# Multiple-Set Split Feasibility Problems forκ-Strictly Pseudononspreading Mapping in Hilbert Spaces

Authors: Jing Quan; Shih-sen Chang; Xiang Zhang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The purpose of this paper is to prove some weak and strong convergence theorems for solving the multiple-set split feasibility problems forκ-strictly pseudononspreading mapping in infinite-dimensional Hilbert spaces by using the proposed iterative method. The main results presented in this paper extend and improve the corresponding results of Xu et al. (2006), of Osilike et al. (2011), and of many other authors.


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## Body

## 1. Introduction and Preliminaries

Censor and Elfving first introduced the split feasibility problem (SFP) [1] in finite dimensional spaces for modeling inverse problems. The SFP can be used in various disciplines such as medical image reconstruction [2], image restoration, computer tomograph, and radiation therapy treatment planning [3–5]. The multiple-set split feasibility problem (MSSFP) was studied in [4–6].In the sequel, we always assume thatH1, H2 are two real Hilbert spaces and denote by “→” and “⇀” the strong and weak convergence, respectively.The so-calledmultiple-set split feasibility problem (MSSFP) is to find x*∈C such that Ax*∈Q, where A:H1→H2 is a bounded linear operator, Si and Ti, i=1,2,…,N are the families of mappings, Si:H1→H1 and Ti:H2→H2, C≔⋂i=1N‍F(Si), and Q≔⋂i=1N‍F(Ti), where F(Si)={x∈H1:Six=x} and F(Ti)={y∈H2:Tiy=y} denote the sets of fixed points of Si and Ti, respectively. In the sequel, we use Γ to denote the set of solutions of the MSSFP; that is,
(1)Γ={x∈C:Ax∈Q}.Recently, Kohsaka and Takahashi [7, 8] introduced an important class of mappings which is called the class of nonspreading mappings.Definition 1 (see [7, 8]).
LetK to be a nonempty closed convex subset of a Hilbert space H. A mapping T:K→K is said to be nonspreading, if
(2)2∥Tx-Ty∥2≤∥Tx-y∥2+∥Ty-x∥2,∀x,y∈K.In [9], Iemoto and Takahashi proved that this definition is equivalent to the following.Definition 2 (see [9]).
LetK be a nonempty closed convex subset of a Hilbert space H. A mapping T:K→K is said to be  nonspreading, if
(3)∥Tx-Ty∥2≤∥x-y∥2+2〈x-Tx,y-Ty〉,∀x,y∈K.Browder and Petryshyn [10] proposed the following κ-strictly pseudononspreading mapping.Definition 3 (see [10]).
LetH be a real Hilbert space. We say that a mapping T:D(T)⊂H→H is κ-strictly pseudononspreading if there exists κ∈[0,1), such that
(4)∥Tx-Ty∥2≤∥x-y∥2+κ∥x-Tx-(y-Ty)∥2+2〈x-Tx,y-Ty〉,∀x,y∈D(T).
Clearly every nonspreading mapping is κ-strictly pseudononspreading.Osilike and Isiogugu [11] introduced a class of nonspreading type mappings which is more general than that of the mappings studied in [12] in Hilbert spaces and proved some weak and strong convergence theorems in real Hilbert spaces. Recently, the split feasibility problem also was considered in the work by Deepho and Kumam [13, 14] and Sunthrayuth et al. [15], and some weak and strong convergence theorems are proved in real Hilbert spaces.The purpose of this paper is to study the multiple-set split feasibility problem (MSSFP) forκ-strictly pseudononspreading mappings in the framework of infinite dimensional Hilbert spaces.In the sequel, we recall some definitions, notations, and conclusions which will be needed in proving our main results.Definition 4 (see [3]).
LetE be a real Banach space. A mapping T with domain D(T) and range R(T) in E is said to be demiclosed at origin if for any sequence {xn} in D(T) which converges weakly to a point x*∈D(T) and ∥(I-T)xn∥ converges strongly to 0, then Tx*=x*.Definition 5.
A Banach spaceE is said to have Opial property if, for any sequence {xn} with xn⇀x*, we have
(5)liminfn→∞∥xn-x*∥<liminfn→∞∥xn-y∥
for all y∈E with y≠x*.Remark 6.
It is well known that each Hilbert space possesses Opial property.Definition 7.
A mappingS:K→K is said to be semicompact, if, for any bounded sequence {xn}⊂K with limn→∞∥xn-Sxn∥=0, there exists a subsequence {xni}⊂{xn} such that {xni} converges strongly to some point x*∈K.Lemma 8 (see [11]).
Let H be a real Hilbert space; then the following results hold.(i)
For allx, y∈H and for all t∈[0,1],
(6)∥tx+(1-t)y∥2=t∥x∥2+(1-t)∥y∥2-t(1-t)∥x-y∥2.(ii)
∥x+y∥2≤∥x∥2+2〈y,x+y〉.(iii)
If{xn}n=1∞ is a sequence in H which converges weakly to z∈H, then
(7)limsupn→∞∥xn-y∥2=limsupn→∞∥xn-z∥2+∥z-y∥2,∀y∈H.Definition 9.
LetK be a nonempty closed convex subset of a real Hilbert space H. The metric projectionPK:H→K is a mapping such that, for each x∈H, PKx is the unique point in K such that ∥x-PKx∥≤∥x-y∥, ∀y∈K. It is known that, for each x∈H,
(8)〈x-PKx,y-PKx〉≤0,∀y∈K.Lemma 10 (see [11]).
LetK be a nonempty closed convex subset of a real Hilbert space H, and let T:K→K be a κ-strictly pseudononspreading mapping. If F(T)≠∅, then it is closed and convex.Lemma 11 (see [11]).
LetK be a nonempty closed convex subset of a real Hilbert space H, and let T:K→K be a κ-strictly pseudononspreading mapping. Then (I-T) is demiclosed at 0.


## 2. Main Results

Theorem 12.
LetH1, H2, A, {Si}, {Ti}, C, Q be the same as aforementioned. For each i=1,2,…,N, let Ti be a κi-strictly pseudononspreading mapping and let Si be a ϱi-strictly pseudononspreading mapping. Let {xn} be the sequence generated by
(9)x1∈H1chosenarbitrarily,un=xn+γA*(Tn(modN)-I)Axn,xn+1=(1-αn)un+αnSn(modN)un,
where γ∈(0,(1-κ)/λ) with λ being the spectral of the operator A*A and κ=max{κ1,κ2,…,κN}∈(0,1), and αn is a sequence in (0,1-ϱ] with ϱ=max{ϱ1,ϱ2,…,ϱN}∈(0,1). If Γ≠∅ (where Γ is the set of solutions of the MSSFP defined by (1)), then the sequence {xn} converges weakly to a point x*∈Γ.Proof.
The proof is divided into four steps.
(I) We first prove thatlimn→∞∥xn-p∥ exists for any p∈Γ.
Sincep∈Γ, p∈C≔⋂i=1N‍F(Si) and Ap∈Q≔⋂i=1N‍F(Ti). It follows from (9) that
(10)∥xn+1-p∥2=∥un-p+αn(Sn(modN)un-un)∥2=∥un-p∥2+2αn〈un-p,Sn(modN)un-un〉+αn2∥un-Sn(modN)un∥2.
BecauseSi is ϱi-strictly pseudononspreading, for each v∈H1, we have
(11)∥Sn(modN)un-Sn(modN)v∥2≤∥un-v∥2+ϱ∥un-Sn(modN)un-(v-Sn(modN)v)∥2+2〈un-Sn(modN)un,v-Sn(modN)v〉.
Takingv=p, we have
(12)∥Sn(modN)un-p∥2≤∥un-p∥2+ϱ∥un-Sn(modN)un∥2.
This implies that(13)∥Sn(modN)un-p∥2=∥Sn(modN)un-un+un-p∥2=∥Sn(modN)un-un∥2+2〈Sn(modN)un-un,un-p〉+∥un-p∥2≤∥un-p∥2+ϱ∥un-Sn(modN)un∥2.
Thus it yields that(14)2αn〈Sn(modN)un-un,un-p〉≤αn(ϱ-1)∥un-Sn(modN)un∥2.
Substituting (14) into (10) and simplifying, we have
(15)∥xn+1-p∥2≤∥un-p∥2+αn(ϱ-1)∥un-Sn(modN)un∥2+αn2∥un-Sn(modN)un∥2=∥un-p∥2-αn(1-ϱ-αn)∥un-Sn(modN)un∥2.
On the other hand,(16)∥un-p∥2=∥xn-p+γA*(Tn(modN)-I)Axn∥2=∥xn-p∥2+2γ〈xn-p,A*(Tn(modN)-I)Axn〉+γ2∥A*(Tn(modN)-I)Axn∥2=∥xn-p∥2+2γ〈xn-p,A*(Tn(modN)-I)Axn〉+γ2〈A*(Tn(modN)-I)Axn,A*(Tn(modN)-I)Axn〉=∥xn-p∥2+2γ〈xn-p,A*(Tn(modN)-I)Axn〉+γ2〈AA*(Tn(modN)-I)Axn,(Tn(modN)-I)Axn〉≤∥xn-p∥2+2γ〈xn-p,A*(Tn(modN)-I)Axn〉+γ2∥A∥2∥(Tn(modN)-I)Axn∥2.
SinceTi is κi-strictly pseudononspreading and noting that Ap∈⋂i=1N‍F(Ti), we have
(17)∥Tn(modN)Axn-Ap∥2=∥Tn(modN)Axn-Axn+Axn-Ap∥2=∥Tn(modN)Axn-Axn∥2+∥Axn-Ap∥2+2〈Tn(modN)Axn-Axn,Axn-Ap〉≤∥Axn-Ap∥2+κ∥Tn(modN)Axn-Axn∥2.
This leads to(18)〈Tn(modN)Axn-Axn,Axn-Ap〉≤κ-12∥Tn(modN)Axn-Axn∥2.
By (18), we have
(19)〈Tn(modN)Axn-Axn,Tn(modN)Axn-Ap〉=〈Tn(modN)Axn-Axn,Tn(modN)Axn+Axn-Axn-Ap〉=∥(Tn(modN)-I)Axn∥2+〈Tn(modN)Axn-Axn,Axn-Ap〉≤∥(Tn(modN)-I)Axn∥2+κ-12∥(Tn(modN)-I)Axn∥2=κ+12∥(Tn(modN)-I)Axn∥2.
It follows from (19) that
(20)2γ〈xn-p,A*(Tn(modN)-I)Axn〉=2γ〈A(xn-p),(Tn(modN)-I)Axn〉=2γ〈A(xn-p)+(Tn(modN)-I)Axn-(Tn(modN)-I)Axn,(Tn(modN)-I)Axn〉=2γ〈Tn(modN)Axn-Ap,(Tn(modN)-I)Axn〉-2γ∥(Tn(modN)-I)Axn∥2≤[γ(1+κ)-2γ]∥(Tn(modN)-I)Axn∥2=[γ(κ-1)]∥(Tn(modN)-I)Axn∥2.
By using (15), (16), (19), and (20), we have
(21)∥xn+1-p∥2≤∥xn-p∥2+γ2∥A∥2∥(Tn(modN)-I)Axn∥2+[γ(κ-1)]∥(Tn(modN)-I)Axn∥2-αn(1-σ-αn)∥un-Sn(modN)un∥2≤∥xn-p∥2-γ(1-κ-γ∥A∥2)∥(Tn(modN)-I)Axn∥2-αn(1-ϱ-αn)∥un-Sn(modN)un∥2≤∥xn-p∥2.
This shows thatlimn→∞∥xn-p∥ exists.
(II) We now prove thatlimn→∞∥un-p∥ exists.
In fact, by (21), we have
(22)[γ(1-κ-γ∥A∥2)]∥(Tn(modN)-I)Axn∥2+αn(1-ϱ-αn)∥un-Sn(modN)un∥2≤∥xn-p∥2-∥xn+1-p∥2.
This implies that(23)limn→∞∥(Tn(modN)-I)Axn∥=0,(24)limn→∞∥un-Sn(modN)un∥=0.
By virtue of (16), (23), and (24), it follows that limn→∞∥un-p∥ exists and limn→∞∥xn-p∥=limn→∞∥un-p∥.
(III) Now, we prove thatlimn→∞∥xn+1-xn∥=0 and  limn→∞∥un+1-un∥=0.
In fact, it follows from (9) that
(25)∥xn+1-xn∥=∥(1-αn)un+αnSn(modN)un-xn∥=∥(1-αn)(xn+γA*(Tn(modN)-I)Axn)+αnSn(modN)un-xn∥=∥(1-αn)(γA*(Tn(modN)-I)Axn)+αn(Sn(modN)un-xn)∥=∥(1-αn)(γA*(Tn(modN)-I)Axn)+αn(Sn(modN)un-un)+αn(un-xn)∥=∥(1-αn)(γA*(Tn(modN)-I)Axn)+αn(Sn(modN)un-un)+αnγA*(Tn(modN)-I)Axn∥=∥γA*(Tn(modN)-I)Axn+αn(Sn(modN)un-un)∥.
This together with (23) and (24) leads to limn→∞∥xn+1-xn∥=0.
Similarly, it follows from (9), (23), and (25) that
(26)∥un+1-un∥=∥xn+1+γA*(Tn+1(modN)-I)Axn+1-[xn+γA*(Tn(modN)-I)Axn]∥≤∥xn+1-xn∥+∥γA*(Tn+1(modN)-I)Axn+1∥+∥[γA*(Tn(modN)-I)Axn]∥⟶0(asn→∞).
(IV) Finally, we prove thatxn⇀x* and un⇀x*, which is a solution of the MSSFP.
In fact, since{un} is bounded, there exists a subsequence {uni}⊂{un} such that {uni}⇀x*∈H1. Hence, for any positive integer j=1,2,…,N, there exists a subsequence {ni(j)}⊂{ni} with ni(j)(modN)=j such that {uni(j)}⇀x*. Again, by (24) we know that ∥uiN+j-SjuiN+j∥→0, as n→∞; therefore, we have that
(27)∥uni(j)-Sjuni(j)∥⟶0,asni(j)⟶∞.
SinceSj is demiclosed at zero, it follows that x∈F(Sj). By the arbitrariness of j=1,2,…,N, we have
(28)x*∈C≔⋂i=1N‍F(Si).
Moreover, from (9) and (24), we have xni=uni-γA*(Tni(modN)-I)Axni⇀x*. Since A is a bounded linear operator, it follows that Axni⇀Ax*. For any positive integer k=1,2,…,N, there exists a subsequence {xni(k)}⊂{xni} with ni(k)(modN)=k such that Axni(k)⇀Ax* and ∥Axni(k)-TkAxni(k)∥→0. Since Tk is demiclosed at zero, we have Ax*∈F(Tk). By the arbitrariness of k∈{1,2,…,N}, it follows that Ax*∈Q≔⋂k=1N‍F(Tk). This together with x*∈C shows that x*∈Γ; that is, x* is a solution to the MSSFP.
Now, we prove thatxn⇀x* and un⇀x*.
Suppose on the contrary that there exists another subsequence{unl}⊂{un} such that {unl}⇀y*∈Γ with y*≠x*. Consequently, by virtue of the existence of limn→∞∥xn-p∥ and the Opial property of Hilbert space, we have
(29)liminfni→∞∥uni-x*∥<liminfni→∞∥uni-y*∥=liminfn→∞∥un-y*∥=liminfnj→∞∥unj-y*∥<liminfnj→∞∥unj-x*∥=liminfn→∞∥un-x*∥=liminfni→∞∥uni-x*∥.
This is a contradiction. Thereforeun⇀x*. By (9) and (24), we have
(30)xn=un-γA*(Tn(modN)n-I)Axn⇀x*.
Therefore, the conclusion follows.
This completes the proof of Theorem12.Theorem 13.
LetH1, H2, A, {Si}, {Ti}, C, Q be the same as in Theorem 12. For each i=1,2,…,N, let Ti be a κi-strictly pseudononspreading mapping, and Si be a ϱi-strictly pseudononspreading mapping. Let {xn} be the sequence generated by
(31)x1∈H1chosenarbitrarily,un=xn+γA*(Tn(modN)-I)Axn,xn+1=(1-αn)un+αnSn(modN)un,
where γ∈(0,(1-κ)/λ) with λ being the spectral of the operator A*A and κ=max{κ1,κ2,…,κN}∈(0,1), and {αn} is a sequence in (0,1-ϱ] with ϱ=max{ϱ1,ϱ2,…,ϱN}∈(0,1). If Γ≠∅ and if there exists a positive integer j such that Sj is semicompact, then the sequence {xn} converges strongly to a point x*∈Γ.Proof.
Without loss of generality, we can assume thatS1 is semicompact. It follows from (27) that
(32)∥uni(1)-S1uni(1)∥⟶0,ni(1)⟶∞.
Therefore, there exists a subsequence of{uni(1)} (for the sake of convenience, we still denote it by {uni(1)}), such that uni(1)→u*∈H1. Since uni(1)⇀x*,  x*=u*, and so uni(1)→x*∈Γ. By virtue of the existence of limn→∞∥xn-p∥, we know that
(33)limn→∞∥un-x*∥=0,limn→∞∥xn-x*∥=0.
That is,{un} and {xn} both converge strongly to the point x*∈Γ. This completes the proof of Theorem 13.



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