# The Split Common Fixed Point Problem for Quasi-Total Asymptotically Nonexpansive Uniformly Lipschitzian Mappings

Authors: Jing Na; Lin Wang; Zhaoli Ma
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We introduce an algorithm for solving the split common fixed point problem for quasi-total asymptotically nonexpansive uniformly Lipschitzian mapping in Hilbert spaces. The results presented in this paper improve and extend some recent corresponding results.


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## Body

## 1. Introduction and Preliminaries

LetH1 and H2 be real Hilbert spaces with inner product 〈·,·〉 and norm ∥·∥. Let C and Q be nonempty closed convex subsets of H1 and H2, respectively. The split feasibility problem (SFP) is formulated as finding a point x* with the property(1.1)x*∈C,Ax*∈Q,
where A:H1→H2 is a bounded linear operator.TheSFP in finite-dimensional Hilbert 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 [1, 3–7]. The SFP in infinite-dimensional Hilbert spaces can be found in [8–16].The split common fixed point problem (SCFP) is a generalization of the split feasibility problem (SFP) and the convex feasibility problem (CFP), see [4]. Let S:H1→H1 and T:H2→H2 be two mappings satisfying F(S)={x∈H1:Sx=x}≠ϕ and F(T)={x∈H2:Tx=x}≠ϕ, respectively. The split common fixed point problem for mappings S and T is to find a point q∈H1 with the property(1.2)q∈F(S),Aq∈F(T),
where A is a bounded linear operator from H1 to H2. We use Γ to denote the set of solutions of SCFP (1.2).We first recall some definitions, notations, and conclusions which will be used 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*.A Banach spaceE is said to satisfy Opial's condition, if for any sequence {xn} in E, xn⇀x* implies that(1.3)liminf⁡n→∞⁡‖xn-x*‖<liminf⁡n→∞⁡‖xn-y‖,∀y∈Ewithy≠x*.
It is well known that every Hilbert space satisfies Opial's condition.Definition 1.1.
LetH be a real Hilbert space, T a mapping from H into itself, and the fixed point set F(T) of T nonempty.(1)
T is said to be quasi-nonexpansive if for all (x,q)∈H×F(T)(1.4)‖Tx-q‖≤‖x-q‖.(2)
T is said to be quasi-asymptotically nonexpansive if there exists a sequence {kn}⊂[1,∞) with kn→1 as n→∞ such that for all (x,q)∈H×F(T)(1.5)‖Tnx-q‖≤kn‖x-q‖,∀n≥1,(3)
T is said to be  ({μn},{ξn},ϕ)-quasi-total asymptotically nonexpansive if for all (x,q)∈H×F(T)(1.6)‖Tnx-q‖2≤‖x-q‖2+μnϕ(‖x-q‖)+ξn,∀n≥1,
where ϕ:[0,∞)→[0,∞) is a continuous and strictly increasing function with ϕ(0)=0 and {μn} and {ξn} are two nonnegative real sequences satisfying μn→0 and ξn→0 as n→∞. The class of mappings was introduced by Alber et al. [17] in 2006.(4)
T is said to be uniformly L-Lipschitzian if there exists a constant L>0 such that for all (x,y)∈H×H(1.7)‖Tnx-Tny‖≤L‖x-y‖.(5)
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 1.2.
From Definition1.1 we can see that a quasi-nonexpansive mapping or an asymptotically quasi-nonexpansive mapping is a ({μn},{ξn},ϕ)-quasi-total asymptotically nonexpansive mapping. But the converse does not hold.
In [9], Moudafi proposed the following iterative algorithm for solving split common fixed problem of quasi-nonexpansive mappings: for arbitrarily chosen x1∈H1,
(1.8){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 quasi-nonexpansive mappings and A:H1→H2 is a bounded linear operator.
Inspired by the work of Moudafi [9, 10], very recently, Qin et al. [12] introduced the following iterative algorithm to study multiples-sets feasibility problem of a finite family of asymptotically quasi-nonexpansive mappings. For arbitrarily chosen x1∈H1, {xn} is defined as follows:
(1.9)yn=xn+γβA*(Tnn-I)Axn,xn+1=(1-αn)yn+αnSnnyn,n≥1,
where Snn=Sn(modN)n, Tnn=Tn(modN)n, n≥1, A:H1→H2 is a bounded linear operator. They also proved that {xn} converges strongly or weakly to a multiple-sets split common fixed point of a finite family of asymptotically quasi-nonexpansive mappings in Hilbert spaces under some suitable conditions.
Motivated and inspired by the work of Moudafi [9, 10] and Qin et al. [12], in this paper, we study the SCFP (1.2) of ({μn},{ξn},ϕ)-quasi-total asymptotically nonexpansive mappings in Hilbert spaces and obtain some of the strong and weak convergence of the presented algorithm to some q∈Γ. The results obtained in this paper improve and extend the result of Qin et al. [12], Xu [14], and Yang [16] and others.
By using the well-known equality〈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. The proof is omitted.Proposition 1.3.
LetT:H→H be a ({μn},{ξn},ϕ)-quasi-total asymptotically nonexpansive mapping. Then for each q∈F(T) and x∈H, the following inequalities hold:
(1.10)〈x-Tnx,x-q〉≥12‖x-Tnx‖2-μn2ϕ(‖x-q‖)-ξn2,(1.11)〈x-Tnx,q-Tnx〉≤12‖x-Tnx‖2+μn2ϕ(‖x-q‖)+ξn2.Lemma 1.4 (18).
Let{an}, {bn}, and {δn} be sequences of nonnegative real numbers satisfying
(1.12)an+1≤(1+δn)an+bn.
If ∑n=1∞δn<∞ and ∑n=1∞bn<∞, then the limn→∞an exists.


## 2. Main Results

For solving split common fixed point problem (1.2), we assume that the following conditions are satisfied:(i)
H1 and H2 are two real Hilbert spaces, and A:H1→H2 is a bounded linear operator;(ii)
S:H1→H1 is uniformly L1-Lipschitzian and ({μn(1)},{ξn(1)},ϕ1)-quasi-total asymptotically nonexpansive mapping and T:H2→H2 is uniformly L2-Lipschitzian and ({μn(2)},{ξn(2)},ϕ2)-quasi-total asymptotically nonexpansive mapping satisfying the following conditions:(a)C:=F(S)≠∅, Q:=F(T)≠∅;(b)μn=max{μn(1),μn(2)}, ξn=max{ξn(1),ξn(2)}, n≥1, and ∑n=1∞μn<∞,∑n=1∞ξn<∞;(c)ϕ=max{ϕ1,ϕ2} and there exists two positive constant M and M* such that   ϕ(λ)≤M*λ2 for all λ≥M.Theorem 2.1.
LetH1,H2,A,S,T,C,Q,L1,L2,{μn},{ξn}, and ϕ be the same as above. Let {xn} be the sequence generated by: arbitrarily chosen x1∈H1(2.1)un=xn+γA*(Tn-I)Axn,xn+1=(1-αn)un+αnSnun,n≥1,
where {αn} is a sequence in [α,1-α] for some α∈(0,1) and γ>0 is a constant satisfying the following condition:
(d)
γ∈(0,1/∥A∥2).
IfT and S both are demi-closed at origin and Γ≠∅, then(I)
the sequence{xn} converges weakly to a split common fixed point x*∈Γ;(II)
in addition, ifS is also semicompact, then {xn} and {un} both converge strongly to a x*∈Γ.Proof.
(I) The proof will be divided into 4 steps.Step 1.
We prove thatlimn→∞∥xn-q∥ and limn→∞∥un-q∥ exist, and limn→∞∥xn-q∥=limn→∞∥un-q∥for each q∈Γ. Since ϕ is a continuous and increasing function, as λ≥0, we can obtain that
(2.2)ϕ(λ)≤ϕ(M)+M*λ2.
Taking q∈Γ, that is q∈F(S)=C and Aq∈F(T)=Q, and using (1.10), (2.1), and (2.2), we have
(2.3)‖xn+1-q‖2≤‖un-q-αn(un-Snun)‖2=‖un-q‖2-2αn〈un-q,un-Snun〉+αn2‖un-Snun‖2≤‖un-q‖2-αn(1-αn)‖un-Snun‖2+αnμn(ϕ(M)+M*(‖un-q‖)2)+αnξn=(1+αnμnM*)‖un-q‖2-αn(1-αn)‖un-Snun‖2+αnμnϕ(M)+αnξn,(2.4)‖un-q‖2=‖xn+γA*(Tn-I)Axn-q‖2=‖xn-q‖2+γ2‖A*(Tn-I)Axn‖2+2γ〈xn-q,A*(Tn-I)Axn〉=‖xn-q‖2+γ2〈(Tn-I)Axn,AA*(Tn-I)Axn〉+2γ〈xn-q,A*(Tn-I)Axn〉,
where
(2.5)γ2〈(Tn-I)Axn,AA*(Tn-I)Axn〉≤‖A‖2γ2〈(Tn-I)Axn,(Tn-I)Axn〉≤‖A‖2γ2‖(Tn-I)Axn‖2,(2.6)2γ〈xn-q,A*(Tn-I)Axn〉=2γ〈A(xn-q),(Tn-I)Axn〉=2γ〈A(xn-q)+(Tn-I)Axn-(Tn-I)Axn,(Tn-I)Axn〉=2γ〈A(xn-q)+(Tn-I)Axn-(Tn-I)Axn,(Tn-I)Axn〉=2γ(〈Tn(Axn)-Aq,(Tn-I)Axn〉-‖(Tn-I)Axn‖2).
Using (1.11), we have
(2.7)2γ(〈Tn(Axn)-Aq,(Tn-I)Axn〉-‖(Tn-I)Axn‖2)≤γ‖(Tn-I)Axn‖2+γμnϕ(‖Axn-Aq‖)+γξn-2γ‖(Tn-I)Axn‖2≤-γ‖(Tn-I)Axn‖2+γμnM*‖A‖2‖xn-q‖+γμnϕ(M)+γξn.
Substituting (2.5) and (2.7) into (2.4) and simplifying, we obtain
(2.8)‖un-q‖2≤‖xn-q‖2+γ2‖A‖2‖(Tn-I)Axn‖2-γ‖(Tn-I)Axn‖2+γμnM*‖A‖2‖xn-q‖2+γμnϕ(M)+γξn≤(1+γμnM*‖A‖2)‖xn-q‖2-γ(1-γ‖A‖2)‖(Tn-I)Axn‖2+γμnϕ(M)+γξn.
Substituting (2.8) into (2.3) and simplifying, we have
(2.9)‖xn+1-q‖2≤(1+αnμnM*){(1+γμnM*‖A‖2)‖xn-q‖2-γ(1-γ‖A‖2)‖(Tn-I)Axn‖2+γμnϕ(M)+γξn}-αn(1-αn)‖un-Snun‖2+αnμnϕ(M)+αnξn=(1+bn)‖xn-q‖2-(1+αnμnM*)γ(1-γ‖A‖2)‖(Tn-I)Axn‖2-αn(1-αn)‖un-Snun‖2+cn,
where
(2.10)bn=αnμnM*+γμnM*‖A‖2+γ‖A‖2αnμn2(M*)2,cn=(1+αnμnM*)(γμnϕ(M)+αnξn)+αnμnϕ(M)+αnξn.
Since ∑n=1∞μn<∞ and ∑n=1∞ξn<∞, so ∑n=1∞bn<∞ and ∑n=1∞cn<∞. By Condition (4), we have
(2.11)‖xn+1-q‖2≤(1+bn)‖xn-q‖2+cn.
Therefore, it follows from Lemma 1.4 that limn→∞∥xn-q∥ exists.We now prove that limn→∞∥un-q∥ exists for each q∈Γ. Since limn→∞∥xn-q∥ exists, from (2.9), we have
(2.12)(1+αnμnM*)γ(1-γ‖A‖2)‖(Tn-I)Axn‖2+αn(1-αn)‖un-Snun‖2≤‖xn-q‖2-‖xn+1-q‖2-bn‖xn-q‖2⟶0,asn⟶∞.
This together with Condition (4) implies that
(2.13)lim⁡n→∞‖u-Snun‖=0,(2.14)lim⁡n→∞‖(Tn-I)Axn‖=0.
Thus, since limn→∞∥xn-q∥ exists, it follows from (2.4) and (2.14) that limn→∞∥un-q∥ exists and limn→∞∥xn-q∥=limn→∞∥un-q∥.Step 2.
Now we prove thatlimn→∞∥xn+1-xn∥=0 and limn→∞∥un+1-un∥=0.
As a matter of fact, it follows from (2.1) that
(2.15)‖xn+1-xn‖=‖(1-αn)un+αnSnun-xn‖=‖(1-αn)(xn+γA*(Tn-I)Axn)+αnSnun-xn‖=‖(1-αn)γA*(Tn-I)Axn+αn(Snun-xn)‖=‖(1-αn)γA*(Tn-I)Axn+αn(Snun-un)+αn(un-xn)‖=‖(1-αn)γA*(Tn-I)Axn+αn(Snun-un)+αnγA*(Tn-I)Axn‖=‖γA*(Tn-I)Axn+αn(Snun-un)‖.
In view of (2.13) and (2.14) we have that
(2.16)lim⁡n→∞⁡‖xn+1-xn‖=0.
Similarly, it follows from (2.1), (2.14), and (2.16) that
(2.17)‖un+1-un‖=‖xn+1+γA*(Tn+1-I)Axn+1-(xn+γA*(Tn-I)Axn)‖≤‖xn+1-xn‖+γ‖A*(Tn+1-I)Axn+1‖+γ‖A*(Tn-I)Axn‖⟶0,(n⟶∞).Step 3.
Next, we prove that∥un-Sun∥→0 and ∥Axn-TAxn∥→0 as n→∞.
Settingηn:=∥un-Snun∥, since S is uniformly L-Lipschitzian continuous, it follows from (2.13), (2.16), and (2.17) that
(2.18)‖un-Sun‖≤‖un-Snun‖+‖Snun-Sun‖≤ηn+L‖Sn-1un-un‖≤ηn+L(‖Sn-1un-Sn-1un-1‖+‖Sn-1un-1-un‖)≤ηn+L2‖un-un-1‖+L(‖Sn-1un-1-un-1‖+‖un-1-un‖)≤ηn+L(1+L)‖un-un-1‖+Lηn-1⟶0,(n⟶∞).
Similarly, we have
(2.19)‖Axn-TAxn‖⟶0,(n⟶∞).Step 4.
Finally, we prove thatxn⇀x* and un⇀x*, where x*∈Γ. Since {un} is bounded, there exists a subsequence {uni} of {un} such that uni⇀x*(some point in H1). From (2.18) we have limi→∞∥uni-Suni∥=0. Since S is demi-closed at zero, we know that x*∈F(S). Moreover, it follows from (2.1) and (2.14) that
(2.20)xni=uni-γA*(Tni-I)Axni⇀x*.
Since A is a linear bounded operator, it gets Axni⇀Ax*. In view of (2.19) we have limi→∞∥Axni-TAxni∥=0.
Again sinceT is demi-closed at zero, we know that Ax*∈F(T). This implies that x*∈Γ.
Assume that there exists another subsequence{unk} of {un} such that {unk} converges weakly to a point y*∈H with y*≠x*. Using the same argument above, we know that y*∈Γ. Since each Hilbert space possesses Opial’s property, we have
(2.21)liminf⁡i→∞⁡‖uni-x*‖<liminf⁡i→∞⁡‖uni-y*‖=lim⁡n→∞⁡‖un-y*‖=liminf⁡k→∞⁡‖unk-y*‖<liminf⁡k→∞⁡‖unk-x*‖=lim⁡n→∞⁡‖un-x*‖=liminf⁡i→∞⁡‖uni-x*‖,
which is a contradiction. This implies that {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 of conclusion (I) is completed.Proof of Conclusion (II).
SinceS is semi-compact, it follows from Step 4 that there exists a subsequence of {uni} (without loss of generality, we still denote it by {uni}) such that {uni}→u*∈H (some point in H). Since {uni}⇀x*, this implies that x*=u*. And so {uni}→x*∈Γ as i→∞. Since limn→∞∥xn-q∥ and limn→∞∥un-q∥ exist, and limn→∞∥xn-q∥=limn→∞∥un-q∥ for each q∈Γ, we know that limn→∞∥un-x*∥=limn→∞∥xn-x*∥=0. This implies that {xn} and {un} both converge strongly to a x*∈Γ. The proof is completed.



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*Source: 768591-2012-05-15.xml*