# Strong Convergence of the Viscosity Approximation Process for the Split Common Fixed-Point Problem of Quasi-Nonexpansive Mappings

Authors: Jing Zhao; Songnian He
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

Very recently, Moudafi (2011) introduced an algorithm with weak convergence for the split common fixed-point problem. In this paper, we will continue to consider the split common fixed-point problem. We discuss the strong convergence of the viscosity approximation method for solving the split common fixed-point problem for the class of quasi-nonexpansive mappings in Hilbert spaces. Our results improve and extend the corresponding results announced by many others.


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## Body

## 1. Introduction and Preliminary

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 C and Q be nonempty closed convex subset of real Hilbert spaces H1 and H2, respectively. The split feasibility problem (SFP) is to find a point (1.1)x∈Csuch thatAx∈Q,
where A:H1→H2 is a bounded linear operator. The SFP 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]. The SFP attracts many authors' attention due to its application in signal processing. Various algorithms have been invented to solve it (see [3–9] and references therein).Note that the split feasibility problem (1.1) can be formulated as a fixed-point equation by using the fact (1.2)PC(I-γA*(I-PQ)A)x*=x*;
that is, x* solves the SFP (1.1) if and only if x* solves the fixed point equation (1.2) (see [10] for the details). This implies that we can use fixed-point algorithms (see [11–13]) to solve SFP. A popular algorithm that solves the SFP (1.1) is due to Byrne's CQ algorithm [2] which is found to be a gradient-projection method (GPM) in convex minimization. Subsequently, Byrne [3] applied KM iteration to the CQ algorithm, and Zhao and Yang [14] applied KM iteration to the perturbed CQ algorithm to solve the SFP. It is well known that the CQ algorithm and the KM algorithm for a split feasibility problem do not necessarily converge strongly in the infinite-dimensional Hilbert spaces.The split common fixed-point problem (SCFP) is a generalization of the split feasibility problem (SFP) and the convex feasibility problem (CFP); see [15]. In this paper, we introduce and study the convergence properties of a viscosity approximation algorithm for solving the SCFP for the class of quasi-nonexpansive operators S such that I-S is demiclosed at the origin.Now let us first recall the definition of quasi-nonexpansive operators which appear naturally when using subgradient projection operator techniques in solving some feasibility problems, and also some definitions of classes of operators often used in fixed-point theory and which are commonly encountered in the literature.LetT:H→H be a mapping. A point x∈H is said to be a fixed point of T provided that Tx=x. In this paper, we use F(T) to denote the fixed-point set and use → and ⇀ to denote the strong convergence and weak convergence, respectively. We use ωw(xk)={x:∃xkj⇀x} stand for the weak ω-limit set of {xk}.(i)
A mappingT:H→H belongs to the general class ΦQ of (possibly discontinuous) quasi-nonexpansive mappings if
(1.3)‖Tx-q‖≤‖x-q‖,∀(x,q)∈H×F(T).(ii)
A mappingT:H→H belongs to the set ΦN of nonexpansive mappings if
(1.4)‖Tx-Ty‖≤‖x-y‖,∀(x,y)∈H×H.(iii)
A mappingT:H→H belongs to the set ΦFN of firmly nonexpansive mappings if
(1.5)‖Tx-Ty‖2≤‖x-y‖2-‖(x-y)-(Tx-Ty)‖2,∀(x,y)∈H×H.(iv)
A mappingT:H→H belongs to the set ΦFQ of firmly quasi-nonexpansive mappings if
(1.6)‖Tx-q‖2≤‖x-q‖2-‖x-Tx‖2,∀(x,q)∈H×F(T).It is easily observed thatΦFN⊂ΦN⊂ΦQ and that ΦFN⊂ΦFQ⊂ΦQ. Furthermore, ΦFN is well known to include resolvents and projection operators, while ΦFQ contains subgradient projection operators (see, e.g., [16] and the reference therein).A mappingT:H→H is called demiclosed at the origin if any sequence {xn} weakly converges to x, and if the sequence {Txn} strongly converges to 0, then Tx=0. A mapping f:H→H is called a contraction of modulus ρ∈[0,1) if (1.7)‖fx-fy‖≤ρ‖x-y‖,∀(x,y)∈H×H.In what follows, we will focus our attention on the following general two-operator split common fixed-point problem:(1.8)findx*∈CsuchthatAx*∈Q,
where A:H1→H2 is a bounded linear operator, U:H1→H1 and S:H2→H2 are two quasi-nonexpansive operators with nonempty fixed-point sets F(U)=C and F(S)=Q, and denote the solution set of the two-operator SCFP by (1.9)Γ={y∈C;Ay∈Q}.Recall thatF(U) and F(S) are nonempty closed convex subsets of H1 and H2, respectively. If Γ≠∅, we have Γ which is close convex subset of H1. To solve (1.8), Censor and Segal [15] proposed and proved, in infinite-dimensional spaces, the convergence of the following algorithm: (1.10)xk+1=U(xk+γAt(S-I)Axk),k∈N,whereγ∈(0,2/λ), with λ being the largest eigenvalue of the matrix AtA (At stands for matrix transposition). Very recently, Moudafi [17] introduced the following relaxed algorithm: (1.11)xk+1=(1-αk)uk+αkU(uk),k∈N,whereuk=xk+γβA*(S-I)Axk, β∈(0,1), αk∈(0,1), and γ∈(0,1/λβ), with λ being the spectral radius of the operator A*A. Moudafi proved weak convergence result of the algorithm in Hilbert spaces.Inspired by their work, we introduce the following viscosity approximation algorithm.Algorithm 1.
Initialization: Letx0∈H be arbitrary.
Iterative step: SetT=U(I+γA*(S-I)A). For k∈N, let
(1.12)xk+1=αkf(xk)+(1-αk)((1-ωk)xk+ωkTxk),
where f:H→H is a contraction of modulus ρ, ωk∈(0,1/2), γ∈(0,1/λ) with λ being the spectral radius of the operator A*A, and αk∈(0,1).
This paper establishes the strong convergence of the sequence given by (1.12) to the unique solution of the variational inequality problem VIP(I-f,Γ):(1.13)findx*∈Γsuchthat〈(I-f)x*,v-x*〉≥0,∀v∈Γ.
Now we give a series of preliminary results needed for the convergence analysis of algorithm (1.12).Lemma 1.1.
LetH be a real Hilbert space and T:H→H a quasi-nonexpansive mapping.  Then, the following properties are reached: (i)
〈x,y〉=-12∥x-y∥2+12∥x∥2+12∥y∥2,∀(x,y)∈H×H;(ii)
〈x-Tx,x-q〉≥12∥x-Tx∥2 and 〈x-Tx,q-Tx〉≤12∥x-Tx∥2,∀(x,q)∈H×F(T).Remark 1.2.
LetF:=I-f, wheref is the contraction defined in (1.7). It is a simple matter to see that the operator F is (1-ρ)  strongly monotone over H;that is,(1.14)〈Fx-Fy,x-y〉≥(1-ρ)‖x-y‖2,∀(x,y)∈H×H.The next result is of fundamental importance for the techniques of analysis used in this paper. It was established in [18], and its proof is given for the sake of completeness.Lemma 1.3 (see [18, Lemma 1.3]).
Let{δn} be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence {δnj}j≥0 of  {δn} which satisfies δnj<δnj+1 for  all j≥0. Also consider the sequence of integers {τ(n)}n≥n0 defined by
(1.15)τ(n)=max⁡{k≤n∣δk<δk+1}.
Then {τ(n)}n≥n0 is a nondecreasing sequence verifying limn→∞τ(n)=∞, and, for all n≥n0, it holds that δτ(n)≤δτ(n)+1 and one has
(1.16)δn≤δτ(n)+1.Proof.
Clearly, we can see that{τ(n)} is a well-defined sequence, and the fact that it is nondecreasing is obvious as well as limn→∞τ(n)=∞ and δτ(n)≤δτ(n)+1. Let us prove (1.16). It is easily observed that τ(n)≤n. Consequently, we prove (1.16) by distinguishing the three cases: (c1) τ(n)=n; (c2) τ(n)=n-1; (c3) τ(n)<n-1. In the first case (i.e., τ(n)=n), (1.16) is immediately given by δτ(n)≤δτ(n)+1. In the second case (i.e., τ(n)=n-1), (1.16) becomes obvious. In the third case (i.e., τ(n)≤n-2), by (1.15) and for any integer n≥n0, we easily observe that δj≥δj+1 for τ(n)+1≤j≤n-1; namely,
(1.17)δτ(n)+1≥δτ(n)+2≥⋯≥δn-1≥δn,
which entails the desired result.


## 2. Main Results

Theorem 2.1.
Given a bounded linear operatorA:H1→H2, let  U:H1→H1 and  S:H2→H2 be  quasi-nonexpansive mappings with nonempty fixed-point set F(U)=C and  F(S)=Q.  Assume that U-I and S-I are demiclosed at origin. Let {xk} be the sequence given by (1.12) with γ∈(0,1/λ),  ωk∈(0,1/2) such that 0<liminfk→∞ωk≤limsupk→∞ωk<1/2 and {αk}⊂(0,1) such that limk→∞αk=0 and  ∑kαk=∞. If  Γ≠∅,  then the sequence {xk} strongly converges to a split common fixed-point x*∈Γ,  verifying x*=PΓf(x*) which equivalently solves the following variational inequality problem:
(2.1)x*∈Γ,〈(I-f)x*,v-x*〉≥0,∀v∈Γ.Proof.
SetTωk=(1-ωk)I+ωkT. Then xk+1=αkf(xk)+(1-αk)Tωkxk.
Firstly, we prove that{xk} is bounded. Taking y∈Γ, that is, y∈F(U), Ay∈F(S). We have
(2.2)‖xk+1-y‖=‖αk(f(xk)-f(y))+αk(f(y)-y)+(1-αk)(Tωkxk-y)‖≤αk‖f(xk)-f(y)‖+αk‖f(y)-y‖+(1-αk)‖Tωkxk-y‖≤αkρ‖xk-y‖+αk‖f(y)-y‖+(1-αk)‖Tωkxk-y‖.
From the definition of Tωk, we get
(2.3)‖Tωkxk-y‖2=‖(1-ωk)xk+ωkTxk-y‖2=‖xk-y+ωk(Txk-xk)‖2=‖xk-y‖2-2ωk〈xk-y,xk-Txk〉+ωk2‖Txk-xk‖2.
On the other hand, we have
(2.4)‖Txk-y‖2=‖U(I+γA*(S-I)A)xk-y‖2≤‖(I+γA*(S-I)A)xk-y‖2=‖xk-y‖2+γ2‖A*(S-I)Axk‖2+2γ〈xk-y,A*(S-I)Axk〉=‖xk-y‖2+γ2〈(S-I)Axk,AA*(S-I)Axk〉+2γ〈xk-y,A*(S-I)Axk〉.
From the definition of λ, it follows that
(2.5)γ2〈(S-I)Axk,AA*(S-I)Axk〉≤λγ2〈(S-I)Axk,(S-I)Axk〉=λγ2‖(S-I)Axk‖2.
Now, by using property (ii) of Lemma 1.1, we obtain
(2.6)2γ〈xk-y,A*(S-I)Axk〉=2γ〈A(xk-y),(S-I)Axk〉=2γ〈A(xk-y)+(S-I)Axk-(S-I)Axk,(S-I)Axk〉=2γ(〈S(Axk)-Ay,(S-I)Axk〉-‖(S-I)Axk‖2)≤2γ(12‖(S-I)Axk‖2-‖(S-I)Axk‖2)=-γ‖(S-I)Axk‖2.
Combining (2.4)–(2.6), we have
(2.7)‖Txk-y‖2≤‖xk-y‖2+λγ2‖(S-I)Axk‖2-γ‖(S-I)Axk‖2=‖xk-y‖2-γ(1-λγ)‖(S-I)Axk‖2≤‖xk-y‖2.
From property (i) of Lemma 1.1, we have
(2.8)〈xk-y,xk-Txk〉=-12‖Txk-y‖2+12‖xk-y‖2+12‖xk-Txk‖2≥12‖xk-Txk‖2.
From (2.3) and (2.8), we have
(2.9)‖Tωkxk-y‖2≤‖xk-y‖2-ωk‖xk-Txk‖2+ωk2‖xk-Txk‖2=‖xk-y‖2-ωk(1-ωk)‖xk-Txk‖2≤‖xk-y‖2,
Combining (2.2), (2.3), and (2.9), it follows that
(2.10)‖xk+1-y‖≤αkρ‖xk-y‖+αk‖f(y)-y‖+(1-αk)‖xk-y‖=[1-αk(1-ρ)]‖xk-y‖+αk‖f(y)-y‖≤max⁡{‖xk-y‖,11-ρ‖f(y)-y‖}.It is obviously that
(2.11)‖xk-y‖≤max⁡{‖x0-y‖,11-ρ‖f(y)-y‖},
and hence {xk} is bounded. Let x*=PΓf(x*). We have
(2.12)xk+1-xk+αk(xk-f(xk))=(1-αk)(Tωkxk-xk),
and hence
(2.13)〈xk+1-xk+αk(I-f)xk,xk-x*〉=-(1-αk)〈xk-Tωkxk,xk-x*〉.
By (2.9) we obtain that
(2.14)〈xk-Tωkxk,xk-x*〉=12‖xk-Tωkxk‖2+12‖xk-x*‖2-12‖Tωkxk-x*‖2≥ωk22‖xk-Txk‖2+12‖xk-x*‖2-12‖xk-x*‖2+ωk2(1-ωk)‖xk-Txk‖2=ωk2‖xk-Txk‖2.
It follows from (2.13) that
(2.15)〈xk+1-xk+αk(I-f)xk,xk-x*〉≤-ωk2(1-αk)‖xk-Txk‖2,
and hence
(2.16)-〈xk-xk+1,xk-x*〉≤-αk〈(I-f)xk,xk-x*〉-ωk2(1-αk)‖xk-Txk‖2.
Setting δk=12∥xk-x*∥2, we have
(2.17)〈xk-xk+1,xk-x*〉=-12‖xk+1-x*‖2+12‖xk-x*‖2+12‖xk-xk+1‖2=-δk+1+δk+12‖xk-xk+1‖2,
so that (2.16) can be rewritten as
(2.18)δk+1-δk-12‖xk-xk+1‖2≤-αk〈(I-f)xk,xk-x*〉-ωk2(1-αk)‖xk-Txk‖2.
Now using (2.12) again, we have
(2.19)‖xk+1-xk‖2=‖αk(f(xk)-xk)+(1-αk)(Tωkxk-xk‖2≤(αk‖f(xk)-xk‖+(1-αk)‖Tωkxk-xk‖)2≤2αk2‖f(xk)-xk‖2+2(1-αk)2‖Tωkxk-xk‖2≤2αk2‖f(xk)-xk‖2+2(1-αk)ωk2‖Txk-xk‖2,
which yields
(2.20)12‖xk+1-xk‖2≤αk2‖f(xk)-xk‖2+(1-αk)ωk2‖Txk-xk‖2.
From (2.18) and (2.20), we obtain
(2.21)δk+1-δk+ωk(1-αk)(12-ωk)‖Txk-xk‖2≤αk[αk‖f(xk)-xk‖2-〈(I-f)xk,xk-x*〉].
It follows from Remark 1.2 that
(2.22)〈(I-f)xk-(I-f)x*,xk-x*〉≥(1-ρ)‖xk-x*‖2=2(1-ρ)δk,
and hence
(2.23)2(1-ρ)δk+〈(I-f)x*,xk-x*〉≤〈(I-f)xk,xk-x*〉.
The rest of the proof will be divided into two parts.
Case 1.
Suppose that there existsk0 such that {δk}k≥k0 is nonincreasing. In this situation, {δk} is convergent because it is nonnegative, so that limk→∞(δk+1-δk)=0; hence, in light of (2.21) together with αk→0, the boundedness of {xk}, and 0<liminfk→∞ωk≤limsupk→∞ωk<1/2, we obtain
(2.24)lim⁡k→∞‖xk-Txk‖=0.
From (2.21) again, we have
(2.25)αk[-αk‖f(xk)-xk‖2+〈(I-f)xk,xk-x*〉]≤δk-δk+1.
By ∑kαk=∞, we deduce that
(2.26)liminf⁡k→∞(-αk‖f(xk)-xk‖2+〈(I-f)xk,xk-x*〉)≤0
and hence (as αk∥f(xk)-xk∥2→0)
(2.27)liminf⁡k→∞〈(I-f)xk,xk-x*〉≤0.
By (2.23) and (2.27), we have
(2.28)liminf⁡k→∞(2(1-ρ)δk+〈(I-f)x*,xk-x*〉)≤0;
recalling that limk→∞δk exists, we obtain
(2.29)2(1-ρ)lim⁡k→∞δk+liminf⁡k→∞〈(I-f)x*,xk-x*〉≤0.
Now we prove that
(2.30)liminf⁡k→∞〈(I-f)x*,xk-x*〉≥0.
It follows from (2.7) and (2.24) that
(2.31)γ(1-λγ)‖(S-I)Axk‖2≤‖xk-y‖2-‖Txk-y‖2=(‖xk-y‖-‖Txk-y‖)(‖xk-y‖+‖Txk-y‖)≤‖xk-Txk‖(‖xk-y‖+‖Txk-y‖)⟶0(k⟶∞),
and hence
(2.32)lim⁡k→∞‖(S-I)Axk‖=0.
Taking y∈ωw(xk), from the demiclosedness of S-I at 0, we obtain
(2.33)S(Ay)=Ay.
Now, by setting uk=xk+γA*(S-I)Axk, it follows that y∈ωw(uk). On the other hand,
(2.34)‖U(uk)-uk‖=‖Txk-xk-γA*(S-I)Axk‖≤‖Txk-xk‖+γ‖A*‖⋅‖(S-I)Axk‖⟶0,
which, combined with the demiclosedness of U-I at 0, yields
(2.35)Uy=y.
Hence, y∈C and y∈Γ. We can take subsequence {xkj} of {xk} such that xkj⇀y as j→∞ and
(2.36)liminf⁡k→∞〈(I-f)x*,xk-x*〉=lim⁡j→∞〈(I-f)x*,xkj-x*〉,
which leads to
(2.37)liminf⁡k→∞〈(I-f)x*,xk-x*〉=〈(I-f)x*,y-x*〉≥0.
By (2.29), we have limk→∞δk=0, and hence {xk} converges strongly to x*.
Case 2.
Suppose there exists a subsequence{δkj}j≥0 of {δk} such that δkj<δkj+1 for all j≥0. In this situation, we consider the sequence of indices {τ(k)} as defined in Lemma 1.3. It follows that δτ(k)+1-δτ(k)>0, which by (2.21) amounts to
(2.38)ωk(1-ατ(k))(12-ωk)‖Txτ(k)-xτ(k)‖2≤ατ(k)[ατ(k)‖f(xτ(k))-xτ(k)‖2-〈(I-f)xτ(k),xτ(k)-x*〉‖f(xτ(k))-xτ(k)‖2].
By the boundedness of {xk} and αk→0, we immediately obtain
(2.39)lim⁡k→∞‖Txτ(k)-xτ(k)‖=0.
Similar to Case 1, we have
(2.40)liminf⁡k→∞〈(I-f)x*,xτ(k)-x*〉≥0.
It follows from (2.38) that
(2.41)〈(I-f)xτ(k),xτ(k)-x*〉≤ατ(k)‖f(xτ(k))-xτ(k)‖2,
which in the light of (2.23) yields
(2.42)2(1-ρ)δτ(k)+〈(I-f)x*,xτ(k)-x*〉≤ατ(k)‖f(xτ(k))-xτ(k)‖2;
hence (as ατ(k)∥f(xτ(k))-xτ(k)∥2→0) it follows that
(2.43)2(1-ρ)lim⁡⁡sup⁡k→∞δτ(k)≤-liminf⁡k→∞〈(I-f)x*,xτ(k)-x*〉.
From (2.40) we have limsupk→∞δτ(k)=0, so that limk→∞δτ(k)=0, and hence limk→∞∥xτ(k)-x*∥=0. On the other hand, it follows that
(2.44)‖xτ(k)+1-xτ(k)‖=‖ατ(k)(f(xτ(k))-xτ(k))+(1-ατ(k))(Tωkxτ(k)-xτ(k))‖≤ατ(k)‖f(xτ(k))-xτ(k)‖+(1-ατ(k))ωk‖Txτ(k)-xτ(k)‖,
which, by (2.39), implies that
(2.45)lim⁡k→∞‖xτ(k)+1-xτ(k)‖=0.
So we have
(2.46)lim⁡k→∞δτ(k)+1=12‖xτ(k)+1-x*‖=0.
Then, recalling that δk≤δτ(k)+1 (by Lemma 1.3), we get limk→∞δk=0, so that the sequence {xk} converges strongly to x*.Theorem 2.2.
Given a bounded linear operatorA:H1→H2, let  U:H1→H1 and S:H2→H2 be quasi-nonexpansive mappings with nonempty fixed-point set F(U)=C and F(S)=Q.  Assume that U-I and  S-I are demiclosed at origin.  Let x0∈H be arbitrary and {xk} the sequence given by
(2.47)xk+1=αkf(xk)+(1-αk)((1-ω)xk+ωTxk),
where T=U(I+γA*(S-I)A), f:H→H a contraction of modulus ρ, γ∈(0,1/λ),  ω∈(0,1/2), and {αk}⊂(0,1) such that limk→∞αk=0 and ∑kαk=∞. If  Γ≠∅, then the sequence {xk} strongly converges to a split common fixed-point x*∈Γ, verifying x*=PΓf(x*) which equivalently solves the following variational inequality problem:
(2.48)x*∈Γ,〈(I-f)x*,v-x*〉≥0,∀v∈Γ.



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