# Strong Convergence of the Iterative Methods for Hierarchical Fixed Point Problems of an Infinite Family of Strictly Nonself Pseudocontractions

Authors: Wei Xu; Yuanheng Wang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper deals with a new iterative algorithm for solving hierarchical fixed point problems of an infinite family of pseudocontractions in Hilbert spaces byyn=βnSxn+(1-βn)xn, xn+1=PC[αnf(xn)+(1-αn)∑i=1∞μi(n)Tiyn], and ∀n≥0, where Ti:C↦H is a nonself ki-strictly pseudocontraction. Under certain approximate conditions, the sequence {xn} converges strongly to x*∈⋂i=1∞F(Ti), which solves some variational inequality. The results here improve and extend some recent results.


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## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈·〉 and norm ∥·∥. Let C be a nonempty closed convex subset of H. A mapping f:C↦H is called a contraction with coefficient γ if there exits a constant γ∈[0,1) such that
(1.1)∥f(x)-f(y)∥≤γ∥x-y∥,∀x,y∈C.A mappingT:C↦C is called nonexpansive if
(1.2)∥Tx-Ty∥≤∥x-y∥,∀x,y∈C.A mappingT:C↦H is called k-strictly pseudocontraction if there exits a constant k∈[0,1) such that
(1.3)∥Tx-Ty∥2≤∥x-y∥2+k∥(I-T)x-(I-T)y∥2,∀x,y∈C.WriteF(T) as the set of fixed points of T, that is, F(T)={x∈C,Tx=x}. In 2000, Moudafi [1] introduced an iterative scheme for nonexpansive mappings
(1.4)xn+1=αnf(xn)+(1-αn)Txn,∀n≥0,
where f be a contraction on H and the sequence {xn} started with arbitrary initial x0∈H. In 2004, Xu [2] proved that the sequence {xn} generated by (1.4) converges strongly to a fixed point of T under certain conditions on the parameters, which also solves the variational inequality
(1.5)〈(I-f)x*,x-x*〉≥0,∀x∈F(T).Recently, some authors studied the problems of fixed points of nonexpansive mappings with strongly positive operators, Lipschitizian, strongly monotone operators, and extragradient methods, and many convergence results were obtained (such as, see [3–9]).In 2008, Yao et al. [10] introduced the following iterative scheme:
(1.6)x0=x∈C,yn=βnxn+(1-βn)Txn,xn+1=αnf(xn)+(1-αn)yn,∀n≥0,
where f is a contraction on C and T:C↦C is nonexpansive mapping. In 2012, Song et al. [11] analyzed the following iterative algorithm:
(1.7)x0=x∈C,yn=PC[βnxn+(1-βn)∑i=1∞‍μi(n)Tixn],xn+1=αnf(xn)+γnxn+((1-γn)I-αnF)yn,∀n≥0,
where Ti is a ki-strictly pseudocontraction, F:C↦C is a lipschitzian and strongly monotone operator, f:C↦C is a contraction, and PC is the metric projection from H onto C. Under certain conditions on the parameters, the sequence {xn} generated by (1.7) converges strongly to a fixed point of a countable family of ki-strictly pseudocontraction, which is the solution of some variational inequality.On the other hand, in 2010, Yao et al. [12] introduced the  iterative algorithm for solving hierarchical fixed point of nonexpansive mappings and gave the following theorem.Theorem YCL
LetC be a nonempty closed convex subset of a real Hilbert space H. Let f:C↦H be a contraction with coefficient γ∈[0,1). Suppose the following conditions are satisfied:(i)
limn→∞αn=0 and Σn=0∞αn=∞;(ii)
limn→∞(βn/αn)=0;(iii)
limn→∞(|αn+1-αn|/αn)=0 and limn→∞(|βn+1-βn|/βn)=0.Then the sequence {xn} generated by
(1.8)x0=x∈C,yn=βnSxn+(1-βn)xn,xn+1=PC[αnf(xn)+(1-αn)Tyn],∀n≥0,
converges strongly to a point of x*∈H, which is the unique solution of the variational inequality
(1.9)x*∈F(T),〈(I-f)x*,x-x*〉≥0,∀x∈F(T).Motivated and inspired by the iterative schemes (1.7) and (1.8), we introduce and study the hybrid iterative algorithm for solving some hierarchical fixed point problem of infinite family of strictly nonself pseudocontractions:
(1.10)x0=x∈C,yn=βnSxn+(1-βn)xn,xn+1=PC[αnf(xn)+(1-αn)∑i=1∞‍μi(n)Tiyn],∀n≥0,
where S, f, and PC are the same in (1.8), Ti:C↦H is a nonself ki-strictly pseudocontraction. Under certain conditions on the parameters, the sequence {xn} generated by (1.10) converges strongly to a common fixed point of infinite family of ki-strictly pseudocontractions, which solves the variational inequality
(1.11)x*∈⋂i=1∞‍F(Ti),〈(I-f)x*,x-x*〉≥0,∀x∈⋂i=1∞‍F(Ti).
So, our results extend and improve some results of other authors (such as [10–12]) from self-mappings to nonself-mappings, from nonexpansive mappings to ki-strictly pseudocontraction, and from one mapping to a infinite family mappings.


## 2. Preliminaries

In this section, we recall some basic facts that will be needed in the proof of the main results.Lemma 2.1 (see [13] demiclosedness principle).
LetC be a nonempty closed convex subset of a real Hilbert space H and let T:C↦C be a nonexpansive mapping with F(T)≠∅. If {xn} is a sequence in C weakly converging to x and if {(I-T)xn} converges strongly to y, then (I-T)x=y; in particular if y=0, then x∈F(T).Lemma 2.2 (see [9]).
Letx∈H and z∈C be any points. The following results hold:(1)
thatz=PCx if and only if there holds the relation:
(2.1)〈x-y,y-z〉≤0,∀y∈C;(2)
thatz=PCx if and only if there holds the relation:
(2.2)∥x-z∥2≤∥x-y∥2-∥y-z∥2,∀y∈C;(3)
there holds the ration:(2.3)〈PCx-PCy,x-y〉≥∥PCx-PCy∥2,∀x,y∈H.Lemma 2.3 (see [14]).
For allx,y∈H, the following inequality holds:
(2.4)∥x+y∥2≤∥x∥2+2〈y,x+y〉.Lemma 2.4 (see [3]).
Letf:C↦H be a contraction with coefficient γ∈[0,1) and let T:C↦C be a nonexpansive mapping. Then for all x,y∈C:(1)
the mapping(I-f) is strongly monotone with coefficient (1-γ), that is,
(2.5)〈x-y,(I-f)x-(I-f)y〉≥(1-γ)∥x-y∥2,(2)
the mapping(I-T) is monotone:
(2.6)〈x-y,(I-T)x-(I-T)y〉≥0.Lemma 2.5 (see [15]).
LetH be a Hilbert space and let C be a nonempty convex subset of H. Let T:C↦H be a k-strictly pseudocontractive mapping with F(T)≠∅. Then F(PCT)=F(T).Lemma 2.6 (see [16]).
LetH be a Hilbert space and let C be a nonempty convex subset of H. Let T:C↦H be a k-strictly pseudocontractive mapping. Define a mapping Jx=δx+(1-δ)Tx for all x∈C. Then as δ∈[k,1), J is a nonexpansive mapping such that F(J)=F(T).Lemma 2.7 (see [11]).
LetH be a Hilbert space and let C be a nonempty convex subset of H. Assume that Ti:C↦H is a countable family of ki-strictly pseudocontraction for some 0≤ki<1 and sup{ki:i∈N}<1 such that ⋂i=1∞‍F(Ti)≠∅. Assume that {μi} is a positive sequence such that ∑i=1∞‍μi=1. Then ∑i=1∞‍μiTi:C↦H is a k-strictly pseudocontraction with coefficient k=sup{ki:i∈N} and F(∑i=1∞‍μiTi)=⋂i=1∞‍F(Ti).Lemma 2.8 (see [17]).
Let{αn} be a sequence of nonnegative real numbers satisfying the following relation: αn+1≤(1-γn)αn+δn, where (1){γn}⊂(0,1), Σn=1∞γn=∞;(2)limsupn→∞(δn/γn)≤0 or Σn=1∞|δn|<∞, then limn→∞αn=0.


## 3. Main Results

In this section, we prove some strong convergence results on the iterative algorithm for solving hierarchical fixed point problem.Theorem 3.1.
LetH be a real Hilbert space and let C be a nonempty closed convex subset of H. Let f:C↦H be a (possibly nonself) contraction with coefficient γ∈[0,1), and let S:C↦C be a nonexpansive mapping. Let Ti:C↦H be a countable family of ki-strictly (possibly nonself) pseudocontraction with 0≤ki≤k<1 such that ⋂i=1∞‍F(Ti)≠∅. Let the sequence {xn} be generated by (1.10) with {αn}, {βn} in [0,1]. Suppose for each n, ∑i=1∞‍μi(n)=1, for all n and μi(n)≥0, for alli∈N. Assume that the parameters satisfy the following conditions:(i)
limn→∞αn=0 and Σn=0∞αn=∞;(ii)
limn→∞(βn/αn)=0;(iii)
limn→∞(|αn+1-αn|/αn)=0, limn→∞(|βn+1-βn|/αn)=0, and limn→∞(∑i=1∞‍|μi(n)-μi(n-1)|/αn)=0;(iv)
limn→∞∑i=1∞‍|μi(n)-μi|=0  and  ∑i=1∞‍μi=1(μi>0).
Then the sequence{xn} converges strongly to x*∈⋂i=1∞‍F(Ti), which solves the variational inequality
(3.1)〈(I-f)x*,x-x*〉≥0,∀x∈⋂i=1∞‍F(Ti).Proof.
The proof is divided into four steps.
Step 1. We show that the sequences {xn} and {yn} are bounded.
For eachn≥0, write Bn=∑i=1∞‍μi(n)Ti and by Lemma 2.7, we have Bn is a k-strictly pseudocontraction on C and F(Bn)=⋂i=1∞‍F(Ti), for all n∈N. Therefore, the iterative algorithm (1.10) can be written as
(3.2)x0=x∈C,yn=βnSxn+(1-βn)xn,xn+1=PC[αnf(xn)+(1-αn)Bnyn],∀n≥0.
By condition (ii), without loss of generality, we may assumeβn≤αn, for all n≥0. Take p∈⋂i=1∞‍F(Ti) and we estimate ∥Bnyn-p∥. For fixed approximate δ∈[k,1), define a mapping Jx=δx+(1-δ)Bnx and by Lemma 2.6, J is a nonexpansive mapping and Fix(J)=Fix(Bn). So
(3.3)∥Bnyn-p∥=∥11-δ(Jyn-δyn)-p∥≤11-δ(∥Jyn-Jp∥+δ∥yn-p∥)≤1+δ1-δ∥yn-p∥.
Together with (3.2) and (3.3), we get
(3.4)∥xn+1-p∥=∥PC[αnf(xn)+(1-αn)Bnyn]-PCp∥≤∥αnf(xn)+(1-αn)Bnyn-p∥≤αn∥f(xn)-f(p)∥+αn∥f(p)-p∥+(1-αn)∥Bnyn-p∥≤αnγ∥xn-p∥+αn∥f(p)-p∥+(1-αn)1+δ1-δ∥yn-p∥≤αnγ∥xn-p∥+αn∥f(p)-p∥+(1-αn)1+δ1-δβn∥Sxn-p∥+(1-αn)1+δ1-δ(1-βn)∥xn-p∥≤αnγ∥xn-p∥+αn∥f(p)-p∥+(1-αn)1+δ1-δβn∥Sxn-Sp∥+(1-αn)1+δ1-δβn∥Sp-p∥+(1-αn)1+δ1-δ(1-βn)∥xn-p∥≤αnγ∥xn-p∥+(1-αn)1+δ1-δ∥xn-p∥+1+δ1-δαn∥Sp-p∥+αn∥f(p)-p∥=[1-αn(1+δ1-δ-γ)]∥xn-p∥+αn(1+δ1-δ-γ)[1((1+δ)/(1-δ))-γ(1+δ1-δ∥Sp-p∥+∥f(p)-p∥)].
Therefore, we obtain(3.5)∥xn-p∥≤max{∥x0-p∥,1((1+δ)/(1-δ))-γ(1+δ1-δ∥Sp-p∥+∥f(p)-p∥)},
which gives the results that the sequence {xn} is bounded and so are {f(xn)}, {yn}, {Bnxn}, {Bnyn}.
Step 2. Now we show that ∥xn+1-xn∥→0 as n→∞. Let
(3.6)vn=αnf(xn)+(1-αn)Bnyn.
Next we estimate ∥xn+1-xn∥. From (3.2),we have
(3.7)∥xn+1-xn∥=∥PC[vn]-PC[vn-1]∥≤∥vn-vn-1∥=∥αnf(xn)+(1-αn)Bnyn-αn-1f(xn-1)-(1-αn-1)Bn-1yn-1∥=∥αn(f(xn)-f(xn-1))+(αn-αn-1)f(xn-1)+(1-αn)(Bnyn-Bnyn-1)+(1-αn)Bnyn-1+(1-αn-1)(Bnyn-1-Bn-1yn-1)[(π2)]∥≤αn∥f(xn)-f(xn-1)∥+|αn-αn-1|∥f(xn-1)-Bnyn-1∥+(1-αn)1+δ1-δ∥yn-yn-1∥+(1-αn-1)∑i=1∞‍|μi(n)-μi(n-1)|∥Tiyn-1∥≤αnγ∥xn-xn-1∥+(1-αn)1+δ1-δ∥yn-yn-1∥+M[|αn-αn-1|+∑i=1∞‍|μi(n)-μi(n-1)|],
where M is a constant such that
(3.8)supn∈N{∥f(xn-1)-Bnyn-1∥+∥S(xn-1)-xn-1∥+∥Tiyn-1∥}≤M.
From (3.2), we also obtain
(3.9)∥yn-yn-1∥=∥βnSxn+(1-βn)xn-βn-1Sxn-1-(1-βn-1)xn-1∥=∥βn(Sxn-Sxn-1)+(βn-βn-1)Sxn-1+(1-βn)(xn-xn-1)+(βn-1-βn)xn-1∥=∥βn(Sxn-Sxn-1)+(1-βn)(xn-xn-1)+(βn-βn-1)(Sxn-1-xn-1)∥≤∥xn-xn-1∥+|βn-βn-1|M.
Together with (3.7) and (3.9), we have
(3.10)∥xn+1-xn∥≤∥vn-vn-1∥≤αnγ∥xn-xn-1∥+(1-αn)1+δ1-δ∥xn-xn-1∥+(1-αn)|βn-βn-1|1+δ1-δM+|αn-αn-1|M+(1-αn-1)∑i=1∞‍|μi(n)-μi(n-1)|M≤[1-αn(1+δ1-δ-γ)]∥xn-xn-1∥+M[|αn-αn-1|+|βn-βn-1|1+δ1-δ+∑i=1∞‍|μi(n)-μi(n-1)|]=[1-αn(1+δ1-δ-γ)]∥xn-xn-1∥+M(αn(1+δ1-δ-γ))×[1αn(((1+δ)/(1-δ))-γ)×(|αn-αn-1|+|βn-βn-1|1+δ1-δ+∑i=1∞‍|μi(n)-μi(n-1)|)].
By Lemma 2.8 and conditions (i)–(iii), we immediately get ∥xn+1-xn∥→0 as n→∞.
Step 3. Next we prove that ∥xn-PC(∑i=1∞‍μiTi)xn∥→0 as n→∞.
LetB=∑i=1∞‍μiTi. By Lemma 2.7 and condition (iv), we get the results that  B:C↦H is a k-strictly pseudocontraction with F(B)=⋂i=1∞‍F(Ti) and Bnx→Bx as n→∞, for any x∈C,
(3.11)∥xn-PCBxn∥≤∥xn-xn+1∥+∥xn+1-PCBxn∥=∥xn-xn+1∥+∥PC[vn]-PCBxn∥≤∥xn-xn+1∥+∥αnf(xn)+(1-αn)Bnyn-Bnxn∥+∥Bnxn-Bxn∥=∥xn-xn+1∥+∥αn(f(xn)-Bnxn)+(1-αn)(Bnyn-Bnxn)∥+∥Bnxn-Bxn∥≤∥xn-xn+1∥+αn∥f(xn)-Bnxn∥+(1-αn)1+δ1-δ∥yn-xn∥+∥Bnxn-Bxn∥=∥xn-xn+1∥+αn∥f(xn)-Bnxn∥+(1-αn)1+δ1-δβn∥Sxn-xn∥+∥Bnxn-Bxn∥.
Because αn→0,βn→0, ∥xn+1-xn∥→0, and Bnx→Bx, so we obtain ∥xn-∑i=1∞‍μiTixn∥→0 as n→∞.
Step 4. Now we show that limsupn→∞〈f(x*)-x*,xn-x*〉≤0, where x*=PF(B)f(x*).
Since the sequence{xn} is bounded, we take s subsequence {xnk} of {xn} such that limsupn→∞〈f(x*)-x*,xn-x*〉=limk→∞〈f(x*)-x*,xnk-x*〉 and xnk⇀x′. Notice that ∥xn-PCBxn∥→0 and by Lemmas 2.1 and 2.5, we have x′∈Fix(PCB)=F(B)=⋂i=1∞‍F(Ti). Then
(3.12)limk→∞〈f(x*)-x*,xnk-x*〉=〈f(x*)-x*,x′-x*〉≤0,x′∈F(B).
Now, by Lemma2.2, we get 〈PC[vn]-vn,PC[vn]-x*〉≤0. Therefore, we have
(3.13)∥xn+1-x*∥2=〈PC[vn]-x*,xn+1-x*〉=〈PC[vn]-vn,xn+1-x*〉+〈vn-x*,xn+1-x*〉=〈PC[vn]-vn,PC[vn]-x*〉+〈vn-x*,xn+1-x*〉≤〈vn-x*,xn+1-x*〉=〈αnf(xn)+(1-αn)Bnyn-x*,xn+1-x*〉=αn〈(f(xn)-f(x*)),xn+1-x*〉+αn〈f(x*)-x*,xn+1-x*〉+(1-αn)〈Bnyn-x*,xn+1-x*〉≤αnγ∥xn-x*∥·∥xn+1-x*∥+αn〈f(x*)-x*,xn+1-x*〉+(1-αn)〈Bn[βnSxn+(1-βn)xn]-Bnx*,xn+1-x*〉≤αnγ∥xn-x*∥·∥xn+1-x*∥+αn〈f(x*)-x*,xn+1-x*〉+(1-αn)∥βn(Sxn-Sx*)+βn(Sx*-x*)+(1-βn)(xn-x*)∥·∥xn+1-x*∥≤αnγ∥xn-x*∥·∥xn+1-x*∥+αn〈f(x*)-x*,xn+1-x*〉+(1-αn)∥xn-x*∥·∥xn+1-x*∥+(1-αn)βn∥Sx*-x*∥·∥xn+1-x*∥=[1-(1-γ)αn]∥xn-x*∥·∥xn+1-x*∥+αn〈f(x*)-x*,xn+1-x*〉+(1-αn)βn∥Sx*-x*∥·∥xn+1-x*∥≤1-(1-γ)αn2(∥xn-x*∥2+∥xn+1-x*∥2)+αn〈f(x*)-x*,xn+1-x*〉+(1-αn)βn∥Sx*-x*∥·∥xn+1-x*∥.
Hence it follows that(3.14)∥xn+1-x*∥2≤[1-2(1-γ)αn1+(1-γ)αn]∥xn-x*∥2+2αn1+(1-γ)αn〈f(x*)-x*,xn+1-x*〉+2(1-αn)βn1+(1-γ)αn∥Sx*-x*∥·∥xn+1-x*∥=[1-2(1-γ)αn1+(1-γ)αn]∥xn-x*∥2+2(1-γ)αn1+(1-γ)αn[1(1-γ)〈f(x*)-x*,xn+1-x*〉+(1-αn)βn(1-γ)αn∥Sx*-x*∥·∥xn+1-x*∥].
Now, by Lemma2.8, conditions (i)–(iii), and
(3.15)limsupn→∞[1(1-γ)〈f(x*)-x*,xn+1-x*〉+(1-αn)βn(1-γ)αn∥Sx*-x*∥·∥xn+1-x*∥]≤0,
we have xn→x*∈⋂i=1∞‍F(Ti) as n→∞ and x* also solves the variational inequality
(3.16)〈(I-f)x*,x-x*〉≥0,∀x∈⋂i=1∞‍F(Ti).
This completes the proof.From Theorem3.1, if we take S=I or βn=0, for all n∈N, we get the following corollary.Corollary 3.2.
LetH be a real Hilbert space and C be a nonempty closed convex subset of H. Let f:C↦H be a (possibly nonself) contraction with coefficient γ∈[0,1) and let  Ti:C↦H be a countable family of ki-strictly (possibly nonself) pseudocontraction with 0≤ki<1 and sup{ki:i∈N}<1 such that ⋂i=1∞‍F(Ti)≠∅. Let the sequence {xn} be generated by
(3.17)x0=x∈C,xn+1=PC[αnf(xn)+(1-αn)∑i=1∞‍μi(n)Tixn],∀n≥0,
with {αn} in [0,1]. Suppose for each n, ∑i=1∞‍μi(n)=1, for all n and μi(n)≥0, for all i∈N. Assume that the parameters satisfied the following conditions:(i)
limn→∞αn=0 and Σn=0∞αn=∞;(ii)
limn→∞(|αn+1-αn|/αn)=0,  and limn→∞((∑i=1∞‍|μi(n)-μi(n-1)|)/αn)=0;(iii)
limn→∞∑i=1∞‍|μi(n)-μi|=0 and  ∑i=1∞‍μi=1(μi>0).
Then the the sequence{xn} converges strongly to x*∈⋂i=1∞‍F(Ti), which solves the variational inequality
(3.18)〈(I-f)x*,x-x*〉≥0,∀x∈⋂i=1∞‍F(Ti).Remark 3.3.
Theorem3.1 extends and improves Theorem YCL in the following way. The nonexpasnsive self-mapping T:C↦C is extended to a infinite family of nonself ki-strictly pseudocontraction Ti:C↦H. If we take ki=0,i∈N in Theorem 3.1, then B=∑i=1∞‍μiTi reduces to a nonexpasnsive (possibly nonself) mapping, thus Theorem 3.1 reduces to Theorem YCL.



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