# Strong Convergence for a Strongly Quasi-Nonexpansive Sequence in Hilbert Spaces

Authors: Satit Saejung; Kanokwan Wongchan
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We prove a strong convergence theorem for strongly quasi-nonexpansive sequence of mappings in Hilbert spaces. Moreover, we can improve the recent results of Tian and Jin (2011). We also give a simple proof of Marino-Xu’s result (2006).


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## Body

## 1. Introduction

LetH be a Hilbert space with inner product 〈·,·〉 and induced norm ∥·∥. Recall that a mapping T:H→H is said to be L-Lipschitzian where L>0 if ∥Tx-Ty∥≤L∥x-y∥ for all x,y∈H. In this paper, we are interested in nonexpansive mappings (that is, 1-Lipschitzian ones) and contractions (that is, L-Lipschitzian ones with L<1). The problem of finding a fixed point of such mappings plays an important role in many nonlinear equations appearing in both pure and applied sciences. The celebrated Banach’s contraction principle is probably known as the major tool for the case of contraction mappings. However, for nonexpansive mappings, the situation is more difficult and different.In 2000, Moudafi [1] introduced the viscosity approximation method, starting with an arbitrary initial x1∈H, and defined a sequence {xn} by
(1)xn+1=ɛn1+ɛnf(xn)+11+ɛnTxn(n≥1),
where T is a nonexpansive mapping, f:H→H is a contraction, and {ɛn} is a sequence in (0,1) satisfying (M1)
limn→∞ɛn=0;(M2)
∑n=1∞‍ɛn=∞;(M3)
limn→∞(1/ɛn)-(1/ɛn+1)=0.It was proved that the sequence {xn} generated by (1) converges to a fixed point z of T and the following inequality holds:
(2)〈f(z)-z,q-z〉≤0∀q∈Fix(T):={x∈H:x=Tx}.
In the literature, Moudafi’s scheme has been widely studied and extended (see [2, 3]). It should be noted that the convergence of Moudafi’s scheme is equivalent to that of its special setting with a constant contraction f (see [4]). In fact, this follows from the role of the nonexpansiveness of T.In the earlier result, the following scheme was studied by Halpern [5]; starting with an arbitrary initial x1∈H and a given u∈H, he defined a sequence {xn} by
(3)xn+1=αnu+(1-αn)Txn(n≥1),
where {αn} is a certain sequence in (0,1). In fact, Halpern proved in 1967 the convergence of the iterative sequence {xn} where αn=n-θ and θ∈(0,1). Many researchers (see, e.g., [6, 7]) have improved Halpern’s result from Hilbert spaces to certain Banach spaces with the following conditions on {αn}: (C1)
limn→∞αn=0;(C2)
∑n=1∞‍αn=∞;(C3)
limn→∞(αn/αn+1)=1 or ∑n=1∞‍|αn-αn+1|<∞.Halpern also showed that conditions (C1) and (C2) are necessary for the convergence of the sequence generated by (3) for any given x1,u∈H.On the other hand, Chidume-Chidume [8] and Suzuki [9] independently discovered that together just conditions (C1) and (C2) are sufficient for the convergence of the following iterative sequence:
(4)x1,u∈C,xn+1=αnu+(1-αn)Tλxn(n≥1),
where Tλ=λI+(1-λ)T and λ∈(0,1). Recently, Saejung [10] proved that the conclusion remains true if T is a strongly nonexpansive mapping. It is noted that in Hilbert spaces the mapping Tλ is strongly nonexpansive whenever λ∈(0,1). Recall that a mapping T:H→H is strongly nonexpansive (see [11, 12]) if it is nonexpansive and limn→∞∥(xn-yn)-(Txn-Tyn)∥=0 whenever {xn}, {yn} are sequences in H such that {xn-yn} is bounded and limn→∞(∥xn-yn∥-∥Txn-Tyn∥)=0.In the aforementioned results, it was assumed thatT has a fixed point; that is, Fix(T)≠Ø. Now we consider the following more general settings. A mapping T:H→H is(i)
quasi-nonexpansive if Fix(T)≠Ø and ∥Tx-q∥≤∥x-q∥ for all x∈H and q∈Fix(T);(ii)
strongly quasi-nonexpansive if it is quasi-nonexpansive and limn→∞∥xn-Txn∥=0 whenever {xn} is a bounded sequence in H such that limn→∞(∥xn-q∥-∥Txn-q∥)=0 for some q∈Fix(T).In 2010, Maingé [2] proved the convergence of the sequence {xn} defined by x1∈H and
(5)xn+1=αnf(xn)+(1-αn)Tωxn,
where Tω=(1-ω)I+ωT, ω∈(0,1/2) and T is a quasi-nonexpansive mapping under the conditions (C1) and (C2). In 2011, Wongchan and Saejung [13] improved Maingé’s result by replacing Tω with a strongly nonexpansive mapping T. Hence, the restriction ω∈(0,1/2) can be extended to ω∈(0,1).There are also some other iterative schemes closely related to the schemes above studied by many authors. For example, inspired by the scheme studied by Yamada [14], Tian and Jin [15, 16] recently proposed the following iterative scheme, starting with an arbitrary initial x1∈H and
(6)xn+1=αnγf(xn)+(I-αnμF)Tωxn(n≥1),
where f and Tω are the same as Maingé’s result but F:H→H is strongly monotone and Lipschitzian.A careful reading shows that there are some connections between them. We will discuss and consolidate them into the following scheme: Started with an arbitrary initialx1∈H and
(7)xn+1=αn(f(xn)+g(Tnxn))+(1-αn)Tnxn(n≥1),
where f, g are Lipschitzian and {Tn} is a certain sequence of quasi-nonexpansive mappings.


## 2. Preliminaries

In this section, we collect together some known lemmas which are our main tool in proving our results. LetC be a closed and convex subset of H. Recall that the metric projectionPC:H→C is defined as follows: for x∈H, PCx is the only one point in C satisfying
(8)∥x-PCx∥=inf{∥x-y∥:y∈C}.Lemma 1 (see [17]).
LetC be a nonempty closed convex subset of a Hilbert space H. Then for x∈H and y∈C, y=PCx if and only if 〈x-y,y-z〉≥0 for all z∈C.Lemma 2.
LetH be a Hilbert space. Then
(9)∥x+y∥2≤∥x∥2+2〈y,x+y〉
for all x,y∈H.We also need the following lemma.Lemma 3 (see [18, Lemma 2.5]).
Let{an}⊂[0,∞), {αn}⊂[0,1), and {bn}⊂(-∞,∞), α^∈[0,1) be such that (i)
{an} is a bounded sequence;(ii)
an+1≤(1-αn)2an+2αnα^anan+1+αnbn for all n∈ℕ;(iii)
whenever{ank} is a subsequence of {an} satisfying liminfk→∞(ank+1-ank)≥0, it follows that limsupk→∞bnk≤0;(iv)
limn→∞αn=0 and ∑n=1∞‍αn=∞.Then limn→∞an=0.Lemma 4 (see [19, Lemma 2.3]).
Let{sn} be a sequence of nonnegative real numbers, {αn} a sequence of real numbers in [0,1] with ∑n=1∞‍αn=∞, {un} a sequence of nonnegative real numbers with ∑n=1∞‍un<∞, and {tn} a sequence of real numbers with limsupn→∞tn≤0. Suppose that
(10)sn+1≤(1-αn)sn+αntn+un∀n∈ℕ.
Then limn→∞sn=0.


## 3. Main Results

Recall that{Tn:H→H} is a strongly quasi-nonexpansive sequence if it satisfies the following conditions:(1)
⋂n=1∞‍Fix(Tn)≠Ø;(2)
∥Tnx-p∥≤∥x-p∥ for all x∈H and p∈⋂n=1∞‍Fix(Tn) and for all n∈ℕ;(3)
limn→∞∥xn-Tnxn∥=0 whenever {xn} is a bounded sequence in H such that limn→∞(∥xn-q∥-∥Tnxn-q∥)=0 for some q∈⋂n=1∞‍Fix(Tn).We also say that {Tn} satisfies the NST-condition if whenever {zn} is a bounded sequence in H such that limn→∞∥zn-Tnzn∥=0 it follows that every weak cluster point of {zn} belongs to ⋂n=1∞‍Fix(Tn).Remark 5.
(1)
Being strongly nonexpansive the sequence and NST-condition are apparently inherited by subsequences.
(2)
Suppose thatTn=T:H→H for all n≥1.
(i)
IfT is a strongly nonexpansive mapping, then {Tn} is a strongly nonexpansive sequence.
(ii)
IfI-T is demiclosed at zero, then {Tn} satisfies NST-condition.Recall that I-T:H→H is demiclosed at zero if {xn} is a sequence in H such that limn→∞∥xn-Txn∥=0 and w-limn→∞xn=p; then p∈Fix(T).We now state our main theorem.Theorem 6.
Let{Tn:H→H} be a strongly quasi-nonexpansive sequence satisfying the NST-condition. Let f,g:H→H be α- and β-Lipschitzian, respectively. Suppose that {xn} is given by x1∈H and
(11)xn+1=αn(f(xn)+g(Tnxn))+(1-αn)Tnxn(n≥1),
where {αn} is a sequence in (0,1) satisfying the conditions (C1) and (C2). Suppose that α+β<1. Then {xn} converges strongly to p=P⋂n=1∞‍Fix(Tn)(f+g)(p).Before we give the proof, we note thatF:=⋂n=1∞‍Fix(Tn) is closed and convex. It follows from α+β<1 that f+g is an (α+β)-contraction. Then the mapping PF(f+g):F→F is a contraction. By Banach’s contraction principle, there exists a unique element p∈F such that p=PF(f+g)(p). It follows then from Lemma 1 that 〈(f+g)(p)-p,z-p〉≤0 for all z∈F.Let us consider the following three lemmas first.Lemma 7.
The sequence{xn} is bounded. Hence, so are the sequences {f(xn)}, {Tnxn}, and {g(Tnxn)}.Proof.
We consider the following inequality:(12)∥xn+1-p∥≤αn∥f(xn)+g(Tnxn)-p∥+(1-αn)∥Tnxn-p∥.
Since each Tn is quasi-nonexpansive and p∈⋂n=1∞‍Fix(Tn), we have
(13)∥Tnxn-p∥≤∥xn-p∥.
It follows from the Lipschitzian conditions of f and g, respectively that,
(14)αn∥f(xn)+g(Tnxn)-p∥≤αn∥f(xn)-f(p)∥+αn∥g(Tnxn)-g(p)∥+αn∥f(p)+g(p)-p∥≤ααn∥xn-p∥+βαn∥xn-p∥+αn∥f(p)+g(p)-p∥.
Then, we have
(15)∥xn+1-p∥≤(1-αn(1-(α+β)))∥xn-p∥+αn(1-(α+β))∥f(p)+g(p)-p∥1-(α+β)≤max{∥xn-p∥,∥f(p)+g(p)-p∥1-(α+β)}.
By induction, for all n≥1, we have
(16)∥xn+1-p∥≤max{∥x1-p∥,∥f(p)+g(p)-p∥1-(α+β)}.
In particular, the sequence {xn} is bounded.Lemma 8.
The following inequality holds for alln≥1:
(17)∥xn+1-p∥2≤(1-αn)2∥xn-p∥2+2(α+β)αn∥xn-p∥×∥xn+1-p∥+2αn〈f(p)+g(p)-p,xn+1-p〉.Proof.
It follows from Lemma2 that
(18)∥xn+1-p∥2=∥αn(f(xn)+g(Tnxn)-p)+(1-αn)(Tnxn-p)∥2≤(1-αn)2∥Tnxn-p∥2+2αn〈f(xn)+g(Tnxn)-p,xn+1-p〉.
Since each Tn is quasi-nonexpansive and p∈⋂n=1∞‍Fix(Tn),
(19)∥Tnxn-p∥2≤∥xn-p∥2.
Next, we consider
(20)〈f(xn)+g(Tnxn)-p,xn+1-p〉=〈f(xn)-f(p),xn+1-p〉+〈g(Tnxn)-g(p),xn+1-p〉+〈f(p)+g(p)-p,xn+1-p〉≤α∥xn-p∥∥xn+1-p∥+β∥xn-p∥×∥xn+1-p∥+〈f(p)+g(p)-p,xn+1-p〉=(α+β)∥xn-p∥∥xn+1-p∥+〈f(p)+g(p)-p,xn+1-p〉.
Hence, the result follows.Lemma 9.
If there is a subsequence{xnk} of {xn} such that liminfk→∞(∥xnk+1-p∥-∥xnk-p∥)≥0, then
(21)limsupk→∞〈f(p)+g(p)-p,xnk+1-p〉≤0.Proof.
We note thatlimk→∞αnk=0. We consider the following inequality:
(22)0≤liminfk→∞(∥xnk+1-p∥-∥xnk-p∥)≤liminfk→∞(αnk∥f(xnk)-g(Tnkxnk)-p∥000000000+(1-αnk)∥Tnkxnk-p∥-∥xnk-p∥)≤liminfk→∞(∥Tnkxnk-p∥-∥xnk-p∥)≤limsupk→∞(∥Tnkxnk-p∥-∥xnk-p∥)≤0.
Then limk→∞(∥Tnkxnk-p∥-∥xnk-p∥)=0. Since {Tn} is strongly quasi-nonexpansive, so is {Tnk}. This implies that limk→∞∥xnk-Tnkxnk∥=0. Moreover,
(23)∥xnk+1-xnk∥≤∥xnk+1-Tnkxnk∥+∥Tnkxnk-xnk∥=αnk∥f(xnk)+g(Tnxnk)-Tnkxnk∥+∥Tnkxnk-xnk∥.
Then limk→∞∥xnk+1-xnk∥=0. Since {xnk} is bounded, there exists a subsequence {xnkl} of {xnk} such that w-liml→∞xnkl=q and
(24)limsupk→∞〈f(p)+g(p)-p,xnk-p〉=liml→∞〈f(p)+g(p)-p,xnkl-p〉.
As limk→∞∥xnk-xnk+1∥=0, we have limsupk→∞〈f(p) +g(p) -p,xnk+1 -p〉=〈f(p)+g(p)- p,q-p〉. Since {Tn} satisfies NST-condition, we have q∈F and hence 〈f(p)+g(p)-p,q-p〉≤0. Therefore,
(25)limsupk→∞〈f(p)+g(p)-p,xnk+1-p〉≤0,
as desired.Proof of Theorem6.
We are ready to apply Lemma3. Set
(26)an:=∥xn-p∥2,bn:=〈f(p)+g(p)-p,xn+1-p〉,α^:=α+β.
It follows that (i)
{an} is a bounded sequence (by Lemma 7);(ii)
an+1≤(1-αn)2an+2αnα^anan+1+αnbn for all n≥1 (by Lemma 8);(iii)
whenever{ank} is a subsequence of {an} satisfying liminfk→∞(ank+1-ank)≥0, it follows that limsupk→∞bnk≤0 (by Lemma 9).Hence, limn→∞∥xn-p∥=limn→∞an=0. This completes the proof.


## 4. Deduced Results

### 4.1. Wongchan and Saejung’s Result

Settingg≡0 and Tn≡T for all n∈ℕ in the proof of Theorem 6, we immediately have the following result of Wongchan and Saejung ([13, Theorem 6]).Corollary 10.
LetC be a closed convex subset of a Hilbert space H and T:C→C a strongly quasi-nonexpansive mapping such that I-T is demiclosed at zero. Suppose that f:C→C is a contraction and a sequence {xn} is generated by x1∈C and
(27)xn+1=αnf(xn)+(1-αn)Txn,
where {αn} is a sequence in (0,1) satisfying the conditions (C1) and (C2). Then {xn} converges strongly to p=PFix(T)f(p).


### 4.2. Tian and Jin’s Result I

Recall that a mappingF:H→H is η-strongly monotone if 〈x-y,Fx-Fy〉≥η∥x-y∥2 for all x,y∈H.Lemma 11.
LetF:H→H be an η-strongly monotone and κ-Lipschitzian mapping. Then ∥(I-μF)x-(I-μF)y∥≤1-2τ∥x-y∥ where τ=μ(η-(μκ2/2)) for all x,y∈H. In particular, if 0<μ<2η/κ2, then I-μF is a contraction.Proof.
Letx,y∈H. Then
(28)∥(I-μF)x-(I-μF)y∥2=∥(x-y)-μ(Fx-Fy)∥2=∥x-y∥2-2μ〈x-y,Fx-Fy〉+μ2∥Fx-Fy∥2≤∥x-y∥2-2μη∥x-y∥2+μ2κ2∥x-y∥2=(1-2μ(η-μκ22))∥x-y∥2=(1-2τ)∥x-y∥2.Theorem 12.
LetT:H→H be a strongly quasi-nonexpansive mapping such that I-T is demiclosed at zero. Let F:H→H be an η-strongly monotone and κ-Lipschitzian mapping. Let f:H→H be an L-Lipschitzian mapping and let a sequence {xn} be generated by x1∈H and
(29)xn+1=αnγf(xn)+(I-αnμF)Txn(n≥1),
where the sequence {αn}⊂(0,1) satisfies the conditions (C1) and (C2). Suppose that 0<μ<2η/κ2 and 0<γL<1-1-2τ, where τ=μ(η-(μκ2/2)). Then {xn} converges to p=PFix(T)(I-μF+γf)p.Proof.
First we rewrite the iteration (29) as follows:
(30)xn+1=αn(f^(xn)+g^(Txn))+(1-αn)Txn,
where f^=γf and g^=I-μF. Note that f^ is a γL-Lipschitzian and g^ is a 1-2τ-Lipschitzian. Using γL+1-2L<1 and putting Tn=T for all n∈ℕ in Theorem 6 imply that {xn} converges to p∈Fix(T), where
(31)p=PFix(T)(f^+g^)(p)=PFix(T)(I-μF+γf)(p).Lemma 13 (see [12]).
IfT:H→H is a quasi-nonexpansive mapping, then the mapping Tω:=(1-ω)I+ωT is strongly quasi-nonexpansive wherever ω∈(0,1).Using Theorem12 and Lemma 13, we immediately have the following result which is an improvement of Tian and Jin’s result ([15, Theorem 3.1]).Theorem 14.
LetT:H→H be a quasi-nonexpansive mapping such that I-T is demiclosed at zero. Let F:H→H be an η-strongly monotone and κ-Lipschitzian mapping. Let f:H→H be an L-Lipschitzian mapping and let the sequence {xn} be generated by x1∈H and
(32)xn+1=αnγf(xn)+(I-αnμF)Tωxn(n≥1),
where Tω=(1-ω)I+ωT, ω∈(0,1) and the sequence {αn}⊂(0,1) satisfies the conditions (C1) and (C2). Suppose that 0<μ<2η/κ2and 0<γL<1-1-2τ where τ=μ(η-(μκ2/2)). Then {xn} converges to p=PFix(T)(I-μF+γf)(p).Remark 15.
Theorem14 improves the result of Tian and Jin ([15, Theorem 3.1]) in the following ways. (i)
We assume thatγL<1-1-2τ while [15, Theorem 3.1] is proved under the assumptions γL<τ. We note that τ<1-1-2τ.(ii)
Our result allows us to chooseω in the wider interval (0,1) while [15, Theorem 3.1] is proved under the assumptions ω∈(0,1/2).


### 4.3. Tian and Jin’s Result II

Recall that a mappingA:H→H is strongly positive with the coefficient γ¯>0 if
(33)〈Ax,x〉≥γ¯∥x∥2
for all x∈H.Lemma 16 (see [20]).
LetA be a strongly positive self-adjoint linear bounded operator with coefficient γ¯>0 on H and 0<ρ≤∥A∥-1. Then ∥I-ρA∥≤1-ργ¯.Theorem 17.
LetT:H→H be a strongly quasi-nonexpansive mapping such that I-T is demiclosed at zero. Let A:H→H be a bounded linear self-adjoint operator and strongly positive with the coefficient γ¯. Let f:H→H be an α-contraction mapping and let a sequence {xn} be generated by x1∈H and
(34)xn+1=αnγf(xn)+(I-αnA)Txn(n≥1),
where the sequence {αn}⊂(0,1) satisfies the conditions (C1) and (C2). Suppose that 0<γα<γ¯. Then {xn} converges to p=PFix(T)(I-A+γf)p.Proof.
By Lemma16, we can choose t∈(0,1) such that ∥I-tA∥≤1-tγ¯. Rewrite the iteration (34) as follows:
(35)xn+1=α^n(f^(xn)+g^(Txn))+(1-α^n)Txn,
where f^:=tγf, g^:=I-tA and α^n≡αn/t for all n∈ℕ. Note that f^ is tγα-Lipschitzian and g^ is (1-tγ¯)-Lipschitzian. It follows from 0<γα<γ¯ that
(36)tγα+1-tγ¯=1-t(γ¯-αγ)<1.
Setting Tn≡T for all n∈ℕ in Theorem 6 implies that {xn} converges to p∈Fix(T) such that p=PFix(T)(f^+g^)p=PFix(T)(tγf+I-tA)p; that is, 〈tγf(p)+p-tAp-p,p-w〉≥0 for all w∈Fix(T). This implies that 〈γf(p)-Ap,p-w〉≥0 for all w∈Fix(T); that is, p=PFix(T)(γf+I-A)p. This completes the proof.Using Lemma13 and Theorem 17, we immediately have the following result which is an improvement of Tian and Jin's result ([16, Theorem 3.1]).Theorem 18.
LetT:H→H be a quasi-nonexpansive mapping such that I-T is demiclosed at zero. Let A:H→H be a bounded linear self-adjoint operator and strongly positive with the coefficient γ¯. Let f:H→H be an α-contraction mapping, and let the sequence {xn} be generated by x1∈H and
(37)xn+1=αnγf(xn)+(I-αnA)Tωxn(n≥1),
where Tω=(1-ω)I+ωT, ω∈(0,1) and the sequence {αn}⊂(0,1) satisfies the conditions (C1) and (C2). Suppose that 0<γα<γ¯. Then {xn} converges to p=PFix(T)(I-A+γf)p.Remark 19.
Theorem18 improves the result of Tian and Jin ([16, Theorem 3.1]). In fact, their result was proved under the assumption ω∈(0,1/2) while our result allows us to choose ω in the wider interval (0,1).


## 4.1. Wongchan and Saejung’s Result

Settingg≡0 and Tn≡T for all n∈ℕ in the proof of Theorem 6, we immediately have the following result of Wongchan and Saejung ([13, Theorem 6]).Corollary 10.
LetC be a closed convex subset of a Hilbert space H and T:C→C a strongly quasi-nonexpansive mapping such that I-T is demiclosed at zero. Suppose that f:C→C is a contraction and a sequence {xn} is generated by x1∈C and
(27)xn+1=αnf(xn)+(1-αn)Txn,
where {αn} is a sequence in (0,1) satisfying the conditions (C1) and (C2). Then {xn} converges strongly to p=PFix(T)f(p).


## 4.2. Tian and Jin’s Result I

Recall that a mappingF:H→H is η-strongly monotone if 〈x-y,Fx-Fy〉≥η∥x-y∥2 for all x,y∈H.Lemma 11.
LetF:H→H be an η-strongly monotone and κ-Lipschitzian mapping. Then ∥(I-μF)x-(I-μF)y∥≤1-2τ∥x-y∥ where τ=μ(η-(μκ2/2)) for all x,y∈H. In particular, if 0<μ<2η/κ2, then I-μF is a contraction.Proof.
Letx,y∈H. Then
(28)∥(I-μF)x-(I-μF)y∥2=∥(x-y)-μ(Fx-Fy)∥2=∥x-y∥2-2μ〈x-y,Fx-Fy〉+μ2∥Fx-Fy∥2≤∥x-y∥2-2μη∥x-y∥2+μ2κ2∥x-y∥2=(1-2μ(η-μκ22))∥x-y∥2=(1-2τ)∥x-y∥2.Theorem 12.
LetT:H→H be a strongly quasi-nonexpansive mapping such that I-T is demiclosed at zero. Let F:H→H be an η-strongly monotone and κ-Lipschitzian mapping. Let f:H→H be an L-Lipschitzian mapping and let a sequence {xn} be generated by x1∈H and
(29)xn+1=αnγf(xn)+(I-αnμF)Txn(n≥1),
where the sequence {αn}⊂(0,1) satisfies the conditions (C1) and (C2). Suppose that 0<μ<2η/κ2 and 0<γL<1-1-2τ, where τ=μ(η-(μκ2/2)). Then {xn} converges to p=PFix(T)(I-μF+γf)p.Proof.
First we rewrite the iteration (29) as follows:
(30)xn+1=αn(f^(xn)+g^(Txn))+(1-αn)Txn,
where f^=γf and g^=I-μF. Note that f^ is a γL-Lipschitzian and g^ is a 1-2τ-Lipschitzian. Using γL+1-2L<1 and putting Tn=T for all n∈ℕ in Theorem 6 imply that {xn} converges to p∈Fix(T), where
(31)p=PFix(T)(f^+g^)(p)=PFix(T)(I-μF+γf)(p).Lemma 13 (see [12]).
IfT:H→H is a quasi-nonexpansive mapping, then the mapping Tω:=(1-ω)I+ωT is strongly quasi-nonexpansive wherever ω∈(0,1).Using Theorem12 and Lemma 13, we immediately have the following result which is an improvement of Tian and Jin’s result ([15, Theorem 3.1]).Theorem 14.
LetT:H→H be a quasi-nonexpansive mapping such that I-T is demiclosed at zero. Let F:H→H be an η-strongly monotone and κ-Lipschitzian mapping. Let f:H→H be an L-Lipschitzian mapping and let the sequence {xn} be generated by x1∈H and
(32)xn+1=αnγf(xn)+(I-αnμF)Tωxn(n≥1),
where Tω=(1-ω)I+ωT, ω∈(0,1) and the sequence {αn}⊂(0,1) satisfies the conditions (C1) and (C2). Suppose that 0<μ<2η/κ2and 0<γL<1-1-2τ where τ=μ(η-(μκ2/2)). Then {xn} converges to p=PFix(T)(I-μF+γf)(p).Remark 15.
Theorem14 improves the result of Tian and Jin ([15, Theorem 3.1]) in the following ways. (i)
We assume thatγL<1-1-2τ while [15, Theorem 3.1] is proved under the assumptions γL<τ. We note that τ<1-1-2τ.(ii)
Our result allows us to chooseω in the wider interval (0,1) while [15, Theorem 3.1] is proved under the assumptions ω∈(0,1/2).


## 4.3. Tian and Jin’s Result II

Recall that a mappingA:H→H is strongly positive with the coefficient γ¯>0 if
(33)〈Ax,x〉≥γ¯∥x∥2
for all x∈H.Lemma 16 (see [20]).
LetA be a strongly positive self-adjoint linear bounded operator with coefficient γ¯>0 on H and 0<ρ≤∥A∥-1. Then ∥I-ρA∥≤1-ργ¯.Theorem 17.
LetT:H→H be a strongly quasi-nonexpansive mapping such that I-T is demiclosed at zero. Let A:H→H be a bounded linear self-adjoint operator and strongly positive with the coefficient γ¯. Let f:H→H be an α-contraction mapping and let a sequence {xn} be generated by x1∈H and
(34)xn+1=αnγf(xn)+(I-αnA)Txn(n≥1),
where the sequence {αn}⊂(0,1) satisfies the conditions (C1) and (C2). Suppose that 0<γα<γ¯. Then {xn} converges to p=PFix(T)(I-A+γf)p.Proof.
By Lemma16, we can choose t∈(0,1) such that ∥I-tA∥≤1-tγ¯. Rewrite the iteration (34) as follows:
(35)xn+1=α^n(f^(xn)+g^(Txn))+(1-α^n)Txn,
where f^:=tγf, g^:=I-tA and α^n≡αn/t for all n∈ℕ. Note that f^ is tγα-Lipschitzian and g^ is (1-tγ¯)-Lipschitzian. It follows from 0<γα<γ¯ that
(36)tγα+1-tγ¯=1-t(γ¯-αγ)<1.
Setting Tn≡T for all n∈ℕ in Theorem 6 implies that {xn} converges to p∈Fix(T) such that p=PFix(T)(f^+g^)p=PFix(T)(tγf+I-tA)p; that is, 〈tγf(p)+p-tAp-p,p-w〉≥0 for all w∈Fix(T). This implies that 〈γf(p)-Ap,p-w〉≥0 for all w∈Fix(T); that is, p=PFix(T)(γf+I-A)p. This completes the proof.Using Lemma13 and Theorem 17, we immediately have the following result which is an improvement of Tian and Jin's result ([16, Theorem 3.1]).Theorem 18.
LetT:H→H be a quasi-nonexpansive mapping such that I-T is demiclosed at zero. Let A:H→H be a bounded linear self-adjoint operator and strongly positive with the coefficient γ¯. Let f:H→H be an α-contraction mapping, and let the sequence {xn} be generated by x1∈H and
(37)xn+1=αnγf(xn)+(I-αnA)Tωxn(n≥1),
where Tω=(1-ω)I+ωT, ω∈(0,1) and the sequence {αn}⊂(0,1) satisfies the conditions (C1) and (C2). Suppose that 0<γα<γ¯. Then {xn} converges to p=PFix(T)(I-A+γf)p.Remark 19.
Theorem18 improves the result of Tian and Jin ([16, Theorem 3.1]). In fact, their result was proved under the assumption ω∈(0,1/2) while our result allows us to choose ω in the wider interval (0,1).


## 5. A Discussion on Marino-Xu’s Result

The following theorem is studied by many authors; for example, see [3].Theorem 20.
LetC be a closed convex subset of a Hilbert space H. Suppose that (i)
T:C→C is a nonexpansive mapping and Fix(T)≠Ø;(ii)
{αn}⊂(0,1) is a sequence satisfying the conditions (C1), (C2), and (C3).Define the following iterative sequence:
(38)u,x1∈C,(39)xn+1=αnu+(1-αn)Txn.
Then {xn} converges to PFix(T)u.Using the technique in [4], we can give a simple proof of the following result proved by Marino and Xu [20].Theorem 21.
Suppose that(i)
A:H→H is a bounded linear self-adjoint operator and it is strongly positive with the coefficient γ¯;(ii)
T:H→H is a nonexpansive mapping and Fix(T)≠Ø;(iii)
f:H→H is an α-contraction;(iv)
γ is a positive number such that 0<γα<γ¯;(v)
{αn}⊂(0,1) is a sequence satisfying the conditions (C1), (C2), and (C3). Define the following iterative sequence:
(40)z1∈H(41)zn+1=αnγf(zn)+(I-αnA)Tzn.
Then {zn} converges to z^∈Fix(T) and 〈Az^-γf(z^),z^-w〉≤0 for all w∈Fix(T).Proof.
Chooset∈(0,1) such that ∥I-tA∥≤1-tγ¯. First we show that I-tA+tγf is a contraction. To see this, let x,y∈H. Then
(42)∥(I-tA+tγf)x-(I-tA+tγf)y∥≤∥(I-tA)x-(I-tA)y∥+tγ∥f(x)-f(y)∥≤∥I-tA∥∥x-y∥+tγ∥f(x)-f(y)∥≤(1-tγ¯)∥x-y∥+tγα∥x-y∥=(1-t(γ¯-γα))∥x-y∥.
It follows from γα<γ¯ that I-tA+tγf is a contraction. Note that PFix(T) is nonexpansive and hence PFix(T)(I-tA+tγf) is a contraction from Fix(T) into itself. It follows from the closedness of Fix(T) and the Banach’s contraction principle that there exists a unique element z^∈Fix(T) such that
(43)z^=PFix(T)(I-tA+tγf)(z^).
Therefore,
(44)〈Az^-γf(z^),z^-w〉≤0∀w∈Fix(T).
Now we define the following iterative sequence:
(45)x1=z1,xn+1=αnt((I-tA)Tz^+tγf(z^))+(1-αnt)Txn.
It follows from Theorem 20 that the sequence {xn} converges to z^=PFix(T)(I-tA+tγf)(z^). Observe that
(46)zn+1=αnt((I-tA)Tzn+tγf(zn))+(1-αnt)Tzn.
We next consider the following expression:
(47)∥zn+1-xn+1∥=∥(1-αnt)(Tzn-Txn)+αnt(I-tA)(Tzn-Tz^)+αnttγ(f(zn)-f(z^))∥≤(1-αnt)∥zn-xn∥+αnt(1-tγ¯)∥zn-z^∥+αnγα∥zn-z^∥=(1-αnt)∥zn-xn∥+(αnt-αn(γ¯-γα))∥zn-z^∥≤(1-αn(γ¯-γα))∥zn-xn∥+(αnt-αn(γ¯-γα))∥xn-z^∥=(1-αn(γ¯-γα))∥zn-xn∥+αn(γ¯-γα)×((1/t)-(γ¯-γα)γ¯-γα)∥xn-z^∥.
It follows from Lemma 4 that limn→∞∥zn-xn∥=0. Therefore, we conclude that {zn} converges to z^∈Fix(T) and 〈Az^-γf(z^),z^-w〉≤0 for all w∈Fix(T). This completes the proof.



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