# Strong Convergence Theorems for the Generalized Split Common Fixed Point Problem

Authors: Cuijie Zhang
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We introduce the generalized split common fixed point problem (GSCFPP) and show that the GSCFPP for nonexpansive operators is equivalent to the common fixed point problem. Moreover, we introduce a new iterative algorithm for finding a solution of the GSCFPP and obtain some strong convergence theorems under suitable assumptions.


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## Body

## 1. Introduction

LetH1 and H2 be real Hilbert spaces and let A:H1→H2 be a bounded linear operator. Given intergers p,r≥1, let us recall that the multiple-set split feasibility problem (MSSFP) was recently introduced [1] and is to find a point:(1.1)x*∈⋂i=1pCi,Ax*∈⋂j=1rQj,
where {Ci}i=1p and {Qj}j=1r are nonempty closed convex subsets of H1 and H2, respectively. If p=r=1, the MSSFP (1.1) becomes the so-called split feasibility problem (SFP) [2] which is to find a point:(1.2)x*∈C,Ax*∈Q,
where C and Q are nonempty closed convex subsets of H1 and H2, respectively. Recently, the SFP (1.2) and MSSFP (1.1) have been investigated by many researchers; see, [3–10].Since every closed convex subset in a Hilbert space is looked as the fixed point set of its associating projection, the MSSFP (1.1) becomes a special case of the split common fixed point problem (SCFPP), which is to find a point:(1.3)x*∈⋂i=1pFix⁡(Ui),Ax*∈⋂j=1rFix⁡(Tj),
where Ui:H1→H1(i=1,2,…,p) and Tj:H2→H2(j=1,2,…,r) are nonlinear operators. If p=r=1, the problem (1.3) reduces to the so-called two-set SCFPP, which is to find a point:(1.4)x*∈Fix⁡(U),Ax*∈Fix⁡(T).Censor and Segal in [11] firstly introduced the concept of SCFPP in finite-dimensional Hilbert spaces and considered the following iterative algorithm for the two-set SCFPP (1.4) for Class-ℑ operators:(1.5)xn+1=U(xn-γA*(I-T)Axn),n≥0,
where x0∈H1,  0<γ<2/∥A∥2 and I is the identity operator. They proved the convergence of the algorithm (1.5) to a solution of problem (1.4). Moreover, they introduced a parallel iterative algorithm, which converges to a solution of the SCFPP (1.3). However, the parallel iterative algorithm does not include the algorithm (1.5) as a special case.Very recently, Wang and Xu in [12] considered the SCFPP (1.3) for Class-ℑ operators and introduced the following iterative algorithm for solving the SCFPP (1.3):(1.6)xn+1=U[n](xn-γA*(I-T[n])Axn),n≥0.
Under some mild conditions, they proved some weak and strong convergence theorems. Their iterative algorithm (1.6) includes Censor and Segal’s algorithm (1.5) as a special case for the two-set SCFPP (1.4). Moreover, they prove that the SCFPP (1.3) for the Class-ℑ operators is equivalent to a common fixed point problem. This is also a classical method. Many problems eventually converted to a common fixed point problem; see [13–15].Motivated and inspired by the aforementioned research works, we introduce a generalized split common fixed point problem (GSCFPP) which is to find a point:(1.7)x*∈⋂i=1∞Fix⁡(Ui),Ax*∈⋂j=1∞Fix⁡(Tj).
Then, we show that the GSCFPP (1.7) for nonexpansive operators is equivalent to the following common fixed point problem:(1.8)x*∈⋂i=1∞Fix⁡(Ui),x*∈⋂j=1∞Fix⁡(Vj),
where Vj=I-γA*(I-Tj)A(0<γ≤1/∥A∥2) for every j∈ℕ. Moreover, we give a new iterative algorithm for solving the GSCFPP (1.7) for nonexpansive operators and obtain some strong convergence theorems.


## 2. Preliminaries

Throughout this paper, we writexn⇀x and xn→x to indicate that {xn} converges weakly to x and converges strongly to x, respectively.An operatorT:H→H is said to be nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for all x,y∈H. The set of fixed points of T is denoted by F(T). It is known that F(T) is closed and convex. An operator f:H→H is called contraction if there exists a constant ρ∈[0,1) such that ∥f(x)-f(y)∥≤ρ∥x-y∥ for all x,y∈H. Let C be a nonempty closed convex subset of H. For each x∈H, there exists a unique nearest point in C, denoted by PCx, such that ∥x-PCx∥≤∥x-y∥ for every y∈C. PC is called a metric projection of H onto C. It is known that for each x∈H,(2.1)〈x-PCx,y-PCx〉≤0
for all y∈C.Let{Tn} be a sequence of operators of H into itself. The set of common fixed points of {Tn} is denoted by F({Tn}), that is, F({Tn})=⋂n=1∞F(Tn). A sequence {Tn} is said to be strongly nonexpansive if each {Tn} is nonexpansive and(2.2)xn-yn-(Tnxn-Tnyn)⟶0
whenever {xn} and {yn} are sequences in C such that {xn-yn} is bounded and ∥xn-yn∥-∥Tnxn-Tnyn∥→0; see [16, 17]. A sequence {zn} in H is said to be an approximate fixed point sequence of {Tn} if zn-Tnzn→0. The set of all bounded approximate fixed point sequences of {Tn} is denoted by F̃({Tn}); see [16, 17]. We know that if {Tn} has a common fixed point, then F̃({Tn}) is nonempty; that is, every bounded sequence in the common fixed point set is an approximate fixed point sequence. A sequence {Tn} with a common fixed point is said to satisfy the condition (Z) if every weak cluster point of {xn} is a common fixed point whenever {xn}∈F̃({Tn}). A sequence {Tn} of nonexpansive mappings of H into itself is said to satisfy the condition (R) if(2.3)lim⁡n→∞sup⁡y∈D‖Tn+1y-Tny‖=0
for every nonempty bounded subset D of H; see [18].In order to prove our main results, we collect the following lemmas in this section.Lemma 2.1 (see [16]).
LetC be a nonempty subset of a Hilbert space  H. Let  {Tn} be a sequence of nonexpansive mappings of  C into  H. Let  {λn} be a sequence in  [0,1] such that  liminfn→∞λn>0. Let  {Un} be a sequence of mappings of  C into  H defined by  Un=λnI+(1-λn)Tn for  n∈ℕ, where  I is the identity mapping on  C. Then  {Un} is a strongly nonexpansive sequence.Lemma 2.2 (see [16]).
LetH be a Hilbert space,  C a nonempty subset of  H, and  {Sn} and  {Tn} sequences of nonexpansive self-mappings of  C. Suppose that  {Sn} or  {Tn} is a strongly nonexpansive sequence and  F̃({Sn})∩F̃({Tn}) is nonempty. Then  F̃({Sn})∩F̃({Tn})=F̃({SnTn}).Lemma 2.3 (see [17]).
LetH be a Hilbert space, and  C a nonempty subset of  H. Both  {Sn} and  {Tn} satisfy the condition  (R) and  {Tny:n∈ℕ,y∈D} is bounded for any bounded subset  D of  C. Then  {SnTn} satisfies the condition  (R).Lemma 2.4 (see [19]).
Let{xn} and  {yn} be bounded sequences in a Banach space  X and let  {βn} be a sequence in  [0,1] with  0<liminfn→∞βn≤limsupn→∞βn<1. Suppose  xn+1=(1-βn)yn+βnxnfor all integers  n≥0 and
(2.4)limsup⁡n→∞(‖yn+1-yn‖-‖xn+1-xn‖)≤0.
Then limn→∞∥yn-xn∥=0.Lemma 2.5 (see [20]).
Assume that{an} is a sequence of nonnegative real numbers such that
(2.5)an+1≤(1-γn)an+δn,n≥0,
where {γn} is a sequence in (0,1) and {δn} is a sequence such that(i)
∑n=1∞γn=∞,(ii)
limsupn→∞δn/γn≤0 or ∑n=1∞|δn|<∞. Then limn→∞an=0.


## 3. Main Results

Now we state and prove our main results of this paper.Lemma 3.1.
LetA:H1→H2 be a given bounded linear operator and let Tn:H2→H2 be a sequence of nonexpansive operators. Assume
(3.1)A-1(Fix⁡({Tn}))={x∈H1:Ax∈Fix⁡({Tn})}≠∅.
For each constant γ>0, Vn is defined by the following:
(3.2)Vn=I-γA*(I-Tn)A.
Then Fix({Vn})=A-1(Fix({Tn})). Moreover, for 0<γ≤1/∥A∥2, Vn is nonexpansive on H1 for n∈ℕ.Proof.
Since the inclusionA-1(Fix({Tn}))⊆Fix({Vn}) is evident, now we only need to show the converse inclusion. If z∈Fix({Vn}), then we have A*(I-Tn)Az=0. Since A-1(Fix({Tn}))≠∅, we take an arbitrary p∈A-1(Fix({Tn})). Hence
(3.3)∥Az-TnAz∥2=〈Az-TnAz,Az-TnAz〉=〈Az-TnAz,Az-Ap+Ap-TnAz〉=〈A*(I-Tn)Az,z-p〉+〈Az-TnAz,Ap-TnAz〉=-12‖Az-Ap‖2+12‖Az-TnAz‖2+12‖Ap-TnAz‖2≤12‖Az-TnAz‖2.
It follows that (1/2)∥Az-TnAz∥2≤0, then Az=TnAz for every n∈ℕ, hence z∈A-1(Fix({Tn})). Next we turn to show that Vn is a nonexpansive operator for n∈ℕ. Since Tn is nonexpansive, we have
(3.4)‖(I-Tn)Ax-(I-Tn)Ay‖2=‖Ax-Ay‖2+‖TnAx-TnAy‖2-2〈Ax-Ay,TnAx-TnAy〉≤2‖Ax-Ay‖2-2〈Ax-Ay,TnAx-TnAy〉≤2〈Ax-Ay,Ax-Ay-(TnAx-TnAy)〉.
Hence
(3.5)‖Vnx-Vny‖2=‖(I-γA*(I-Tn)A)x-(I-γA*(I-Tn)A)y‖2=‖x-y‖2+γ2‖A‖2‖(I-Tn)Ax-(I-Tn)Ay‖2-2γ〈Ax-Ay,(I-Tn)Ax-(I-Tn)Ay〉≤‖x-y‖2+γ(γ‖A‖2-1)‖(I-Tn)Ax-(I-Tn)Ay‖2.
For 0<γ≤1/∥A∥2, we can immediately obtain that Vn is a nonexpansive operator for every n∈ℕ.From Lemma3.1, we can obtain that the solution set of GSCFPP (1.7) is identical to the solution set of problem (1.8).Theorem 3.2.
Let{Un} and  {Vn} be sequences of nonexpansive operators on Hilbert space  H1. Both  {Un} and  {Vn} satisfy the conditions  (R) and  (Z). Let  f:H1→H1 be a contraction with coefficient  ρ∈[0,1). Suppose  Ω=Fix(Un)⋂Fix(Vn)≠∅. Take an initial guess  x1∈H1 and define a sequence  {xn} by the following algorithm:
(3.6)yn=λnxn+(1-λn)Vnxn,xn+1=αnf(xn)+βnxn+γnUnyn,
where {αn}, {βn}, {γn}, and {λn} are sequences in [0,1]. If the following conditions are satisfied:(i)
αn+βn+γn=1,foralln≥1;(ii)
limn→∞αn=0  and   Σn=1∞αn=∞;(iii)
0<liminfn→∞βn≤limsupn→∞βn<1;(iv)
0<liminfn→∞λn≤limsupn→∞λn<1;(v)
limn→∞|λn+1-λn|=0, then {xn} converges strongly to w∈Ω where w=PΩf(w).Proof.
We proceed with the following steps.
Step 1.
First show that there existsw∈Ω such that w=PΩf(w).
In fact, sincef is a contraction with coefficient ρ, we have
(3.7)‖PΩf(x)-PΩf(y)‖≤‖f(x)-f(y)‖≤ρ‖x-y‖
for every x, y. Hence PΩf is also a contraction. Therefore, there exists a unique w∈Ω such that w=PΩf(w).Step 2.
Now we show that{xn} is bounded.
Letp∈Ω, then p∈Fix({Un}) and p∈Fix({Vn}). Hence
(3.8)∥Unyn-p∥≤∥yn-p∥≤λn∥xn-p∥+(1-λn)∥Vnxn-p∥≤∥xn-p∥.
Then
(3.9)‖xn+1-p‖≤αn‖f(xn)-p‖+βn‖xn-p‖+γn‖Unyn-p‖≤αnρ‖xn-p‖+αn‖f(p)-p‖+βn‖xn-p‖+γn‖xn-p‖≤(1-αn(1-ρ))‖xn-p‖+αn(1-ρ)11-ρ‖f(p)-p‖≤max⁡{‖xn-p‖,11-ρ‖f(p)-p‖}.
By induction on n,
(3.10)‖Vnxn-p‖≤‖xn-p‖≤max⁡{‖x1-p‖,11-ρ‖f(p)-p‖}
for every n∈ℕ. This shows that {xn} and {Vnxn} are bounded, and hence, {Unyn}, {yn}, and {f(xn)} are also bounded.Step 3.
We claim thatF̃({An})=F̃({Vn}) and F̃({UnAn})=F̃({Un})∩F̃({Vn}), where An=λnI+(1-λn)Vn.
We first show the former equality. Let{zn} be a bounded sequence in H1. If {zn}∈F̃({Vn}), then
(3.11)‖Anzn-zn‖=‖λnzn+(1-λn)Vnzn-zn‖=(1-λn)‖Vnzn-zn‖⟶0.
Hence {zn}∈F̃({An}). On the other hand, if {zn}∈F̃({An}), combining (3.11) and limsupn→∞λn<1, we obtain that ∥Vnzn-zn∥→0. Hence {zn}∈F̃({Vn}). Therefore, F̃({An})=F̃({Vn}).
Next, we show the latter equality. Using Lemma2.1, we know that {An} is a strongly nonexpansive sequence. Thus, since F̃({Un})∩F̃({An})=F̃({Un})∩F̃({Vn})≠∅, from Lemma 2.2 we have
(3.12)F̃({UnAn})=F̃({Un})∩F̃({An})=F̃({Un})∩F̃({Vn}).Step 4.
{Sn} satisfies the condition (R), where Sn=UnAn.
LetD be a nonempty bounded subset of H1. From the definition of {An}, we have, for all y∈D,
(3.13)‖An+1y-Any‖=‖λn+1y+(1-λn+1)Vn+1y-λny-(1-λn)Vny‖≤|λn+1-λn|‖y‖+‖Vn+1y-Vny‖+‖λn+1Vn+1y-λnVny‖≤|λn+1-λn|‖y‖+‖Vn+1y-Vny‖+‖λn+1Vn+1y-λnVn+1y‖+‖λnVn+1y-λnVny‖=|λn+1-λn|‖y‖+‖Vn+1y-Vny‖+|λn+1-λn|‖Vn+1y‖+λn‖Vn+1y-Vny‖=|λn+1-λn|(‖y‖+‖Vn+1y‖)+(1+λn)‖Vn+1y-Vny‖.
It follows that
(3.14)sup⁡y∈D‖An+1y-Any‖≤|λn+1-λn|sup⁡y∈D(‖y‖+‖Vn+1y‖)+(1+λn)sup⁡y∈D‖Vn+1y-Vny‖.
Since {Vn} satisfies the condition (R) and limn→∞|λn+1-λn|=0, we have
(3.15)lim⁡n→∞sup⁡y∈D‖An+1y-Any‖=0,
that is, {An} satisfies the condition (R). Since {Any:n∈ℕ,y∈D} is bounded for any bounded subset D of H1, by using Lemma 2.3, we have that {VnAn} satisfies the condition (R), that is, {Sn} satisfies the condition (R).Step 5.
We show∥xn+1-xn∥→0.
We can write (3.6) as xn+1=βnxn+(1-βn)zn where zn=(αnf(xn)+γnSnxn)/1-βn. It follows that
(3.16)zn+1-zn=αn+1f(xn+1)+γn+1Sn+1xn+11-βn+1-αnf(xn)+γnSnxn1-βn=αn+11-βn+1(f(xn+1)-f(xn))+(αn+11-βn+1-αn1-βn)f(xn)+γn+11-βn+1(Sn+1xn+1-Snxn)+(γn+11-βn+1-γn1-βn)Snxn.
From Step 2, we may assume that {xn}⊂D′, where D′ is a bounded set of H1. Then from (3.16), we obtain
(3.17)∥zn+1-zn∥≤|αn+11-βn+1-αn1-βn|(‖f(xn)‖+‖Snxn‖)+αn+11-βn+1ρ‖xn+1-xn‖+γn+11-βn+1‖Sn+1xn+1-Snxn+1‖+γn+11-βn+1‖Snxn+1-Snxn‖≤|αn+11-βn+1-αn1-βn|(‖f(xn)‖+‖Snxn‖)+[1-αn+11-βn+1(1-ρ)]‖xn+1-xn‖+γn+11-βn+1sup⁡y∈D′‖Sn+1y-Sny‖.
It follows that
(3.18)‖zn+1-zn‖-‖xn+1-xn‖≤|αn+11-βn+1-αn1-βn|(‖f(xn)‖+‖Snxn‖)+γn+11-βn+1sup⁡y∈D′‖Sn+1y-Sny‖-αn+11-βn+1(1-ρ)‖xn+1-xn‖.
Since {Sn} satisfies the condition (R), combining αn→0 as n→∞, we have
(3.19)limsup⁡n→∞(‖zn+1-zn‖-‖xn+1-xn‖)≤0.
Hence by Lemma 2.4, we get ∥zn-xn∥→0 as n→∞. Consequently,
(3.20)lim⁡n→∞‖xn+1-xn‖=lim⁡n→∞(1-βn)‖zn-xn‖=0.Step 6.
We claim that{xn}∈F̃({Un})∩F̃({Vn}).
From (3.6), we have
(3.21)‖Snxn-xn‖≤‖Snxn-xn+1‖+‖xn+1-xn‖=‖Snxn-αnf(xn)-βnxn-γnSnxn‖+‖xn+1-xn‖≤αn‖Snxn-f(xn)‖+βn‖Snxn-xn‖+‖xn+1-xn‖,
and hence
(3.22)(1-βn)‖Snxn-xn‖≤αn‖Snxn-f(xn)‖+‖xn+1-xn‖.
Since ∥xn+1-xn∥→0, αn→0 and limsupn→∞βn<1, we derive
(3.23)‖Snxn-xn‖⟶0.
Thus (3.23) and Steps 2 and 3 imply that
(3.24){xn}∈F̃({Sn})=F̃({Un})∩F̃({Vn}).Step 7.
Showlimsupn→∞〈f(w)-w,xn-w〉≤0, where w=PΩf(w).
Since{xn} is bounded, there exist a point v∈H1 and a subsequence {xni} of {xn} such that
(3.25)limsup⁡n→∞〈f(w)-w,xn-w〉=lim⁡i→∞〈f(w)-w,xni-w〉
and xni⇀v. Since {Un} and {Vn} satisfy the condition (Z), from Step 6, we have v∈F({Un})∩F({Vn}). Using (2.1), we get
(3.26)limsup⁡n→∞〈f(w)-w,xn-w〉=lim⁡i→∞〈f(w)-w,xni-w〉=〈f(w)-w,v-w〉≤0.Step 8.
Showxn→w=PΩf(w).
Sincew∈Ω, using (3.8), we have
(3.27)‖xn+1-w‖2=〈αn(f(xn)-w)+βn(xn-w)+γn(Unyn-w),xn+1-w〉≤αn〈f(xn)-f(w),xn+1-w〉+αn〈f(w)-w,xn+1-w〉+βn‖xn-w‖⋅‖xn+1-w‖+γn‖yn-w‖⋅‖xn+1-w‖≤12αnρ(‖xn-w‖2+‖xn+1-w‖2)+αn〈f(w)-w,xn+1-w〉+12βn(‖xn-w‖2+‖xn+1-w‖2)+12γn(‖xn-w‖2+‖xn+1-w‖2)≤12[1-αn(1-ρ)]‖xn-w‖2+12‖xn+1-w‖2+αn〈f(w)-w,xn+1-w〉,
which implies that
(3.28)‖xn+1-w‖2≤[1-αn(1-ρ)]‖xn-w‖2+2αn(1-ρ)11-ρ〈f(w)-w,xn+1-w〉,
for every n∈ℕ. Consequently, according to Step 7, ρ∈[0,1), and Lemma 2.5, we deduce that {xn} converges strongly to w=PΩ(w). This completes the proof.Combining Lemma3.1 and Theorem 3.2, we can obtain the following strong convergence theorem for solving the GSCFPP (1.7).Theorem 3.3.
Let{Un} and  {Tn} be sequences of nonexpansive operators on Hilbert space  H1 and  H2, respectively. Both  {Un} and  {Tn} satisfy the conditions  (R) and  (Z). Let  f:H1→H1 be a contraction with coefficient  ρ∈[0,1). Suppose that the solution set  Ω of GSCFPP (1.7) is nonempty. Take an initial guess  x1∈H1 and define a sequence  {xn} by the following algorithm:
(3.29)yn=xn-γ(1-λn)A*(I-Tn)Axn,xn+1=αnf(xn)+βnxn+γnUnyn,
where γ∈(0,1/∥A∥2), and {αn}, {βn}, {γn}, {λn} are sequences in [0,1]. If the following conditions are satisfied:(i)
αn+βn+γn=1,foralln≥1;(ii)
limn→∞αn=0  and  Σn=1∞αn=∞;(iii)
0<liminfn→∞βn≤limsupn→∞βn<1;(iv)
0<liminfn→∞λn≤limsupn→∞λn<1;(v)
limn→∞|λn+1-λn|=0, then {xn} converges strongly to w∈Ω where w=PΩf(w).Proof.
SetVn=I-γA*(I-Tn)A. By Lemma 3.1, Vn is a nonexpansive operator for every n∈ℕ. We can rewrite (3.29) as
(3.30)yn=λnxn+(1-λn)Vnxn,xn+1=αnf(xn)+βnxn+γnUnyn.
We only need to prove that{Vn} satisfies the conditions (R) and (Z). Assume that D is a nonempty bounded subset of H1. For every y∈D, we have
(3.31)‖(I-γA*(I-Tn+1)A)y-(I-γA*(I-Tn)A)y‖≤γ‖A*(I-Tn+1)Ay-A*(I-Tn)Ay‖≤γ‖A‖‖Tn+1(Ay)-Tn(Ay)‖.
Since {Tn} satisfies the condition (R), and D′={Ay:y∈D} is bounded, it follows from (3.31) that
(3.32)sup⁡y∈D‖(I-γA*(I-Tn+1)A)y-(I-γA*(I-Tn)A)y‖≤γ‖A‖sup⁡y∈D‖Tn+1(Ay)-Tn(Ay)‖=γ‖A‖sup⁡z∈D′‖Tn+1z-Tnz‖⟶0.
Therefore, {Vn} satisfies the condition (R).
Assume thatxn⇀z and xn-Vnxn→0; we next show that Vnz=z. By using xn-Vnxn→0, we have A*(I-Tn)Axn→0. Since A-1(Fix({Tn}))≠∅, we choose an arbitrary point p∈A-1(Fix({Tn})); then for every n∈ℕ,
(3.33)∥Axn-TnAxn∥2=〈Axn-TnAxn,Axn-Ap+Ap-TnAxn〉=〈A*(I-Tn)Axn,xn-p〉+〈Axn-TnAxn,Ap-TnAxn〉=〈A*(I-Tn)Axn,xn-p〉-12‖Axn-Ap‖2+12‖Axn-TnAxn‖2+12‖Ap-TnAxn‖2≤〈A*(I-Tn)Axn,xn-p〉+12‖Axn-TnAxn‖2.
Hence
(3.34)12‖Axn-TnAxn‖2≤〈A*(I-Tn)Axn,xn-p〉⟶0.
Then we get Axn∈F̃({Tn}). Since {Tn} satisfies the condition (Z) and Axn⇀Az, we have Az∈F({Tn}). From Lemma 3.1, we have z∈Fix({Vn}).LetT:H→H be a nonexpansive mapping with a fixed point, and define Tn=T for all n∈ℕ. Then {Tn} satisfies the conditions (R) and (Z). Thus, one obtains the algorithm for solving the two-set SCFPP (1.4).Corollary 3.4.
LetU and T be nonexpansive operators on Hilbert space H1 and H2, respectively. Let f:H1→H1 be a contraction with coefficient ρ∈[0,1). Suppose that the solution set Ω of SCFPP (1.4) is nonempty. Take an initial guess x1∈H1 and define a sequence {xn} by the following algorithm in (3.29),  where γ∈(0,1/∥A∥2), and {αn}, {βn}, {γn}, {λn} are sequences in [0,1]. If the following conditions are satisfied:(i)
αn+βn+γn=1,foralln≥1;(ii)
limn→∞αn=0 and Σn=1∞αn=∞;(iii)
0<liminfn→∞βn≤limsupn→∞βn<1;(iv)
0<liminfn→∞λn≤limsupn→∞λn<1;(v)
limn→∞|λn+1-λn|=0. Then {xn} converges strongly to w∈Ω where w=PΩf(w).Remark 3.5.
By adding more operators to the families{Un} and {Tn} by setting Ui=I for i≥p+1 and Tj=I for j≥r+1, the SCFPP (1.3) can be viewed as a special case of the GSCFPP (1.7).



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*Source: 575014-2012-05-08.xml*