# Viscosity Projection Algorithms for Pseudocontractive Mappings in Hilbert Spaces

Authors: Xiujuan Pan; Shin Min Kang; Young Chel Kwun
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

An explicit projection algorithm with viscosity technique is constructed for finding the fixed points of the pseudocontractive mapping in Hilbert spaces. Strong convergence theorem is demonstrated. Consequently, as an application, we can approximate to the minimum-norm fixed point of the pseudocontractive mapping.


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## Body

## 1. Introduction

Letℍ be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥, respectively. Let ℂ be a nonempty closed convex subset of ℍ.Recall that a mapping𝕋:ℂ→ℂ is said to be(i)
L-Lipschitz ⇔ there exists a constant L>0 such that ∥𝕋u-𝕋v∥≤L∥u-v∥ for all u,v∈ℂ; if L∈(0,1), then 𝕋 is said to be contractive; if L=1, then 𝕋 is said to be nonexpansive;(ii)
pseudocontractive
⇔〈𝕋u-𝕋v,u-v〉≤∥u-v∥2;⇔∥𝕋u-𝕋v∥2≤∥u-v∥2+∥(I-𝕋)u-(I-𝕋)v)∥2;⇔〈u-v,(I-𝕋)u-(I-𝕋)v〉≥0,for allu,v∈ℂ.Interest in pseudocontractive mappings stems mainly from their firm connection with the class of nonlinear accretive operators. It is a classical result, see Deimling [1], that if 𝕋 is an accretive operator, then the solutions of the equations 𝕋x=0 correspond to the equilibrium points of some evolution systems. This explains the importance, from this point of view, of the improvement brought by the Ishikawa iteration which was introduced by Ishikawa [2] in 1974. The original result of Ishikawa is stated in the following.Theorem 1.
Letℂ be a convex compact subset of a Hilbert space ℍ and let 𝕋:ℂ→ℂ be an L-Lipschitzian pseudocontractive mapping with Fix⁡(𝕋)≠∅. For any x0∈ℂ, define the sequence {xn} iteratively by
(1)yn=(1-ηn)xn+ηn𝕋xn,xn+1=(1-ξn)xn+ξn𝕋yn,
for all n∈ℕ, where {ξn}⊂[0,1]and   {ηn}⊂[0,1] satisfy the conditions: lim⁡n→∞ηn=0 and ∑n=1∞‍ξnηn=∞. Then the sequence {xn} generated by (1) converges strongly to a fixed point of 𝕋.The iteration (1) is now referred to as the Ishikawa iterative sequence. However, we note that C is compact subset. Now, we know that strong convergence has not been achieved without compactness assumption on the involved operation or the underlying spaces. A counter example can be found in Chidume and Mutangadura [3].In order to obtain a strong convergence result for pseudocontractive mappings without the compactness assumption, in [4], Zhou coupled the Ishikawa algorithm with the hybrid technique and presented the following algorithm for Lipschitz pseudocontractive mappings:
(2)yn=(1-ξn)xn+ξn𝕋xn,zn=(1-ηn)xn+ηn𝕋yn,ℂn={z∈ℂ:∥zn-z∥2≤∥xn-z∥2ℂn=hhhhhh-ξnηn(1-2ξn-ξn2L2)∥xn-𝕋xn∥2},ℚn={z∈ℂ:〈xn-z,x0-xn〉≥0},xn+1=projℂn∩ℚnx0,n∈ℕ.
Zhou proved that the sequence {xn} generated by (2) converges strongly to the fixed point of 𝕋. Further, in [5], Yao et al. introduced the hybrid Mann algorithm as follows.Letℂ be a nonempty closed convex subset of a real Hilbert space ℍ. Let {ξn} be a sequence in (0,1). Let x0∈ℍ. For ℂ1=ℂ and x1=projℂ1x0, define a sequence {xn} of ℂ as follows:
(3)yn=(1-ξn)xn+ξn𝕋xn,ℂn+1={z∈ℂn:∥ξn(I-𝕋)yn∥2ℂn+1hhh≤2ξn〈xn-z,(I-𝕋)yn〉∥ξn(I-𝕋)yn∥2},xn+1=projℂn+1x0,n∈ℕ.
Note that, in iterations (2) and (3), we need to compute the half-spaces ℂn (and/or ℚn). Very recently, Zegeye et al. [6] further studied the convergence analysis of the Ishikawa iteration (1). They proved ingeniously the strong convergence of the Ishikawa iteration (1). However, we have to assume that the interior of Fix⁡(𝕋) is nonempty. This appears very restrictive since even in ℝ with the usual norm, Lipschitz pseudocontractive maps with finite number of fixed points do not enjoy this condition that intFix(𝕋)≠∅. For some related works, please refer to [7–19].On the other hand, we notice that it is quite often to seek a particular solution of a given nonlinear problem, in particular, the minimum-norm solution. For instance, given a closed convex subsetℂ of a Hilbert space ℍ1 and a bounded linear operator 𝔹:ℍ1→ℍ2, where ℍ2 is another Hilbert space. The ℂ-constrained pseudoinverse of 𝔹, 𝔹ℂ†, is then defined as the minimum-norm solution of the constrained minimization problem
(4)𝔹ℂ†(b):=argmin⁡x∈ℂ∥𝔹x-b∥
which is equivalent to the fixed point problem
(5)u=projℂ(u-μ𝔹*(𝔹u-b)),
where 𝔹* is the adjoint of 𝔹, μ>0 is a constant, and b∈ℍ2 is such that proj𝔹(ℂ)¯(b)∈𝔹(ℂ).It is, therefore, an interesting problem to invent iterative algorithms that can generate sequences which converge strongly to the minimum-norm solution of a given fixed point problem. The purpose of this paper is to solve such a problem for pseudocontractions. More precisely, we will introduce an explicit projection algorithm with viscosity technique for finding the fixed points of a Lipschitzian pseudocontractive mapping. Strong convergence theorem is demonstrated. Consequently, as an application, we can find the minimum-norm fixed point of the pseudocontractive mapping.


## 2. Preliminaries

Recall that the metric projectionprojℂ:ℍ→ℂ satisfies ∥u-projℂu∥=inf⁡{∥u-v∥:v∈ℂ}. The metric projection projℂ is a typical firmly nonexpansive mapping. The characteristic inequality of the projection is 〈u-projℂu,v-projℂu〉≤0 for all u∈ℍ, v∈ℂ.In the sequel we will use the following expressions:(i)
Fix⁡(𝕋) denotes the set of fixed points of 𝕋;(ii)
xn⇀x† denotes the weak convergence of xn to x†;(iii)
xn→x† denotes the strong convergence of xn to x†.The following lemmas will be useful for the next section.Lemma 2 (see [20]).
Letℂ be a nonempty closed convex subset of a real Hilbert space ℍ. Let 𝕋:ℂ→ℂ be a nonexpansive mapping with Fix⁡(𝕋)≠∅. Then,
(6){ℂ⊃un⇀u‡(𝕀-𝕋)un⟶ν}⟹(𝕀-𝕋)u‡=ν.Lemma 3 (see [21]).
Letℂ be a nonempty closed convex subset of a real Hilbert space ℍ. Assume that a mapping 𝔸:ℂ→ℍ is monotone and weakly continuous along segments (i.e., 𝔸(x+ty)→𝔸(x) weakly, as t→0, whenever x+ty∈ℂ for x,y∈ℂ). Then the variational inequality
(7)x‡∈C,〈𝔸x‡,x-x‡〉≥0,∀x∈ℂ,
is equivalent to the dual variational inequality
(8)x‡∈ℂ,〈𝔸x,x-x‡〉≥0,∀x∈ℂ.Lemma 4 (see [22]).
Assume that the sequence{an} satisfies an≥0 and an+1≤(1-νn)an+ςnνn where {νn} is a sequence in (0,1) and {ςn} is a sequence such that ∑n=1∞‍νn=∞ and limsup⁡n→∞ςn≤0 (or ∑n=1∞‍|ςnνn|<∞). Then lim⁡n→∞an=0.


## 3. Main Results

In order to prove our main result, we need the following proposition.Proposition 5.
Letℂ be a nonempty closed convex subset of a real Hilbert space ℍ. Let 𝕎:ℂ→ℂ be a nonexpansive mapping with Fix⁡(𝕎)≠∅. Let ϱ:ℂ→ℍ be a κ-contraction.
For eacht∈(0,1), let the net {ut} be defined by
(9)ut=𝕎projℂ[tϱ(ut)+(1-t)ut].
Then, as t→0+, the net {ut} converges strongly to a point x‡∈Fix⁡(𝕎) which solves the following variational inequality:
(10)x‡∈Fix⁡(𝕎),〈(I-ϱ)x‡,z-x‡〉≥0,z∈Fix⁡(𝕎).Proof.
Fort∈(0,1), define a mapping 𝕎t:ℂ→ℂ by
(11)𝕎tu=𝕎projℂ[tϱ(u)+(1-t)u],u∈ℂ.
For any u,v∈ℂ, we have
(12)∥𝕎tu-𝕎tv∥=∥𝕎projℂ[tϱ(u)+(1-t)u]hhh-𝕎projℂ[tϱ(v)+(1-t)v]∥≤t∥ϱ(u)-ϱ(v)∥+(1-t)∥u-v∥≤[1-(1-κ)t]∥u-v∥.
Hence, 𝕎t is a 1-(1-κ)t-contraction on ℂ with ut∈ℂ as its unique fixed point. So, {ut} is well defined.
Letu∈Fix⁡(𝕎). From (9), we have
(13)∥ut-u∥=∥𝕎projℂ[tϱ(ut)+(1-t)ut]-𝕎projℂu∥≤t∥ϱ(ut)-ϱ(u)∥+t∥ϱ(u)-u∥+(1-t)∥ut-u∥≤[1-(1-κ)t]∥ut-u∥+t∥ϱ(u)-u∥.
It follows that
(14)∥ut-u∥≤∥ϱ(u)-u∥1-κ.
Thus, {ut} and {ϱ(ut)} are bounded.
Again from (9), we get
(15)∥ut-𝕎ut∥=∥𝕎projℂ[tϱ(ut)+(1-t)ut]-𝕎projℂut∥≤t∥ϱ(ut)-ut∥⟶0ast⟶0+.
Let {tn}⊂(0,1) be a sequence such that tn→0+ as n→∞. Put un:=utn. From (15), we have
(16)∥un-𝕎un∥⟶0.
From (9), we obtain
(17)∥ut-u∥2=∥𝕎projℂ[tϱ(ut)+(1-t)ut]-𝕎projℂu∥2≤∥ut-u+t(ϱ(ut)-ut)∥2=∥ut-u∥2+2t〈ϱ(ut)-ut,ut-u〉+t2∥ϱ(ut)-ut∥2=∥ut-u∥2+2t〈ϱ(ut)-ϱ(u),ut-u〉+2t〈ϱ(u)-u,ut-u〉+2t〈u-ut,ut-u〉+t2∥ϱ(ut)-ut∥2≤[1-2(1-κ)t]∥ut-u∥2+2t〈ϱ(u)-u,ut-u〉+t2∥ϱ(ut)-ut∥2.
It follows that
(18)∥ut-u∥2≤11-κ〈ϱ(u)-u,ut-u〉+tM,
where M>0 is a constant such that
(19)M>12(1-κ)sup⁡{∥ϱ(ut)-ut∥2:t∈(0,1)}.
In particular, we have
(20)∥un-u∥2≤11-κ〈ϱ(u)-u,un-u〉+tnM,u∈Fix⁡(𝕎).
Noting that {un} is bounded, without loss of generality, we assume that un⇀x‡. It is obvious that x‡∈ℂ. From (16) and Lemma 2, we deduce x‡∈Fix⁡(𝕎). Substitute x‡ for u in (20) to get
(21)∥un-x‡∥2≤11-κ〈ϱ(x‡)-x‡,xn-x‡〉+tnM.
Since un⇀x‡, we deduce from (21) that un→x‡. The net {ut} is, therefore, relatively compact, as t→0+, in the norm topology.
In (20), letting n→∞, we get
(22)∥x‡-u∥2≤11-κ〈ϱ(u)-u,x‡-u〉,u∈Fix⁡(𝕎).
Therefore, x‡ solves the variational inequality
(23)x‡∈Fix⁡(𝕎),〈(I-ϱ)u,u-x‡〉≥0,u∈Fix⁡(𝕎).
By Lemma 3, (23) equals its dual variational inequality
(24)x‡∈Fix⁡(𝕎),〈(𝕀-ϱ)x‡,u-x‡〉≥0,u∈Fix⁡(𝕎).
This indicates that x‡=(projFix⁡(𝕎)ϱ)x‡. That is, x‡ is the unique fixed point in Fix⁡(𝕎) of the contraction projFix⁡(𝕎)ϱ. So, the entire net {ut} converges in norm to x‡ as t→0+. This completes the proof.Corollary 6.
Letℂ be a nonempty closed convex subset of a real Hilbert space ℍ. Let 𝕎:ℂ→ℂ be a nonexpansive mapping with Fix⁡(𝕎)≠∅.
For eacht∈(0,1), let the net {ut} be defined by
(25)ut=𝕎projℂ[(1-t)ut].
Then, as t→0+, the net {ut} converges strongly to the minimum-norm fixed point of 𝕎.Proof.
As a matter of fact, in (9), we choose ϱ=0, and then (9) reduces to (25). From Proposition 5, (24) is reduced to
(26)0≤〈x‡,u-x‡〉,u∈Fix⁡(𝕎).
Equivalently,
(27)∥x‡∥2≤〈x‡,u〉,u∈Fix⁡(𝕎).
This implies that
(28)∥x‡∥≤∥u∥,u∈Fix⁡(𝕎).
Therefore, x‡ is the minimum-norm fixed point of 𝕎. This completes the proof.Algorithm 7.
Letℂ be a nonempty closed subset of a real Hilbert space ℍ. Let 𝕋:ℂ→ℂ be a pseudocontraction and let ϱ:ℂ→ℍ be a κ-contraction. Let {ξn}⊂[0,1] and {ηn}⊂[0,1] be two real number sequences. For x0∈ℂ, we define a sequence {xn} iteratively by
(29)xn+1=projℂ[ξnϱ(xn)+(1-ξn-ηn)xn+ηn𝕋xn],n≥0.Theorem 8.
Letℂ be a nonempty closed convex subset of a real Hilbert space ℍ. Let 𝕋:ℂ→ℂ be an L-Lipschitzian and pseudocontraction with Fix⁡(𝕋)≠∅ and ϱ:ℂ→ℍ a κ-contraction. Suppose the following conditions are satisfied: (C1)
lim⁡n→∞ξn=0 and ∑n=0∞‍ξn=∞;(C2)
lim⁡n→∞(ξn/ηn)=lim⁡n→∞(ηn2/ξn)=0;(C3)
lim⁡n→∞((ξnηn-1-ξn-1ηn)/ξn2ηn-1)=0.Then the sequence {xn} generated by the algorithm (29) converges strongly to x‡=(projFix⁡(𝕋)ϱ)x‡.Proof.
First, we prove that the sequence{xn} is bounded. We will show this fact by induction. According to conditions (C1) and (C2), there exists a sufficiently large positive integer m such that
(30)1-21-κ(L+1)(L+3)(ξn+2ηn+ηn2ξn)>0,n≥m.
Fix a p∈Fix⁡(𝕋) and take a constant M1>0 such that
(31)max⁡{∥x0-p∥,∥x1-p∥,…,∥xm-p∥,2∥ϱ(p)-p∥}hhh≤M1.
Next, we show that∥xm+1-p∥≤M1.
Set(32)ym=ξmϱ(xm)+(1-ξm-ηm)xm+ηm𝕋xm;thus,xm+1=projℂ[ym].
Then we have
(33)〈xm+1-ym,xm+1-p〉≤0.
Since 𝕋 is pseudocontractive, 𝕀-𝕋 is monotone. So, we have
(34)〈(𝕀-𝕋)xm+1-(𝕀-𝕋)p,xm+1-p〉≥0.
From (29), (33), and (34), we obtain
(35)∥xm+1-p∥2=〈xm+1-p,xm+1-p〉=〈xm+1-ym,xm+1-p〉+〈ym-p,xm+1-p〉≤〈ym-p,xm+1-p〉=〈xm-p,xm+1-p〉-ξm〈xm-ϱ(xm),xm+1-p〉+ηm〈𝕋xm-xm,xm+1-p〉=〈xm-p,xm+1-p〉+ξm〈xm+1-xm,xm+1-p〉+ξm〈ϱ(xm)-ϱ(p),xm+1-p〉+ξm〈f(p)-p,xm+1-p〉-ξm〈xm+1-p,xm+1-p〉+ηm〈𝕋xm-𝕋xm+1,xm+1-p〉+ηm〈xm+1-xm,xm+1-p〉-ηm〈xm+1-𝕋xm+1,xm+1-p〉≤∥xm-p∥∥xm+1-p∥+ξm∥xm+1-xm∥×∥xm+1-p∥+ξmκ∥xm-p∥∥xm+1-p∥+ξm∥ϱ(p)-p∥∥xm+1-p∥-ξm∥xm+1-p∥2+ηm(∥𝕋xm-𝕋xm+1∥+∥xm+1-xm∥)×∥xm+1-p∥≤(1+ξmκ)∥xm-p∥×∥xm+1-p∥+ξm∥ϱ(p)-p∥×∥xm+1-p∥-ξm∥xm+1-p∥2+(L+1)(ξm+ηm)∥xm+1-xm∥∥xm+1-p∥.
It follows that
(36)(1+ξm)∥xm+1-p∥≤(1+ξmκ)∥xm-p∥+ξm∥ϱ(p)-p∥+(L+1)(ξm+ηm)∥xm+1-xm∥.
By (29), we have
(37)∥xm+1-xm∥=∥projℂ[ξmϱ(xm)+(1-ξm-ηm)xm+ηm𝕋xm]-projℂ[xm]∥≤∥ξmϱ(xm)+(1-ξm-ηm)xm+ηm𝕋xm-xm∥≤ξm(∥ϱ(xm)-ϱ(p)∥+∥ϱ(p)-p∥+∥xm-p∥)+ηm(∥𝕋xm-p∥+∥xm-p∥)≤ξm(∥ϱ(p)-p∥+(1+κ)∥xm-p∥)+(L+1)ηm∥xm-p∥≤(L+1+κ)(ξm+ηm)∥xm-p∥+ξm∥ϱ(p)-p∥≤(L+3)(ξm+ηm)M1.
Substitute (37) into (36) to obtain
(38)(1+ξm)∥xm+1-p∥≤(1+ξmκ)∥xm-p∥+ξm∥ϱ(p)-p∥+(L+1)(L+3)(ξm+ηm)2M1≤(1+1+κ2ξm)M1+(L+1)(L+3)(ξm+ηm)2M1;
that is,
(39)∥xm+1-p∥≤[1-((1-κ)ξm/2)-(L+1)(L+3)(ξm+ηm)21+ξm]M1={1-((((1-κ)ξm2)[(ξm+2ηm+(ηm2ξm))1-21-κ(L+1)(L+3)hhhhhhhhhhhhhhhhhhhhhhh×(ξm+2ηm+(ηm2ξm))]((((1-κ)ξm2)[1-21-κ(L+1)(L+3))hhhhhhhhhh×(1+ξm)-1(((1-κ)ξm2)[(ξm+2ηm+(ηm2ξm))1-21-κ(L+1)(L+3)){1-(((1-κ)ξm2)[1-21-κ(L+1)(L+3)}M1≤M1.
By induction, we get
(40)∥xn-p∥≤M1,∀n≥0,
which implies that {xn} is bounded and so is {𝕋xn}. Now, we take a constant M2>0 such that
(41)M2=sup⁡n{∥xn∥+∥𝕋xn-xn∥}.
Set𝕊=(2𝕀-𝕋)-1 (i.e., 𝕊 is a resolvent of the monotone operator 𝕀-𝕋). We then have that 𝕊 is a nonexpansive self mapping of ℂ and Fix⁡(𝕊)=Fix⁡(𝕋).
By Proposition5, we know that, whenever {γn}⊂(0,1) and γn→0+, the sequence {zn} defined by
(42)zn=𝕊projℂ[γnϱ(zn)+(1-γn)zn]
converges strongly to the fixed point x‡ of 𝕊 (and of 𝕋 as Fix⁡(𝕊)=Fix⁡(𝕋)). Without loss of generality, we may assume that ∥zn∥≤M2 for all n.
It suffices to prove that∥xn+1-zn∥→0 as n→∞ (for some γn→0+). To this end, we rewrite (42) as
(43)(2𝕀-𝕋)zn=projℂ[γnϱ(zn)+(1-γn)zn],n≥0.
By using the property of the metric projection, we have
(44)〈γnϱ(zn)+(1-γn)zn-(2zn-𝕋zn),xn+1-(2zn-𝕋zn)〉≤0⟹〈-γn(zn-ϱ(zn)),xn+1-zn-(zn-𝕋zn)〉+〈𝕋zn-zn,xn+1-zn-(zn-𝕋zn)〉≤0⟹〈-γn(zn-ϱ(zn))+𝕋zn-zn,xn+1-zn〉+∥zn-𝕋zn∥2≤〈γn(zn-ϱ(zn)),𝕋zn-zn〉⟹〈-γn(zn-ϱ(zn))+𝕋zn-zn,xn+1-zn〉≤γn∥zn-ϱ(zn)∥∥𝕋zn-zn∥⟹〈-(zn-ϱ(zn))+𝕋zn-znγn,xn+1-zn〉≤∥zn-ϱ(zn)∥∥𝕋zn-zn∥.
Note that
(45)∥zn-𝕋zn∥=∥projℂ[γnϱ(zn)+(1-γn)zn]-zn∥≤∥γnϱ(zn)+(1-γn)zn-zn∥=γn∥zn-ϱ(zn)∥.
Hence, we get
(46)〈-(zn-ϱ(zn))+𝕋zn-znγn,xn+1-zn〉≤γn∥zn-ϱ(zn)∥2.
From (42), we have
(47)∥zn+1-zn∥=∥𝕊projℂ[γn+1ϱ(zn+1)+(1-γn+1)zn+1]-𝕊projℂ[γnϱ(zn)+(1-γn)zn]∥≤∥γn+1ϱ(zn+1)+(1-γn+1)zn+1-γnϱ(zn)-(1-γn)zn∥=∥(1-γn+1)(zn+1-zn)+(γn-γn+1)×(zn-ϱ(zn))+γn+1(ϱ(zn+1)-ϱ(zn))∥≤[1-(1-κ)γn+1]∥zn+1-zn∥+|γn+1-γn|∥zn-ϱ(zn)∥.
It follows that
(48)∥zn+1-zn∥≤|γn+1-γn|(1-κ)γn+1∥zn-ϱ(zn)∥.
Set
(49)γn:=ξnηn.
By condition (C2), γn→0+ and γn∈(0,1), for n large enough. Hence, by (46) and (48), we have
(50)〈-(zn-ϱ(zn))+ηn(𝕋zn-zn)ξn,xn+1-zn〉≤ξnηn∥zn-ϱ(zn)∥2≤ξnηnM22,∥zn-zn-1∥≤|ξnηn-1-ξn-1ηn|ξnηn-1M2.
By (29), we have
(51)∥xn+1-xn∥=∥projℂ[ξnϱ(xn)+(1-ξn-ηn)xn+ηn𝕋xn]-projℂxn∥≤ξn∥xn-ϱ(xn)∥+ηn∥𝕋xn-xn∥≤(ξn+ηn)M2.
Next, we estimate ∥xn+1-zn+1∥. Since xn+1=projℂ[yn], 〈xn+1-yn,xn+1-zn〉≤0. Then, we have
(52)∥xn+1-zn∥2=〈xn+1-zn,xn+1-zn〉=〈xn+1-yn,xn+1-zn〉+〈yn-zn,xn+1-zn〉≤〈yn-zn,xn+1-zn〉=〈[ξnϱ(xn)+(1-ξn-ηn)xn+ηn𝕋xn]-zn,xn+1-zn〉=(1-ξn-ηn)〈xn-zn,xn+1-zn〉+ηn〈𝕋xn-𝕋xn+1,xn+1-zn〉+ηn〈𝕋xn+1-𝕋zn,xn+1-zn〉+〈ξn(ϱ(xn)-zn)hhhh+ηn(𝕋zn-zn),xn+1-zn〉.
It follows that(53)∥xn+1-zn∥2≤(1-ξn-ηn)∥xn-zn∥∥xn+1-zn∥+ηnL∥xn-xn+1∥∥xn+1-zn∥+ηn∥xn+1-zn∥2+ξn〈ϱ(zn)-zn+ηnξn(𝕋zn-zn),xn+1-zn〉≤1-ξn-ηn2(∥xn-zn∥2+∥xn+1-zn∥2)+ηn22∥xn+1-zn∥2+L22∥xn-xn+1∥2+ηn∥xn+1-zn∥2+ξn2ηn∥zn-ϱ(zn)∥2.
Thus,
(54)∥xn+1-zn∥2≤1-ξn-ηn1+ξn-ηn∥xn-zn∥2+L21+ξn-ηn∥xn+1-xn∥2+2ξn2(1+ξn-ηn)ηn∥zn-ϱ(zn)∥2+ηn21+ξn-ηn∥xn+1-zn∥2≤(1-2ξn1+ξn-ηn)∥xn-zn∥2+(ξn+ηn)21+ξn-ηnL2M22+2ξn2(1+ξn-ηn)ηnM22+ηn21+ξn-ηn4M22≤(1-2ξn1+ξn-ηn)×(∥xn-zn-1∥+∥zn-zn-1∥)2+{(ξn+ηn)21+ξn-ηn+2ξn2(1+ξn-ηn)ηn+ηn21+ξn-ηn}M≤(1-2ξn1+ξn-ηn)∥xn-zn-1∥2+11+ξn-ηn∥zn-zn-1∥×(2∥xn-zn-1∥+∥zn-zn-1∥)+{(ξn+ηn)21+ξn-ηn+2ξn2(1+ξn-ηn)ηn+ηn21+ξn-ηn}M≤(1-2ξn1+ξn-ηn)∥xn-zn-1∥2+11+ξn-ηn|ξnηn-1-ξn-1ηn|ξnηn-1M+{(ξn+ηn)21+ξn-ηn+2ξn2(1+ξn-ηn)ηn+ηn21+ξn-ηn}M,
where the finite constant M>0 is given by
(55)M:=max⁡{M2sup⁡n(2∥xn-zn-1∥+∥zn-zn-1∥)}L2M22,4M22,hhhhhhhM2sup⁡n(2∥xn-zn-1∥+∥zn-zn-1∥)}.
Set
(56)δn=2ξn1+ξn-ηn≈2ξn(asn⟶∞),θn={ξnηn-1-ξn-1ηn2ξn2ηn-1+12(ξn+2ηn+ηn2ξn)+ξnηn+ηn22ξn}M.
Then (54) can be rewritten as
(57)∥xn+1-zn∥2≤(1-δn)∥xn-zn-1∥2+δnθn.
By the conditions (C1)–(C3), we deuce that
(58)lim⁡n→∞δn=0,∑n=1∞‍δn=∞,lim⁡n→∞θn=0.
From Lemma 4 and (57), we get ∥xn+1-zn∥2→0 as n→∞. This completes the proof.Algorithm 9.
Letℂ be a nonempty closed subset of a real Hilbert space ℍ. Let 𝕋:ℂ→ℂ be a pseudocontraction. Let {ξn}⊂[0,1] and {ηn}⊂[0,1] be two real number sequences. For x0∈ℂ, we define a sequence {xn} iteratively by
(59)xn+1=projℂ[(1-ξn-ηn)xn+ηn𝕋xn],n≥0.Corollary 10.
Letℂ be a nonempty closed convex subset of a real Hilbert space ℍ. Let 𝕋:ℂ→ℂ be an L-Lipschitzian and pseudocontraction with Fix⁡(𝕋)≠∅. Suppose the following conditions are satisfied:(C1)
lim⁡n→∞ξn=0 and ∑n=0∞‍ξn=∞;(C2)
lim⁡n→∞(ξn/ηn)=lim⁡n→∞(ηn2/ξn)=0;(C3)
lim⁡n→∞((ξnηn-1-ξn-1ηn)/ξn2ηn-1)=0.Then the sequence {xn} generated by the algorithm (59) converges strongly to the minimum-norm fixed point of 𝕋.



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*Source: 791209-2014-02-09.xml*