# General Split Feasibility Problems in Hilbert Spaces

Authors: Mohammad Eslamian; Abdul Latif
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Introducing a general split feasibility problem in the setting of infinite-dimensional Hilbert spaces, we prove that the sequence generated by the purposed new algorithm converges strongly to a solution of the general split feasibility problem. Our results extend and improve some recent known results.


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## Body

## 1. Introduction

LetH and K be infinite-dimensional real Hilbert spaces, and let A:H→K be a bounded linear operator. Let {Ci}i=1p and {Qi}i=1r be the families of nonempty closed convex subsets of H and K, respectively.(a) Theconvex feasibility problem (CFP) is formulated as the problem of finding a point x⋆ with the property:
(1)x⋆∈⋂i=1pCi.(b) Thesplit feasibility problem (SEP) is formulated as the problem of finding a point x⋆ with the property:
(2)x⋆∈C,Ax⋆∈Q,
where C and Q are nonempty closed convex subsets of H and K, respectively.(c) Themultiple-set split feasibility problem (MSSFP) is formulated as the problem of finding a point x⋆ with the property:
(3)x⋆∈⋂i=1pCi,Ax⋆∈⋂i=1rQi.
Note that (MSSFP) reduces to (SEP) if we take p=r=1.There is a considerable investigation on CFP in view of its applications in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment planning [1]. The split feasibility problem SFP in the setting of finite-dimensional Hilbert spaces was first introduced by Censor and Elfving [2] for modelling inverse problems which arise from phase retrievals and in medical image reconstruction [3]. Since then, a lot of work has been done on finding a solution of SFP and MSSFP; see, for example, [2–25]. Recently, it is found that the SFP can also be applied to study the intensity-modulated radiation therapy; see, for example, [6, 16] and the references therein. Very recently, Xu [8] considered the SFP in the setting of infinite-dimensional Hilbert spaces.The original algorithm given in [2] involves the computation of the inverse A-1 provided it exists. In [8], Xu studied some algorithm and its convergence. In particular, he applied Mann’s algorithm to the SFP and purposed an algorithm which is proved to be weakly convergent to a solution of the SFP. He also established the strong convergence result, which shows that the minimum-norm solution can be obtained. In [7], Wang and Xu purposed the following cyclic algorithm to solve MSSFP:
(4)xn+1=PC[n](xn+γA*(PQ[n]-I)Axn),
where [n]∶=n(modp), (mod function take values in {1,2,…,p}), and γ∈(0,2/∥A∥2). They show that the sequence {xn} convergence weakly to a solution of MSSFP provided the solution exists. To study strong convergence to a solution of MSSFP, first we introduce a general form of the split feasibility problem for infinite families as follows.(d)General split feasibility problem (GSFP) is to find a point x* such that
(5)x⋆∈⋂i=1∞Ci,Ax⋆∈⋂i=1∞Qi.
We denote by Ω the solution set of GSFP.In this paper, using viscosity iterative method defined by Moudafi [21], we propose an algorithm for finding the solutions of the general split feasibility problem in a Hilbert space. We establish the strong convergence of the proposed algorithm to a solution of GSFP.


## 2. Preliminaries

Throughout the paper, we denote byH a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥. Let {xn} be a sequence in H and x∈H. Weak convergence of {xn} to x is denoted by xn⇀x, and strong convergence by xn→x. Let C be a closed and a convex subset of H. For every point x∈H, there exists a unique nearest point in C, denoted by PCx. This point satisfies
(6)∥x-PCx∥≤∥x-y∥,∀y∈C.
The operator PC is called the metric projection or the nearest point mapping of H onto C. The metric projection PC is characterized by the fact that PC(x)∈C and
(7)〈y-PC(x),x-PC(x)〉≤0,∀x∈H,y∈C.
Recall that a mapping T:C→C is called nonexpansive if
(8)∥Tx-Ty∥≤∥x-y∥,∀x,y∈C.
It is well known that PC is a nonexpansive mapping. It is also known that H satisfies Opial’s condition, that is, for any sequence {xn} with xn⇀x, the inequality
(9)liminfn→∞∥xn-x∥<liminfn→∞∥xn-y∥
holds for every y∈H with y≠x.Lemma 1.
LetH be a Hilbert space. Then, for all x,y∈H(10)∥x+y∥2≤∥x∥2+2〈y,x+y〉.Lemma 2 (see [22]).
LetH be a Hilbert space, and let {xn} be a sequence in H. Then, for any given sequence {λn}n=1∞⊂(0,1) with ∑n=1∞λn=1 and for any positive integer i,j with i<j,
(11)∥∑n=1∞λnxn∥2≤∑n=1∞λn∥xn∥2-λiλj∥xi-xj∥2.Lemma 3 (see [23]).
Assume that{an} is a sequence of nonnegative real numbers such that
(12)an+1≤(1-γn)an+γnδn+βn,n≥0,
where {γn}, {βn}, and {δn} satisfy the following conditions: (i)
γn⊂[0,1], ∑n=1∞γn=∞,(ii)
limsupn→∞δn≤0 or ∑n=1∞|γnδn|<∞,(iii)
βn≥0 for all n≥0 with ∑n=0∞βn<∞.Then, limn→∞an=0.Lemma 4 (see [24]).
Let{tn} be a sequence of real numbers such that there exists a subsequence {ni} of {n} such that tni<tni+1 for all i∈ℕ. Then, there exists a nondecreasing sequence {τ(n)}⊂ℕ such that τ(n)→∞, and the following properties are satisfied by all (sufficiently large) numbers n∈ℕ:
(13)tτ(n)≤tτ(n)+1,tn≤tτ(n)+1.
In fact
(14)τ(n)=max{k≤n:tk<tk+1}.Lemma 5 (demiclosedness principle [25]).
LetC be a nonempty closed and convex subset of a real Hilbert space H. Let T:C→C be a nonexpansive mapping such that Fix(T)≠∅. Then, T is demiclosed on C, that is, if yn⇀z∈C, and (yn-Tyn)→y, then (I-T)z=y.


## 3. Main Result

In the following result, we propose an algorithm and prove that the sequence generated by the proposed method converges strongly to a solution of the GSFP.Theorem 6.
LetH and K be real Hilbert spaces, and let A:H→K be a bounded linear operator. Let {Ci}i=1∞ and {Qi}i=1∞ be the families of nonempty closed convex subsets of H and K, respectively. Assume that GSFP (5) has a nonempty solution set Ω. Suppose that f is a self k-contraction mapping of H, and let {xn} be a sequence generated by x0∈H as
(15)xn+1=αnxn+βnf(xn)+∑i=1∞γn,iPCi(I-λn,iA*(I-PQi)A)xn,n≥0,
where αn+βn+∑i=1∞γn,i=1. If the sequences {αn}, {βn}, {γn,i}, and {λn,i} satisfy the following conditions: (i)
limn→∞βn=0 and ∑n=0∞βn=∞,(ii)
for eachi∈ℕ, liminfnαnγn,i>0,(iii)
for eachi∈ℕ, {λn,i}⊂(0,2/∥A∥2) and 0<liminfn→∞λn,i≤limsupn→∞λn,i<2/∥A∥2,then, the sequence {xn} converges strongly to x⋆∈Ω, where x⋆=PΩf(x⋆).Proof.
First, we show that{xn} is bounded. In fact, let z∈Ω. Since {λn,i}⊂(0,2/∥A∥2), the operators PCi(I-λn,iA*(I-PQi)A) are nonexpansive, and hence we have
(16)∥xn+1-z∥=∥∑i=1∞γn,iPCi(I-λn,iA*(I-PQi)A)xn-zαnxn+βnf(xn)=k+∑i=1∞γn,iPCi(I-λn,iA*(I-PQi)A)xn-z∥≤αn∥xn-z∥+βn∥f(xn)-z∥+∑i=1∞γn,i∥PCi(I-λn,iA*(I-PQi)A)xn-z∥≤αn∥xn-z∥+βn∥f(xn)-z∥+∑i=1∞γn,i∥xn-z∥≤(1-βn)∥xn-z∥+βn∥f(xn)-z∥≤(1-βn)∥xn-z∥+βn∥f(xn)-f(z)∥+βn∥f(z)-z∥≤(1-βn)∥xn-z∥+βnk∥xn-z∥+βn∥f(z)-z∥≤(1-(1-k))βn∥xn-z∥+(1-k)βn1-k∥f(z)-z∥≤max{∥xn-z∥,11-k∥f(z)-z∥}⋮≤max{∥x0-z∥,11-k∥f(z)-z∥},
which implies that {xn} is bounded, and we also obtain that {f(xn)} is bounded. Next, we show that for each i∈ℕ,
(17)limn→∞∥xn-PCi(I-λn,iA*(I-PQi)A)xn∥=0.
By using Lemma 2, for every z∈Ω and i∈ℕ, we have that
(18)∥xn+1-z∥2=∥∑j=1∞γn,jPCj(I-λn,jA*(I-PQj)A)xn-zαnxn+βnf(xn)=k+∑j=1∞γn,jPCj(I-λn,jA*(I-PQj)A)xn-z∥2≤αn∥xn-z∥2+βn∥f(xn)-z∥2+∑j=1∞γn,j∥PCj(I-λn,jA*(I-PQj)A)xn-z∥2-αnγn,i∥PCi(I-λn,iA*(I-PQi)A)xn-xn∥2≤αn∥xn-z∥2+βn∥f(xn)-z∥2+∑j=1∞γn,j∥xn-z∥2-αnγn,i∥PCi(I-λn,iA*(I-PQi)A)xn-xn∥2≤(1-βn)∥xn-z∥2+βn∥f(xn)-z∥2-αnγn,i∥PCi(I-λn,iA*(I-PQi)A)xn-xn∥2.
Hence, for each i∈ℕ, we have
(19)αnγn,i∥PCi(I-λn,iA*(I-PQi)A)xn-xn∥2≤∥xn-z∥2-∥xn+1-z∥2+βn∥f(xn)-z∥2.
Next, we show that there exists a uniquex⋆∈Ω such that x⋆=PΩf(x⋆). We observe that for each n≥0, x⋆∈Ω solves the GSFP (5) if and only if x⋆ solves the fixed point equation
(20)x⋆=PCi(I-λn,iA*(I-PQi)A)x⋆,i∈ℕ,
that is, the solution sets of fixed point equation (20) and GSFP (5) are the same (see for details [8]). Note that if {λn,i}⊂(0,2/∥A∥2), then the operators PCi(I-λn,iA*(I-PQi)A) are nonexpansive. Since the fixed point set of nonexpansive operators is closed and convex, the projection onto the solution set Ω is well defined whenever Ω≠∅. We observe that PΩ(f) is a contraction of H into itself. Indeed, since PΩ is nonexpansive,
(21)∥PΩ(f)(x)-PΩ(f)(y)∥≤∥f(x)-f(y)∥≤k∥x-y∥.
Hence, there exists a unique element x⋆∈Ω such that x⋆=PΩf(x⋆).
In order to prove thatxn→x⋆ as n→∞, we consider two possible cases.
Case1. Assume that {∥xn-x⋆∥} is a monotone sequence. In other words, for n0 large enough, {∥xn-x⋆∥}n≥n0 is either nondecreasing or nonincreasing. Since ∥xn-x⋆∥ is bounded we have ∥xn-x⋆∥ is convergent. Since limn→∞βn=0 and {f(xn)} is bounded, from (19) we get that
(22)limn→∞αnγn,i∥PCi(I-λn,iA*(I-PQi)A)xn-xn∥2=0.
By assuming that liminfnαnγn,i>0, we obtain
(23)limn→∞∥PCi(I-λn,iA*(I-PQi)A)xn-xn∥=0,∀i∈ℕ.
Now, we show that
(24)limsupn→∞〈f(x⋆)-x⋆,xn-x⋆〉≤0.
To show this inequality, we choose a subsequence {xnk} of {xn} such that
(25)limk→∞〈f(x⋆)-x⋆,xnk-x⋆〉=limsupn→∞〈f(x⋆)-x⋆,xn-x⋆〉.
Since {xnk} is bounded, there exists a subsequence {xnkj} of {xnk} which converges weakly to w. Without loss of generality, we can assume that xnk⇀w and λn,i→λi∈(0,2/∥A∥2) for each i∈ℕ. From (23), we have
(26)∥PCi(I-λiA*(I-PQi)A)xn-xn∥≤∥PCi(I-λiA*(I-PQi)A)xn≤k-PCi(I-λn,iA*(I-PQi)A)xn∥+∥PCi(I-λn,iA*(I-PQi)A)xn-xn∥≤∥(I-λiA*(I-PQi)A)xn≤k-(I-λn,iA*(I-PQi)A)xn∥+∥PCi(I-λn,iA*(I-PQi)A)xn-xn∥≤|λi-λn,i|∥A*(I-PQi)Axn∥+∥PCi(I-λn,iA*(I-PQi)A)xn-xn∥→0asn→∞.
Notice that for each i∈ℕ, PCi(I-λiA*(I-PQi)A) is nonexpansive. Thus, from Lemma 5, we have w∈Ω. Therefore, it follows that
(27)limsupn→∞〈f(x⋆)-x⋆,xn-x⋆〉=limk→∞〈f(x⋆)-x⋆,xnk-x⋆〉=〈f(x⋆)-x⋆,w-x⋆〉≤0.
Finally, we show that xn→PΩf(x⋆). Applying Lemma 1, we have that
(28)∥xn+1-x⋆∥2=∥∑i=1∞γn,i(PCi(I-λn,iA*(I-PQi)A)xn-x⋆)αn(xn-x⋆)=k+∑i=1∞γn,i(PCi(I-λn,iA*(I-PQi)A)xn-x⋆)∥2+2βn〈f(xn)-x⋆,xn+1-x⋆〉≤(1-βn)2∥xn-x⋆∥2+2βn〈f(xn)-f(x⋆),xn+1-x⋆〉+2βn〈f(x⋆)-x⋆,xn+1-x⋆〉≤(1-βn)2∥xn-x⋆∥2+2βnk∥xn-x⋆∥∥xn+1-x⋆∥+2βn〈f(x⋆)-x⋆,xn+1-x⋆〉≤(1-βn)2∥xn-x⋆∥2+βnk{∥xn-x⋆∥2+∥xn+1-x⋆∥2}+2βn〈f(x⋆)-x⋆,xn+1-x⋆〉.
This implies that
(29)∥xn+1-x⋆∥2≤(1-βn)2+βnk1-βnk∥xn-x⋆∥2+2βn1-βnk〈f(x⋆)-x⋆,xn+1-x⋆〉=1-2βn+βnk1-βnk∥xn-x⋆∥2+βn21-βnk∥xn-x⋆∥2+2βn1-βnk〈f(z)-x⋆,xn+1-x⋆〉≤(1-2(1-k)βn1-βnk)∥xn-x⋆∥2+2(1-k)βn1-βnk{βnM2(1-k)+2(1-k)βn1-βnkm+11-k〈f(x⋆)-x⋆,xn+1-x⋆〉}≤(1-ηn)∥xn-x⋆∥2+ηnδn,
where
(30)δn=βnM2(1-k)+11-k〈f(x⋆)-x⋆,xn+1-x⋆〉,M=sup{∥xn-x⋆∥2:n≥0} and ηn=2(1-k)βn/(1-βnk). It is easy to see that ηn→0, ∑n=1∞ηn=∞ and limsupn→∞δn≤0. Hence, by Lemma 3, the sequence {xn} converges strongly to x⋆=PΩf(x⋆).
Case2. Assume that {∥xn-x⋆∥} is not a monotone sequence. Then, we can define an integer sequence {τ(n)} for all n≥n0 (for some n0 large enough) by
(31)τ(n)=max{k∈ℕ;k≤n:∥xk-x⋆∥<∥xk+1-x⋆∥}.
Clearly, τ(n) is a nondecreasing sequence such that τ(n)→∞ as n→∞ and for all n≥n0,
(32)∥xτ(n)-x⋆∥<∥xτ(n)+1-x⋆∥.
From (19), we obtain that
(33)limn→∞∥PCi(I-λτ(n),iA*(I-PQi)A)xτ(n)-xτ(n)∥=0.
Following an argument similar to that in Case 1, we have
(34)limsupn→∞〈f(x⋆)-x⋆,xτ(n)+1-x⋆〉≤0.
And by similar argument, we have
(35)∥xτ(n)+1-x⋆∥2≤(1-ητ(n))∥xτ(n)-x⋆∥2+ητ(n)δτ(n),
where ητ(n)→0, ∑n=1∞ητ(n)=∞ and limsupn→∞δτ(n)≤0. Hence, by Lemma 3, we obtain limn→∞∥xτ(n)-x⋆∥=0 and limn→∞∥xτ(n)+1-x⋆∥=0. Now, from Lemma 4, we have
(36)0≤∥xn-x⋆∥≤max{∥xτ(n)-x⋆∥,∥xn-x⋆∥}≤∥xτ(n)+1-x⋆∥.
Therefore, {xn} converges strongly to x⋆=PΩf(x⋆).For finite collections we have the following consequence of Theorem6.Theorem 7.
LetH and K be real Hilbert spaces, and let A:H→K be a bounded linear operator. Let{Ci}i=1p be a family of nonempty closed convex subsets in H, and let {Qi}i=1p be a family of nonempty closed convex subsets in K. Assume that MSSFP has a nonempty solution set Ω. Let u be an arbitrary element in H, and let {xn} be a sequence generated by x0∈H and
(37)xn+1=αnxn+βnu+∑i=1pγn,iPCi(I-λn,iA*(I-PQi)A)xn,n≥0,
where αn+βn+∑i=1pγn,i=1. If the sequences {αn}, {βn}, {γn,i}, and {λn,i} satisfy the following conditions: (i)
limn→∞βn=0 and ∑n=0∞βn=∞,(ii)
for alli∈{1,2,…,p}, liminfnαnγn,i>0,(iii)
for alli∈{1,2,…,p}, {λn,i}⊂(0,2/∥A∥2) and
(38)0<liminfn→∞λn,i≤limsupn→∞λn,i<2∥A∥2,then the sequence {xn} converges strongly to x⋆∈Ω, where x⋆=PΩu.



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