# The Split Feasibility Problems for Countable Families of Asymptotically Strict Pseudocontractions

Authors: Wei-Qi Deng
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

An up-to-date algorithm for solving the split feasibility problem for countable families of asymptotically strict pseudocontractions is introduced in the framework of Hilbert spaces. Our results greatly improve and extend those of other authors whose related research studies are restricted to the situation of at most finitely many such mappings.


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## Body

## 1.  Introduction

The split feasibility problem (SFP) in finite dimensional spaces was first introduced by Censor and Elfving [1] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [2]. Recently, it has been found that the SFP can also be used in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment planning [3–5].The split feasibility problem in an infinite dimensional Hilbert space can be found in [2, 4, 6–8].LetH1 and H2 be two real Hilbert spaces with inner product 〈·,·〉 and the corresponding norm ∥·∥. Let C and Q be nonempty closed convex subsets of H1 and H2, respectively. The purpose of this paper is to introduce and study the following multiple-set split feasibility problem for an infinite family of asymptotically strict pseudocontractions (MSSFP) in the framework of infinite-dimensional Hilbert spaces. Find x*∈C such that
(1)Ax*∈Q,
where A:H1→H2 is a bounded linear operator.In the sequel, we useΓ to denote the set of solutions of the problem (MSSFP), that is,
(2)Γ={x∈C,Ax∈Q}.


## 2. Preliminaries

We first recall some definitions, notations, and conclusions which will be needed in proving our main results.LetE be a Banach space. A mapping T:E→E is said to be demiclosed at origin, if for any sequence {xn}⊂E with xn⇀x* and ∥(I-T)xn∥→0, then x*=Tx*, where xn⇀x* denotes that {xn} converges weakly to x*.A Banach spaceE is said to satisfy Opial’s condition, if for any sequence {xn} in E, xn⇀x* implies that
(3)liminfn→∞∥xn-x*∥<liminfn→∞∥xn-y∥,∀y∈Ewithy≠x*.
It is well known that every Hilbert space satisfies Opial's condition.Definition 1.
LetH be a real Hilbert space, T be a mapping from H into itself and the fixed point set F(T) of T be nonempty.(1)
Tis called a (γ,{kn})-asymptotically strict pseudocontraction if there exists a constant γ∈[0,1) and a sequence {kn}⊂[1,∞) with kn→1 such that
(4)∥Tnx-Tny∥2≤kn∥x-y∥2+γ∥(I-Tn)x-(I-Tn)y∥2,∀x,y∈H.
Especially, if kn=1 for each n≥1 in (4) and there exists a γ∈[0,1) such that
(5)∥Tx-Ty∥2≤∥x-y∥2+γ∥(I-T)x-(I-T)y∥2,∀x,y∈H,
then T:H→H is called a γ-strict pseudocontraction.(2)
T is said to be uniformlyL-Lipschitzian if there exists a constant L>0 such that
(6)∥Tnx-Tny∥2≤L∥x-y∥,∀x,y∈H,n≥1.(3)
T is said to be semicompact if for any bounded sequence {xn}⊂H with limn→∞∥xn-Txn∥=0, there exists a subsequence {xni} of {xn} such that {xni} converges strongly to a point x*∈H.Remark 2.
(
1
) If we put γ=0 in (4), then the mapping T:H→H is asymptotically nonexpansive.
(
2
) If we put γ=0 in (5), then the mapping T:H→H is nonexpansive.
(
3
) Each (γ,{kn})-asymptotically strict pseudocontraction and each γ-strict pseudocontraction both are demiclosed at origin [9].In 2011, Moudafi [10] proposed the following iterative algorithm for solving split common fixed problem of quasinonexpansive mappings: for arbitrarily chosen x1∈H1,
(7)un=xn+γβA*(T-I)Axn,xn+1=(1-αn)un+αnUun,n∈ℕ,
and proved that {xn} converges weakly to a split common fixed point x*∈Γ, where U:H1→H1 and T:H2→H2 are two quasinonexpansive mappings, A:H1→H2 is a bounded linear operator, and A* denotes the adjoint of A.Motivated and inspired by the studies of Moudafi [10, 11] and Chang et al. [12], in this paper, we introduce an algorithm for solving the split feasibility problems for countable families of asymptotically strict pseudocontractions and prove some strong and weak convergence theorems for such mappings in Hilbert spaces. The results extend those of the authors [12] whose related research studies are restricted to the situation of at most finite families of such mappings.By using the well-known inequality〈x,y〉=(1/2)∥x∥2+(1/2)∥y∥2-(1/2)∥x-y∥2 in Hilbert spaces, we can easily show the following proposition, whose proof is omitted.Proposition 3 (see [12]).
LetT:H→H be a (γ,{kn})-asymptotically strict pseudocontraction. If Γ≠∅, then for each p∈F(T) and x∈H, the following inequalities hold and they are equivalent:
(8)∥Tnx-p∥2≤kn∥x-p∥2+γ∥x-Tnx∥2;(9)〈x-Tnx,x-p〉≥1-γ2∥x-Tnx∥2-kn-12∥x-p∥2;(10)〈x-Tnx,p-Tnx〉≤1+γ2∥x-Tnx∥2+kn-12∥x-p∥2.Lemma 4 (see [13]).
Let{an}, {bn}, and {δn} be the sequences of nonnegative real numbers satisfying
(11)an+1≤(1+δn)an+bn.
If ∑n=1∞δn<∞ and ∑n=1∞bn<∞, then the limn→∞an exists.Lemma 5 (see [14]).
LetK be a nonempty closed convex subset of a real Hilbert space H and T a nonexpansive mapping from K into itself. If T has a fixed point, then I-T is demiclosed at zero, where I is the identity mapping of H.Lemma 6 (see [15]).
The unique solutions to the positive integer equation(12)n=i+(m-1)m2,m≥i,n=1,2,3,…
are
(13)i=n-(m-1)m2,m=-[12-2n+14],n=1,2,3,…,
where [x] denotes the maximal integer that is not larger than x.


## 3. Main Results

In the sequel, we assume that the following conditions are satisfied:(a)
H1 and H2 are two real Hilbert spaces, A:H1→H2 is a bounded linear operator, and A* denotes the adjoint of A;(b)
{Si}:H1→H1 is a sequence of uniformly L1-Lipschitzian and (βi,{k1,n(i)})-asymptotically strict pseudocontractions and {Ti}:H2→H2 is a sequence of uniformly L2-Lipschitzian and (μi,{k2,n(i)})-asymptotically strict pseudocontractions satisfying the following conditions:
(1)
C:=∩i=1∞F(Si)≠∅, Q:=∩i=1∞F(Ti)≠∅;(2)
β:=supi≥1{βi}<1 and μ:=supi≥1{μi}<1;(3)
for eachi≥1, kn(i):=max{k1,n(i),k2,n(i)}, and ∑i=1∞∑n=1∞(kn(i)-1)<∞.The multiple-set split feasibility problem for infinite families of nonlinear mappings{Si} and {Ti} is to find a point
(14)q∈CsuchthatAq∈Q,
whose set of solutions is denoted by Γ.Lemma 7.
LetH1,H2,A,{Si},{Ti},C,Q,β,μ,L1,L2 and {kn(i)} be the same as those mentioned above. Let {xn} be the following sequence generated by an arbitrarily chosen x1∈H1(15)un=xn+γA*((Tn*)mn-I)Axn,xn+1=(1-αn)un+αn(Sn*)mnun,n∈ℕ,
where Tn*=Tin, Sn*=Sin with in and mn being the solutions to the positive integer equation: n=i+(m-1)m/2(m≥i,n=1,2,…); that is, for each n≥1, there exist unique in and mn such that
(16)i1=1,i2=1,i3=2,i4=1,i5=2,i6=3,i7=1,i8=2,…;m1=1,m2=2,m3=2,m4=3,m5=3,m6=3,m7=4,m8=4,…;{αn} is a sequence in [0,1], and γ is a constant satisfying the following condition: (4)
:αn∈(δ,1-β) for all n≥1 and γ∈(0,(1-μ)/∥A∥2), where δ∈(0,1-β) is a positive constant. If Γ≠∅, for any q∈Γ, then
(I)
limn→∞∥xn-q∥ and limn→∞∥un-q∥ exist and have the same values;(II)
for eachi≥1, there exists a corresponding subsequence {xn}n∈ℕi of {xn} such that
(17)limℕi∋n→∞∥un-Siun∥=∥Axn-TiAxn∥=0,where ℕi:={n∈ℕ:n=i+(m-1)m/2,m≥i,m∈ℕ}.Proof.
(I) Takingq∈Γ, that is, q∈C and Aq∈Q, and using (15) and (9), we have
(18)∥xn+1-q∥2≤∥un-q-αn(un-(Sn*)mnun)∥2=∥un-q∥2-2αn〈un-q,un-(Sn*)mnun〉+αn2∥un-(Sn*)mnun∥2≤∥un-q∥2-αn[(kmn(in)-1)(1-β)∥un-(Sn*)mnun∥2-(kmn(in)-1)∥un-q∥2∥un-(Sn*)mnun∥2]+αn2∥un-(Sn*)mnun∥2=(1+αn(kmn(in)-1))∥un-q∥2-αn(1-β-αn)∥un-(Sn*)mnun∥2,(19)∥un-q∥2=∥xn+γA*((Tn*)mn-I)Axn-q∥2=∥xn-q∥2+γ2∥A*((Tn*)mn-I)Axn∥2+2γ〈xn-q,A*((Tn*)mn-I)Axn〉=∥xn-q∥2+γ2〈((Tn*)mn-I)Axn,AA*((Tn*)mn-I)Axn〉+2γ〈xn-q,A*((Tn*)mn-I)Axn〉,
where
(20)γ2〈((Tn*)mn-I)Axn,AA*((Tn*)mn-I)Axn〉≤∥A∥2γ2〈((Tn*)mn-I)Axn,((Tn*)mn-I)Axn〉≤∥A∥2γ2∥((Tn*)mn-I)Axn∥2,(21)2γ〈xn-q,A*((Tn*)mn-I)Axn〉=2γ〈A(xn-q),((Tn*)mn-I)Axn〉=2γ〈A(xn-q)+((Tn*)mn-I)Axn-((Tn*)mn-I)Axn,((Tn*)mn-I)Axn〉=2γ(∥((Tn*)mn-I)Axn∥2〈(Tn*)mnAxn-Aq,((Tn*)mn-I)Axn〉-∥((Tn*)mn-I)Axn∥2).
Further, letting x=Axn, Tn=(Tn*)mn, p=Aq, γ=μ in (10) and noting Aq∈F(Tn*), we have
(22)〈(Tn*)mnAxn-Aq,((Tn*)mn-I)Axn〉≤1+μ2∥((Tn*)mn-I)Axn∥2+kmn(in)-12∥Axn-Aq∥2≤1+μ2∥((Tn*)mn-I)Axn∥2+(kmn(in)-1)∥A∥22∥xn-q∥2.
Substituting (22) into (21) and simplifying it, we have
(23)2γ〈xn-q,A*((Tn*)mn-I)Axn〉≤γ(μ-1)∥((Tn*)mn-I)Axn∥2+(kmn(in)-1)γ∥A∥2∥xn-q∥2.
Substituting (20) and (23) into (19) and simplifying it, we have
(24)∥un-q∥2≤∥xn-q∥2+γ2∥A∥2∥((Tn*)mn-I)Axn∥2+γ(μ-1)∥((Tn*)mn-I)Axn∥2+(kmn(in)-1)γ∥A∥2∥xn-q∥2=∥xn-q∥2-γ(1-μ-γ∥A∥2)×∥((Tn*)mn-I)Axn∥2+(kmn(in)-1)γ∥A∥2∥xn-q∥2.
Again, substituting (24) into (18) and simplifying it, we have
(25)∥xn+1-q∥2≤(1+αn(kmn(in)-1))×{(kmn(in)-1)∥xn-q∥2-γ(1-μ-γ∥A∥2)×∥((Tn*)mn-I)Axn∥2+(kmn(in)-1)γ∥A∥2∥xn-q∥2}-αn(1-β-αn)∥un-(Sn*)mnun∥2≤(1+αn(kmn(in)-1))∥xn-q∥2-γ(1-μ-γ∥A∥2)∥((Tn*)mn-I)Axn∥2+(1+αn(kmn(in)-1))(kmn(in)-1)γ∥A∥2∥xn-q∥2-αn(1-β-αn)∥un-(Sn*)mnun∥2.
By condition (4), we have(26)∥xn+1-q∥2≤(1+αn(kmn(in)-1))∥xn-q∥2+(1+αn(kmn(in)-1))(kmn(in)-1)×γ∥A∥2∥xn-q∥2≤(1+K(kmn(in)-1))∥xn-q∥2,
where
(27)K=supn≥1{αn+(1+αn(kmn(in)-1))γ∥A∥2}<∞.
Note that ∑n=1∞‍(kmn(in)-1)=∑i=1∞‍∑n=i∞‍(kn(i)-1)≤∑i=1∞‍∑n=1∞‍(kn(i)-1)<∞. Hence, from Lemma 4, we know that the following limit exists:
(28)limn→∞∥xn-q∥.
We now prove that for each q∈Γ, the limit
(29)limn→∞∥un-q∥
exists. In fact, from (25) and (28), it follows that
(30)γ(1-μ-γ∥A∥2)∥((Tn*)mn-I)Axn∥2+αn(1-β-αn)∥un-(Sn*)mnun∥2≤∥xn-q∥2-∥xn+1-q∥2+K(kmn(in)-1)∥xn-q∥2⟶0(n⟶∞).
This, combined with condition (4), implies that
(31)limn→∞∥un-(Sn*)mnun∥=0,(32)limn→∞∥((Tn*)mn-I)Axn∥=0.
Therefore, it follows from (19), (28), and (32) that limn→∞∥un-q∥ exists.
(II) We firstly prove thatlimn→∞∥xn+1-xn∥=0 and limn→∞∥un+1-un∥=0. As a matter of fact, it follows from (15) that
(33)∥xn+1-xn∥=∥(1-αn)un+αn(Sn*)mnun-xn∥=∥(1-αn)(xn+γA*((Tn*)mn-I)Axn)+αn((Sn*)mnun-xn)∥=∥(1-αn)γA*((Tn*)mn-I)Axn+αn((Sn*)mnun-xn)∥=∥(1-αn)γA*((Tn*)mn-I)Axn+αn((Sn*)mnun-un)+αn(un-xn)(Tn*)mn∥=∥(1-αn)γA*((Tn*)mn-I)Axn+αn((Sn*)mnun-un)+αnγA*((Tn*)mn-I)Axn∥=∥γA*((Tn*)mn-I)Axn+αn((Sn*)mnun-un)∥.
In view of (31) and (32), we have that
(34)limn→∞∥xn+1-xn∥=0.
Similarly, it follows from (15), (32), and (34) that
(35)∥un+1-un∥=∥(Tn+1*)mn+1xn+1+γA*((Tn+1*)mn+1-I)Axn+1-(xn+γA*((Tn*)mn-I)Axn)∥=∥xn+1-xn∥+γ∥A*((Tn+1*)mn+1-I)Axn+1∥+γ∥A*((Tn*)mn-I)Axn∥⟶0(n⟶∞).
Next, for each i∈ℕ, we consider the corresponding subsequence {un}n∈ℕi of {un}. For example, by Lemma 6 and the definition of ℕ1, we have ℕ1={1,2,4,7,11,16,…} and i1=i2=i4=i7=i11=i16=⋯=1. Note that {mn}n∈ℕi={i,i+1,i+2,…}, that is, mn-1=mn-1, and Sn*=Sin=Si whenever n∈ℕi. Set ηn:=∥un-Simnun∥. Since {Si} are uniformly L1-Lipschitzian and mn≥1 for n≥1, we have, for each n∈ℕi and n≥2,
(36)∥un-Siun∥≤∥un-Simnun∥+∥Simnun-Siun∥≤ηn+L1∥Simn-1un-un∥≤ηn+L1(∥Simn-1un-Simn-1un-1∥+∥Simn-1un-1-un∥)≤ηn+L1∥Simn-1un-Simn-1un-1∥+L1(∥Simn-1un-1-un-1∥+∥un-un-1∥∥Simn-1un-1-un-1∥)≤ηn+L12∥un-un-1∥+L1(∥Simn-1un-1-un-1∥+∥un-1-un∥)≤ηn+L1(1+L1)∥un-un-1∥+L1ηn-1.
Thus, it follows from (31) and (35) that, for each i≥1,
(37)limℕi∋n→∞∥un-Siun∥=0.
Similarly, we have, for each i≥1,
(38)limℕi∋n→∞∥Axn-TiAxn∥=0.
This completes the proof.Theorem 8.
LetH1,H2,A,{Si},{Ti},C,Q,β,μ,L1,L2 and {kn(i)} be the same as those in Lemma 7. Suppose that {xn} is a sequence defined by (15). If Γ≠∅ and there exist mappings Si0∈{Si}i=1∞ and Ti0∈{Ti}i=1∞ and nondecreasing functions f,h:[0,∞)→[0,∞) with f(0)=h(0)=0 and f(r),h(r)>0 for all r∈(0,∞) such that f(d(xn,Γ))≤∥un-Si0un∥ and h(d(Axn,Q))≤∥Axn-Ti0Axn∥ for all n≥1, then {xn} converges strongly to some member of Γ.Proof.
By Lemma7, there exists a subsequence {un}n∈ℕi0 of {un} such that
(39)limℕi0∋n→∞∥un-Si0un∥=0.
Since for all n∈ℕi0,
(40)f(d(xn,Γ))≤∥un-Si0un∥,
by taking limsup as ℕi0∋n→∞ on both sides in the inequality above, we have
(41)limℕi0∋n→∞f(d(xn,Γ))=0,
which implies limℕi0∋n→∞d(xn,Γ)=0 by the definition of the function f.
Now we will show that{xn}n∈ℕi0 is a Cauchy sequence. By Lemma 7, there exists a constant M>0 such that ∥xn-q∥2≤M∥xm-q∥2 for any q∈Γ and all n>m. And for any ϵ>0, there exists a positive integer N such that d2(xn,Γ)<ϵ/4M for all n≥N and n∈ℕi0. Then, for any q∈Γ and n,m≥N and n,m∈ℕi0, we have
(42)∥xn-xm∥2≤2(∥xn-q∥2+∥xm-q∥2)≤4M∥xN-q∥2.
Taking the infimum in the above inequalities for all q∈Γ yields that
(43)∥xn-xm∥2≤4Md2(xN,Γ)<ϵ,
which implies that {xn}n∈ℕi0 is a Cauchy sequence. Therefore, there exists a p∈H1 such that xn→p as ℕi0∋n→∞ since H1 is complete. Firstly, we show that p∈C. limℕi0∋n→∞d(xn,Γ)=0 shows that d(p,Γ)=0, which implies that p∈C since Γ⊂C. Secondly, we show that Ap∈Q. Since {xn}n∈ℕi0 converges to p and h(d(Axn,Q))≤∥Axn-Ti0Axn∥ for all n∈ℕi0, then d(Ap,Q)=0. This implies that Ap∈Q because of the closedness of Q, and so p∈Γ. It finally follows from the existence of limn→∞∥xn-p∥ that xn→p as n→∞. This completes the proof.Example 9.
LetH1=H2=ℝ1 with the standard norm ∥·∥=|·| and C=[-1,1]. Let {Si}={Ti}:C→C be two sequences of mappings defined by
(44)Six={xi+1,x∈(0,1],x,x∈[-1,0].
It is easily shown that {Si} is uniformly L-Lipschitzian and a sequence of (0,{kn=1})-asymptotically strict pseudocontractions. We now prove that the sequence {xn} defined by (15) converges strongly to some member of Γ. Let Ax=A*x=x/2 for all x∈C with ∥A∥=1/2 and γ=3∈(0,1/∥A∥2). If {xn}⊂(0,1], we then have
(45)|un-S1un|=12|un|=12|xn+32[(1in+1)mnxn2-xn2]|=κn|xn|,
where κn:=(1/2)|1+(3/4)[(1/(in+1))mn-1]| with κ:=infn≥1κn>0. Define a nondecreasing function f:[0,∞)→[0,∞) by f(x)=κx. Since Γ=[-1,0], we then have
(46)f(d(xn,Γ))=f(|xn|)=κ|xn|≤|un-S1un|.
Similarly, we also can define a nondecreasing function h:[0,∞)→[0,∞) with h(0)=0 such that
(47)h(d(Axn,Q))≤|Axn-Ti0Axn|
for some i0≥1, which implies that, by Lemma 7 and Theorem 8, xn→x*∈Γ as n→∞.Theorem 10.
LetH1,H2,A,C and Q be the same as those in Lemma 7. Let {Si}:H1→H1 and {Ti}:H2→H2 be two sequences of nonexpansive mappings. Let {xn} be the following sequence generated by an arbitrarily chosen x1∈H1(48)un=xn+γA*(Tn*-I)Axn,xn+1=(1-αn)un+αnSn*un,n≥1,
where {αn} is a sequence in [α,1-α] for some α∈(0,1); γ∈(0,1/∥A∥2); Tn*=Tin,Sn*=Sin with in satisfying the positive integer equation: n=i+(m-1)m/2(m≥i,n=1,2,…). Then {xn} converges weakly to an x*∈Γ.Proof.
It is clear that both{Si} and {Ti} are asymptotically strict pseudocontractions. Then, by the proof of Lemma 7, we have
(49)limn→∞∥un-Sn*un∥=0,(50)limn→∞∥(Tn*-I)Axn∥=0.
In addition, we also have
(51)limn→∞∥un+1-un∥=∥xn+1-xn∥=0,
which implies that, by induction, for any nonnegative integer k,
(52)limn→∞∥un+k-un∥=limn→∞∥xn+k-xn∥=0.
For each k≥1, since
(53)∥un-Sn+k*un∥≤∥un-un+k∥+∥un+k-Sn+k*un∥≤∥un-un+k∥+∥un+k-Sn+k*un+k∥+∥Sn+k*un+k-Sn+k*un∥≤2∥un-un+k∥+∥un+k-Sn+k*un+k∥,
it follows from (49) and (52) that
(54)limn→∞∥un-Sn+k*un∥=0.
Now, for each i≥1, we claim that
(55)limn→∞∥un-Siun∥=0.
As a matter of fact, setting
(56)n=Nm+i,
where Nm=(m-1)m/2, m≥i, we obtain that
(57)∥un-Siun∥≤∥un-uNm∥+∥uNm-Siun∥≤∥un-uNm∥+∥uNm-SNm+i*uNm∥+∥SNm+i*uNm-Siun∥=∥un-uNm∥+∥uNm-SNm+i*uNm∥+∥SiuNm-Siun∥≤2∥un-uNm∥+∥uNm-SNm+i*uNm∥=2∥un-un-i∥+∥uNm-SNm+i*uNm∥.
Then, since Nm→∞ as n→∞, it follows from (52) and (54) that (55) holds obviously. Similarly, we have, for each i≥1,
(58)limn→∞∥Axn-TiAxn∥=0.
Next, since {un} is bounded, there exists a subsequence {uni}⊂{un} such that uni⇀x* (some point in H1). From (55) we have limi→∞∥uni-Sjuni∥=0 for each j≥1. By Lemma 5, each Sj is demiclosed at zero, so we know that x*∈∩j=1∞F(Sj). Moreover, it follows from (48) and (50) that
(59)xni=uni-γA*(Tni*-I)Axni⇀x*(i⟶∞).
Since A is a linear bounded operator, it yields that Axni⇀Ax*. In view of (58) we have
(60)limi→∞∥Axni-TjAxni∥=0,∀j≥1.
Again since each Tj is demiclosed at zero, we know that Ax*∈∩j=1∞F(Tj). This implies that x*∈Γ.
Note that each Hilbert space possesses Opial property, which guarantees that the weakly subsequential limit of{un} is unique. Consequently, {un} converges weakly to the point x*∈Γ. Since xn=un-γA*(Tn*-I)Axn, we know that {xn} converges weakly to x*∈Γ. The proof is completed.Remark 11.
Note that, from Remark2(3), the class of (γ,{kn})-asymptotically strict pseudocontractions is demiclosed at zero. Then, together with nonexpansiveness replaced by Lipschitz continuity, the two sequences of nonexpansive mappings {Si} and {Ti} in Theorem 10 can be extended to (γ,{kn})-asymptotically strict pseudocontractions as in Lemma 7.



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*Source: 817943-2013-09-25.xml*