# The Split Common Fixed Point Problem forϱ-Strictly Pseudononspreading Mappings

Authors: Shubo Cao
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We introduce and analyze the viscosity approximation algorithm for solving the split common fixed point problem for the strictly pseudononspreading mappings in Hilbert spaces. Our results improve and develop previously discussed feasibility problems and related results.


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## Body

## 1. Introduction

Throughout this paper, we always assume thatH is a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥. Let I denote the identity operator on H. Let H1 and H2 be two real Hilbert spaces and let A:H1→H2 be a bounded linear operator. Given closed convex subsets C and Q of H1 and H2, respectively.The split feasibility problem (SFP) (Censor and Elfving 1994 [1]), modeling phase retrieval problems, is to find a point x* with the property
(1)x*∈C,Ax*∈D.
Recently, it has been found that the SFP can also be used to model the intensity-modulated radiation therapy [2–8]. A special case of the SFP (1) is the convexly constrained linear problem:
(2)Ax=b,x∈C.
This problem, due to its applications in many applied disciplines, has extensively been investigated in the literature ever since Landweber [9] introduced his iterative method in 1951.Note that the split feasibility problem (1) can be formulated as fixed point equation by using the fact
(3)x*=PC(I-γA*(I-PQ)A)x*,
where PC and PQ are the projections onto C and Q, respectively, γ>0 is any positive constant, and A* denotes the adjoint of A; that is, x* solves the SFP (1) if and only if x* solves the fixed point equation (3) (see [10] for more details). This implies that we can use fixed point algorithms to solve SFP.In 2002, Byrne [2] proposed his CQ algorithm to solve (1). The sequence {xn} is generated by the following iteration scheme:
(4)xn+1=PC(I-γA*(I-PQ)A)xn,n∈N,
where γ∈(0,2/λ), with λ being the spectral radius of the operator A*A.The CQ algorithm (4) is a special case of the K-M algorithm. Due to the fixed point formulation (2) of the SFP, Moudafi [11] applied the K-M algorithm to the operator PC(I-γA*(I-PQ)A) to obtain a sequence given by
(5)xn+1=(1-αn)xn-αnPC(I-γA*(I-PQ)A)xn,n∈N,
where γ∈(0,2/λ), with λ being the spectral radius of the operator A*A, and the sequence {αn} satisfies the condition ∑n=1∞‍αn(1-αn)=+∞; he proved weak convergence result of the algorithm (5) in Hilbert spaces.In 2009, Censor and Segal [12] considered the following algorithm to be solved (1).Algorithm 1.
Initialization: letx*∈H1=Rn be arbitrary.
Iterative step: forn∈N let
(6)xn+1=U(xn+γA*((V-I)Axn)),n∈N,
where γ∈(0,2/λ) with λ being the spectral radius of the operator A*A and U,V be a single pair of directed operators.In 2010, Moudafi [13] extended the Algorithm 1 and introduced the following algorithm with weak convergence for the split common fixed point problem.Algorithm 2.
Initialization: letx*∈H1=Rn be arbitrary.
Iterative step: forn∈N let
(7)un=xn+γβA*(V-I)Axn,xn+1=(1-αn)un+αnU(un),n∈N,
where β∈(0,1), αn∈(0,1), and γ∈(0,1/λβ) with λ being the spectral radius of the operator A*A and U,V be a pair of quasi-nonexpansive operators.In 2012, Zhao and He [14] continue to consider the split common fixed point problem with quasi-nonexpansive operators and to use the following algorithm to obtain the strong convergence of the viscosity method for solving the split common fixed point problem.Algorithm 3.
Initialization: letx*∈H1=Rn be arbitrary.
Iterative step: forn∈N let
(8)T=U(I+γA*(V-I)A)xn+1=αnf(xn)+(1-αn)×((1-ωn)xn+ωnTxn),n∈N,
where f:H→H is a contractive mapping with constant β∈(0,1), αn∈(0,1), and γ∈(0,1/λ) with λ being the spectral radius of the operator A*A and U,V be a pair of quasi-nonexpansive operators.Motivated and inspired by Censor and Segal [12], Moudafi [11], and Zhao and He [14], we introduce the following relaxed algorithm.Algorithm 4.
Initialization: letx*∈H1=Rn be arbitrary.
Iterative step: forn∈N let
(9)T=U(I+γA*(V-I)A)xn+1=αnσf(xn)+(I-μαnB)×((1-ωn)xn+ωnTxn),n∈N,
where f:H→H is a contractive mapping with constant β∈(0,1), B:H→H is η-strongly monotone and boundedly Lipschitzian, αn∈(0,1), and γ∈(0,1/λ) with λ being the spectral radius of the operator A*A and U,V be a pair of ϱi-strictly pseudononspreading mappings i=1,2.This paper establishes the strong convergence of the sequence given by (9) to the unique solution of solving the split common fixed point problem and the following variational inequality problem VIP(μB-σf,T):
(10)findx*∈Γsuchthat〈(μB-σf)x*,v-x*〉≥0,v∈Γ.


## 2. Preliminaries

In this section, we introduce the concepts of contraction mappings, nonexpansive mappings, quasi-nonexpansive mappings, andϱ-strictly pseudononspreading mappings and some Lemmas.Assume thatC is a nonempty closed and convex subset of Hilbert space H. Recall that the (nearest point or metric) projection from H onto K, that denoted PC, assigns, to each x∈H, the unique point PCx∈C with the property
(11)∥x-PCx∥=inf{∥x-y∥:y∈C}.Definition 5.
A mappingT:H→H is said to be (1)
contraction, if∥Tx-Ty∥≤β∥x-y∥, ∀x,y∈H and β∈(0,1);(2)
nonexpansive, if∥Tx-Ty∥≤∥x-y∥, ∀x,y∈H;(3)
quasi-nonexpansive,∥Tx-q∥≤∥x-q∥, ∀(x,q)∈H×Fix(T).Remark 6.
From the Definition5, It is easy to see that(i)
iterative methods for quasi-nonexpansive mappings have been extensively investigated; see [13–17];(ii)
a nonexpansive mapping is a quasi-nonexpansive mapping.Following the terminology of Browder and Petryshyn [18], we obtain the following definitions.Definition 7.
A mappingT:D(T)⊆H→H is ϱ-strictly pseudononspreading if there exists ϱ∈[0,1) such that
(12)∥Tx-Ty∥2≤∥x-y∥2+ϱ∥x-Tx-(y-Ty)∥2+2〈x-Tx,y-Ty〉,
for all x,y∈D(T).Iterative methods for strictly pseudononspreading mapping have been extensively investigated; see [19–23].Lemma 8 (see [24]).
LetH be a Hilbert spaces, and f:H→H is a contractive mapping with constant β∈(0,1). B:H→H is k-Lipschitzian and η-strongly monotone operator with k>0, η>0. Then for 0<σ<μη/β,
(13)〈x-y,(μB-σf)x-(μB-σf)y〉≥(μη-σβ)∥x-y∥2,x,y∈H.
That is, μB-σf is strongly monotone with coefficient μη-σβ.Lemma 9.
LetH be a real Hilbert space. Then the following well-known results hold: for all x,y∈H and t∈[0,1](i)
∥x+y∥2≤∥x∥2+2〈y,x+y〉;(ii)
∥tx+(1-t)y∥2=t∥x∥2+(1-t)∥y∥2-t(1-t)∥x-y∥2;(iii)
〈x,y〉=-(1/2)∥x-y∥2+(1/2)∥x∥2+(1/2)∥y∥2.Lemma 10.
LetT be a ϱ-strictly pseudononspreading mapping with ϱ∈(0,1), and set Tα=(1-α)I+αT, α∈(ϱ,1). The following properties are reached for each (x,p)∈H×Fix(T): (1)
〈x-Tx,x-p〉≥((1-ϱ)/2)∥x-Tx∥2 and 〈x-Tx,p-Tx〉≤((1+ϱ)/2)∥x-Tx∥2;(2)
∥Tαx-p∥2≤∥x-p∥2-α(1-ϱ-α)∥x-Tx∥2;(3)
〈x-Tαx,x-p〉≥(α(1-ϱ)/2)∥x-Tx∥2.Proof.
Note that property (1) is easily deduced from the Lemma8(iii) and the fact that T is ϱ-strictly pseudononspreading mapping, we obtain
(14)〈x-Tx,x-p〉=-12∥Tx-p∥2+12∥x-Tx∥2+12∥x-p∥2=-12∥Tx-p∥2+1-ϱ2∥x-Tx∥2+12(∥x-p∥2+ϱ∥x-Tx∥2)≥-12∥Tx-p∥2+12∥Tx-p∥2+1-ϱ2∥x-Tx∥2≥1-ϱ2∥x-Tx∥2,〈x-Tx,p-Tx〉=-12∥x-p∥2+12∥x-Tx∥2+12∥Tx-p∥2≤-12∥Tx-p∥2+∥Tx-p∥2+(12+ϱ)∥Tx-p∥2≤1+ϱ2∥x-Tx∥2.
Property (2) is obtained from property (1) and by(15)∥Tαx-p∥2=∥x-p∥2-2α〈x-p,x-Tx〉+α2∥Tx-x∥2.
Property (3) is given byI-Tα=α(I-T) and property (1).Lemma 11 (see [25]).
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
(16)δ(n)=max{k≤n∣𝒯k<𝒯k+1}.
Then {δ(n)}n≥n0 is a nondecreasing sequence verifying limn→∞δ(n)=∞, for all n≥n0; it holds that 𝒯δ(n)<𝒯δ(n)+1 and one has
(17)𝒯n<𝒯δ(n)+1.Lemma 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
(18)〈x-y,y-z〉≥0,∀z∈K.


## 3. Main Results

In what follows, we will focus our attention on the following general two operator split common fixed point problem in real Hilbert spaceH:
(19)findx*∈CsuchthatAx*∈D,
where A:H1→H2 is a bounded linear operator, U:H1→H1 and V:H2→H2 are two ϱi-strictly pseudononspreading mappings i=1,2 with nonempty fixed point sets Fix(U)=C and Fix(V)=Q, and denote the solution set of the two-operator SCFP by
(20)Γ={y∈C;Ay∈Q}.
On the other hand, x*∈Γ is also unique solution of solving the variational inequality problem VIP(μB-σf,T):
(21)x*∈Γ,〈(μB-σf)x*,y-x*〉≥0,y∈Γ,
where B:H→H is η-strongly monotone and k-Lipschitzian on H with k>0, η>0. Let 0<μ<2η/k2, 0<σ<μ(η-(μk2/2))/β=τ/β.Before stating our main convergence result, we establish the boundedness of the iterates given by (9).Lemma 13.
The sequence{xn} is generated by (9), and let U:H1→H1 and V:H2→H2 be two ϱi-strictly pseudononspreading mappings on H, i=1,2, and f:H→H is a contractive mapping with constant β∈(0,1), {αn}⊂(0,1) and 0<ϱi<ωn<1/2, i=1,2. Then {xn} is bounded.Proof.
SetTωn=(1-ωn)I+ωnT. Then xn+1=αnσf(xn)+(I-μαnB)Tωnxn.
Takingy∈Γ, that is, y∈Fix(U) and Ay∈Fix(V). We obtain
(22)∥xn+1-y∥=∥(Tωnxn-y)αnσ(f(xn)-f(y))+αn(σf(y)-μBy)+(I-μαnB)(Tωnxn-y)∥≤αnσ∥f(xn)-f(y)∥+αn∥σf(y)-μBy∥+(1-αnτ)∥Tωnxn-y∥≤αnσβ∥xn-y∥+αn∥σf(y)-μBy∥+(1-αnτ)∥Tωnxn-y∥.
From the definition of Tωn, we have
(23)∥Tωnxn-y∥2=∥(1-ωn)xn+ωnTxn-y∥2=∥xn-y+ωn(Txn-xn)∥2=∥xn-y∥2-2ωn〈xn-y,xn-Txn〉+ωn2∥Txn-xn∥2.
On the other hand, we obtain
(24)∥Txn-y∥2=∥U(I+γA*(V-I)A)xn-y∥2≤∥(I+γA*(V-I)A)xn-y∥2=∥xn-y∥2+γ2∥A*(V-I)Axn∥2+2γ〈xn-y,A*(V-I)Axn-y〉=∥xn-y∥2+γ2〈(V-I)Axn,AA*(V-I)Axn〉+2γ〈xn-y,A*(V-I)Axn〉.
According to the definition of λ, we have
(25)γ2〈(V-I)Axn,AA*(V-I)Axn〉≤λγ2〈(V-I)Axn,(V-I)Axn〉=λγ2∥(V-I)Axn∥2.
Now, by using property (1) of Lemma 9, we obtain
(26)2γ〈xn-y,A*(V-I)Axn〉=2γ〈A(xn-y),(V-I)Axn〉=2γ〈A(xn-y)+(V-I)Axn-(V-I)Axn,(V-I)Axn〉=2γ(∥(V-I)Axn∥2〈VAxn-Ay,(V-I)Axn〉-∥(V-I)Axn∥2)≤2γ(1-ϱ22∥(V-I)Axn∥2-∥(V-I)Axn∥21-ϱ221-ϱ22)≤-γ(1-ϱ2)∥(V-I)Axn∥2.
Combining (24)–(26), we obtain
(27)∥Txn-y∥2=∥xn-y∥2+λγ2∥(V-I)Axn∥2-γ(1-ϱ2)∥(V-I)Axn∥2=∥xn-y∥2-γ((1-ϱ2)-λγ)×∥(V-I)Axn∥2≤∥xn-y∥2.
From property (i) of Lemma 9 and (23), we get
(28)∥Tωnxn-y∥2≤∥xn-y∥2-(1-ϱ2)ωn∥xn-Txn∥2+ωn2∥Txn-xn∥2=∥xn-y∥2-ωn(1-ϱ2-ωn)×∥xn-Txn∥2≤∥xn-y∥2.
Combining (22), (23), and (28), we have
(29)∥xn+1-y∥≤αnσβ∥xn-y∥+αn∥f(y)-μBy∥+(1-αnτ)∥Tωnxn-y∥=[1-αn(τ-σβ)]∥xn-y∥+αn∥f(y)-μBy∥≤max{∥xn-y∥,1τ-σβ∥f(y)-μBy∥}.
It follows from (29) and induction that
(30)∥xn+1-y∥≤max{∥x0-y∥,1τ-σβ∥f(y)-μBy∥},
and hence {xn} is bounded.Now we are in position to claim the main convergence result.Theorem 14.
Given a bounded linear operatorA:H1→H2, let U:H1→H1 and V:H2→H2 be two ϱi-strictly pseudononspreading mappings, and i=1,2 with fixed point Fix(U)=C and Fix(V)=Q. Assume that U-I and V-I are demiclosed at origin. Let B:H→H be η-strongly monotone and k-Lipschitzian on H with k>0, η>0, and f:H→H is a contractive mapping with constant β∈(0,1). Assume that {xn} is the sequence given by Algorithm 4 with γ∈(0,1/λ), 0<ϱi<ωn<1/2, and i=1,2 such that 0<liminfn→∞ωn≤limsupn→∞ωn<1/2 and αn∈(0,1) such that limn→∞αn=0 and ∑n=1∞‍αn=∞. If Γ≠∅, then the sequence {xn} strongly converges to a split common fixed point y∈Γ, verifying y=PΓ(I-μB+σf)(y) which equivalently solves the following variational inequality problem:
(31)y∈Γ,〈(μB-σf)y,x*-y〉≥0,x*∈Γ.Proof.
Lety be the solution of (31). From (9) we obtain that
(32)xn+1-xn+αn(μBxn-σf(xn))=(I-μαnB)(Tωnxn-xn),
hence
(33)〈xn+1-xn+αn(μBxn-σf(xn)),xn-y〉=〈(I-μαnB)(Tωnxn-xn),xn-y〉=(1-αn)〈Tωnxn-xn,xn-y〉+αn〈(I-μB)(Tωnxn-xn),xn-y〉≤(1-αn)〈Tωnxn-xn,xn-y〉+αn∥(I-μB)(Tωnxn-xn)∥∥xn-y∥≤(1-αn)〈Tωnxn-xn,xn-y〉+αn(1-τ)∥Tωnxn-xn∥∥xn-y∥=(1-αn)〈Tωnxn-xn,xn-y〉+ωnαn(1-τ)∥Txn-xn∥∥xn-y∥.
By (28), we obtain that
(34)〈xn-Tωnxn,xn-y〉=ωn22∥xn-Tωnxn∥2+12∥xn-y∥2-12∥Tωnxn-y∥2≥ωn22∥xn-Txn∥2+12∥xn-y∥2-12∥xn-y∥2+ωn2(1-ϱ2-ωn)∥xn-Txn∥2=ωn(1-ϱ2)2∥xn-Txn∥2.
It follows from (33) that
(35)〈xn+1-xn+αn(μBxn-σf(xn)),xn-y〉≤-ωn2(1-ϱ2)(1-αn)∥xn-Txn∥2+ωnαn(1-τ)∥Txn-xn∥∥xn-y∥
or equivalently
(36)-〈xn-xn+1,xn-y〉≤-αn〈(μB-σf)xn,xn-y〉-ωn2(1-ϱ2)(1-αn)∥xn-Txn∥2+ωnαn(1-τ)∥Txn-xn∥∥xn-y∥.
Furthermore, using the classical equality (iii) in Lemma 10 and setting 𝒯n=(1/2)∥xn-y∥2, we have
(37)〈xn-xn+1,xn-y〉=𝒯n-𝒯n+1+12∥xn-xn+1∥2.
So that (36) can be equivalently rewritten as
(38)𝒯n+1-𝒯n-12∥xn-xn+1∥2≤-αn〈(μB-σf)xn,xn-y〉-ωn2(1-ϱ2)(1-αn)∥xn-Txn∥2+ωnαn(1-τ)∥Txn-xn∥∥xn-y∥.
Now using (32) again, we have
(39)∥xn+1-xn∥2=∥(Tωnxn-xn)αn(σf(xn)-μBxn)+(I-μαnB)(Tωnxn-xn)∥2.
Since B:H→H is η-strongly monotone and k-Lipschitzian on H, hence it is a classical matter to see that
(40)∥xn+1-xn∥2≤2αn2∥σf(xn)-μBxn∥2+2(1-αnτ)2∥Tωnxn-xn∥2,
which by ∥Tωnxn-xn∥=ωn∥xn-Txn∥ yields
(41)12∥xn+1-xn∥2≤αn2∥σf(xn)-μBxn∥2+(1-αnτ)2ωn2∥xn-Txn∥2.
Then from (38) and (41), we have
(42)𝒯n+1-𝒯n+[ωn2(1-ϱ2)(1-αn)-ωn2(1-αnτ)2]×∥xn-Txn∥2≤αn[αn∥σf(xn)-μBxn∥2-〈(μB-σf)xn,xn-y〉+ωn(1-τ)∥Txn-xn∥∥xn-y∥∥σf(xn)-μBxn∥2].
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 (42) together with limn→∞αn=0, the boundedness of {xn} and 0<liminfn→∞ωn≤limsupn→∞ωn<1/2, we obtain
(43)limn→∞∥xn-Txn∥=0.
It also follows from (42) that
(44)𝒯n-𝒯n+1≥αn(-αn∥σf(xn)-μBxn∥2+〈(μB-σf)xn,xn-y〉-ωn(1-τ)∥Txn-xn∥∥xn-y∥∥σf(xn)-μBxn∥2).
Then, by ∑n=0∞‍αn=∞, we obviously deduce that
(45)liminfn→∞αn(-αn∥σf(xn)-μBxn∥2+〈(μB-σf)xn,xn-y〉-ωn(1-τ)∥Txn-xn∥∥xn-y∥∥σf(xn)-μBxn∥2)≤0
or equivalently (as αn∥σf(xn)-μBxn∥2→0 and limn→∞(𝒯n+1-𝒯n)=0). From (45), we get
(46)liminfn→∞〈(μB-σf)xn,xn-y〉≤0.
Moreover, by Lemma 8, we have
(47)2(μη-σβ)𝒯n+〈(μB-σf)y,xn-y〉≤〈(μB-σf)xn,xn-y〉,
which by (46) entails
(48)liminfn→∞〈2(μη-σβ)𝒯n+(μB-σf)y,xn-y〉≤0.
Hence, recalling that limn→∞𝒯n exists, we equivalently obtain
(49)2(μη-σβ)limn→∞𝒯n+liminfn→∞〈(μB-σf)y,xn-y〉≤0,
Namely,
(50)2(μη-σβ)limn→∞𝒯n≤-liminfn→∞〈(μB-σf)y,xn-y〉.
Now we prove that
(51)liminfn→∞〈(μB-σf)y,xn-y〉≤0.
It follows from (27) and (43) that
(52)γ((1-ϱ2)-λγ)∥(V-I)Axn∥2≤∥xn-y∥2-∥Txn-y∥2=(∥xn-y∥-∥Txn-y∥)×(∥xn-y∥+∥Txn-y∥)=(∥xn-Txn∥)(∥xn-y∥+∥Txn-y∥)⟶0,(n⟶∞),
and hence
(53)limn→∞∥(V-I)Axn∥=0.
Taking x*∈ωw(xn), from the demiclosedness of V-I at 0, we have
(54)V(Ax*)=Ax*.
Now, by setting un=xn+γA*V-IAxn, it follows that x*∈ωw(un). On the other hand,
(55)∥U(un)-un∥=∥Txn-xn-γA*V-IAxn∥≤∥Txn-xn∥+γ∥A*∥∥V-IAxn∥⟶0,
which, combined with the demiclosedness of U-I at 0, yields
(56)Ux*=x*.
Hence, x*∈C and x*∈Γ. We can take subsequence {xnj} of {xn} such that limn→∞xnj=x* and
(57)liminfn→∞〈(σf-μB)y,xn-y〉=limj→∞〈(σf-μB)y,xnj-y〉,
which leads to
(58)liminfn→∞〈(σf-μB)y,xn-y〉=〈(σf-μB)y,x*-y〉≤0.
By (50), we have limn→∞Γn=0, and hence {xn} converges strongly to y.
Case  2.  Suppose that 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 11. It follows that 𝒯δ(n+1)-𝒯δ(n)>0, which by (42) amounts to
(59)[ωn2(1-ϱ2)(1-αδ(n))-ωn2(1-αδ(n)τ)2]∥xδ(n)-Txδ(n)∥2≤αδ(n)[αδ(n)∥σf(xδ(n))-μBxn∥2-〈(μB-σf)xn,xn-y〉+ωn(1-τ)∥Txδ(n)-xδ(n)∥∥xδ(n)-y∥∥σf(xδ(n))-μBxn∥2].
By the boundedness of {xn} and limn→∞αn=0, we immediately obtain
(60)limn→∞∥xδ(n)-Txδ(n)∥=0.
Using (9), we have
(61)∥xδ(n)+1-xδ(n)∥≤αδ(n)∥σf(xδ(n))-μBxδ(n)∥+|1-αδ(n)τ|∥Tωnxδ(n)-xδ(n)∥≤αδ(n)∥σf(xδ(n))-μBxδ(n)∥+|1-αδ(n)ωnτ|∥Txδ(n)-xδ(n)∥,
which together with (60) and limn→∞αn=0 yields
(62)limn→∞∥xδ(n)+1-xδ(n)∥=0.
Similar to Case 1, we have
(63)liminfn→∞〈(μB-σf)y,xδ(n)-y〉≥0.
Now by (59) we clearly have
(64)αδ(n)∥σf(xδ(n))-μBxδ(n)∥2+α(1-τ)×∥Txδ(n)-xδ(n)∥∥xδ(n)-y∥≥〈(μB-σf)xδ(n),xδ(n)-y〉,
which in the light of (47) yields
(65)2(μη-σβ)𝒯δ(n)+〈(μB-σf)y,xδ(n)-y〉≤αδ(n)∥σf(xδ(n))-μBxδ(n)∥2+α(1-τ)∥Txδ(n)-xδ(n)∥∥xδ(n)-y∥.
Hence (as limn→∞αδ(n)∥σf(xδ(n))-μBxδ(n)∥2=0 and limn→∞∥Txδ(n)-xδ(n)∥=0) it follows that
(66)2(μη-σβ)limsupn→∞𝒯δ(n)≤-liminfn→∞〈(μB-σf)y,xδ(n)-y〉.
From (59) and (63), we obtain
(67)limn→∞〈(μB-σf)y,xδ(n)-y〉≥0,
which by (60) yields limsupn→∞𝒯δ(n)=0, so that limn→∞𝒯δ(n)=0. Combining (62), we have limn→∞𝒯δ(n)+1=0. Then, recalling that 𝒯n<𝒯δ(n)+1 (by Lemma 11), we get limn→∞𝒯n=0, so that xn→y strongly.
In addition, the variational inequality (50) and (67) can be written as
(68)〈(I-μB+σf)y-y,x*-y〉≤0,x*∈Γ.
So, by the Lemma 12, it is equivalent to the fixed point equation
(69)PΓ(I-μB+σf)y=y.


## 4. Application in Other Nonlinear Operators

In order to define our motivations, we recall some definitions of classed of operators as followsDefinition 15.
T
:
D
(
T
)
⊆
H
→
H is said to be (1)
nonspreading in [26, 27], if
(70)∥Tx-Ty∥2≤∥Tx-y∥2+∥Ty-x∥2,∀x,y∈C,(2)
demicontractive in [28], if there exists a constant α<1 such that
(71)∥Tx-q∥2≤∥x-q∥2+α∥x-Tx∥2,∀(x,q)∈H×Fix(T).Remark 16.
Iemoto and Takahashi [29] proved that (70) is equivalent to
(72)∥Tx-Ty∥2≤∥x-y∥2+2〈x-Tx,y-Ty〉,∀x,y∈C.
Iterative methods for nonspreading mapping have been extensively investigated; see [30–34].Remark 17.
From the Definition5 (3), Definition 7, and Definition 15, we have the following facts. (i)
Observe that every nonspreading mapping is 0-strictly pseudononspreading.(ii)
IfT is nonspreading mapping and the set of fixed point is nonempty, then T is quasi-nonexpansive mapping.(iii)
Every pseudononspreading mapping with a nonempty fixed point setFix(T) is demicontractive (see [28]).Corollary 18.
Given a bounded linear operatorA:H1→H2, let U:H1→H1 and V:H2→H2 be two nonspreading mappings with fixed point Fix(U)=C and Fix(V)=Q. Assume that U-I and V-I are demiclosed at origin. Let B:H→H be η-strongly monotone and k-Lipschitzian on H with k>0, η>0, and let f:H→H be a contractive mapping with constant β∈(0,1). Let {xn} be the sequence given by (9) with γ∈(0,1/λ), 0<ϱi<ωn<1/2, and i=1,2 such that 0<liminfn→∞ωn≤limsupn→∞ωn<1/2 and αn∈(0,1) such that limn→∞αn=0 and ∑n=1∞‍αn=∞. If Γ≠∅, then the sequence {xn} strongly converges to a split common fixed point x*∈Γ, verifying x*=PΓ(I-μB+σf)(x*) which equivalently solves the following variational inequality problem:
(73)x*∈Γ,〈(μB-σf)x*,y-x*〉≥0,y∈Γ.Proof.
Form the proof of the Theorem14, we can easily certify this theorem by nonspreading mapping (i.e., nonspreading is 0-strictly pseudononspreading).From the Remark17(ii) and the Corollary 18, we have the following corollary.Corollary 19.
Given a bounded linear operatorA:H1→H2, let U:H1→H1 and V:H2→H2 be two quasi-nonexpansives with fixed point Fix(U)=C and Fix(V)=Q. Assume that U-I and V-I are demiclosed at origin. Let B:H→H be η-strongly monotone and k-Lipschitzian on H with k>0, η>0, and let f:H→H be a contractive mapping with constant β∈(0,1). Let {xn} be the sequence given by (9) with γ∈(0,1/λ), ωn∈(0,1/2) such that 0<liminfn→∞ωn≤limsupn→∞ωn<1/2 and αn∈(0,1) such that limn→∞αn=0 and ∑n=1∞‍αn=∞. If Γ≠∅, then the sequence {xn} strongly converges to a split common fixed point x*∈Γ, verifying x*=PΓf(x*) which equivalently solves the following variational inequality problem:
(74)x*∈Γ,〈(μB-σf)x*,y-x*〉≥0,y∈Γ.Ifσ=μ=1 and B=I in (9), thus k=η=1 and β∈(0,1/2), and then we obtain (8) and the following corollary. On the other hand, this corollary was proven by Zhao and He [14].Corollary 20.
Given a bounded linear operatorA:H1→H2, let U:H1→H1 and V:H2→H2 be two quasi-nonexpansives with fixed point Fix(U)=C and Fix(V)=Q. Assume that U-I and V-I are demiclosed at origin. Let f:H→H be a contractive mapping with constant β∈(0,1/2). Let {xn} be the sequence given by (8) with γ∈(0,1/λ), ωn∈(0,1/2) such that 0<liminfn→∞ωn≤limsupn→∞ωn<1/2, and αn∈(0,1) such that limn→∞αn=0 and ∑n=1∞‍αn=∞. IfΓ≠∅, then the sequence {xn} strongly converges to a split common fixed point x*∈Γ, verifying x*=PΓ(I-μB+σf)(x*) which equivalently solves the following variational inequality problem:
(75)x*∈Γ,〈(I-f)x*,y-x*〉≥0,y∈Γ.



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*Source: 241789-2013-05-08.xml*