prove (b), we can assumeN=2. Set Tω1=(1-ω1)I+ω1T1 and Tω2=(1-ω2)I+ω1T2, 0<ki<ωi<1/2, i=1,2. Obviously
(2.13)Fix(Tω1)∩Fix(Tω2)⊂Fix(Tω1Tω2).
Now we prove
(2.14)Fix(Tω1)∩Fix(Tω2)⊃Fix(Tω1Tω2),
for all q∈Fix(Tω1Tω2) and Tω1Tω2q=q. If Tω2q=q, then Tω1q=q; the conclusion holds. From Lemma 2.10, we can know that Fix(Tω1)∩Fix(Tω2)=Fix(T1)∩Fix(T2)≠∅. Taking p∈Fix(Tω1)∩Fix(Tω2), then
(2.15)‖p-q‖2=‖p-Tω1Tω2q‖2=‖p-[(1-ω1)(Tω2q)+ω1T1Tω2q]‖2=‖(1-ω1)(p-Tω2q)+ω1(p-T1Tω2q)‖2=(1-ω1)‖p-Tω2q‖2+ω1‖p-T1Tω2q‖2-ω1(1-ω1)‖Tω2q-T1Tω2q‖2≤(1-ω1)‖p-Tω2q‖2-ω1(1-ω1)‖Tω2q-T1Tω2q‖2+ω1[‖p-Tω2q‖2+k1‖Tω2q-T1Tω2q‖2+2〈p-T1Tω2p,Tω2q-T1Tω2q〉]=(1-ω1)‖p-Tω2q‖2-ω1(1-ω1)‖Tω2q-T1Tω2q‖2+ω1[‖p-Tω2q‖2+k1‖Tω2q-T1Tω2q‖2]≤‖p-Tω2q‖2-ω1(1-ω1-k1)‖Tω2q-T1Tω2q‖2≤‖p-q‖2-ω1(1-ω1-k1)‖Tω2q-T1Tω2q‖2.
Since 0<k1<ω1<1/2, we obtain
(2.16)‖Tω2q-T1Tω2q‖2≤0.
Namely, Tω2q=T1Tω2q, that is:
(2.17)Tω2q∈Fix(T1)=Fix(Tω1),Tω2q=Tω1Tω2q.
By induction, we also claim that the Lemma2.11(b) holds.Lemma 2.12.
LetK be a closed convex subset of a real Hilbert space H, given x∈H and y∈K. Then y=PKx if and only if there holds the inequality
(2.18)〈x-y,y-z〉≥0,∀z∈K.


## 3. Cyclic Algorithm

In this section, we are concerned with the problem of finding a pointp such that
(3.1)p∈⋂i=1N‍Fix(Tωi)=⋂i=1N‍Fix(Ti),N≥1,
where Tωi=(1-ωi)I+ωiTi, {ωi}i=1N∈(0,1/2] and {Ti}i=1N are ki-strictly pseudononspreading mappings with ki∈[0,ωi), (i=1,2,…,N), defined on a closed convex subset C in Hilbert space H. Here Fix(Tωi)={q∈C:Tωiq=q} is the set of fixed points of Ti, 1≤i≤N.LetH be a real Hilbert space, and let B:H→H be η-strongly monotone and ρ-Lipschitzian on H with ρ>0, η>0. Let 0<μ<2η/ρ2, 0<γ<μ(η-(μρ2/2))/L=τ/L. Let N be a positive integer, and let   Ti:H→H be a ki-strictly pseudononspreading mapping for some ki∈[0,1), (i=1,2,…,N), such that ⋂i=1N‍Fix(Ti)≠∅. We consider the problem of finding p∈⋂i=1N‍Fix(Ti) such that
(3.2)〈(γf-μB)p,v-p〉≤0,∀v∈⋂i=1N‍Fix(Ti).Since⋂i=1N‍Fix(Ti) is a nonempty closed convex subset of H, VI (3.2) has a unique solution. The variational inequality has been extensively studied in literature; see, for example, [12–16].Remark 3.1.
LetH be a real Hilbert space. Let B be a ρ-Lipschitzian and η-strongly monotone operator on H with ρ>0,η>0. Leting 0<μ<2η/ρ2 and  leting S=(I-tμB) and μ(η-(μρ2/2))=τ, then for ∀x,y∈H and t∈(0,min{1,1/τ}), S is a contraction.Proof.
Consider(3.3)‖Sx-Sy‖2=‖(I-tμB)x-(I-tμB)y‖2=〈(I-tμB)x-(I-tμB)y,(I-tμB)x-(I-tμB)y〉=‖x-y‖2+t2μ2‖Bx-By‖2-2tμ〈x-y,Bx-By〉≤∥x-y∥2+t2μ2ρ2‖x-y‖2-2tμη‖x-y‖2≤[1-2tμ(η-μρ22)]‖x-y‖2=(1-2tτ)‖x-y‖2≤(1-tτ)2‖x-y‖2.
It follows that
(3.4)‖Sx-Sy‖≤(1-tτ)‖x-y‖.
So S is a contraction.Next, we consider the cyclic algorithm (1.15), respectively, for solving the variational inequality over the set of the common fixed points of finite strictly pseudononspreading mappings.Lemma 3.2.
Assume that{xn} is defined by (1.15); if p is solution of (3.2) with T:H→H being strictly pseudononspreading mapping and demiclosed and {yn}⊂H is a bounded sequence such that ∥Tyn-yn∥→0, then
(3.5)liminf⁡n→∞〈(γf-μB)p,yn-p〉≤0.Proof.
By∥Tyn-yn∥→0 and T:H→H demi-closed, we know that any weak cluster point of {yn} belongs to ⋂i=1N‍Fix(Ti). Furthermore, we can also obtain that there exists y- and a subsequence {ynj} of {yn} such that ynj⇀y- as j→∞ (hence y-∈⋂i=1N‍Fix(Ti)) and
(3.6)liminf⁡n→∞〈(γf-μB)p,yn-p〉=lim⁡j→∞〈(γf-μB)p,ynj-p〉.
From (3.2), we can derive that (3.7)liminf⁡n→∞〈(γf-μB)p,yn-p〉=〈(γf-μB)p,y--p〉≤0.
It is the desired result. In addition, the variational inequality (3.7) can be written as
(3.8)〈(I-μB+γf)p-p,y--p〉≤0,y-∈⋂i=1N‍Fix(Ti).
So, by Lemma 2.12, it is equivalent to the fixed point equation
(3.9)P⋂i=1NFix(Ti)(I-μB+γf)p=p.Theorem 3.3.
LetC be a nonempty closed convex subset of H and for 1≤i≤N. Let Ti:H→H be ki-strictly pseudononspreading mappings for some ki∈[0,ωi), ωi∈(0,1/2), (i=1,2,…,N), and k=max{ki:1≤i≤N}. Let f be L-Lipschitz mapping on H with coefficient L>0, and let B:H→H be η-strongly monotone and ρ-Lipschitzian on H with ρ>0, η>0. Let {αn} being a sequence in (0,min{1,1/τ}) satisfying the following conditions:  (c1)
limn→∞αn=0,(c2)
∑n=0∞‍αn=∞.
Givenx0∈C, let {xn}n=1∞ be the sequence generated by the cyclic algorithm (1.15). Then {xn} converges strongly to the unique element p in ⋂i=1N‍Fix(Ti) verifying
(3.10)p=P⋂i=1NFix(Ti)(I-μB+γf)p,
which equivalently solves the variational inequality problem (3.2).Proof.
Take ap∈⋂i=1N‍Fix(Ti). Let Tωx=(1-ω)x+ωTx and 0<k<ω<1/2. Then ∀x,y∈C, we have
(3.11)∥Tωx-Tωy∥2=ω‖x-y‖2+(1-ω)‖Tx-Ty‖2-ω(1-ω)‖x-Tx-(y-Ty)‖2≤ω‖x-y‖2+(1-ω)[‖x-y‖2+k‖x-Tx-(y-Ty)‖2+2〈x-Tx,y-Ty〉]-ω(1-ω)‖x-Tx-(y-Ty)‖2=‖x-y‖2+2(1-ω)〈x-Tx,y-Ty〉-(1-ω)(ω-k)‖x-Tx-(y-Ty)‖2≤‖x-y‖2+2(1-ω)〈x-Tx,y-Ty〉=‖x-y‖2+2(1-ω)ω2〈x-Tωx,y-Tωy〉.
Fromp∈Fix(T) and (3.11), we also have
(3.12)‖Tωxn-p‖≤‖xn-p‖.
Using (1.15) and (3.12), we obtain
(3.13)‖xn+1-p‖=‖αnγ(f(xn)-f(p))+αn(γf(p)-p)+(I-μαnB)(Tω[n]xn-p)‖≤αnγ‖f(xn)-f(p)‖+αn‖γf(p)-p‖+(1-αnτ)‖xn-p‖,
which combined with (3.12) and ∥f(xn)-f(p)∥≤L∥xn-p∥ amounts to
(3.14)‖xn+1-p‖≤(1-αn(τ-γL))‖xn-p‖+αn‖γf(p)-p‖.
Putting M1=max{∥x0-p∥,∥γf(p)-p∥}, we clearly obtain ∥xn-p∥≤M1. Hence {xn} is bounded. We can also prove that the sequences {f(xn)} and {Tω[n]xn} are all bounded.
From (1.15) we obtain that
(3.15)xn+1-xn+αn(μBxn-γf(xn))=(I-μαnB)(Tω[n]xn-xn),
hence
(3.16)〈xn+1-xn+αn(μBxn-γf(xn)),xn-p〉=(1-αn)〈Tω[n]xn-xn,xn-p〉+αn〈(I-μB)(Tω[n]xn-xn),xn-p〉=(1-αn)〈Tω[n]xn-xn,xn-p〉+αn‖(I-μB)(Tω[n]xn-xn)‖‖xn-p‖=(1-αn)〈Tω[n]xn-xn,xn-p〉+αn(1-τ)‖Tω[n]xn-xn‖‖xn-p‖=(1-αn)〈Tω[n]xn-xn,xn-p〉+ω[n]αn(1-τ)‖T[n]xn-xn‖‖xn-p‖.
Moreover, by p∈⋂i=1N‍Fix(Ti) and using Remark 2.5, we obtain
(3.17)〈xn-Tω[n]xn,xn-p〉≥12ω[n](1-k[n])‖xn-T[n]xn‖2,
which combined with (3.16) entails
(3.18)〈xn+1-xn+αn(μB-γf)xn,xn-p〉≤-12ω[n](1-k[n])(1-αn)‖xn-T[n]xn‖2+ω[n]αn(1-τ)‖T[n]xn-xn‖‖xn-p‖,
or equivalently
(3.19)-〈xn-xn+1,xn-p〉≤-αn〈(μB-γf)xn,xn-p〉-12ω[n](1-k[n])(1-αn)‖xn-T[n]xn‖2+ω[n]αn(1-τ)‖T[n]xn-xn‖‖xn-p‖.
Furthermore, using the following classical equality
(3.20)〈u,v〉=12‖u‖2-12‖u-v‖2+12‖v‖2,∀u,v∈C,
and setting 𝒯n=1/2∥xn-p∥2, we have
(3.21)〈xn-xn+1,xn-p〉=Tn-Tn+1+12‖xn-xn+1‖2.
So (3.19) can be equivalently rewritten as
(3.22)Tn+1-Tn-12‖xn-xn+1‖2≤-αn〈(μB-γf)xn,xn-p〉-12ω[n](1-k[n])(1-αn)‖xn-T[n]xn‖2+ω[n]αn(1-τ)‖T[n]xn-xn‖‖xn-p‖.
Now using (3.15) again, we have
(3.23)‖xn+1-xn‖2=‖αn(γf(xn)-μBxn)+(I-μαnB)(Tω[n]xn-xn)‖2.
Since B:H→H is η-strongly monotone and k-Lipschitzian on H, hence it is a classical matter to see that
(3.24)‖xn+1-xn‖2≤2αn2‖γf(xn)-μBxn‖2+2(1-αnτ)2‖Tω[n]xn-xn‖2,
which by ∥Tω[n]xn-xn∥=ω[n]∥xn-T[n]xn∥ and (1-αnτ)2≤(1-αnτ) yields
(3.25)12‖xn+1-xn‖2≤αn2‖γf(xn)-μBxn‖2+(1-αnτ)ω[n]2‖xn-T[n]xn‖2.
Then from (3.22) and (3.25), we have
(3.26)Tn+1-Tn+ω[n](12(1-k[n])(1-αn)-ω[n](1-αnτ))‖xn-T[n]xn‖2≤αn(αn‖γf(xn)-μBxn‖2-〈(μB-γf)xn,xn-p〉+ω[n](1-τ)‖T[n]xn-xn‖‖xn-p‖).
The rest of the proof will be divided into two parts.
Case  1. Suppose that there exists n0 such that {𝒯n}n≥n0 is nonincreasing. In this situation, {𝒯n} is then convergent because it is also nonnegative (hence it is bounded from below), so that limn→∞(𝒯n+1-𝒯n)=0; hence, in light of (3.26) together with limn→∞αn=0 and the boundedness of {xn}, we obtain
(3.27)lim⁡n→∞‖xn-T[n]xn‖=0.
It also follows from (3.26) that
(3.28)Tn-Tn+1≥αn(-αn‖γf(xn)-μBxn‖2+〈(μB-γf)xn,xn-p〉+ω[n](1-τ)‖T[n]xn-xn‖‖xn-p‖‖γf(xn)-μBxn‖2).
Then, by ∑n=0∞‍αn=∞, we obviously deduce that
(3.29)liminf⁡n→∞αn(-αn‖γf(xn)-μBxn‖2+〈(μB-γf)xn,xn-p〉+ω[n](1-τ)‖T[n]xn-xn‖‖xn-p‖‖γf(xn)-μBxn‖2)≤0,
or equivalently (as αn∥γf(xn)-μBxn∥2→0)
(3.30)liminf⁡n→∞〈(μB-γf)xn,xn-p〉≤0.
Moreover, by Remark 1.2, we have
(3.31)2(μη-γL)Tn+〈(μB-γf)p,xn-p〉≤〈(μB-γf)xn,xn-p〉,
which by (3.30) entails
(3.32)liminf⁡n→∞〈2(μη-γL)Tn+(μB-γf)p,xn-p〉≤0,
hence, recalling that limn→∞𝒯n exists, we equivalently obtain
(3.33)2(μη-γL)lim⁡n→∞Tn+liminf⁡n→∞〈(μB-γf)p,xn-p〉≤0,
namely
(3.34)2(μη-γL)lim⁡n→∞Tn≤-liminf⁡n→∞〈(μB-γf)p,xn-p〉.
From (3.27) and Lemma 3.2, we obtain
(3.35)liminf⁡n→∞〈(μB-γf)p,xn-p〉≥0,
which yields limn→∞𝒯n=0, so that {xn} converges strongly to p.
Case  2.  Suppose there exists a subsequence {𝒯nk}k≥0 of {𝒯n}n≥0 such that 𝒯nk≤𝒯nk+1 for all k≥0. In this situation, we consider the sequence of indices {δ(n)} as defined in Lemma 2.8. It follows that 𝒯δ(n+1)-𝒯δ(n)>0, which by (3.26) amounts to
(3.36)ω[n](12(1-k[n])(1-αδ(n))-ω[n](1-αδ(n)τ))‖xδ(n)-T[n]xδ(n)‖2<αδ(n)(αδ(n)‖γf(xδ(n))-μBxδ(n)‖2-〈(μB-γf)xδ(n),xδ(n)-p〉+ω[n](1-τ)‖T[n]xδ(n)-xδ(n)‖‖xδ(n)-p‖).
By the boundedness of {xn} and limn→∞αn=0, we immediately obtain
(3.37)lim⁡n→∞‖xδ(n)-T[n]xδ(n)‖=0.
Using (1.15), we have
(3.38)∥xδ(n)+1-xδ(n)∥≤αδ(n)‖γf(xδ(n))-μBxδ(n)‖+(1-αδ(n)τ)‖Tω[n]xδ(n)-xδ(n)‖≤αδ(n)‖γf(xδ(n))-μBxδ(n)‖+ω[n](1-αδ(n)τ)‖T[n]xδ(n)-xδ(n)‖,
which together with (3.37) and limn→∞αn=0 yields
(3.39)lim⁡n→∞‖xδ(n)+1-xδ(n)‖=0.
Now by (3.36) we clearly have
(3.40)αδ(n)‖γf(xδ(n))-μBxδ(n)‖2+ω[n](1-τ)‖T[n]xδ(n)-xδ(n)‖‖xδ(n)-p‖≥〈(μB-γf)xδ(n),xδ(n)-p〉,
which in the light of (3.31) yields
(3.41)2(μη-γL)Tδ(n)+〈(μB-γf)p,xδ(n)-p〉≤αδ(n)‖γf(xδ(n))-μBxδ(n)‖2+ω[n](1-τ)‖T[n]xδ(n)-x[n]‖‖xδ(n)-p‖,
hence (as limn→∞αδ(n)∥γf(xδ(n))-μBxδ(n)∥2=0 and (3.37)) it follows that
(3.42)2(μη-γL)limsup⁡n→∞Tδ(n)≤-liminf⁡n→∞〈(μB-γf)p,xδ(n)-p〉.
From (3.37) and Lemma 3.2, we obtain
(3.43)lim⁡n→∞〈(μB-γf)p,xδ(n)-p〉≥0,
which by (3.42) yields limsupn→∞𝒯δ(n)=0, so that limn→∞𝒯δ(n)=0. Combining (3.39), we have limn→∞𝒯δ(n)+1=0. Then, recalling that 𝒯n<𝒯δ(n)+1 (by Lemma 2.8), we get limn→∞𝒯n=0, so that xn→p strongly.Takingki=0, we know that ki-strictly pseudononspreading mapping is nonspreading mapping and i=n (mod N), 0≤i≤N-1. According to the proof Theorem 3.3, we obtain the following corollary.Corollary 3.4.
LetC be a nonempty closed convex subset of H. Let Ti:C→C be nonspreading mappings and ωi∈(0,1/2), (i=1,2,…,N). Let f be L-Lipschitz mapping on H with coefficient L>0 and let B:H→H be η-strongly monotone and ρ-Lipschitzian on H with ρ>0, η>0. Let {αn} be a sequence in (0,min{1,1/τ}) satisfying the following conditions:      (c1)
limn→∞αn=0,(c2)
∑n=0∞‍αn=∞.
Givenx0∈C, let {xn}n=1∞ be the sequence generated by the cyclic algorithm (1.15). Then {xn} converges strongly to the unique element p in ⋂i=1N‍Fix(Ti) verifying
(3.44)p=P⋂i=1NFix(Ti)(I-μB+γf)p,
which equivalently solves the variational inequality problem (3.2).



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*Source: 435676-2012-07-18.xml*