# Hierarchical Fixed Point Problems in Uniformly Smooth Banach Spaces

Authors: Lu-Chuan Ceng; Ching-Feng Wen; Chin-Tzong Pang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We propose some relaxed implicit and explicit viscosity approximation methods for hierarchical fixed point problems for a countable family of nonexpansive mappings in uniformly smooth Banach spaces. These relaxed viscosity approximation methods are based on the well-known viscosity approximation method and hybrid steepest-descent method. We obtain some strong convergence theorems under mild conditions.


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## Body

## 1. Introduction

LetX be a real Banach space and U the unit sphere of X; that is, U={x∈X:∥x∥=1}. Recall that X is said to be smooth if the limit
(1)lim⁡t→0∥x+ty∥-∥x∥t
exists for all x,y∈U; in this case, X is also said to have a Gâteaux differentiable norm. X is said to have a uniformly Gâteaux differentiable norm if for each y∈U, the limit is attained uniformly for x∈U. Moreover, it is said to be uniformly smooth if this limit is attained uniformly for x,y∈U. The norm of X is said to be the Fréchet differential if for each x∈U, this limit is attained uniformly for y∈U. In addition, we define a function ρ:[0,∞)→[0,∞) called the modulus of smoothness of X as follows:
(2)ρ(τ)=sup⁡{12(∥x+y∥+∥x-y∥)-1:x,y∈X,∥x∥=1,∥y∥=τ12}.
It is known that X is uniformly smooth if and only if lim⁡τ→0ρ(τ)/τ=0.LetX be a real Banach space and let J denote the normalized duality mapping from X to  2X* given by
(3)J(x)={x*∈X*:〈x,x*〉=∥x∥2=∥x*∥2},∀x∈X,
where X* denotes the dual space of X and 〈·,·〉 denotes the generalized duality pairing. We use Fix⁡(T) to denote the set of fixed points of the mapping T. It is well known that if X is smooth, then J is single-valued and norm-to-weak* continuous, whereas if X is a Banach space with a uniformly Gâteaux differentiable norm, then J is single-valued and norm-to-weak* uniformly continuous on bounded subsets of X. Further, if X is a uniformly smooth Banach space, then J is single-valued and norm-to-norm uniformly continuous on bounded subsets of X. In what follows, we still denote by J the single-valued normalized duality mapping.LetC be a nonempty closed convex subset of X. Recall that a mapping T:C→C is said to be L-Lipschitzian if there exists a constant L>0 such that
(4)∥Tx-Ty∥≤L∥x-y∥,∀x,y∈C.
In particular, if L=1, then T is said to be nonexpansive; that is,
(5)∥Tx-Ty∥≤∥x-y∥,∀x,y∈C.
We use the notation ⇀ to indicate the weak convergence and the one → to indicate the strong convergence.Definition 1.
LetA:C→X be a mapping of C into X. Then A is said to be(i)
accretive if for eachx,y∈C, there exists j(x-y)∈J(x-y) such that
(6)〈Ax-Ay,j(x-y)〉≥0,whereJ is the normalized duality mapping;(ii)
α-strongly accretive if for each x,y∈C, there exists j(x-y)∈J(x-y) such that
(7)〈Ax-Ay,j(x-y)〉≥α∥x-y∥2,for someα∈(0,1);(iii)
pseudocontractive if for eachx,y∈C, there exists j(x-y)∈J(x-y) such that
(8)〈Ax-Ay,j(x-y)〉≤∥x-y∥2;(iv)
β-strongly pseudocontractive if for each x,y∈C, there exists j(x-y)∈J(x-y) such that
(9)〈Ax-Ay,j(x-y)〉≤β∥x-y∥2,for someβ∈(0,1);(v)
λ-strictly pseudocontractive if for each x,y∈C, there exists j(x-y)∈J(x-y) such that
(10)〈Ax-Ay,j(x-y)〉≤∥x-y∥2-λ∥x-y-(Ax-Ay)∥2,for someλ∈(0,1).In a real smooth Banach spaceX we say that an operator A is strongly positive [1] if there exists a constant γ¯>0 with the property
(11)〈Ax,J(x)〉≥γ¯∥x∥2,∥aI-bA∥=sup⁡∥x∥≤1|〈(aI-bA)x,J(x)〉|,ppppppppppa∈[0,1],b∈[-1,1],
where I is the identity mapping.Recently, the problem of convergence of implicit iterative algorithms to a common fixed point for a family of nonexpansive mappings and its extensions to Hilbert spaces or Banach spaces have been considered by many authors; see [1–9] and the references therein.Yao et al. [10] introduced the following Halpern-type implicit iterative algorithm,
(12)xn=αnu+βnxn-1+γnTxn,∀n≥1,
and proved a strong convergence theorem under suitable conditions.On the other hand, letC be a nonempty closed convex subset of a real Hilbert space H and let A:C→H be a nonlinear mapping. The classical variational inequality problem (VIP) is to find x*∈C such that
(13)〈Ax*,x-x*〉≥0,∀x∈C.If we assume thatC is the fixed point set of a nonexpansive mapping T and S is another nonexpansive mapping (not necessarily with fixed points), the problem (13) becomes the VIP of finding x*∈Fix⁡(T) such that
(14)〈(I-S)x*,x-x*〉≥0,∀x∈Fix⁡(T),
introduced first by Moudafi and Maingé in [11], which is called hierarchical fixed point problem.In particular, wheneverFix⁡(S)≠∅, all elements of Fix⁡(S) are solutions of VIP (14). If S is a ρ-contraction (i.e., ∥Sx-Sy∥≤ρ∥x-y∥ for some ρ∈(0,1)), the set of solutions of VIP (14) is a singleton and it is well known as a viscosity problem, which was first introduced by Moudafi [12] and then developed by several authors [13, 14].Very recently, Cai and Bu [1] investigated a general hierarchical fixed point problem for a countable family of continuous pseudocontractions, which covers as a special case of the problem considered in [10]. For this purpose, they first established strong convergence of an implicit iterative scheme for solving a hierarchical fixed point problem for a continuous pseudocontractive mapping in a uniformly smooth Banach space.In this paper, letC be a nonempty closed convex subset of a uniformly smooth Banach space X such that C±C⊂C. Let T:C→C be a nonexpansive mapping with Fix⁡(T)≠∅ and let f:C→C be a fixed contractive mapping with contractive coefficient β∈(0,1). Let F:C→C be α-strongly accretive and λ-strictly pseudocontractive with α+λ>1 and let A:C→C be a γ¯-strongly positive linear bounded operator. First of all, we introduce a relaxed implicit viscosity scheme for solving a hierarchical fixed point problem for a nonexpansive mapping T:
(15)xt=(I-θtF)Txt+θt[f(xt)-t(Af(xt)-Txt)],
where lim⁡t→0θt=0. It is proven that as t→0,{xt} converges strongly to a point z∈Fix⁡(T), which is the unique solution in Fix⁡(T) to the VIP:
(16)〈(F-f)z,J(z-p)〉≤0,∀p∈Fix⁡(T).
On the other hand, let {Tn}n=0∞ be a countable family of nonexpansive mappings from C to itself such that Ω=⋂i=0∞‍Fix⁡(Ti)≠∅. We propose a relaxed implicit viscosity iterative algorithm for solving a hierarchical fixed point problem for a countable family of nonexpansive mappings {Tn}:
(17)yn=αnf(yn)+βnxn+((1-βn)I-αnA)(I-ϵnF)Tnyn,xn+1=σnf(yn)+(I-σnA)Tnyn,∀n≥0,
where {αn},{βn},{ϵn}, and {σn} are four sequences in (0,1). It is proven that under mild conditions {xn} converges strongly to a point z∈Ω, which is the unique solution in Ω to the VIP:
(18)〈(A-f)z,J(z-p)〉≤0,∀p∈Ω.
Furthermore, we also propose a relaxed explicit viscosity iterative algorithm for solving another hierarchical fixed point problem for a countable family of nonexpansive mappings {Tn}:
(19)x0∈Cchosenarbitrarily,xn+1=(I-βnF)Tnxn+βn[f(xn)-αn(Af(xn)-Tnxn)],∀n≥0,
where {αn} and {βn} are two sequences in (0,1). It is proven that under appropriate assumptions, {xn} converges strongly to a point z∈Ω, which is the unique solution in Ω to the VIP:
(20)〈(F-f)z,J(z-p)〉≤0,∀p∈Ω.
The above relaxed viscosity algorithms are based on the well-known viscosity approximation method (see, e.g., [4–6, 9]) and hybrid steepest-descent method (see, e.g., [14–17]). Our results extend, improve, supplement, and develop the recent results announced by many authors.


## 2. Preliminaries

We list some lemmas that will be used in the sequel. Lemma2 can be found in [18]. Lemma 3 is an immediate consequence of the subdifferential inequality of the function (1/2)∥·∥2.Lemma 2.
Let{sn} be a sequence of nonnegative real numbers satisfying
(21)sn+1≤(1-αn)sn+αnβn+γn,∀n≥0,
where {αn},{βn}, and {γn} satisfy the following conditions:(i)
{αn}⊂[0,1],∑n=0∞‍αn=∞;(ii)
lim⁡sup⁡n→∞βn≤0;(iii)
γn≥0(∀n≥0),∑n=0∞‍γn<∞.Then lim⁡sup⁡n→∞sn=0.Lemma 3.
In a smooth Banach spaceX, there holds the following inequality:
(22)∥x+y∥2≤∥x∥2+2〈y,J(x+y)〉,∀x,y∈X.LetLIM be a continuous linear functional on l∞ and (a0,a1,…)∈l∞. We write LIMan instead of LIM((a0,a1,…)). LIM is said to be Banach limit if LIM satisfies ∥LIM∥=LIM1=1 and LIMan+1=LIMan for all (a0,a1,…)∈l∞. It is well known that for Banach limit LIM the following holds:(i)
for alln≥1,an≤cn implies that LIMan≤LIMcn;(ii)
LIMan+N=LIMan for any fixed positive integer N;(iii)
lim⁡inf⁡n→∞an≤LIMan≤lim⁡sup⁡n→∞an for all (a0,a1,…)∈l∞.It is easy to see that there holds the following conclusion.Lemma 4 (see [19]).
Let(a0,a1,…)∈l∞. If LIMan=0, then there exists a subsequence {ank} of {an} such that ank→0 as k→∞.Recall that a Banach spaceX is said to satisfy Opial’s condition, if whenever {xn} is a sequence in X which converges weakly to x as n→∞, then
(23)lim⁡sup⁡n→∞∥xn-x∥<lim⁡sup⁡n→∞∥xn-y∥,∀y∈X,y≠x.Lemma 5 (Demiclosedness principle; see [20, Theorem 10.3]).
LetX be a reflexive Banach space satisfying Opial’s condition, C a nonempty closed convex subset of X, and T:C→C a nonexpansive mapping. Then the mapping I-T is demiclosed on C, where I is the identity mapping; that is, if {xn} is a sequence of C such that xn⇀x and (I-T)xn→y, then (I-T)x=y.The following lemma can be derived by the standard argument and hence its proof will be omitted.Lemma 6.
LetC be a nonempty closed convex subset of a real smooth Banach space X and let F:C→X be a mapping.(i)
IfF:C→X is  α-strongly accretive and λ-strictly pseudocontractive with α+λ≥1, then I-F nonexpansive and F is Lipschitz continuous with constant 1+1/λ;(ii)
IfF:C→X is  α-strongly accretive and λ-strictly pseudocontractive with α+λ>1, then for any fixed τ∈(0,1),I-τF is contractive with coefficient 1-τ(1-(1-α)/λ).


## 3. Relaxed Implicit Viscosity Scheme for Hierarchical Fixed Point Problem for a Nonexpansive Mapping

In this section, we introduce our relaxed implicit viscosity scheme for solving hierarchical fixed point problem for a nonexpansive mapping and show the strong convergence theorem. First, we list several useful and helpful lemmas.Lemma 7 (see [21]).
LetX be a Banach space, C a nonempty closed and convex subset of X, and T:C→C a continuous and strong pseudocontraction. Then T has a unique fixed point in C.Lemma 8 (see [19]).
Let{an} be a sequence of nonnegative real numbers satisfying the property an+1≤(1-γn)an+γnβn,∀n≥0, where {γn}⊂(0,1) and {βn}⊂R such that (i)  ∑n=0∞‍γn=∞ and (ii)   lim⁡sup⁡n→∞βn≤0. Then {an} converges to zero as n→∞.Lemma 9 (see [22]).
LetC be a nonempty closed convex subset of a real Banach space X and T:C→C a continuous pseudocontractive map. We denote B=(2I-T)-1. Then the following holds.(i)
The mapB is a nonexpansive self-mapping on C.(ii)
Iflim⁡n→∞∥xn-Txn∥=0, then lim⁡n→∞∥xn-Bxn∥=0.Lemma 10 (see [23]).
Assume thatA is a strongly positive linear bounded operator on a smooth Banach space X with coefficient γ¯>0 and 0<ρ≤∥A∥-1. Then ∥I-ρA∥≤1-ργ¯.We now state and prove our first result.Theorem 11.
LetC be a nonempty closed convex subset of a uniformly smooth Banach space X such that C±C⊂C. Let T:C→C be a nonexpansive mapping with Fix⁡(T)≠∅ and F:C→C  α-strongly accretive and λ-strictly pseudocontractive with α+λ>1. Let f:C→C be a fixed contractive mapping with contractive coefficient β∈(0,γ0),γ0=1-(1-α)/λ. Let A:C→C be a γ¯-strongly positive linear bounded operator with γ¯β<1. Let {xt} be defined by
(24)xt=(I-θtF)Txt+θt[f(xt)-t(Af(xt)-Txt)],
where lim⁡t→0θt=0. Then, as t→0,{xt} converges strongly to some fixed point z of  T, which is the unique solution in Fix⁡(T) to the VIP:
(25)〈(F-f)z,J(z-p)〉≤0,∀p∈Fix⁡(T).Proof.
First, we claim thatγ0<α. Indeed, it is known that strongly accretive constant α∈(0,1) and strictly pseudocontractive constant λ∈(0,1). Moreover, observe that
(26)(1-α)λ<1⟺1-α<1-αλ⟺γ0<α.
Let us show that the net{xt} is defined well. As a matter of fact, we define the mapping St:C→C as follows:
(27)Stx=(I-θtF)Tx+θt[f(x)-t(Af(x)-Tx)],∀x∈C.
Since lim⁡t→0θt=0, we may assume, without loss of generality, that θt∈(0,1) for all t∈(0,ϵ0), where ϵ0=min⁡{(γ0-β)/2(1-γ¯β),∥A∥-1}. Utilizing Lemmas 6 and 10, we obtain that for each t∈(0,ϵ0)(28)〈Stx-Sty,J(x-y)〉=〈(I-θtF)Tx-(I-θtF)Ty,J(x-y)〉+θt〈(I-tA)f(x)-(I-tA)f(y),J(x-y)〉+θtt〈Tx-Ty,J(x-y)〉≤∥(I-θtF)Tx-(I-θtF)Ty∥∥x-y∥+θt∥(I-tA)f(x)-(I-tA)f(y)∥∥x-y∥+θtt∥Tx-Ty∥∥x-y∥≤(1-θt(1-1-αλ))∥Tx-Ty∥∥x-y∥+θt(1-tγ¯)∥f(x)-f(y)∥∥x-y∥+tθt∥x-y∥2≤(1-θtγ0)∥x-y∥2+θt(1-tγ¯)β∥x-y∥2+tθt∥x-y∥2=[1-θt(γ0-β-t(1-γ¯β))]∥x-y∥2≤[1-θt(γ0-β-γ0-β2(1-γ¯β)(1-γ¯β))]∥x-y∥2=(1-12θt(γ0-β))∥x-y∥2.
It follows that for each t∈(0,ϵ0), St:C→C is a continuous and strongly pseudocontractive mapping with pseudocontractive coefficient 1-(1/2)θt(γ0-β). Hence, by Lemma 7 we know that there exists a unique fixed point in C, denoted by xt, which uniquely solves the fixed point equation:
(29)xt=(I-θtF)Txt+θt[f(xt)-t(Af(xt)-Txt)].
Let us show the uniqueness of the solution of VIP (25). Suppose both z1∈Fix⁡(T) and z2∈Fix⁡(T) are solutions to VIP (25). Then we have
(30)〈(F-f)z1,J(z1-z2)〉≤0,〈(F-f)z2,J(z2-z1)〉≤0.
Adding up the above two inequalities, we obtain
(31)〈(F-f)z1-(F-f)z2,J(z1-z2)〉≤0.
Note that
(32)〈(F-f)z1-(F-f)z2,J(z1-z2)〉=〈Fz1-Fz2,J(z1-z2)〉-〈f(z1)-f(z2),J(z1-z2)〉≥α∥z1-z2∥2-β∥z1-z2∥2=(α-β)∥z1-z2∥2≥0.
Taking into account α-β>0, we have z1=z2, and hence the uniqueness is proved. We use z~ to denote the unique solution of VIP (25).
Next, we prove that{xt:t∈(0,ϵ0)} is bounded. Indeed, we note that 0<θt<1,∀t∈(0,ϵ0). Take a fixed p∈Fix⁡(T) arbitrarily. Utilizing Lemma 10 we deduce that for all t∈(0,ϵ0)(33)∥xt-p∥2=〈(I-θtF)Txt+θt[f(xt)-t(Af(xt)-Txt)]-p,J(xt-p)〉=〈(I-θtF)Txt-(I-θtF)Tp,J(xt-p)〉+θt〈(I-tA)f(xt)-(I-tA)f(p),J(xt-p)〉+θtt〈Txt-p,J(xt-p)〉-θt〈(F-f)p,J(xt-p)〉+θtt〈(I-Af)p,J(xt-p)〉≤∥(I-θtF)Txt-(I-θtF)Tp∥∥xt-p∥+θt∥(I-tA)f(xt)-(I-tA)f(p)∥∥xt-p∥+θtt∥Txt-p∥∥xt-p∥+θt∥(F-f)p∥∥xt-p∥+θtt∥(I-Af)p∥∥xt-p∥≤(1-θt(1-1-αλ))∥Txt-Tp∥∥xt-p∥+θt(1-tγ¯)∥f(xt)-f(p)∥∥xt-p∥+θtt∥Txt-p∥∥xt-p∥+θt∥(F-f)p∥∥xt-p∥+θtt∥(I-Af)p∥∥xt-p∥≤(1-θtγ0)∥xt-p∥2+θt(1-tγ¯)β∥xt-p∥2+θtt∥xt-p∥2+θt∥(F-f)p∥∥xt-p∥+θtt∥(I-Af)p∥∥xt-p∥=[1-θt(γ0-β-t(1-γ¯β))]∥xt-p∥2+θt∥(F-f)p∥∥xt-p∥+θtt∥(I-Af)p∥∥xt-p∥≤[1-θt(γ0-β-γ0-β2(1-γ¯β)(1-γ¯β))]×∥xt-p∥2+θt∥(F-f)p∥∥xt-p∥+θtt∥(I-Af)p∥∥xt-p∥=(1-12θt(γ0-β))∥xt-p∥2+θt∥(F-f)p∥∥xt-p∥+θtt∥(I-Af)p∥∥xt-p∥,
which immediately yields
(34)∥xt-p∥≤2γ0-β(∥(F-f)p∥+t∥(I-Af)p∥)≤2γ0-β(∥(F-f)p∥+∥A∥-1∥(I-Af)p∥).
Thus {xt:t∈(0,ϵ0)} is bounded.
Assume that{tn}⊂(0,ϵ0) and tn→0 as n→∞. Set θn=θtn and xn:=xtn, and define μ:C→R by μ(x)=LIM∥xn-x∥2,∀x∈C, where LIM is a Banach limit on l∞. Let
(35)K={x∈C:μ(x)=min⁡y∈CLIM∥xn-y∥2}.
We see easily that K is a nonempty closed convex subset of X. Note that ∥xn-Txn∥=θn∥f(xn)-tn(Af(xn)-Txn)-FTxn∥→0 as n→∞. In terms of Lemma 9, we know that the mapping B=(2I-T)-1:C→C is nonexpansive and Fix⁡(T)=Fix⁡(B) and lim⁡n→∞∥xn-Bxn∥=0, where I denotes the identity operator. It follows that
(36)μ(Bx)=LIM∥xn-Bx∥2=LIM∥Bxn-Bx∥2≤LIM∥xn-x∥2=μ(x),
which implies that B(K)⊂K; that is, K is invariant under B. Since a uniformly smooth Banach space has the fixed point property for nonexpansive mapping, B has a fixed point, say z∈K. Since z is also a minimizer of μ over C, we have that, for t∈(0,ϵ0) and x∈C,
(37)0≤μ(z+t(x-Fz))-μ(z)t=LIM∥xn-z+t(Fz-x)∥2-∥xn-z∥2t=LIM(-∥xn-z∥2)t-1)(-∥xn-z∥2)〈xn-z,J(xn-z+t(Fz-x))〉ppppppplpl+t〈Fz-x,J(xn-z+t(Fz-x))〉pppppplpp-∥xn-z∥2)t-1).
Since X is uniformly smooth, we conclude that the duality mapping J is norm-to-norm uniformly continuous on any bounded subset of X. Letting t→0, we find that the two limits above can be interchanged and obtain
(38)LIM〈x-Fz,J(xn-z)〉≤0,∀x∈C.
On the other hand, we have
(39)xn-z=(I-θnF)Txn-(I-θnF)Tz+θn[(I-tnA)f(xn)-(I-tnA)f(z)+tn(Txn-z)]+θn(f-F)z+θntn(I-Af)z.
It follows that
(40)∥xn-z∥2=〈(I-θnF)Txn-(I-θnF)Tz,J(xn-z)〉+θn[〈(I-tnA)(f(xn)-f(z)),J(xn-z)〉pppppppp+tn〈Txn-z,J(xn-z)〉〈(I-tnA)(f(xn)-f(z)),J(xn-z)〉]+θn〈(f-F)z,J(xn-z)〉+θntn〈(I-Af)z,J(xn-z)〉≤(1-θnγ0)∥Txn-Tz∥∥xn-z∥+θn[(1-tnγ¯)∥f(xn)-f(z)∥∥xn-z∥ppppppppl+tn∥Txn-z∥∥xn-z∥]+θn〈(f-F)z,J(xn-z)〉+θntn∥(I-Af)z∥∥xn-z∥≤(1-θnγ0)∥xn-z∥2+θn[(1-tnγ¯)β∥xn-z∥2+tn∥xn-z∥2]+θn〈(f-F)z,J(xn-z)〉+θntn∥(I-Af)z∥∥xn-z∥=[1-θn(γ0-β-tn(1-γ¯β))]∥xn-z∥2+θn〈(f-F)z,J(xn-z)〉+θntn∥(I-Af)z∥∥xn-z∥≤[1-θn(γ0-β-γ0-β2(1-γ¯β)(1-γ¯β))]∥xn-z∥2+θn〈(f-F)z,J(xn-z)〉+θntn∥(I-Af)z∥∥xn-z∥=(1-12θn(γ0-β))∥xn-z∥2+θn〈(f-F)z,J(xn-z)〉+θntn∥(I-Af)z∥∥xn-z∥.
Therefore,
(41)∥xn-z∥2≤2γ0-β×(〈(f-F)z,J(xn-z)〉+tn∥(I-Af)z∥∥xn-z∥).
Combining (38) and (41), we get
(42)LIM∥xn-z∥2≤2γ0-βLIM〈(f-F)z,J(xn-z)〉≤0,
which leads to LIM∥xn-z∥2=0. Hence there exists a subsequence which is still denoted as {xn} such that xn→z as n→∞.
Next, we prove thatz solves VIP (25). Since
(43)xt=(I-θtF)Txt+θt[f(xt)-t(Af(xt)-Txt)],
we can deduce that
(44)xt-Txt=θt(f(xt)-FTxt)+θtt(Txt-Af(xt)).
Since T is nonexpansive, I-T is accretive. So, from the accretivity of I-T, it follows that, for any fixed p∈Fix⁡(T),
(45)0≤〈(I-T)xt-(I-T)p,J(xt-p)〉=〈(I-T)xt,J(xt-p)〉=θt〈f(xt)-FTxt,J(xt-p)〉+θtt〈Txt-Af(xt),J(xt-p)〉=θt〈(f-F)xt,J(xt-p)〉+θt〈Fxt-FTxt,J(xt-p)〉+θtt〈Txt-Af(xt),J(xt-p)〉.
This implies that
(46)〈(F-f)xt,J(xt-p)〉≤〈Fxt-FTxt,J(xt-p)〉+t〈Txt-Af(xt),J(xt-p)〉.
Now replacing t with tn, letting n→∞, and noticing the boundedness of {Txtn-Af(xtn)} and the fact that Fxtn-FTxtn→Fz-FTz=0 for z∈Fix⁡(T), we have that
(47)〈(F-f)z,J(z-p)〉≤0,∀p∈Fix⁡(T).
That is, z∈Fix⁡(T) is a solution of VIP (25). Then z=z~. In summary, we infer that each cluster point of {xn} is equal to z as tn→0. This completes the proof.


## 4. Relaxed Viscosity Algorithms for Hierarchical Fixed Point Problems for a Countable Family of Nonexpansive Mappings

In this section, we propose relaxed implicit and explicit viscosity algorithms for solving hierarchical fixed point problems for a countable family of nonexpansive mappings and show strong convergence theorems. For this purpose, we will use the following lemmas in the sequel.Lemma 12 (see [24]).
LetC be a nonempty closed convex subset of a Banach space X. Let T1,T2,… be a sequence of mappings of C into itself. Suppose that ∑n=1∞‍sup⁡{∥Tn+1x-Tnx∥:x∈C}<∞. Then, for each y∈C,{Tny} converges strongly to some point of C. Moreover, let T be a mapping of C into itself defined by Ty=lim⁡n→∞Tny, for all y∈C. Then lim⁡n→∞sup⁡{∥Tx-Tnx∥:x∈C}=0.Lemma 13 (see [1, Lemma 2.6]).
LetC be a nonempty closed convex subset of a real Banach space X which has uniformly Gateaux differentiable norm. Let T:C→C be a continuous pseudocontractive mapping with Fix⁡(T)≠∅ and let f:C→C be a fixed Lipschitzian strongly pseudocontractive mapping with pseudocontractive coefficient β∈(0,1) and Lipschitzian constant L>0. Let A:C→C be a γ¯-strongly positive linear bounded operator with coefficient γ¯>0. Assume that C±C⊂C and that {xt} converges strongly to z∈Fix⁡(T) as t→0, where xt is defined by xt=tf(xt)+(I-tA)Txt. Suppose that {xn}⊂C is bounded and that lim⁡n→∞∥xn-Txn∥=0. Then limsup⁡n→∞〈(f-A)z,J(xn-z)〉≤0.Theorem 14.
LetC be a nonempty closed convex subset of a uniformly smooth Banach space X such that C±C⊂C. Let {Ti}i=0∞ be a countable family of nonexpansive mappings from C to itself such that Ω=⋂i=0∞‍Fix(Ti)≠∅. Let F:C→C be α-strongly accretive and λ-strictly pseudocontractive with α+λ>1, and let f:C→C be a fixed contractive mapping with contractive coefficient β∈(0,1). Let A:C→C be a γ¯-strongly positive linear bounded operator with γ¯∈(β,1+β). For arbitrarily given x0∈C, let the sequence {xn} be generated iteratively by
(48)yn=αnf(yn)+βnxn+((1-βn)I-αnA)(I-ϵnF)Tnyn,xn+1=σnf(yn)+(I-σnA)Tnyn,∀n≥0,
where {αn},{βn},{ϵn}, and {σn} are four sequences in (0,1) satisfying the following conditions:(i)
lim⁡n→∞αn=lim⁡n→∞βn=0;(ii)
lim⁡n→∞(ϵn/αn)=0,lim⁡sup⁡n→∞(σn/αn)<∞, and ∑n=0∞‍(αn/(αn+βn))=∞.Assume that ∑n=0∞‍sup⁡x∈D∥Tn+1x-Tnx∥<∞ for any bounded subset D of C, let T be a mapping of C into itself defined by Tx=lim⁡n→∞Tnx for all x∈C, and suppose that Fix⁡(T)=⋂i=0∞‍Fix⁡(Ti). Then, {xn} converges strongly to a point z of Ω such that z is a unique solution in Ω to the VIP:
(49)〈(f-A)z,J(p-z)〉≤0,∀p∈Ω.Proof.
By condition (i), we may assume, without loss of generality, thatαn≤(1-βn)∥A∥-1. Since A is a γ¯-strongly positive linear bounded operator on C, from (11) we have
(50)∥A∥=sup⁡{|〈Au,J(u)〉|:u∈C,∥u∥=1}.
Observe that
(51)〈((1-βn)I-αnA)u,J(u)〉=1-βn-αn〈Au,J(u)〉≥1-βn-αn∥A∥≥0.
It follows that
(52)∥(1-βn)I-αnA∥=sup⁡⁡{〈((1-βn)I-αnA)u,J(u)〉:u∈C,∥u∥=1}=sup⁡⁡{1-βn-αn〈Au,J(u)〉:u∈C,∥u∥=1}≤1-βn-αnγ¯.
Next, we show that{yn} is well defined. For each n≥0, define a mapping Sn:C→C by
(53)Snx=αnf(x)+βnxn+((1-βn)I-αnA)(I-ϵnF)Tnx,Snx=αnf(x)+βnxn+((1-βn)I-Tnx,αnAϵn)∀x∈C.
For every x,y∈C, we have
(54)〈Snx-Sny,J(x-y)〉=αn〈f(x)-f(y),J(x-y)〉+〈((1-βn)I-αnA)pppppp×((I-ϵnF)Tnx-(I-ϵnF)Tny),J(x-y)〉≤αnβ∥x-y∥2+(1-βn-αnγ¯)×∥(I-ϵnF)Tnx-(I-ϵnF)Tny∥∥x-y∥≤αnβ∥x-y∥2+(1-βn-αnγ¯)(1-ϵnγ0)∥Tnx-Tny∥∥x-y∥≤αnβ∥x-y∥2+(1-βn-αnγ¯)∥x-y∥2=[1-βn-αn(γ¯-β)]∥x-y∥2,
where γ0=1-(1-α)/λ. Therefore, Sn is a continuous strong pseudocontraction for each n≥0. By Lemma 7, we see that there exists a unique fixed point yn for each n≥0 such that
(55)yn=αnf(yn)+βnxn+((I-βn)I-αnA)(I-ϵnF)Tnyn.
That is, the sequence {yn} is well defined. Next, we prove that {xn} is bounded. Take a fixed p∈Ω arbitrarily. Taking into account lim⁡n→∞(ϵn/αn)=0, we may assume that there exists a constant τ∈(0,1) such that ϵn≤ταn for all n≥0. Then we have
(56)∥yn-p∥2=αn〈f(yn)-Ap,J(yn-p)〉+βn〈xn-p,J(yn-p)〉+〈((1-βn)I-αnA)((I-ϵnF)Tnyn-(I-ϵnF)p),J(yn-p)〉-ϵn〈((1-βn)I-αnA)Fp,J(yn-p)〉≤αn〈f(yn)-f(p),J(yn-p)〉+αn〈f(p)-Ap,J(yn-p)〉+βn∥xn-p∥∥yn-p∥+(1-βn-αnγ¯)∥(I-ϵnF)Tnyn-(I-ϵnF)p∥×∥yn-p∥+ϵn(1-βn-αnγ¯)∥Fp∥∥yn-p∥≤αnβ∥yn-p∥2+αn〈f(p)-Ap,J(yn-p)〉+βn∥xn-p∥∥yn-p∥+(1-βn-αnγ¯)(1-ϵnγ0)∥Tnyn-p∥∥yn-p∥+ϵn(1-βn-αnγ¯)∥Fp∥∥yn-p∥≤αnβ∥yn-p∥2+(1-βn-αnγ¯)∥yn-p∥2+αn〈f(p)-Ap,J(yn-p)〉+βn∥xn-p∥∥yn-p∥+ϵn∥Fp∥∥yn-p∥=(1-βn-αn(γ¯-β))∥yn-p∥2+βn∥xn-p∥∥yn-p∥+αn∥f(p)-Ap∥∥yn-p∥+ϵn∥Fp∥∥yn-p∥≤(1-βn-αn(γ¯-β))∥yn-p∥2+βn∥xn-p∥∥yn-p∥+(αn+ϵn)(∥f(p)-Ap∥+∥Fp∥)∥yn-p∥≤(1-βn-αn(γ¯-β))∥yn-p∥2+βn∥xn-p∥∥yn-p∥+αn(1+τ)(∥f(p)-Ap∥+∥Fp∥)∥yn-p∥,
which implies that
(57)∥yn-p∥≤βnβn+αn(γ¯-β)∥xn-p∥+αn(γ¯-β)βn+αn(γ¯-β)·(1+τ)(∥f(p)-Ap∥+∥Fp∥)γ¯-β.
Therefore, we have
(58)∥xn+1-p∥=∥σnf(yn)+(I-σnA)Tnyn-p∥=∥σn(f(yn)-f(p))+(I-σnA)Tnyn-(I-σnA)Tnp+σn(f(p)-Ap)∥≤σn∥f(yn)-f(p)∥+∥(I-σnA)(Tnyn-Tnp)∥+σn∥f(p)-Ap∥≤σnβ∥yn-p∥+(1-σnγ¯)∥yn-p∥+σn∥f(p)-Ap∥=(1-σn(γ¯-β))∥yn-p∥+σn∥f(p)-Ap∥≤(1-σn(γ¯-β))×[βnβn+αn(γ¯-β)∥xn-p∥+αn(γ¯-β)βn+αn(γ¯-β)·(1+τ)(∥f(p)-Ap∥+∥Fp∥)γ¯-β]+σn∥f(p)-Ap∥≤(1-σn(γ¯-β))×max⁡⁡{∥xn-p∥,(1+τ)(∥f(p)-Ap∥+∥Fp∥)γ¯-β}+σn∥f(p)-Ap∥=(1-σn(γ¯-β))×max⁡⁡{∥xn-p∥,(1+τ)(∥f(p)-Ap∥+∥Fp∥)γ¯-β}+σn(γ¯-β)∥f(p)-Ap∥γ¯-β≤max⁡⁡{∥xn-p∥,(1+τ)(∥f(p)-Ap∥+∥Fp∥)γ¯-β,ppppppl∥f(p)-Ap∥γ¯-β}≤max⁡⁡{∥xn-p∥,(1+τ)(∥f(p)-Ap∥+∥Fp∥)γ¯-β}.
By induction, we get
(59)∥xn-p∥≤max⁡{∥x0-p∥,(1+τ)(∥f(p)-Ap∥+∥Fp∥)γ¯-β},{∥x0-p∥≤,(1+τ)(∥f(p)-Ap∥+∥Fp∥)γ¯-β}∀n≥0.
Therefore, {xn} is bounded and so are the sequences {yn},{Tnyn}. We observe that
(60)∥yn-Tnyn∥=∥αn(f(yn)-ATnyn)+βn(xn-Tnyn)-ϵn((1-βn)I-αnA)FTnyn∥≤αn∥f(yn)-ATnyn∥+βn∥xn-Tnyn∥+ϵn∥((1-βn)I-αnA)FTnyn∥≤αn∥f(yn)-ATnyn∥+βn∥xn-Tnyn∥+ϵn(1-βn-αnγ¯)∥FTnyn∥≤αn∥f(yn)-ATnyn∥+βn∥xn-Tnyn∥+ϵn∥FTnyn∥,
which go together with condition (i) and ϵn≤ταn,∀n≥0, implying that
(61)lim⁡n→∞∥yn-Tnyn∥=0.
On the other hand, we have
(62)∥yn-Tyn∥≤∥yn-Tnyn∥+∥Tnyn-Tyn∥.
Utilizing Lemma 12, we immediately derive
(63)lim⁡n→∞∥yn-Tyn∥=0.
Let xt=tf(xt)+(I-tA)Txt. Utilizing [1, Lemma 2.5] and Lemma 13, we conclude that {xt} converges strongly to z∈Fix⁡(T)=⋂i=0∞‍Fix⁡(Ti)=Ω and
(64)lim⁡sup⁡n→∞〈(f-A)z,J(yn-z)〉≤0.
Finally, we show that xn→z as n→∞. We observe that
(65)∥yn-z∥2=αn〈f(yn)-Az,J(yn-z)〉+βn〈xn-z,J(yn-z)〉+〈((1-βn)I-αnA)(Tnyn-z),J(yn-z)〉-ϵn〈((1-βn)I-αnA)FTnyn,J(yn-z)〉≤(1-βn-αnγ¯)∥yn-z∥2+βn∥xn-z∥∥yn-z∥+αn〈f(yn)-f(z),J(yn-z)〉+αn〈f(z)-Az,J(yn-z)〉+ϵn∥((1-βn)I-αnA)FTnyn∥∥yn-z∥≤(1-βn-αnγ¯)∥yn-z∥2+βn∥xn-z∥∥yn-z∥+αnβ∥yn-z∥2+αn〈f(z)-Az,J(yn-z)〉+ϵn(1-βn-αnγ¯)∥FTnyn∥∥yn-z∥≤(1-βn-αnγ¯)∥yn-z∥2+βn2∥xn-z∥2+βn2∥yn-z∥2+αnβ∥yn-z∥2+αn〈f(z)-Az,J(yn-z)〉+ϵn∥FTnyn∥∥yn-z∥=(1-βn2-αn(γ¯-β))∥yn-z∥2+βn2∥xn-z∥2+αn〈f(z)-Az,J(yn-z)〉+ϵn∥FTnyn∥∥yn-z∥,
which implies that
(66)∥yn-z∥2≤βnβn+2αn(γ¯-β)∥xn-z∥2+2αnβn+2αn(γ¯-β)〈f(z)-Az,J(yn-z)〉+2ϵnβn+2αn(γ¯-β)∥FTnyn∥∥yn-z∥=(1-2αn(γ¯-β)βn+2αn(γ¯-β))∥xn-z∥2+2αn(γ¯-β)βn+2αn(γ¯-β)×(〈f(z)-Az,J(yn-z)〉γ¯-β+ϵnαn·∥FTnyn∥∥yn-z∥γ¯-β).
Furthermore, utilizing Lemma 3 from the last relation we have
(67)∥xn+1-z∥2=∥σn(f(yn)-f(z))+(I-σnA)Tnyn-(I-σnA)Tnz+σn(f(z)-F(z))∥2≤∥σn(f(yn)-f(z))+(I-σnA)Tnyn-(I-σnA)Tnz∥2+2σn〈f(z)-Az,J(xn+1-z)〉≤[σnβ∥yn-z∥+(1-σnγ¯)∥Tnyn-Tnz∥]2+2σn∥f(z)-Az∥∥xn+1-z∥≤[σnβ∥yn-z∥+(1-σnγ¯)∥yn-z∥]2+2σn∥f(z)-Az∥∥xn+1-z∥=(1-σn(γ¯-β))2∥yn-z∥2+2σn∥f(z)-Az∥∥xn+1-z∥≤∥yn-z∥2+2σn∥f(z)-Az∥∥xn+1-z∥≤(1-2αn(γ¯-β)βn+2αn(γ¯-β))∥xn-z∥2+2αn(γ¯-β)(βn+2αn(γ¯-β))×(〈f(z)-Az,J(yn-z)〉γ¯-β+ϵnαn·∥FTnyn∥∥yn-z∥γ¯-β)+2σn∥f(z)-Az∥∥xn+1-z∥=(1-2αn(γ¯-β)βn+2αn(γ¯-β))∥xn-z∥2+2αn(γ¯-β)βn+2αn(γ¯-β)×{〈f(z)-Az,J(yn-z)〉γ¯-β+ϵnαn·∥FTnyn∥∥yn-z∥γ¯-βppppppl+βn+2αn(γ¯-β)γ¯-β·σnαnppppppl·∥f(z)-Az∥∥xn+1-z∥{〈f(z)-Az,J(yn-z)〉γ¯-β+ϵnαn·∥FTnyn∥∥yn-z∥γ¯-β}.
We note that
(68)2αn(γ¯-β)βn+2αn(γ¯-β)>2αn(γ¯-β)2βn+2αn=(γ¯-β)αnαn+βn.
Therefore, condition (ii) leads to ∑n=0∞‍(2αn(γ¯-β)/(βn+2αn(γ¯-β)))=∞. In addition, since αn→0,βn→0,(ϵn/αn)→0, and lim⁡sup⁡n→∞(σn/αn)<∞, we get the following from (64)
(69)lim⁡sup⁡n→∞{〈f(z)-Az,J(yn-z)〉γ¯-β+ϵnαn·∥FTnyn∥∥yn-z∥γ¯-βpppppppp+βn+2αn(γ¯-β)γ¯-β·σnαnpppppppp·∥f(z)-F(z)∥∥xn+1-z∥βn+2αn(γ¯-β)γ¯-β}≤0.
Applying Lemma 2, we have xn→z as n→∞. This completes the proof.Remark 15.
Putαn=σn=1/n and βn=ϵn=1/n2. Then {αn},{βn},{ϵn}, and {σn} satisfy conditions (i) and (ii) of Theorem 14. But we note that αn/βn=n→∞.Remark 16.
In the iterative scheme of Theorem14, the first iterative step yn=αnf(yn)+βnxn+((1-βn)I-αnA)(I-ϵnF)Tnyn is a predictor step and the second iterative step xn+1=σnf(yn)+(I-σnA)Tnyn is a corrector step. Hence our iteration process is the predictor-corrector method.Remark 17.
Theorem14 extends and improves Theorem 3.1 of [10] to a great extent in the following aspects:(i)
u is replaced by a fixed contractive mapping;(ii)
one continuous pseudocontractive mapping (including nonexpansive mapping) is replaced by a countable family of nonexpansive mappings;(iii)
conditionαn/βn→0 is weakened to the one αn→0 and βn→0 as n→∞;(iv)
we add a strongly positive linear bounded operatorA and a strongly accretive and strictly pseudocontractive mapping F in our iterative algorithm.Theorem 18.
LetC be a nonempty closed convex subset of a uniformly smooth Banach space X which has the weakly sequentially continuous duality mapping J. Assume that C±C⊂C. Let {Ti}i=0∞ be a countable family of nonexpansive mappings from C to itself such that Ω=⋂i=0∞‍Fix⁡(Ti)≠∅. Let F:C→C be α-strongly accretive and λ-strictly pseudocontractive with α+λ>1, and let f:C→C be a fixed contractive mapping with contractive coefficient β∈(0,γ0),γ0=1-(1-α)/λ. Let A:C→C be a γ¯-strongly positive linear bounded operator with γ¯β<1. For arbitrarily given x0∈C, let the sequence {xn} be generated iteratively by
(70)xn+1=(I-βnF)Tnxn+βn[f(xn)-αn(Af(xn)-Tnxn)],=(I-βnF)Tnxn+βn[f(xn)-αn(Af(xn))],∀n≥0,
where {αn} and {βn} are two sequences in (0,1) satisfying the following conditions:(i)
lim⁡n→∞αn=lim⁡n→∞βn=0 and ∑n=0∞‍βn=∞;(ii)
∑n=1∞‍|βn-βn-1|<∞ or lim⁡n→∞βn-1/βn=1.Assume that ∑n=0∞‍sup⁡x∈D∥Tn+1x-Tnx∥<∞ for any bounded subset D of C, let T be a mapping of C into itself defined by Tx=lim⁡n→∞Tnx for all x∈C, and suppose that Fix⁡(T)=⋂i=0∞‍Fix⁡(Ti). Then, {xn} converges strongly to a point z of Ω such that z is a unique solution in Ω to the VIP:
(71)〈(F-f)z,J(z-p)〉≤0,∀p∈Ω.Proof.
First, sinceA is a γ¯-strongly positive linear bounded operator on C, from (11) we have
(72)∥A∥=sup⁡{|〈Au,J(u)〉|:u∈C,∥u∥=1}.
Let us show that{xn} is bounded. Indeed, since lim⁡n→∞αn=0, without loss of generality, we may assume that 0<αn≤min⁡{(γ0-β)/2(1-γ¯β),∥A∥-1},∀n≥0. Take p∈Ω. Then it follows that p=Tnp,∀n≥0, and
(73)xn+1-p=(I-βnF)Tnxn-(I-βnF)Tnpxn+1-p=+βn[(I-αnA)f(xn)-(I-αnA)f(p)xn+1-p=+βn+αn(Tnxn-p)]xn+1-p=+βn(f-F)p+βnαn(I-Af)p.
Hence we deduce the following 0<αn≤min⁡{(γ0-β)/2(1-γ¯β),∥A∥-1} that
(74)∥xn+1-p∥=∥(I-βnF)Tnxn-(I-βnF)Tnp+βn[(I-αnA)f(xn)-(I-αnA)f(p)ppppppppl+αn(Tnxn-p)]+βn(f-F)p+βnαn(I-Af)p∥≤∥(I-βnF)Tnxn-(I-βnF)Tnp∥+βn[∥I-αnA∥∥f(xn)-f(p)∥+αn∥Tnxn-p∥]+βn∥(f-F)p∥+βnαn∥(I-Af)p∥≤(1-βnγ0)∥xn-p∥+βn[(1-αnγ¯)β∥xn-p∥+αn∥xn-p∥]+βn∥(f-F)p∥+βnαn∥(I-Af)p∥=[1-βn(γ0-β-αn(1-γ¯β))]∥xn-p∥+βn∥(f-F)p∥+βnαn∥(I-Af)p∥≤[1-βn(γ0-β-γ0-β2(1-γ¯β)(1-γ¯β))]∥xn-p∥+βn∥(f-F)p∥+βnαn∥(I-Af)p∥≤(1-12βn(γ0-β))∥xn-p∥+βn(∥(f-F)p∥+∥(I-Af)p∥)=(1-12βn(γ0-β))∥xn-p∥+12βn(γ0-β)2(∥(f-F)p∥+∥(I-Af)p∥)γ0-β≤max⁡⁡{∥xn-p∥,2(∥(f-F)p∥+∥(I-Af)p∥)γ0-β}.
By induction
(75)∥xn-p∥≤max⁡{∥x0-p∥,2(∥(f-F)p∥+∥(I-Af)p∥)γ0-β},{∥x0-p∥,≤2(∥(f-F)p∥+∥(I-Af)p∥)γ0-β}∀n≥0.
This implies that {xn} is bounded and so are {Tnxn},{f(xn)} and {FTnxn}.
Now we claim that(76)lim⁡n→∞∥xn+1-xn∥=0.
Indeed, first of all, (70) can be rewritten as follows:
(77)yn=(I-αnA)f(xn)+αnTnxn,xn+1=(I-βnF)Tnxn+βnyn,∀n≥0.
Observe that
(78)∥yn-yn-1∥=∥(I-αnA)f(xn)+αnTnxn-(I-αn-1A)f(xn-1)-αn-1Tn-1xn-1∥=∥αn(Tnxn-Tn-1xn-1)+(αn-αn-1)(Tn-1xn-1-Af(xn-1))+(I-αnA)f(xn)-(I-αnA)f(xn-1)∥≤αn∥Tnxn-Tn-1xn-1∥+|αn-αn-1|∥Tn-1xn-1-Af(xn-1)∥+∥I-αnA∥∥f(xn)-f(xn-1)∥≤αn(∥Tnxn-Tnxn-1∥+∥Tnxn-1-Tn-1xn-1∥)+|αn-αn-1|∥Tn-1xn-1-Af(xn-1)∥+(1-αnγ¯)β∥xn-xn-1∥≤αn(∥xn-xn-1∥+∥Tnxn-1-Tn-1xn-1∥)+|αn-αn-1|∥Tn-1xn-1-Af(xn-1)∥+(1-αnγ¯)β∥xn-xn-1∥=(β-αn(γ¯β-1))∥xn-xn-1∥+|αn-αn-1|∥Tn-1xn-1-Af(xn-1)∥+αn∥Tnxn-1-Tn-1xn-1∥,
and hence
(79)∥xn+1-xn∥=∥(I-βnF)Tnxn+βnyn-(I-βn-1F)Tn-1xn-1-βn-1yn-1∥≤∥βn(yn-yn-1)+(βn-βn-1)(yn-1-FTn-1xn-1)+(I-βnF)Tnxn-(I-βnF)Tn-1xn-1∥≤βn∥yn-yn-1∥+|βn-βn-1|∥yn-1-FTn-1xn-1∥+(1-βnγ0)∥Tnxn-Tn-1xn-1∥≤βn∥yn-yn-1∥+|βn-βn-1|∥yn-1-FTn-1xn-1∥+(1-βnγ0)(∥Tnxn-Tnxn-1∥+∥Tnxn-1-Tn-1xn-1∥)≤βn∥yn-yn-1∥+|βn-βn-1|∥yn-1-FTn-1xn-1∥+(1-βnγ0)(∥xn-xn-1∥+∥Tnxn-1-Tn-1xn-1∥)≤βn[(β-αn(γ¯β-1))∥xn-xn-1∥+|αn-αn-1|×∥Tn-1xn-1-Af(xn-1)∥+αn∥Tnxn-1-Tn-1xn-1∥(β-αn(γ¯β-1))]+|βn-βn-1|∥yn-1-FTn-1xn-1∥+(1-βnγ0)(∥xn-xn-1∥+∥Tnxn-1-Tn-1xn-1∥)≤βn[(β-αn(γ¯β-1))∥xn-xn-1∥+|αn-αn-1|M+αn∥Tnxn-1-Tn-1xn-1∥]+|βn-βn-1|M+(1-βnγ0)(∥xn-xn-1∥+∥Tnxn-1-Tn-1xn-1∥)=[1-βn(γ0-β-αn(1-γ¯β))]∥xn-xn-1∥+M(βn|αn-αn-1|+|βn-βn-1|)+(βnαn+(1-βnγ0))∥Tnxn-1-Tn-1xn-1∥≤[1-βn(γ0-β-γ0-β2(1-γ¯β)(1-γ¯β))]×∥xn-xn-1∥+M(βn|αn-αn-1|+|βn-βn-1|)+2∥Tnxn-1-Tn-1xn-1∥=[1-12βn(γ0-β)]∥xn-xn-1∥+M(βn|αn-αn-1|+|βn-βn-1|)+2∥Tnxn-1-Tn-1xn-1∥,
where sup⁡n≥0{∥Tnxn-Af(xn)∥+∥yn-FTnxn∥}≤M for some M>0 (it is easy to see that {yn} is bounded due to the boundedness of {xn}). Utilizing Lemma 2, we conclude that ∥xn+1-xn∥→0 as n→∞ from conditions (i)-(ii) and the property imposed on {Tn}.
Next let us show that(80)lim⁡n→∞∥xn-Txn∥=0.
Indeed, from (76), (77), and βn→0, it follows that
(81)∥xn-Tnxn∥≤∥xn-xn+1∥+∥xn+1-Tnxn∥=∥xn-xn+1∥+βn∥yn-FTnxn∥→0asn→∞.
That is,
(82)lim⁡n→∞∥xn-Tnxn∥=0.
Also, it is clear that
(83)∥xn-Txn∥≤∥xn-Tnxn∥+∥Tnxn-Txn∥.
By Lemma 12, we conclude from (82) and (83) that (80) holds. Let xt=(I-θtF)Txt+θt[f(xt)-t(Af(xt)-Txt)]. According to Theorem 11, we know that {xt} converges strongly to z∈Fix⁡(T)=⋂i=0∞‍Fix⁡(Ti)=Ω, which is the unique solution in Ω to the VIP:
(84)〈(F-f)z,J(z-p)〉≤0,∀p∈Ω.
Further, let us show that(85)lim⁡sup⁡n→∞〈(f-F)z,J(xn-z)〉≤0.
Indeed, take a subsequence {xni} of {xn} such that
(86)limsup⁡n→∞〈(f-F)z,J(xn-z)〉=lim⁡i→∞〈(f-F)z,J(xni-z)〉.
Without loss of generality, we may assume that xni⇀x~. Utilizing Lemma 5 we obtain from (80) that x~∈Fix⁡(T). Hence from (84) and (86) we get
(87)limsup⁡n→∞〈(f-F)z,J(xn-z)〉=〈(f-F)z,J(x~-z)〉≤0.
As required, let us show thatxn→z as n→∞.
As a matter of fact, we observe that(88)∥xn+1-z∥2=∥(I-βnF)Tnxn-(I-βnF)Tnz+βn[(I-αnA)f(xn)-(I-αnA)f(z)pppppppl+αn(Tnxn-z)]+βn(f-F)z+βnαn(I-Af)z∥2≤∥(I-βnF)Tnxn-(I-βnF)Tnz+βn[(I-αnA)(f(xn)-f(z))+αn(Tnxn-z)]∥2+2βn〈(f-F)z,J(xn+1-z)〉+2βnαn〈(I-Af)z,J(xn+1-z)〉≤[∥(I-βnF)Tnxn-(I-βnF)Tnz∥+βn(∥I-αnA∥∥f(xn)-f(z)∥+αn∥Tnxn-z∥)]2+2βn〈(f-F)z,J(xn+1-z)〉+2βnαn∥(I-Af)z∥∥xn+1-z∥≤{(1-βnγ0)∥Tnxn-Tnz∥+βn[(1-αnγ¯)β∥xn-z∥+αn∥Tnxn-z∥]}2+2βn〈(f-F)z,J(xn+1-z)〉+2βnαn∥(I-Af)z∥∥xn+1-z∥≤{(1-βnγ0)∥xn-z∥+βn[(1-αnγ¯)β∥xn-z∥+αn∥xn-z∥]}2+2βn〈(f-F)z,J(xn+1-z)〉+2βnαn∥(I-Af)z∥∥xn+1-z∥=[1-βn(γ0-β-αn(1-γ¯β))]2∥xn-z∥2+2βn〈(f-F)z,J(xn+1-z)〉+2βnαn∥(I-Af)z∥∥xn+1-z∥≤[1-βn(γ0-β-αn(1-γ¯β))]∥xn-z∥2+2βn〈(f-F)z,J(xn+1-z)〉+2βnαn∥(I-Af)z∥∥xn+1-z∥≤[1-βn(γ0-β-γ0-β2(1-γ¯β)(1-γ¯β))]∥xn-z∥2+2βn〈(f-F)z,J(xn+1-z)〉+2βnαn∥(I-Af)z∥∥xn+1-z∥=[1-12βn(γ0-β)]∥xn-z∥2+2βn〈(f-F)z,J(xn+1-z)〉+2βnαn∥(I-Af)z∥∥xn+1-z∥=(1-μn)∥xn-z∥2+μnνn,
where μn=(1/2)βn(γ0-β) and
(89)νn=4(〈(f-F)z,J(xn+1-z)〉+αn∥(I-Af)z∥∥xn+1-z∥)γ0-β.
It can be easily seen from (85) and conditions (i) and (ii) that
(90)∑n=0∞‍μn=∞,lim⁡sup⁡n→∞νn≤0.
In terms of Lemma 8, we infer that xn→z as n→∞.Finally, we provide an example to illustrate Theorem18.Example 19.
LetX=R2 with inner product 〈·,·〉 and norm ∥·∥ which are defined by
(91)〈x,y〉=ac+bd,∥x∥=a2+b2,
for all x,y∈R2 with x=(a,b) and y=(c,d). Let C={(a,a):a∈R}. Clearly, C is a nonempty closed convex subset of a uniformly smooth Banach space X=R2 such that C±C⊂C. Let {Tn}n=0∞ be a countable family of nonexpansive mappings from C to itself such that Ω=⋂n=0∞‍Fix⁡(Tn)≠∅, for instance, putting Tn=(1-1/2n+1)T with T={3/52/52/53/5}. Then ∥T∥=1 and ∥Tn∥=1-1/2n+1,∀n≥0. It is clear that Tn and T are nonexpansive mappings with Ω=⋂n=0∞‍Fix⁡(Tn)={0}≠∅, and {Tn} satisfies the assumption in Theorem 18. Let F:C→C be α-strongly accretive and λ-strictly pseudocontractive with α+λ>1, and f:C→C be a fixed contractive mapping with contractive coefficient β∈(0,γ0),γ0=1-(1-α)/λ, for instance, putting S={2/31/31/32/3},  F=(1/2)S, and f={3/252/252/253/25}, we know that ∥F∥=(1/2)∥S∥=1/2,∥f∥=1/5 and that F is a (1/2)-strongly accretive and (8/9)-strictly pseudocontractive mapping and f is a (1/5)-contraction with (1/5)∈(0,γ0) and γ0=1/4. Let A:C→C be a γ¯-strongly positive linear bounded operator with γ¯β<1; for instance, putting A=(7/6)S, we know that A is a (7/6)-strongly positive linear bounded operator with γ¯β=(7/6)×(1/5)<1. In this case, from iterative scheme (70) in Theorem 18, we obtain that for any given x0∈C,
(92)x1=(I-β0F)T0x0+β0[f(x0)-α0(Af(x0)-T0x0)]=(1-12β0)(1-120+1)x0+β0[15x0-α0(76·15x0-(1-120+1)x0)]=[(1-12β0)(1-120+1)+15β0-α0β0(730-(1-120+1))]x0∈C.
It can be readily seen that
(93)xn+1=[(1-12βn)(1-12n+1)+15βn-αnβn(730-(1-12n+1))]xn,∀n≥0.
We claim that xn converges to the unique point 0 in Ω if αn=(6/23)βn and ∑n=0∞‍βn=∞. Indeed, observe that
(94)∥xn+1∥=[(1-12βn)(1-12n+1)+15βn-αnβn(730-(1-12n+1))]∥xn∥≤[(1-12βn)+15βn-αnβn(730-1)]∥xn∥=(1-310βn+2330αnβn)∥xn∥=(1-310βn+2330·623βnβn)∥xn∥≤(1-310βn+15βn)∥xn∥=(1-110βn)∥xn∥≤∏i=0n‍(1-110βi)∥x0∥.
Thus, we conclude from ∑n=0∞‍βn=∞ that xn converges to the unique point 0 in Ω. It is clear that z=0 is a unique solution in Ω for the following variational inequality problem (VIP):
(95)〈(f-F)z,J(p-z)〉≤0,∀p∈Ω.



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*Source: 173461-2014-01-22.xml*