# Strong Convergence by a Hybrid Algorithm for Finding a Common Fixed Point of Lipschitz Pseudocontraction and Strict Pseudocontraction in Hilbert Spaces

Authors: Kasamsuk Ungchittrakool
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We prove a strong convergence theorem by using a hybrid algorithm in order to find a common
fixed point of Lipschitz pseudocontraction andκ-strict pseudocontraction in Hilbert spaces. Our results extend the
recent ones announced by Yao et al. (2009) and many others.


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## Body

## 1. Introduction

LetH be a real Hilbert space, and let C be a nonempty closed convex subset of H. Let T:C→C. Recall that T is said to be a pseudocontraction if (1.1)‖Tx-Ty‖2⩽‖x-y‖2+‖(I-T)x-(I-T)y‖2
is equivalent to (1.2)〈x-y,(I-T)x-(I-T)y〉⩾0,
for all x,y∈C, and T is said to be a strict pseudocontraction if there exists a constant 0⩽κ<1 such that (1.3)‖Tx-Ty‖2⩽‖x-y‖2+κ‖(I-T)x-(I-T)y‖2,
for all x,y∈C. For the second case, we say that T is a κ-strict pseudocontraction. We use F(T) to denote the set of fixed points of T.The class of strict pseudocontractions extend the class of nonexpansive mapping. (A mappingT is said to be nonexpansive if ∥Tx-Ty∥⩽∥x-y∥, for all x,y∈C) that is, T is nonexpansive if and only if T is a 0-strict pseudocontraction. The pseudocontractive mapping includes the strict pseudocontractive mapping.Iterative methods for finding fixed points of nonexpansive mappings are an important topic in the theory of nonexpansive mappings and have wide applications in a number of applied areas, such as the convex feasibility problem [1–4], the split feasibility problem [5–7] and image recovery and signal processing [3, 8, 9], and so forth. However, the Picard sequence {Tnx}n=0∞ often fails to converge even in the weak topology. Thus, averaged iterations prevail. The Mann iteration [10] is one of the types and is defined by(1.4)xn+1=αnxn+(1-αn)Txn,n⩾0,
where x0∈C is chosen arbitrarily and {αn}⊂[0,1]. Reich [11] proved that if E is a uniformly convex Banach space with a Fréchet differentiable norm and if {αn} is chosen such that ∑n=0∞αn(1-αn)=∞, then the sequence {xn} defined by (1.4) converges weakly to a fixed point of T. However, we note that Mann iterations have only weak convergence even in a Hilbert space (see e.g., [12]). From a practical point of view, strict pseudocontractions have more powerful applications than nonexpansive mappings do in solving inverse problems (see [13]). Therefore, it is important to develop theory of iterative methods for strict pseudocontractions. Indeed, Browder and Petryshyn [14] prove that if the sequence {xn} is generated by the following: (1.5)xn+1=αxn+(1-α)Txn,n⩾0,
for any starting point x0∈C, α is a constant such that κ<α<1, {xn} converges weakly to a fixed point of strict pseudocontraction. Marino and Xu [15] extended the result of Browder and Petryshyn [14] to Mann iteration (1.4); they proved {xn} converges weakly to a fixed point of T, provided the control sequence {αn} satisfies the conditions that κ<αn<1 for all n and ∑n=0∞(αn-k)(1-αn)=∞.The well-known strong convergence theorem for pseudocontractive mapping was proved by Ishikawa [16] in 1974. More precisely, he got the following theorem.Theorem 1.1 (see [16]).
LetC be a convex compact subset of a Hilbert space H and let T:C→C be a Lipschitzian pseudocontractive mapping. For any x1∈C, suppose the sequence {xn} is defined by
(1.6)xn+1=(1-αn)xn+αnTyn,yn=(1-βn)xn+βnTxn,n⩾1,
where {αn}, {βn} are two real sequences in [0,1] satisfying  (i)
αn⩽βn, n⩾1,(ii)
limn→∞βn=0,(iii)
∑n=1∞αnβn=∞.Then {xn} converges strongly to a fixed point of T.Remark 1.2.
(i) Since0⩽αn⩽βn⩽1, n⩾1 and ∑n=1∞αnβn=∞, the iterative sequence (1.6) could not be reduced to a Mann iterative sequence (1.4). Therefore, the iterative sequence (1.6) has some particular cases.
(ii) The iterative sequence (1.6) is usually called the Ishikawa iterative sequence.
(iii) Chidume and Mutangadura [17] gave an example to show that the Mann iterative sequence failed to be convergent to a fixed point of Lipschitzian pseudocontractive mapping.In an infinite-dimensional Hilbert spaces, Mann and Ishikawa's iteration algorithms have only weak convergence, in general, even for nonexpansive mapping. In order to obtain a strong convergence theorem for the Mann iteration method (1.4) to nonexpansive mapping, Nakajo and Takahashi [18] modified (1.4) by employing two closed convex sets that are created in order to form the sequence via metric projection so that strong convergence is guaranteed. Later, it is often referred as the hybrid algorithm or the CQ algorithm. After that the hybrid algorithm have been studied extensively by many authors (see e.g., [19–23]). Particularly, Martinez-Yanes and Xu [24] and Plubtieng and Ungchittrakool [20] extended the same results of Nakajo and Takahashi [18] to the Ishikawa iteration process. In 2007, Marino and Xu [15] further generalized the hybrid algorithm from nonexpansive mappings to strict pseudocontractive mappings. In 2008, Zhou [25] established the hybrid algorithm for pseudocontractive mapping in the case of the Ishikawa iteration process.Recently, Yao et al. [26] introduced the hybrid iterative algorithm which just involved one closed convex set for pseudocontractive mapping in Hilbert spaces as follows.LetC be a nonempty closed convex subset of a real Hilbert space H. Let T:C→C be a pseudocontraction. Let {αn} be a sequence in (0,1). Let x0∈H. For C1=C and x1=PC1(x0), define a sequence {xn} of C as follows.(1.7)yn=(1-αn)xn+αnTzn,Cn+1={v∈Cn:‖αn(I-T)yn‖2⩽2αn〈xn-v,(I-T)yn〉},xn+1=PCn+1(x0).Theorem 1.3 (see [26]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T:C→C be a L-Lipschitz pseudocontraction such that F(T)≠∅. Assume the sequence {αn}⊂[a,b] for some a,b∈(0,1/(L+1)). Then the sequence {xn} generated by (1.7) converges strongly to PF(T)(x0).Very recently, Tang et al. [27] generalized the hybrid algorithm (1.7) in the case of the Ishikawa iterative precess as follows:(1.8)yn=(1-αn)xn+αnTzn,zn=(1-βn)xn+βnTxn,Cn+1={v∈Cn:‖αn(I-T)yn‖2⩽2αn〈xn-v,(I-T)yn〉+2αnβnL‖xn-Txn‖‖yn-xn+αn(I-T)yn‖‖αn(I-T)yn‖2},xn+1=PCn+1(x0).
Under some appropriate conditions of {αn} and {βn}, they proved that (1.8) converges strongly to PF(T)(x0).Motivated and inspired by the above works, in this paper, we generalize (1.7) to the Ishikawa iterative process in the case of finding the common fixed point of Lipschitz pseudocontraction and κ-strict pseudocontraction. More precisely, we provide some applications of the main theorem to find the common zero point of the Lipshitz monotone mapping and γ-inverse strongly monotone mapping in Hilbert spaces.


## 2. Preliminaries

LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥, and let C be a closed convex subset of H. For every point x∈H, there exists a unique nearest point in C, denoted by PC(x), such that (2.1)‖x-PCx‖⩽‖x-y‖,∀y∈C,
where PC is called the metric projection of H onto C. We know 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 (2.2)lim⁡inf⁡n→∞‖xn-x‖<lim⁡inf⁡n→∞‖xn-y‖
holds for every y∈H with y≠x.For a given sequence{xn}⊂C, let ωw(xn)={x:∃xnj⇀x} denote the weak ω-limit set of {xn}.Now we collect some Lemmas which will be used in the proof of the main result in the next section. We note that Lemmas2.1 and 2.2 are well known.Lemma 2.1.
LetH be a real Hilbert space. There holds the following identities:  (i)
∥x-y∥2=∥x∥2-∥y∥2-2〈x-y,y〉,forallx,y∈H,(ii)
∥λx+(1-λ)y∥2=λ∥x∥2+(1-λ)∥y∥2-λ(1-λ)∥x-y∥2,forallx,y∈H and λ∈[0,1].Lemma 2.2.
LetC be a closed convex subset of real Hilbert space H. Given x∈H and z∈C, then z=PCx if and only if there holds the relation
(2.3)〈x-z,y-z〉⩽0,∀y∈C.Proposition 2.3 (see [15, Proposition  2.1]).
AssumeC is a closed convex subset of a Hilbert space H; let T:C→C be a self-mapping of C. If T is a κ-strict pseudocontraction, then T satisfies the Lipschitz condition
(2.4)‖Tx-Ty‖⩽1+κ1-κ‖x-y‖,∀x,y∈C.Lemma 2.4 (see [28]).
LetH be a real Hilbert space, let C be a closed convex subset of H, and let T:C→C be a continuous pseudocontractive mapping, then  (i)
F(T) is closed convex subset of C,(ii)
I-T is demiclosed at zero, that is, if {xn} is a sequence in C such that xn⇀z and (I-T)xn→0, then (I-T)z=0.Lemma 2.5 (see [24]).
LetC be a closed convex subset of H. Let {xn} be a sequence in H, and let u∈H. Let q=PCu. If {xn} is such that ωw(xn)⊂C and satisfies the condition
(2.5)‖xn-u‖⩽‖u-q‖,∀n,
then xn→q.


## 3. Main Result

Theorem 3.1.
LetC be a nonempty closed convex subset of a real Hilbert space H, let T:C→C be LT-Lipschitz pseudocontraction, and let S:C→C be κ-strict pseudocontraction with F̃:=F(S)∩F(T)≠∅. Let x0∈H. For C1=C and x1=PC1(x0), define a sequence {xn} of C as follows:
(3.1)yn=(1-αn)xn+αnTzn,zn=(1-βn)xn+βnSxn,Cn+1={((2βn1-κ)2-1)v∈Cn:‖αn(I-T)yn‖2+(1-κ)‖(I-S)xn‖2⩽2αn〈xn-v,(I-T)yn〉+2〈xn-v,(I-S)zn+(I-S)xn〉+2αnβnLT‖xn-Sxn‖‖yn-xn+αn(I-T)yn‖+βn((2βn1-κ)2-1)‖(I-S)xn‖2},xn+1=PCn+1(x0).
Assume the sequence {αn}, {βn} be such that 0<a⩽αn⩽b<1/(LT+1)<1 and 0<βn⩽1 for all n∈ℕ with limn→∞βn=0. Then {xn} converges strongly to PF̃(x0).Proof.
By Lemma2.4(i), we see that F(S) and F(T) are closed and convex, then F̃ is as well. Hence, PF̃ is well defined. Next, we will prove by induction that F̃⊂Cn for all n∈ℕ. Note that F̃⊂C=C1. Assume that F̃⊂Ck holds for k⩾1. Let p∈F̃, thus p∈Ck, and we observe that
(3.2)‖xk-p-αk(I-T)yk‖2=‖xk-p‖2-‖αk(I-T)yk‖2-2αk〈(I-T)yk,xk-p-αk(I-T)yk〉=‖xk-p‖2-‖αk(I-T)yk‖2-2αk〈(I-T)yk-(I-T)p,yk-p〉-2αk〈(I-T)yk,xk-yk-αk(I-T)yk〉⩽‖xk-p‖2-‖αk(I-T)yk‖2-2αk〈(I-T)yk,xk-yk-αk(I-T)yk〉=‖xk-p‖2-‖(xk-yk)+(yk-xk+αk(I-T)yk)‖2-2αk〈(I-T)yk,xk-yk-αk(I-T)yk〉=‖xk-p‖2-‖xk-yk‖2-‖yk-xk+αk(I-T)yk‖2-2〈xk-yk,yk-xk+αk(I-T)yk〉-2αk〈(I-T)yk,xk-yk-αk(I-T)yk〉⩽‖xk-p‖2-‖xk-yk‖2-‖yk-xk+αk(I-T)yk‖2+2|〈xk-yk-αk(I-T)yk,xk-yk-αk(I-T)yk〉|.
Consider the last term of (3.2), we obtain
(3.3)|〈xk-yk-αk(I-T)yk,yk-xk+αk(I-T)yk〉|=αk|〈xk-Tzk-(I-T)yk,yk-xk+αk(I-T)yk〉|=αk|〈xk-Txk+Txk-Tzk-(I-T)yk,yk-xk+αk(I-T)yk〉|=αk|〈(I-T)xk-(I-T)yk,yk-xk+αk(I-T)yk〉+〈Txk-Tzk,yk-xk+αk(I-T)yk〉|⩽αk(LT+1)‖xk-yk‖‖yk-xk+αk(I-T)yk‖+αkLT‖xk-zk‖‖yk-xk+αk(I-T)yk‖=αk(LT+1)‖xk-yk‖‖yk-xk+αk(I-T)yk‖+αkβkLT‖xk-Sxk‖‖yk-xk+αk(I-T)yk‖⩽αk(LT+1)2(‖xk-yk‖2+‖yk-xk+αk(I-T)yk‖2)+αkβkLT‖xk-Sxk‖‖yk-xk+αk(I-T)yk‖.
Substituting (3.3) into (3.2), we obtain
(3.4)‖xk-p-αk(I-T)yk‖2⩽‖xk-p‖2-‖xk-yk‖2-‖yk-xk+αk(I-T)yk‖2+αk(LT+1)(‖xk-yk‖2+‖yk-xk+αk(I-T)yk‖2)+2αkβkLT‖xk-Sxk‖‖yk-xk+αk(I-T)yk‖⩽‖xk-p‖2+2αkβkLT‖xk-Sxk‖‖yk-xk+αk(I-T)yk‖.
Notice that
(3.5)‖xk-p-αk(I-T)yk‖2=‖xk-p‖2-2αk〈xk-p,(I-T)yk〉+‖αk(I-T)yk‖2.
Therefore, from (3.4) and (3.5), we get
(3.6)‖αk(I-T)yk‖2⩽2αk〈xk-p,(I-T)yk〉+2αkβkLT‖xk-Sxk‖‖yk-xk+αk(I-T)yk‖.
On the other hand, we found that
(3.7)‖xk-p-βk(I-S)zk‖2=‖xk-p‖2-‖βk(I-S)zk‖2-2βk〈(I-S)zk,xk-p-βk(I-S)zk〉=‖xk-p‖2-‖βk(I-S)zk‖2-2βk〈(I-S)zk-(I-S)p,zk-p〉-2βk〈(I-S)zk,xk-zk-βk(I-S)zk〉⩽‖xk-p‖2-‖βk(I-S)zk‖2-2〈βk(I-S)zk,xk-zk-βk(I-S)zk〉=‖xk-p‖2-‖βk(I-S)zk‖2+(‖βk(I-S)zk‖2-‖xk-zk‖2+‖xk-zk-βk(I-S)zk‖2)=‖(xk-zk)+(zk-p)‖2-‖xk-zk‖2+‖βk(I-S)xk-βk(I-S)zk‖2=‖xk-zk‖2+2〈xk-zk,zk-p〉+‖zk-p‖2-‖xk-zk‖2+‖βk(I-S)xk-βk(I-S)zk‖2=2〈xk-zk,(zk-xk)+(xk-p)〉+‖(1-βk)(xk-p)+βk(Sxk-p)‖2+‖βk(I-S)xk-βk(I-S)zk‖2⩽2〈xk-p,βk(I-S)xk〉+(1-βk)‖xk-p‖2+βk‖Sxk-p‖2-βk(1-βk)‖xk-Sxk‖2-2βk2‖(I-S)xk‖2+βk2(1+κ1-κ+1)2‖xk-zk‖2⩽2〈xk-p,βk(I-S)xk〉+(1-βk)‖xk-p‖2+βk‖xk-p‖2+βkκ‖(I-S)xk‖2-βk(1-βk)‖(I-S)xk‖2-2βk2‖(I-S)xk‖2+βk4(21-κ)2‖(I-S)xk‖2=2〈xk-p,βk(I-S)xk〉+‖xk-p‖2-βk(1-κ)‖(I-S)xk‖2-βk2‖(I-S)xk‖2+βk4(21-κ)2‖(I-S)xk‖2.
Notice that
(3.8)‖xk-p-βk(I-S)zk‖2=‖xk-p‖2-2βk〈xk-p,(I-S)zk〉+βk2‖(I-S)zk‖2.
Combining (3.7) and (3.8) and then it implies that
(3.9)βk(1-κ)‖xk-Sxk‖2⩽2βk〈xk-p,(I-S)zk+(I-S)xk〉+βk2((2βk1-κ)2-1)‖(I-S)xk‖2.
Since βn>0 for all n, so we get
(3.10)(1-κ)‖xk-Sxk‖2⩽2〈xk-p,(I-S)zk+(I-S)xk〉+βk((2βk1-κ)2-1)‖(I-S)xk‖2.
It follows from (3.6) and (3.10) that we obtain
(3.11)‖αk(I-T)yk‖2+(1-κ)‖(I-S)xk‖2⩽2αk〈xk-v,(I-T)yk〉+2〈xk-v,(I-S)zk+(I-S)xk〉+2αkβkLT‖xk-Sxk‖‖yk-xk+αk(I-T)yk‖+βk((2βk1-κ)2-1)‖(I-S)xk‖2.
Therefore, p∈Ck+1. By mathematical induction, we have F̃⊂Cn for all n∈ℕ. It is easy to check that Cn is closed and convex, and then {xn} is well defined. From xn=PCn(x0), we have 〈x0-xn,xn-y〉⩾0 for all y∈Cn. Using F̃⊂Cn, we also have 〈x0-xn,xn-u〉⩾0 for all u∈F̃. So, for u∈F̃, we have
(3.12)0⩽〈x0-xn,xn-u〉=〈x0-xn,xn-x0+x0-u〉=-‖x0-xn‖2+〈x0-xn,x0-u〉⩽-‖x0-xn‖2+‖x0-xn‖‖x0-u‖.
Hence, ∥x0-xn∥⩽∥x0-u∥,forallu∈F̃. In particular,
(3.13)‖x0-xn‖⩽‖x0-q‖,whereq=PF̃(x0).
This implies that {xn} is bounded, and then {yn}, {Tyn}, {zn}, {Szn}, and {Sxn} are as well.
Fromxn=PCn(x0) and xn+1=PCn+1(x0)∈Cn+1⊂Cn, we have
(3.14)〈x0-xn,xn-xn+1〉⩾0.
Hence
(3.15)0⩽〈x0-xn,xn-xn+1〉=〈x0-xn,xn-x0+x0-xn+1〉=-‖x0-xn‖2+〈x0-xn,x0-xn+1〉⩽-‖x0-xn‖2+‖x0-xn‖‖x0-xn+1‖,
and; therefore,
(3.16)‖x0-xn‖⩽‖x0-xn+1‖,
which implies that limn→∞∥xn-x0∥ exists. From Lemma 2.1 and (3.14), we obtain
(3.17)‖xn+1-xn‖2=‖(xn+1-x0)-(xn-x0)‖2=‖xn+1-x0‖2-‖xn-x0‖2-2〈xn+1-xn,xn-x0〉⩽‖xn+1-x0‖2-‖xn-x0‖2⟶0.
Since xn+1∈Cn+1⊂Cn, we have
(3.18)‖αn(I-T)yn‖2+(1-κ)‖(I-S)xn‖2⩽2αn〈xn-xn+1,(I-T)yn〉+2〈xn-xn+1,(I-S)zn+(I-S)xn〉+2αnβnLT‖xn-Sxn‖‖yn-xn+αn(I-T)yn‖+βn((2βn1-κ)2-1)‖(I-S)xn‖2⟶0asn⟶∞,
therefore, we obtain
(3.19)‖yn-Tyn‖⟶0,‖xn-Sxn‖⟶0.
We note that
(3.20)‖xn-Txn‖⩽‖xn-yn‖+‖yn-Tyn‖+‖Tyn-Txn‖⩽(LT+1)‖xn-yn‖+‖yn-Tyn‖⩽αn(LT+1)‖xn-Tzn‖+‖yn-Tyn‖⩽αn(LT+1)‖xn-Txn‖+αn(LT+1)‖Txn-Tzn‖+‖yn-Tyn‖⩽αn(LT+1)‖xn-Txn‖+αnβnLT(LT+1)‖xn-Sxn‖+‖yn-Tyn‖,
that is,
(3.21)‖xn-Txn‖⩽αnβnLT(LT+1)1-αn(LT+1)‖xn-Sxn‖+11-αn(LT+1)‖yn-Tyn‖⟶0,asn⟶∞.
By Lemma2.4(ii), I-T and I-S are demiclosed at zero. Together with the fact that {xn} is bounded, which guarantees that every weak limit point of {xn} is a fixed point of T and S, that is ωw(xn)⊂F(T)∩F(S)=F̃, therefore, by inequality (3.13) and Lemma 2.5, we know that {xn} converges strongly to q=PF̃(x0). This completes the proof.IfS=I, then we obtain the following corollary.Corollary 3.2 (Yao et al. [26, Theorem  3.1]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T:C→C be L-Lipschitz pseudocontraction such that F(T)≠∅. Assume the sequence {αn} be such that 0<a⩽αn⩽b<1/(L+1)<1 for all n. Then the sequence {xn} generated by (1.7) converges strongly to PF(T)(x0).IfT and S are nonexpansive, then we also have the following corollary.Corollary 3.3.
LetC be a nonempty closed convex subset of a real Hilbert space H, and let S,T:C→C be nonexpansive mappings. Suppose that F̃:=F(S)∩F(T)≠∅. Assume the sequence {αn}  be such that 0<a⩽αn⩽b<1/2 and 0<βn⩽1 for all n∈ℕ with limn→∞βn=0. Let x0∈H. For C1=C and x1=PC1(x0), define a sequence {xn} of C as follows:
(3.22)yn=(1-αn)xn+αnTzn,zn=(1-βn)xn+βnSxn,Cn+1={v∈Cn:‖αn(I-T)yn‖2+‖(I-S)xn‖2⩽2αn〈xn-v,(I-T)yn〉+2〈xn-v,(I-S)zn+(I-S)xn〉+2αnβn‖xn-Sxn‖‖yn-xn+αn(I-T)yn‖+βn(4βn2-1)‖(I-S)xn‖2},xn+1=PCn+1(x0).
Then {xn} converges strongly to PF̃(x0).Recall that a mappingA is said to be monotone if 〈x-y,Ax-Ay〉⩾0 for all x,y∈H and inverse strongly monotone if there exists a real number γ>0 such that 〈x-y,Ax-Ay〉⩾γ∥Ax-Ay∥2 for all x,y∈H. For the second case, A is said to be γ-inverse strongly monotone. It follows immediately that if A is γ-inverse strongly monotone, then A is monotone and Lipschitz continuous, that is, ∥Ax-Ay∥⩽(1/γ)∥x-y∥. It is well known (see e.g., [29]) that if A is monotone, then the solutions of the equation Ax=0 correspond to the equilibrium points of some evolution systems. Therefore, it is important to focus on finding the zero point of monotone mappings. The pseudocontractive mapping and strictly pseudocontractive mapping are strongly related to the monotone mapping and inverse strongly monotone mapping, respectively. It is well known that  (i)
A is monotone ⇔T:=(I-A) is pseudocontractive,(ii)
A is inverse strongly monotone ⇔T:=(I-A) is strictly pseudocontractive.Indeed, for (ii), we notice that the following equality always holds in a real Hilbert space:(3.23)‖(I-A)x-(I-A)y‖2=‖x-y‖2+‖Ax-Ay‖2-2〈x-y,Ax-Ay〉,∀x,y∈H.
Without loss of generality, we can assume that γ∈(0,1/2], and then it yields (3.24)〈x-y,Ax-Ay〉⩾γ‖Ax-Ay‖2⟺-2〈x-y,Ax-Ay〉⩽-2γ‖Ax-Ay‖2⟺‖(I-A)x-(I-A)y‖2⩽‖x-y‖2+(1-2γ)‖Ax-Ay‖2(via(3.23))⟺‖Tx-Ty‖2⩽‖x-y‖2+κ‖(I-T)x-(I-T)y‖2(whereT:=(I-A),κ:=1-2γ).
Due to Theorem 3.1, we have the following corollary which generalize the corresponding results of Yao et al. [26].Corollary 3.4.
LetA:H→H be LA-Lipschitz monotone mapping and let B:H→H be an γ̂-inverse strongly monotone which A-1(0)∩B-1(0)≠∅. Assume the sequence {αn} be such that 0<a⩽αn⩽b<1/(LA+2), 0<βn⩽1 for all n∈ℕ with limn→∞βn=0 and γ∈(0,1/2] such that γ̂⩾γ. Let x0∈H. For C1=H and x1=PC1(x0)=x0, define a sequence {xn} as follows:
(3.25)yn=xn-αn(xn-zn)-αnAzn,zn=xn-βnBxn,Cn+1={((βnγ)2-1)v∈Cn:‖αnAyn‖2+2γ‖Bxn‖2⩽2αn〈xn-v,Ayn〉+2〈xn-v,Bzn+Bxn〉+2αnβn(LA+1)‖Bxn‖‖yn-xn+αnAyn‖+βn((βnγ)2-1)‖Bxn‖2},xn+1=PCn+1(x0).
Then {xn} converges strongly to PA-1(0)∩B-1(0)(x0).Proof.
LetT:=(I-A) and let S:=(I-B). Then T and S are pseudocontractive and (1-2γ)-pseudocontractive, respectively. Moreover, T is also (LA+1)-Lipschitz, and if we set κ:=1-2γ, S is also ((1-γ)/γ)-Lipschitz, and then (2/(1-κ))2=1/γ2. Hence, it follows from Theorem 3.1 that we have the desired result.IfB=0 (zero mapping), then zn=xn and B-1(0)=H. So, we obtain the following corollary.Corollary 3.5 (Yao et al. [26, Corollary  3.2]).
LetA:H→H be a LA-Lipschitz monotone mapping for which A-1(0)≠∅. Assume that the sequence {αn} be as in Corollary 3.4. Then the sequence {xn} generated by
(3.26)yn=xn-αnAzn,Cn+1={v∈Cn:‖αnAyn‖2⩽2αn〈xn-v,Ayn〉},xn+1=PCn+1(x0)
strongly converges to PA-1(0)(x0).



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*Source: 530683-2011-08-04.xml*