# Strong Convergence Theorems of Modified Ishikawa Iterative Method for an Infinite Family of  Strict  Pseudocontractions in Banach Spaces

Authors: Phayap Katchang; Wiyada Kumam; Usa Humphries; Poom Kumam
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

We introduce a new modified Ishikawa iterative process and a new W-mapping for comput-
ing fixed points of an infinite family of strict pseudocontractions mapping in the framework of q-uniformly
smooth Banach spaces. Then, we establish the strong convergence theorem of the proposed iterative scheme
under some mild conditions. The results obtained in this paper extend and improve the recent results of
Cai and Hu 2010, Dong et al. 2010, Katchang and Kumam 2011 and many others in the literature.


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## Body

## 1. Introduction

LetE be a real Banach  space with norm ∥·∥ and C a nonempty closed convex subset of E. Let E* be the dual space of E, and let 〈·,·〉 denotes the generalized duality  pairing between E and E*. For q>1, the generalized duality mappingJq:E→2E* is defined by (1.1)Jq(x)={f∈E*:〈x,f〉=‖x‖q,‖f‖=‖x‖q-1},
for all x∈E. In particular, if q=2, the mapping J=J2 is called the normalized duality mapping and Jq(x)=∥x∥q-2J2(x) for x≠0. It is well known that if E is smooth, then Jq is single-valued, which is denoted by jq.A mappingT:C→C is called nonexpansive if(1.2)‖Tx-Ty‖≤‖x-y‖,∀x,y∈C.
We use F(T) to denote the set of fixed points of T; that is, F(T)={x∈C:Tx=x}.T is said to be a λ-strict pseudocontraction in the terminology of Browder and Petryshyn [1] if there exists a constant λ>0 and for some jq(x-y)∈Jq(x-y) such that (1.3)〈Tx-Ty,jq(x-y)〉≤‖x-y‖q-λ‖(I-T)x-(I-T)y‖q,∀x,y∈C.T is said to be a strong pseudocontraction if there exists k∈(0,1) such that (1.4)〈Tx-Ty,jq(x-y)〉≤k∥x-y∥q,∀x,y∈C.Remark 1.1 (see [2]).
LetT be a λ-strict pseudocontraction in a Banach space. Let x∈C and p∈F(T). Then,
(1.5)‖Tx-p‖≤(1+1λ1/q-1)‖x-p‖.Recall that a self mappingf:C→C is contraction on C if there exists a constant α∈(0,1) and x,y∈C such that (1.6)‖f(x)-f(y)‖≤α‖x-y‖.
We use ΠC to denote the collection of all contractions on C. That is, ΠC={f∣f:C→Cacontraction}. Note that each f∈ΠC has a unique fixed point in C.Very recently, Cai and Hu [3] also proved the strong convergence theorem in Banach spaces. They considered the following iterative algorithm:(1.7)x1=x∈Cchosenarbitrarily,yn=PC[βnxn+(1-βn)∑i=iNηi(n)Tixn],xn+1=αnγf(xn)+γnxn+((1-γn)I-αnA)yn,∀n≥1,
where Ti is a non-self-λi-strictly pseudocontraction, f is a contraction, and A is a strongly positive linear bounded operator.Dong et al. [2] proved the sequence {xn} converges strongly in Banach spaces under certain appropriate assumptions and used the Wn mapping defined by (1.11). Let the sequences {xn} be generated by(1.8)x0=x∈Cchosenarbitrarily,yn=δnxn+(1-δn)Wnxn,xn+1=αnf(xn)+βnxn+γnWnyn,∀n≥0.On the other hand, Katchang and Kumam [4, 5] introduced the following new modified Ishikawa iterative process for computing fixed points of an infinite family nonexpansive mapping in the framework of Banach spaces; let the sequences {xn} be generated by(1.9)x0=x∈Cchosenarbitrarily,zn=γnxn+(1-γn)Wnxn,yn=βnxn+(1-βn)Wnzn,xn+1=αnγf(x)+(I-αnA)yn,∀n≥0,
where f is a contraction, A is a strongly positive linear bounded self-adjoint operator, and Wn mapping (see [6, 7]) is defined by(1.10)Un,n+1=I,Un,n=λnTnUn,n+1+(1-λn)I,Un,n-1=λn-1Tn-1Un,n+(1-λn-1)I,⋮Un,k=λkTkUn,k+1+(1-λk)I,Un,k-1=λk-1Tk-1Un,k+(1-λk-1)I,⋮Un,2=λ2T2Un,3+(1-λ2)I,Wn=Un,1=λ1T1Un,2+(1-λ1)I,
where T1,T2,… is an infinite family of nonexpansive mappings of C into itself and λ1,λ2,… is real numbers such that 0≤λn≤1 for every n∈ℕ. In 2010, Cho [8] considered and proved the strong convergence of the implicit iterative process for an infinite family of strict pseudocontractions in an arbitrary real Banach space.In this paper, motivated and inspired by Cai and Hu [9], we consider the mapping Wn defined by(1.11)Un,n+1=I,Un,n=tnTn,nUn,n+1+(1-tn)I,⋮Un,k=tkTn,kUn,k+1+(1-tk)I,Un,k-1=tk-1Tn,k-1Un,k+(1-tk-1)I,⋮Un,2=t2Tn,2Un,3+(1-t2)I,Wn=Un,1=t1Tn,1Un,2+(1-t1)I,
where t1,t2,… are real numbers such that 0≤tn≤1.Tn,k=θn,kSk+(1-θn,k)I, where Sk is a λk-strict pseudocontraction of C into itself and θn,k∈(0,μ],μ=min{1,{qλ/Cq}1/q-1}, where λ=infλk for all k∈ℕ. By Lemma 2.3, we know that Tn,k is a nonexpansive mapping, and therefore, Wn is a nonexpansive mapping. We note that the W-mapping (1.10) is a special case of a W-mapping (1.11) when θn,k=θk is constant for all n≥1.Throughout this paper, we will assume thatinfλi>0, 0<tn≤b<1 for all n∈ℕ and {θn,k} satisfies(H1)
θn,k∈(0,μ],μ=min{1,infi{qλi/Cq}1/q-1} for all k∈ℕ,(H2)
|θn+1,k-θn,k|≤an for all n∈ℕ and 1≤k≤n, where {an} satisfies ∑n=1∞an<∞.The hypothesis (H2) secures the existence oflimn→∞θn,k for all k∈ℕ. Set θ1,k:=limn→∞θn,k for all k∈ℕ. Furthermore, we assume that(H3)
θ1,k>0 for all k∈ℕ.It is obvious thatθ1,k satisfy (H1). Using condition (H3), from Tn,k=θn,kSk+(1-θn,k)I, we define mappings T1,kx:=limn→∞Tn,kx=θ1,kSkx+(1-θ1,k)x for all x∈C.Our results improve and extend the recent ones announced by Cai and Hu [3], Dong et al. [2], Katchang and Kumam [4, 5], and many others.


## 2. Preliminaries

Recall thatU={x∈E:∥x∥=1}. A Banach space E is said to be uniformly convex if, for any ϵ∈(0,2], there exists δ>0 such that for any x,y∈U, ∥x-y∥≥ϵ implies ∥(x+y)/2∥≤1-δ. It is known that a uniformly convex Banach space is reflexive and strictly convex (see also [10]). A Banach space E is said to be smooth if the limit limt→0(∥x+ty∥-∥x∥)/t exists for all x,y∈U. It is also said to be uniformly smooth if the limit is attained uniformly for x,y∈U.In a smooth Banach space, we define an operatorA as strongly positive if there exists a constant γ¯>0 with the property(2.1)〈Ax,J(x)〉≥γ¯‖x‖2,‖aI-bA‖=sup⁡∥x∥≤1|〈(aI-bA)x,J(x)〉|a∈[0,1],b∈[-1,1],
where I is the identity mapping and J is the normalized duality mapping.IfC and D are nonempty subsets of a Banach space E such that C is a nonempty closed convex and D⊂C, then a mapping Q:C→D is sunny [11, 12] provided that Q(x+t(x-Q(x)))=Q(x) for all x∈C and t≥0 whenever x+t(x-Q(x))∈C. A mapping Q:C→C is called a retraction if Q2=Q. If a mapping Q:C→C is a retraction, then Qz=z for all z in the range of Q. A subset D of C is said to be a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction Q of C onto D. A sunny nonexpansive retraction is a sunny retraction, which is also nonexpansive. Sunny nonexpansive retractions play an important role in our argument. They are characterized as follows [11, 12]: if E is a smooth Banach space, then Q:C→D is a sunny nonexpansive retraction if and only if there holds the inequality(2.2)〈x-Qx,J(y-Qx)〉≤0,∀x∈C,y∈D.We need the following lemmas for proving our main results.Lemma 2.1 (see [13]).
In a Banach space E, the following holds:(2.3)‖x+y‖2≤‖x‖2+2〈y,j(x+y)〉,∀x,y∈E,
where j(x+y)∈J(x+y).Lemma 2.2 (see [14]).
LetE be a real q-uniformly smooth Banach space, then there exists a constant Cq>0 such that
(2.4)‖x+y‖q≤‖x‖q+q〈y,jqx〉+Cq‖y‖q,∀x,y∈E.
In particular, if E be a real 2-uniformly smooth Banach space with the best smooth constant K, then the following inequality holds:
(2.5)‖x+y‖2≤‖x‖2+2〈y,jx〉+2‖Ky‖2,∀x,y∈E.The relation between theλ-strict pseudocontraction and the nonexpansive mapping can be obtained from the following lemma.Lemma 2.3 (see [15]).
LetC be a nonempty convex subset of a real q-uniformly smooth Banach space E and S:C→C a λ-strict pseudocontraction. For α∈(0,1), one defines Tx=(1-α)x+αSx. Then, as α∈(0,μ],μ=min{1,{qλ/Cq}1/q-1}, T:C→C is nonexpansive such that F(T)=F(S), where Cq is the constant in Lemma 2.2.ConcerningWn, we have the following lemmas, which are important to prove the main results.Lemma 2.4 (see [2]).
LetC be a nonempty closed convex subset of a q-uniformly smooth and strictly convex Banach space E. Let Si,i=1,2,…, be a λi-strict pseudocontraction from C into itself such that ∩n=1∞F(Sn)≠∅, and let infλi>0. Let tn,n=1,2,…, be real numbers such that 0<tn≤b<1 for any n≥1. Assume that the sequence {θn,k} satisfies (H1) and (H2). Then, for every x∈C and k∈ℕ, the limit limn→∞Un,kx exists.Using Lemma2.4, we define the mappings U1,k and W:C→C as follows: (2.6)U1,kx:=lim⁡n→∞Un,kx,Wx:=lim⁡n→∞Wnx=lim⁡n→∞Un,1x,
for all x∈C. Such W is called the W-mapping generated by S1,S2,…,t1,t2,… and θn,k,foralln∈ℕ and 1≤k≤n.Lemma 2.5 (see [2]).
Let{xn} be a bounded sequence in a q-uniformly smooth and strictly convex Banach space E. Under the assumptions of Lemma 2.4, it holds
(2.7)lim⁡n→∞‖Wnxn-Wxn‖=0.Lemma 2.6 (see [2]).
LetC be a nonempty closed convex subset of a q-uniformly smooth and strictly convex Banach space E. Let Si,i=1,2,…, be a λi-strict pseudocontraction from C into itself such that ∩n=1∞F(Sn)≠∅, and let infλi>0. Let tn,n=1,2,…, be real numbers such that 0<tn≤b<1 for any n≥1. Assume that the sequence {θn,k} satisfies (H1)–(H3). Then, F(W)=∩n=1∞F(Sn).Lemma 2.7 (see [16]).
Assume that{an} is a sequence of nonnegative real numbers such that
(2.8)an+1≤(1-αn)an+δn,n≥0,
where {αn} is a sequence in (0,1) and {δn} is a sequence in ℝ such that (i)
∑n=1∞αn=∞,(ii)
limsupn→∞δn/αn≤0 or ∑n=1∞|δn|<∞. Then, limn→∞an=0.Lemma 2.8 (see [17]).
Let{xn} and {yn} be bounded sequences in a Banach space X, and let {βn} be a sequence in [0,1] with 0<liminfn→∞βn≤limsupn→∞βn<1. Suppose that xn+1=(1-βn)yn+βnxn for all integers n≥0 and limsupn→∞(∥yn+1-yn∥-∥xn+1-xn∥)≤0. Then, limn→∞∥yn-xn∥=0.Lemma 2.9 (see [3]).
Assume thatA is a strong positive linear bounded operator on a smooth Banach space E with coefficient γ¯>0 and 0<ρ≤∥A∥-1. Then, ∥I-ρA∥≤1-ργ¯.


## 3. Main Results

In this section, we prove a strong convergence theorem.Theorem 3.1.
LetE be a real q-uniformly smooth and strictly convex Banach space which admits a weakly sequentially continuous duality mapping J from E to E*. Let C be a nonempty closed and convex subset of E which is also a sunny nonexpansive retraction of E such that C+C⊂C. Let A be a strongly positive linear bounded operator on E with coefficient γ¯>0 such that 0<γ<γ¯/α, and let f be a contraction of C into itself with coefficient α∈(0,1). Let Si,i=1,2,…, be λi-strict pseudocontractions from C into itself such that ∩n=1∞F(Sn)≠∅ and infλi>0. Assume that the sequences {αn},{βn}, {γn}, and {δn} in (0,1) satisfy the following conditions:(i)
∑n=0∞αn=∞; and limn→∞αn=0,(ii)
0<liminfn→∞βn≤limsupn→∞βn<1,(iii)
limn→∞|γn+1-γn|=0,(iv)
limn→∞|δn+1-δn|=0,(v)
δn(1+γn)-2γn>a for some a∈(0,1),and the sequence {θn,k} satisfies (H1)–(H3). Then, the sequence {xn} generated by
(3.1)x0∈Cchosenarbitrarily,zn=δnxn+(1-δn)Wnxn,yn=γnxn+(1-γn)Wnzn,xn+1=αnγf(xn)+βnxn+((1-βn)I-αnA)yn,∀n≥0
converges strongly to x*∈∩n=1∞F(Sn), which solves the following variational inequality:
(3.2)〈γf(x*)-Ax*,J(p-x*)〉≤0,∀f∈ΠC,p∈⋂n=1∞F(Sn).Proof.
By (i), we may assume, without loss of generality, thatαn≤(1-βn)∥A∥-1 for all n. Since A is a strongly positive bounded linear operator on E and by (2.1), we have
(3.3)‖A‖=sup⁡{|〈Ax,J(x)〉|:x∈E,‖x‖=1}.
Observe that
(3.4)〈((1-βn)I-αnA)x,J(x)〉=1-βn-αn〈Ax,J(x)〉≥1-βn-αn∥A∥≥0,∀x∈E.
This shows that (1-βn)I-αnA is positive. It follows that
(3.5)‖(1-βn)I-αnA‖=sup⁡{|〈((1-βn)I-αnA)x,J(x)〉|:x∈E,‖x‖=1}=sup⁡{1-βn-αn〈Ax,J(x)〉:x∈E,‖x‖=1}≤1-βn-αnγ¯.
First, we show that{xn} is bounded. Let p∈∩n=1∞F(Sn). By the definition of {zn},{yn}, and {xn}, we have
(3.6)‖zn-p‖=‖δnxn+(1-δn)Wnxn-p‖≤δn‖xn-p‖+(1-δn)‖Wnxn-p‖≤δn‖xn-p‖+(1-δn)‖xn-p‖=‖xn-p‖,
and from this, we have
(3.7)‖yn-p‖=‖γnxn+(1-γn)Wnzn-p‖≤γn‖xn-p‖+(1-γn)‖Wnzn-p‖≤γn‖xn-p‖+(1-γn)‖zn-p‖≤γn‖xn-p‖+(1-γn)‖xn-p‖=‖xn-p‖.
It follows that(3.8)‖xn+1-p‖=‖αnγf(xn)+βnxn+((1-βn)I-αnA)yn-p‖=‖αn(γf(xn)-Ap)+βn(xn-p)+((1-βn)I-αnA)(yn-p)‖≤αn‖γf(xn)-Ap‖+βn‖xn-p‖+(1-βn-αnγ¯)‖yn-p‖≤αn‖γf(xn)-Ap‖+βn‖xn-p‖+(1-βn-αnγ¯)‖xn-p‖≤αn‖γf(xn)-γf(p)‖+αn‖γf(p)-Ap‖+(1-αnγ¯)‖xn-p‖≤αnγα‖xn-p‖+αn‖γf(p)-Ap‖+(1-αnγ¯)‖xn-p‖=(1-(γ¯-γα)αn)‖xn-p‖+(γ¯-γα)αn‖γf(p)-Ap‖γ¯-γα≤max⁡{‖x1-p‖,‖γf(p)-Ap‖γ¯-γα}.
By induction on n, we obtain ∥xn-p∥≤max{∥x0-p∥,∥γf(p)-Ap∥/γ¯-γα∥} for every n≥0 and x0∈C, then {xn} is bounded. So, {yn},{zn},{Ayn},{Wnxn}, {Wnzn}, and {f(xn)} are also bounded.
Next, we claim that∥xn+1-xn∥→0 as n→∞. Let x∈C and p∈∩n=1∞F(Sn). Fix k∈ℕ for any n∈ℕ with n≥k, and since Tn,k and Un,k are nonexpansive, we have ∥Tn,kx-p∥≤∥x-p∥ and ∥Un,kx-p∥≤∥x-p∥, respectively. From (1.5), it follows that ∥Skx-p∥≤(1+(1/λk1/q-1))supn∥x-p∥. We can set
(3.9)M1=inf⁡i(2+1λi1/q-1)sup⁡n‖xn-p‖<∞,M2=inf⁡i(2+1λi1/q-1)sup⁡n‖zn-p‖<∞.
From (1.11), we have
(3.10)‖Wn+1xn-Wnxn‖=‖Un+1,1xn-Un,1xn‖=‖t1Tn+1,1Un+1,2xn+(1-t1)xn-t1Tn,1Un,2xn-(1-t1)xn‖=t1‖Tn+1,1Un+1,2xn-Tn,1Un,2xn‖=t1‖(θn+1,1S1+(1-θn+1,1))Un+1,2xn-Tn,1Un,2xn‖=t1‖(θn,1S1+(1-θn,1))Un+1,2xn-Tn,1Un,2xn+(θn+1,1-θn,1)(S1Un+1,2xn-Un+1,2xn)‖≤t1‖Tn,1Un+1,2xn-Tn,1Un,2xn‖+t1|θn+1,1-θn,1|‖S1Un+1,2xn-Un+1,2xn‖≤t1‖Un+1,2xn-Un,2xn‖+t1|θn+1,1-θn,1|M1≤t1‖Un+1,2xn-Un,2xn‖+t1anM1⋮≤∏i=1nti‖Un+1,n+1xn-Un,n+1xn‖+anM1∑j=1n∏i=1jti≤∏i=1nti‖tn+1Tn+1,n+1xn+(1-tn+1)xn-xn‖+anM1b1-b≤∏i=1n+1ti‖Tn+1,n+1xn-xn‖+anM1b1-b≤(bn+1+anb1-b)M1,
for all n≥0. Similarly, we also have ∥Wn+1zn-Wnzn∥≤(bn+1+an(b/(1-b)))M2 for all n≥0. We compute that
(3.11)‖zn+1-zn‖=‖(δn+1xn+1+(1-δn+1)Wn+1xn+1)-(δnxn+(1-δn)Wnxn)‖≤(1-δn+1)‖Wn+1xn+1-Wn+1xn‖+|δn+1-δn|‖xn-Wn+1xn‖+δn+1‖xn+1-xn‖+(1-δn)‖Wn+1xn-Wnxn‖≤(1-δn+1)‖xn+1-xn‖+|δn+1-δn|‖Wn+1xn-xn‖+δn+1‖xn+1-xn‖+(1-δn)‖Wn+1xn-Wnxn‖=‖xn+1-xn‖+|δn+1-δn|‖Wn+1xn-xn‖+(1-δn)‖Wn+1xn-Wnxn‖≤‖xn+1-xn‖+|δn+1-δn|‖Wn+1xn-xn‖+‖Wn+1xn-Wnxn‖≤‖xn+1-xn‖+|δn+1-δn|‖Wn+1xn-xn‖+(bn+1+anb1-b)M1,
and
(3.12)‖yn+1-yn‖=‖(γn+1xn+1+(1-γn+1)Wn+1zn+1)-(γnxn+(1-γn)Wnzn)‖=‖γn+1xn+1+(1-γn+1)Wn+1zn+1-(1-γn+1)Wn+1zn+(1-γn+1)Wn+1zn-γnxn-(1-γn)Wnzn-(1-γn)Wn+1zn+(1-γn)Wn+1zn-γn+1xn+γn+1xn‖=‖(1-γn+1)(Wn+1zn+1-Wn+1zn)+(γn-γn+1)Wn+1zn+γn+1(xn+1-xn)+(γn+1-γn)xn+(1-γn)(Wn+1zn-Wnzn)‖≤(1-γn+1)‖Wn+1zn+1-Wn+1zn‖+|γn+1-γn|‖xn-Wn+1zn‖+γn+1‖xn+1-xn‖+(1-γn)‖Wn+1zn-Wnzn‖≤(1-γn+1)‖zn+1-zn‖+|γn+1-γn|‖Wn+1zn-xn‖+γn+1‖xn+1-xn‖+‖Wn+1zn-Wnzn‖≤(1-γn+1)(‖xn+1-xn‖+|δn+1-δn|‖Wn+1xn-xn‖+(bn+1+anb1-b)M1)+|γn+1-γn|‖Wn+1zn-xn‖+γn+1‖xn+1-xn‖+(bn+1+anb1-b)M2≤‖xn+1-xn‖+|δn+1-δn|‖Wn+1xn-xn‖+|γn+1-γn|‖Wn+1zn-xn‖+2(bn+1+anb1-b)M,
where M=infi(2+(1/λi1/q-1))supn(∥xn-p∥+∥zn-p∥)<∞. Observe that we put ln=(xn+1-βnxn)/1-βn, then
(3.13)xn+1=(1-βn)ln+βnxn,∀n≥0.
Now, we have
(3.14)‖ln+1-ln‖=‖αn+1γf(xn+1)+((1-βn+1)I-αn+1A)yn+11-βn+1-αnγf(xn)+((1-βn)I-αnA)yn1-βn‖=‖αn+1γf(xn+1)1-βn+1+(1-βn+1)yn+11-βn+1-αn+1Ayn+11-βn+1-αnγf(xn)1-βn-(1-βn)yn1-βn+αnAyn1-βn‖=‖αn+11-βn+1(γf(xn+1)-Ayn+1)+αn1-βn(Ayn-γf(xn))+yn+1-yn‖≤αn+11-βn+1‖γf(xn+1)-Ayn+1‖+αn1-βn‖Ayn-γf(xn)‖+‖yn+1-yn‖≤αn+11-βn+1‖γf(xn+1)-Ayn+1‖+αn1-βn‖Ayn-γf(xn)‖+‖xn+1-xn‖+|δn+1-δn|‖Wn+1xn-xn‖+|γn+1-γn|‖Wn+1zn-xn‖+2(bn+1+anb1-b)M.
Therefore, we have
(3.15)‖ln+1-ln‖-‖xn+1-xn‖≤αn+11-βn+1‖γf(xn+1)-Ayn+1‖+αn1-βn‖Ayn-γf(xn)‖+|δn+1-δn|‖Wn+1xn-xn‖+|γn+1-γn|‖Wn+1zn-xn‖+2(bn+1+anb1-b)M.
From the conditions (i)–(iv), (H2), 0<b<1 and the boundedness of {xn}, {f(xn)}, {Ayn}, {Wnxn}, and {Wnzn}, we obtain
(3.16)limsup⁡n→∞(‖ln+1-ln‖-‖xn+1-xn‖)≤0.
It follows from Lemma 2.8 that limn→∞∥ln-xn∥=0. Noting (3.13), we see that
(3.17)‖xn+1-xn‖=(1-βn)‖ln-xn‖⟶0,
as n→∞. Therefore, we have
(3.18)lim⁡n→∞‖xn+1-xn‖=0.
We also have ∥yn+1-yn∥→0 and ∥zn+1-zn∥→0 as n→∞. Observing that
(3.19)‖xn-yn‖≤‖xn-xn+1‖+‖xn+1-yn‖≤‖xn-xn+1‖+αn‖γf(xn)-Ayn‖+βn‖xn-yn‖,
it follows that
(3.20)(1-βn)‖xn-yn‖≤‖xn-xn+1‖+αn‖γf(xn)-Ayn‖.
By the conditions (i), (ii), (3.18), and the boundedness of {xn}, {f(xn)}, and {Ayn}, we obtain
(3.21)lim⁡n→∞‖xn-yn‖=0.
Consider
(3.22)‖yn-Wnzn‖=‖γnxn+(1-γn)Wnzn-Wnzn‖=γn‖xn-Wnzn‖,‖zn-xn‖=‖δnxn+(1-δn)Wnxn-xn‖=(1-δn)‖Wnxn-xn‖.
It follows that
(3.23)‖xn-Wnxn‖≤‖xn-yn‖+‖yn-Wnzn‖+‖Wnzn-Wnxn‖≤‖xn-yn‖+γn‖xn-Wnzn‖+‖zn-xn‖≤‖xn-yn‖+γn‖xn-Wnxn‖+γn‖Wnxn-Wnzn‖+‖zn-xn‖≤‖xn-yn‖+γn‖xn-Wnxn‖+(1+γn)‖zn-xn‖=‖xn-yn‖+γn‖xn-Wnxn‖+(1+γn)(1-δn)‖Wnxn-xn‖.
This implies that
(3.24)(δn(1+γn)-2γn)‖Wnxn-xn‖≤‖xn-yn‖.
From the condition (v) and (3.21), we get
(3.25)lim⁡n→∞‖Wnxn-xn‖=0.
On the other hand,
(3.26)‖Wxn-xn‖≤‖Wxn-Wnxn‖+‖Wnxn-xn‖.
From the boundedness of {xn} and using (2.7), we have ∥Wxn-Wnxn∥→0 as n→∞. It follows that
(3.27)lim⁡n→∞‖Wxn-xn‖=0.
Next, we prove that(3.28)limsup⁡n→∞〈γf(x*)-Ax*,J(xn-x*)〉≤0,
where x*=limt→0xt with xt being the fixed point of contraction x↦tγf(x)+(1-tA)Wx. Noticing that xt solves the fixed point equation xt=tγf(xt)+(1-tA)Wxt, it follows that
(3.29)‖xt-xn‖=‖(I-tA)(Wxt-xn)+t(γf(xt)-Axn)‖.
It follows from Lemma 2.1 that(3.30)‖xt-xn‖2=‖(I-tA)(Wxt-xn)+t(γf(xt)-Axn)‖2≤(1-γ¯t)2‖Wxt-xn‖2+2t〈γf(xt)-Axn,J(xt-xn)〉=(1-γ¯t)2‖Wxt-Wxn+Wxn-xn‖2+2t〈γf(xt)-Axn,J(xt-xn)〉≤(1-γ¯t)2[‖Wxt-Wxn‖2+2〈Wxn-xn,J(Wxt-xn)〉]+2t〈γf(xt)-Axn,J(xt-xn)〉≤(1-γ¯t)2[‖xt-xn‖2+2‖Wxn-xn‖‖Wxt-xn‖]+2t〈γf(xt)-Axn,J(xt-xn)〉≤(1-γ¯t)2[‖xt-xn‖2+2‖Wxn-xn‖(‖xt-xn‖+‖Wxn-xn‖)]+2t〈γf(xt)-Axn,J(xt-xn)〉=(1-2γ¯t+(γ¯t)2)‖xt-xn‖2+2(1-γ¯t)2‖Wxn-xn‖(‖xt-xn‖+‖Wxn-xn‖)+2t〈γf(xt)-Axt,J(xt-xn)〉+2t〈Axt-Axn,J(xt-xn)〉,
where
(3.31)fn(t)=2(1-γ¯t)2‖Wxn-xn‖(‖xt-xn‖+‖Wxn-xn‖)⟶0asn⟶∞.
Since A is linearly strong and positive and using (2.1), we have
(3.32)〈Axt-Axn,J(xt-xn)〉=〈A(xt-xn),J(xt-xn)〉≥γ¯‖xt-xn‖2.
Substituting (3.32) in (3.30), we have
(3.33)〈Axt-γf(xt),J(xt-xn)〉≤(γ¯2t2-γ¯)‖xt-xn‖2+12tfn(t)+〈Axt-Axn,J(xt-xn)〉≤(γ¯t2-1)〈Axt-Axn,J(xt-xn)〉+12tfn(t)+〈Axt-Axn,J(xt-xn)〉=γ¯t2〈Axt-Axn,J(xt-xn)〉+12tfn(t).
Letting n→∞ in (3.33) and noting (3.31) yield that
(3.34)limsup⁡n→∞〈Axt-γf(xt),J(xt-xn)〉≤t2M3,
where M3>0 is a constant such that M3≥γ¯〈Axt-Axn,J(xt-xn)〉 for all t∈(0,1) and n≥0. Taking t→0 from (3.34), we have
(3.35)limsup⁡t→0limsup⁡n→∞〈Axt-γf(xt),J(xt-xn)〉≤0.
On the other hand, we have
(3.36)〈γf(x*)-Ax*,J(xn-x*)〉=〈γf(x*)-Ax*,J(xn-x*)〉-〈γf(x*)-Ax*,J(xn-xt)〉+〈γf(x*)-Ax*,J(xn-xt)〉-〈γf(x*)-Axt,J(xn-xt)〉+〈γf(x*)-Axt,J(xn-xt)〉-〈γf(xt)-Axt,J(xn-xt)〉+〈γf(xt)-Axt,J(xn-xt)〉=〈γf(x*)-Ax*,J(xn-x*)-J(xn-xt)〉+〈Axt-Ax*,J(xn-xt)〉+〈γf(x*)-γf(xt),J(xn-xt)〉+〈γf(xt)-Axt,J(xn-xt)〉,
which implies that
(3.37)limsup⁡n→∞〈γf(x*)-Ax*,J(xn-x*)〉≤limsup⁡n→∞〈γf(x*)-Ax*,J(xn-x*)-J(xn-xt)〉+‖A‖‖xt-x*‖limsup⁡n→∞‖xn-xt‖+γα‖x*-xt‖limsup⁡n→∞‖xn-xt‖+limsup⁡n→∞〈γf(xt)-Axt,J(xn-xt)〉.
Noticing thatJ is norm-to-norm uniformly continuous on bounded subsets of C, it follows from (3.35) that
(3.38)limsup⁡n→∞〈γf(x*)-Ax*,J(xn-x*)〉=limsup⁡t→0limsup⁡n→∞〈γf(x*)-Ax*,J(xn-x*)〉≤0.
Therefore, we obtain that (3.28) holds.
Finally, we prove thatxn→x* as n→∞. Now, from Lemma 2.1, we have
(3.39)‖xn+1-x*‖2=‖αnγf(xn)+βnxn+[(1-βn)I-αnA]yn-x*‖2=‖[(1-βn)I-αnA](yn-x*)+αn(γf(xn)-Ax*)+βn(xn-x*)‖2≤(1-βn-αnγ¯)2‖yn-x*‖2+2〈αn(γf(xn)-Ax*)+βn(xn-x*),J(xn+1-x*)〉=(1-βn-αnγ¯)2‖yn-x*‖2+2〈αn(γf(xn)-Ax*),J(xn+1-x*)〉+2βn〈xn-x*,J(xn+1-x*)〉=(1-βn-αnγ¯)2‖yn-x*‖2+2αnγ〈f(xn)-f(x*),J(xn+1-x*)〉+2αn〈γf(x*)-Ax*,J(xn+1-x*)〉+2βn〈xn-x*,J(xn+1-x*)〉≤(1-βn-αnγ¯)2‖xn-x*‖2+αnγα(‖xn+1-x*‖2+‖xn-x*‖2)+2αn〈γf(x*)-Ax*,J(xn+1-x*)〉+βn(‖xn+1-x*‖2+‖xn-x*‖2)=[(1-βn-αnγ¯)2+αnγα+βn]‖xn-x*‖2+(αnγα+βn)‖xn+1-x*‖2+2αn〈γf(x*)-Ax*,J(xn+1-x*)〉,
and consequently,
(3.40)‖xn+1-x*‖2≤(1-βn-αnγ¯)2+αnγα+βn1-αnγα-βn‖xn-x*‖2+2αn1-αnγα-βn〈γf(x*)-Ax*,J(xn+1-x*)〉=[1-2αn(γ¯-γα)1-αnγα-βn]‖xn-x*‖2+βn2+2βnαnγ¯+αn2γ¯21-αnγα-βn‖xn-x*‖2+2αn1-αnγα-βn〈γf(x*)-Ax*,J(xn+1-x*)〉=[1-2αn(γ¯-γα)1-αnγα-βn]‖xn-x*‖2+2αn(γ¯-γα)1-αnγα-βn×[βn2+2βnαnγ¯+αn2γ¯22αn(γ¯-γα)M4+1γ¯-γα〈γf(x*)-Ax*,J(xn+1-x*)〉],
where M4 is an appropriate constant such that M4≥supn≥0∥xn-x*∥2. Setting cn=2αn(γ¯-γα)/(1-αnγα-βn) and bn=(βn2+2βnαnγ¯+αn2γ¯2)/(2αn(γ¯-γα))M4+(1/(γ¯-γα))〈γf(x*)-Ax*,J(xn+1-x*)〉, then we have
(3.41)‖xn+1-x*‖2≤(1-cn)‖xn-x*‖2+cnbn.
By (3.28), (i) and applying Lemma 2.7 to (3.41), we have xn→x* as n→∞. This completes the proof.Corollary 3.2.
LetE be a real q-uniformly smooth and strictly convex Banach space which admits a weakly sequentially continuous duality mapping J from E to E*. Let C be a nonempty closed and convex subset of E which is also a sunny nonexpansive retraction of E such that C+C⊂C. Let A be a strongly positive linear bounded operator on E with coefficient γ¯>0 such that 0<γ<γ¯/α, and let f be a contraction of C into itself with coefficient α∈(0,1). Let Si,i=1,2,…, be λi-strict pseudocontractions from C into itself such that ∩n=1∞F(Sn)≠∅ and infλi>0. Assume that the sequences {αn},{βn}, {γn}, and {δn} in (0,1) satisfy the following conditions:(i)
∑n=0∞αn=∞; and limn→∞αn=0,(ii)
0<liminfn→∞βn≤limsupn→∞βn<1,(iii)
limn→∞|γn+1-γn|=0,(iv)
limn→∞|δn+1-δn|=0,(v)
δn(1+γn)-2γn>a for some a∈(0,1),and the sequence {θn} satisfies (H1). Then, the sequence {xn} generated by
(3.42)x0∈Cchosenarbitrarily,zn=δnxn+(1-δn)Wnxn,yn=γnxn+(1-γn)Wnzn,xn+1=αnγf(xn)+βnxn+((1-βn)I-αnA)yn,∀n≥0
converges strongly to x*∈∩n=1∞F(Sn), which solves the following variational inequality:
(3.43)〈γf(x*)-Ax*,J(p-x*)〉≤0,∀f∈ΠC,p∈⋂n=1∞F(Sn).Corollary 3.3.
LetE be a real q-uniformly smooth and strictly convex Banach space which admits a weakly sequentially continuous duality mapping J from E to E*. Let C be a nonempty closed and convex subset of E which is also a sunny nonexpansive retraction of E such that C+C⊂C. Let A be a strongly positive linear bounded operator on E with coefficient γ¯>0 such that 0<γ<γ¯/α, and let f be a contraction of C into itself with coefficient α∈(0,1). Let Si,i=1,2,…, be a nonexpansive mapping from C into itself such that ∩n=1∞F(Sn)≠∅ and infλi>0. Assume that the sequences {αn},{βn}, {γn}, and {δn} in (0,1) satisfy the following conditions:(i)
∑n=0∞αn=∞; and limn→∞αn=0,(ii)
0<liminfn→∞βn≤limsupn→∞βn<1,(iii)
limn→∞|γn+1-γn|=0,(iv)
limn→∞|δn+1-δn|=0,(v)
δn(1+γn)-2γn>a for some a∈(0,1).Then, the sequence {xn} generated by
(3.44)x0∈Cchosenarbitrarily,zn=δnxn+(1-δn)Wnxn,yn=γnxn+(1-γn)Wnzn,xn+1=αnγf(xn)+βnxn+((1-βn)I-αnA)yn,∀n≥0
converges strongly to x*∈∩n=1∞F(Sn), which solves the following variational inequality:
(3.45)〈γf(x*)-Ax*,J(p-x*)〉≤0,∀f∈ΠC,p∈⋂n=1∞F(Sn).Remark 3.4.
Theorem3.1, Corollaries 3.2, and 3.3, improve and extend the corresponding results of Cai and Hu [3], Dong et al. [2], and Katchang and Kumam [4, 5] in the following senses.(i)
For the mappings, we extend the mappings from an infinite family of nonexpansive mappings to an infinite family of strict pseudocontraction mappings.(ii)
For the algorithms, we propose new modified Ishikawa iterative algorithms, which are different from the ones given in [2–5] and others.



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*Source: 549364-2011-05-10.xml*