# Computing the Fixed Points of Strictly Pseudocontractive Mappings by the Implicit and Explicit Iterations

Authors: Yeong-Cheng Liou
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

It is known that strictly pseudocontractive mappings have more powerful applications than nonexpansive mappings in solving inverse problems. In this paper, we devote to study computing the fixed points of strictly pseudocontractive mappings by the iterations. Two iterative methods (one implicit and another explicit) for finding the fixed point of strictly pseudocontractive mappings have been constructed in Hilbert spaces. As special cases, we can use these two methods to find the minimum norm fixed point of strictly pseudocontractive mappings.


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## Body

## 1. Introduction

In this paper, we devote to study computing the fixed points of strictly pseudocontractive mappings by the iterations. Our motivations are mainly in two respects.Motivation 1
Iterative methods for finding fixed points of nonexpansive mappings have received vast investigations due to its extensive applications in a variety of applied areas of inverse problem, partial differential equations, image recovery, and signal processing; see [1–35] and the references therein. It is known [36] that strictly pseudocontractive mappings have more powerful applications than nonexpansive mappings in solving inverse problems. Therefore it is interesting to develop the algorithms for strictly pseudocontractive mappings.Motivation 2
In many problems, it is needed to find a solution with minimum norm. In an abstract way, we may formulate such problems as finding a pointx† with the property
(1.1)x†∈C,‖x†‖=min⁡x∈C‖x‖,
where C is a nonempty closed convex subset of a real Hilbert space H. A typical example is the least-squares solution to the constrained linear inverse problem [37]. Some related works for finding the minimum-norm solution (or fixed point of nonexpansive mappings) have been considered by some authors. The reader can refer to [38–41].
In the present paper, we present two iterative methods (one implicit and another explicit) for finding the fixed point of strictly pseudocontractive mappings in Hilbert spaces. As special cases, we can use these two methods to find the minimum norm fixed point of strictly pseudocontractive mappings.


## 2. Preliminaries

LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥, respectively. Let C be a nonempty closed convex subset of H.


### 2.1. Some Concepts

Recall that a mappingT:C→C is called nonexpansive, if
(2.1)‖Tx-Ty‖≤‖x-y‖,
for all x,y∈C. And a mapping T:C→C is said to be strictly pseudocontractive if there exists a constant 0≤λ<1 such that
(2.2)‖Tx-Ty‖2≤‖x-y‖2+λ‖(I-T)x-(I-T)y‖2,
for all x,y∈C. For such a case, we also say that T is a λ-strictly pseudocontractive mapping. It is clear that, in a real Hilbert space H, (2.2) is equivalent to
(2.3)〈Tx-Ty,x-y〉≤‖x-y‖2-1-λ2‖(I-T)x-(I-T)y‖2,
for all x,y∈C. It is clear that the class of strictly pseudocontractive mappings strictly includes the class of nonexpansive mappings.Recall that the nearest point (or metric) projection fromH onto C is defined as follows: for each point x∈H, PC[x] is the unique point in C with the property:
(2.4)‖x-PCx‖≤‖x-y‖,y∈C.
Note that PC is characterized by the inequality:
(2.5)PCx∈C,〈x-PCx,y-PCx〉≤0,y∈C.
Consequently, PC is nonexpansive.


### 2.2.  Several Useful Lemmas

Lemma 2.1 (see [42]).
LetH be a real Hilbert space. There holds the following identity:
(2.6)‖x+y‖2=‖x‖2+2〈x,y〉+‖y‖2,
for all x,y∈H.Lemma 2.2 (see [43]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T:C→C be a λ-strict pseudocontraction. Then,  (i)
F(T) is closed convex so that the projection PF(T) is well defined;(ii)
κI+(1-κ)T for κ∈[λ,1), is nonexpansive.Lemma 2.32.3 (see [42]).
LetC be a nonempty closed convex of a real Hilbert space H. Let T:C→C be a λ-strictly pseudocontractive mapping. Then I-T is demiclosed at 0 that is if xn⇀x∈C and xn-Txn→0, then x=Tx.Lemma 2.4 (see [44]).
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
(2.7)xn+1=(1-βn)yn+βnxn
for all n≥0 and
(2.8)limsup⁡n→∞(‖yn+1-yn‖-‖xn+1-xn‖)≤0.
Then limn→∞∥yn-xn∥=0.Lemma 2.5 (see [45]).
Let{an}n=0∞ be a sequence of nonnegative real numbers satisfying
(2.9)an+1≤(1-γn)an+γnσn,n≥0,
where {γn}n=0∞⊂(0,1) and {σn}n=0∞ satisfy  (i)
∑n=0∞‍γn=∞,(ii)
eitherlimsupn→∞σn≤0 or ∑n=0∞‍|γnσn|<∞. Then {an}n=0∞ converges to 0.We use the following notation:(i)
Fix(T) stands for the set of fixed points of T;(ii)
xn⇀x stands for the weak convergence of (xn) to x;(iii)
xn→x stands for the strong convergence of (xn) to x.


## 2.1. Some Concepts

Recall that a mappingT:C→C is called nonexpansive, if
(2.1)‖Tx-Ty‖≤‖x-y‖,
for all x,y∈C. And a mapping T:C→C is said to be strictly pseudocontractive if there exists a constant 0≤λ<1 such that
(2.2)‖Tx-Ty‖2≤‖x-y‖2+λ‖(I-T)x-(I-T)y‖2,
for all x,y∈C. For such a case, we also say that T is a λ-strictly pseudocontractive mapping. It is clear that, in a real Hilbert space H, (2.2) is equivalent to
(2.3)〈Tx-Ty,x-y〉≤‖x-y‖2-1-λ2‖(I-T)x-(I-T)y‖2,
for all x,y∈C. It is clear that the class of strictly pseudocontractive mappings strictly includes the class of nonexpansive mappings.Recall that the nearest point (or metric) projection fromH onto C is defined as follows: for each point x∈H, PC[x] is the unique point in C with the property:
(2.4)‖x-PCx‖≤‖x-y‖,y∈C.
Note that PC is characterized by the inequality:
(2.5)PCx∈C,〈x-PCx,y-PCx〉≤0,y∈C.
Consequently, PC is nonexpansive.


## 2.2.  Several Useful Lemmas

Lemma 2.1 (see [42]).
LetH be a real Hilbert space. There holds the following identity:
(2.6)‖x+y‖2=‖x‖2+2〈x,y〉+‖y‖2,
for all x,y∈H.Lemma 2.2 (see [43]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T:C→C be a λ-strict pseudocontraction. Then,  (i)
F(T) is closed convex so that the projection PF(T) is well defined;(ii)
κI+(1-κ)T for κ∈[λ,1), is nonexpansive.Lemma 2.32.3 (see [42]).
LetC be a nonempty closed convex of a real Hilbert space H. Let T:C→C be a λ-strictly pseudocontractive mapping. Then I-T is demiclosed at 0 that is if xn⇀x∈C and xn-Txn→0, then x=Tx.Lemma 2.4 (see [44]).
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
(2.7)xn+1=(1-βn)yn+βnxn
for all n≥0 and
(2.8)limsup⁡n→∞(‖yn+1-yn‖-‖xn+1-xn‖)≤0.
Then limn→∞∥yn-xn∥=0.Lemma 2.5 (see [45]).
Let{an}n=0∞ be a sequence of nonnegative real numbers satisfying
(2.9)an+1≤(1-γn)an+γnσn,n≥0,
where {γn}n=0∞⊂(0,1) and {σn}n=0∞ satisfy  (i)
∑n=0∞‍γn=∞,(ii)
eitherlimsupn→∞σn≤0 or ∑n=0∞‍|γnσn|<∞. Then {an}n=0∞ converges to 0.We use the following notation:(i)
Fix(T) stands for the set of fixed points of T;(ii)
xn⇀x stands for the weak convergence of (xn) to x;(iii)
xn→x stands for the strong convergence of (xn) to x.


## 3. Iterations and Convergence Analysis

Theorem 3.1.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T:C→C a λ-strictly pseudocontractive mapping with Fix(T)≠∅. Let κ∈(0,1) be a constant. For u∈H and any x0∈C, let {xn} be the sequence defined by the following implicit manner:
(3.1)xn=κTxn+(1-κ)PC[αnu+(1-αn)xn],n≥0.
Then the sequence {xn} converges strongly to PFix(T)(u).Proof.
Step1. The sequence {xn} is well defined.
Setβ=κ/(1-(1-κ)λ). It is easily to check that β∈(0,1). Then, we can rewrite (3.1) as
(3.2)xn=β(1-λ)1-βλTxn+1-β1-βλPC[αnu+(1-αn)xn],n≥0,
which is equivalent to the following:
(3.3)xn=β(λxn+(1-λ)Txn)+(1-β)PC[αnu+(1-αn)xn],n≥0.
Note that λI+(1-λ)T is nonexpansive (see Lemma 2.2). For fix n, we define a mapping Sn:C→C by
(3.4)Snx=β(λx+(1-λ)Tx)+(1-β)PC[αnu+(1-αn)x],x∈C.
For x,y∈C, we have
(3.5)‖Snx-Sny‖=‖β(λI+(1-λ)T)x+(1-β)PC[αnu+(1-αn)x]=-β(λI+(1-λ)T)y-(1-β)PC[αnu+(1-αn)y]‖≤β‖x-y‖+(1-β)(1-αn)‖x-y‖=[1-(1-β)αn]‖x-y‖,
which implies that Sn is a self-contraction of C for every n. Hence Sn has a unique fixed point xn∈C which is the unique solution of the fixed point equation (3.3).
Step2. The sequence {xn} is bounded.
Pick up anyx*∈Fix(T). From (3.3), we have
(3.6)‖xn-x*‖=‖β(λxn+(1-λ)Txn)+(1-β)PC[αnu+(1-αn)xn]-x*‖≤β‖λxn+(1-λ)Txn-x*‖+(1-β)‖PC[αnu+(1-αn)xn]-x*‖≤β‖xn-x*‖+(1-β)‖αn(u-x*)+(1-αn)(xn-x*)‖≤β‖xn-x*‖+(1-β)[(1-αn)‖xn-x*‖+αn‖u-x*‖].
It follows that
(3.7)‖xn-x*‖≤‖u-x*‖.
Hence,{xn} is bounded and so is {Txn}.
Step3. limn→∞∥xn-Txn∥=0.
From (3.3), we have
(3.8)‖xn-Txn‖≤βλ‖xn-Txn‖+(1-β)‖PC[αnu+(1-αn)xn]-PC[Txn]‖≤βλ‖xn-Txn‖+(1-β)(αn‖u-Txn‖+(1-αn)‖xn-Txn‖)=[1-(1-αn-λ)β]‖xn-Txn‖+(1-β)αn‖u-Txn‖.
It follows that(3.9)‖xn-Txn‖≤1-β(1-αn-λ)βαn‖u-Txn‖⟶0.
Step  4.  xn→x-∈PFix(T)(u).
Since{xn} is bounded, there exists a subsequence {xni} of {xn}, which converges weakly to a point x-∈C. Noticing (3.9) we can use Lemma 2.3 to get x-∈Fix(T).
By using the convexity of the norm and Lemma2.1, for any x~∈Fix(T), we have
(3.10)‖xn-x~‖2=‖β(λxn+(1-λ)Txn-x~)+(1-β)(PC[αnu+(1-αn)xn]-x~)‖2≤β‖λxn+(1-λ)Txn-x~‖2+(1-β)‖PC[αnu+(1-αn)xn]-x~‖2≤β‖xn-x~‖2+(1-β)‖αn(u-x~)+(1-αn)(xn-x~)‖2=β‖xn-x~‖2+(1-β)[(1-αn)2‖xn-x~‖2+2αn(1-αn)〈u-x~,xn-x~〉+αn2‖u-x~‖2]=β‖xn-x~‖2+(1-β)[‖xn-x~‖2-2αn‖xn-x~‖2+2αn〈u-x~,xn-x~〉=β‖xn-x~‖2+(1-β)+αn2(‖u-x~‖2+‖xn-x~‖2-2〈u-x~,xn-x~〉)].
It turns out that
(3.11)‖xn-x~‖2≤〈u-x~,xn-x~〉+αn2(‖u-x~‖2+‖xn-x~‖2+2‖u-x~‖‖xn-x~‖)≤〈u-x~,xn-x~〉+αnM,
where M>0 is some constant such that
(3.12)sup⁡12{‖u-x~‖2+‖xn-x~‖2+2‖u-x~‖‖xn-x~‖}≤M.
Therefore we can substitute x~ for x- in (3.11) to get
(3.13)‖xn-x-‖2≤〈u-x-,xn-x-〉+αnM.
However, xn⇀x-. This together with (3.13) guarantees that xn→x-. It is clear that x-=PFix(T)(u). As a matter of fact, in (3.11), if we let n→∞, then we get
(3.14)〈u-x~,x--x~〉≥0,∀x~∈Fix⁡(T).
This is equivalent to
(3.15)〈u-x-,x--x~〉≥0,∀x~∈Fix⁡(T).
Hence, x-=PFix(T)(u). Therefore, xn→x-=PFix(T)(u). This completes the proof.Corollary 3.2.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T:C→C a nonexpansive mapping with Fix(T)≠∅. Let κ∈(0,1) be a constant. For u∈H and any x0∈C, let {xn} be the sequence defined by the following implicit manner:
(3.16)xn=κTxn+(1-κ)PC[αnu+(1-αn)xn],n≥0.
Then the sequence {xn} converges strongly to PFix(T)(u).Corollary 3.3.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T:C→C a λ-strictly pseudocontractive mapping with Fix(T)≠∅. Let κ∈(0,1) be a constant. For any x0∈C, let {xn} be the sequence defined by the following implicit manner:
(3.17)xn=κTxn+(1-κ)PC[(1-αn)xn],n≥0.
Then the sequence {xn} converges strongly to PFix(T)(0) which is the minimum norm fixed point of T.Corollary 3.4.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T:C→C a nonexpansive mapping with Fix(T)≠∅. Let κ∈(0,1) be a constant. For any x0∈C, let {xn} be the sequence defined by the following implicit manner:
(3.18)xn=κTxn+(1-κ)PC[(1-αn)xn],n≥0.
Then the sequence {xn} converges strongly to PFix(T)(0) which is the minimum norm fixed point of T.Next, we introduce an explicit algorithm for finding the fixed point ofT.Theorem 3.5.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T:C→C a λ-strictly pseudocontractive mapping with Fix(T)≠∅. Let β and δ be two constants in (0,1) satisfying β+δ<1. For u∈H and any x0∈C, let {xn} be the sequence defined by the following explicit manner:
(3.19)xn+1=(βλ+δ)xn+β(1-λ)Txn+(1-β-δ)PC[αnu+(1-αn)xn],n≥0,
where αn∈(0,1) satisfies the following conditions:  (C1)
limn→∞αn=0,(C2)
∑n=0∞‍αn=∞.
Then the sequence{xn} converges strongly to PFix(T)(u).Proof.
Step1. The sequence {xn} is bounded.
First, we can rewrite (3.19) as
(3.20)xn+1=β(λxn+(1-λ)Txn)+δxn+(1-β-δ)PC[αnu+(1-αn)xn],n≥0.
Take x*∈Fix(T). From (3.20), we have
(3.21)‖xn+1-x*‖=‖β(λxn+(1-λ)Txn-x*)+δ(xn-x*)+(1-β-δ)(PC[αnu+(1-αn)xn]-x*)‖≤β‖λxn+(1-λ)Txn-x*‖+δ‖xn-x*‖22β+(1-β-δ)‖PC[αnu+(1-αn)xn]-x*‖≤β‖xn-p‖+δ‖xn-x*‖+(1-β-δ)(αn‖u-x*‖+(1-αn)‖xn-x*‖)=[1-(1-β-δ)αn]‖xn-x*‖+(1-β-δ)αn‖u-x*‖≤max⁡⁡{‖xn-x*‖,‖u-x*‖}.
By induction,
(3.22)‖xn+1-x*‖≤max⁡{‖xn-x*‖,‖u-x*‖}.
Hence, the sequence {xn} is bounded and {Txn} is also bounded.
Step2.limn→∞∥xn-Txn∥=0.
We can rewrite (3.20) as
(3.23)xn+1=δxn+(1-δ)yn,
where
(3.24)yn=β1-δ(λxn+(1-λ)Txn)+1-β-δ1-δPC[αnu+(1-αn)xn],n≥0.
It follows that
(3.25)‖yn+1-yn‖≤β1-δ‖(λxn+1+(1-λ)Txn+1)-(λxn+(1-λ)Txn)‖+1-β-δ1-δ‖PC[αn+1u+(1-αn+1)xn+1]-PC[αnu+(1-αn)xn]‖≤β1-δ‖xn+1-xn‖+1-β-δ1-δ|αn+1-αn|(‖u‖+‖xn‖)+1-β-δ(1-δ)(1-αn+1)‖xn+1-xn‖≤‖xn+1-xn‖+1-β-δ1-δ|αn+1-αn|(‖u‖+‖xn‖).
Thus,
(3.26)limsup⁡n→∞(‖yn+1-yn‖-‖xn+1-xn‖)≤limsup⁡n→∞1-β-δ1-δ|αn+1-αn|(‖u‖+‖xn‖)=0.
This together with Lemma 2.4 implies that
(3.27)lim⁡n→∞‖yn-xn‖=0.
Note that
(3.28)‖xn-Txn‖≤‖xn-yn‖+‖yn-Txn‖=‖xn-yn‖+‖β1-δ(λxn+(1-λ)Txn)+1-β-δ1-δPC[αnu+(1-αn)xn]-Txn‖≤‖xn-yn‖+β1-δλ‖xn-Txn‖+1-β-δ1-δ‖PC[αnu+(1-αn)xn]-Txn‖≤‖xn-yn‖+β1-δλ‖xn-Txn‖+1-β-δ1-δαn‖u-Txn‖+1-β-δ1-δ(1-αn)‖xn-Txn‖.
It follows that
(3.29)‖xn-Txn‖≤1-δβ(1-λ)‖xn-yn‖+1-β-δβ(1-λ)αn‖u-Txn‖.
Thus,
(3.30)lim⁡n→∞‖xn-Txn‖=0.
Step3.  limsupn→∞〈u-x~,xn-x~〉≤0, where x~=PFix(T)(u).
To see this, we can take a subsequence{xnk} of {xn} satisfying the properties
(3.31)limsup⁡n→∞〈u-x~,xn-x~〉=lim⁡k→∞〈u-x~,xnk-x~〉,(3.32)xnk⇀x*ask⟶∞.
By the demiclosed principle (see Lemma 2.3) and (3.30), we have that x*∈Fix(T). So,
(3.33)limsup⁡n→∞〈u-x~,xn-x~〉=〈u-x~,x*-x~〉≤0.Step4. xn→x~.
From (3.20), we get
(3.34)‖xn+1-x~‖2≤β‖λxn+(1-λ)Txn-x~‖2+δ‖xn-x~‖2+(1-β-δ)‖PC[αnu+(1-αn)xn]-x~‖2≤(β+δ)‖xn-x~‖2+(1-β-δ)‖αn(u-x~)+(1-αn)(xn-x~)‖2=(β+δ)‖xn-x~‖2+(1-β-δ)[(1-2αn)‖xn-x~‖2+2αn〈u-x~,xn-x~〉(1-2αn)‖xn-x~‖2+(1-β-δ)+αn2(‖u-x~‖2+‖xn-x~‖2-2〈u-x~,xn-x~〉)]=[1-2(1-β-δ)αn]‖xn-x~‖2+2(1-β-δ)αn(〈u-x~,xn-x~〉+αn2(‖u-x~‖2+‖xn-x~‖2-2〈u-x~,xn-x~〉))=(1-δn)‖xn-x~‖2+δnθn,
where δn=2(1-β-δ)αn and
(3.35)θn=〈u-x~,xn-x~〉+αn2(‖u-x~‖2+‖xn-x~‖2-2〈u-x~,xn-x~〉).
It is easy to see that limn→∞δn=0 and limsupn→∞θn≤0. We can therefore apply Lemma 2.5 to (3.34) and conclude that xn+1→x~ as n→∞. This completes the proof.Corollary 3.6.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T:C→C a nonexpansive mapping with Fix(T)≠∅. Let β and δ be two constants in (0,1) satisfying β+δ<1. For u∈H and any x0∈C, let {xn} be the sequence defined by the following explicit manner:
(3.36)xn+1=(βλ+δ)xn+β(1-λ)Txn+(1-β-δ)PC[αnu+(1-αn)xn],n≥0,
where αn∈(0,1) satisfies the following conditions:  (C1)
limn→∞αn=0,(C2)
∑n=0∞‍αn=∞. Then the sequence {xn} converges strongly to PFix(T)(u).Corollary 3.7.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T:C→C a λ-strictly pseudocontractive mapping with Fix(T)≠∅. Let β and δ be two constants in (0,1) satisfying β+δ<1. For any x0∈C, let {xn} be the sequence defined by the following explicit manner:
(3.37)xn+1=(βλ+δ)xn+β(1-λ)Txn+(1-β-δ)PC[(1-αn)xn],n≥0,
where αn∈(0,1) satisfies the following conditions:  (C1)
limn→∞αn=0,(C2)
∑n=0∞‍αn=∞. Then the sequence {xn} converges strongly to PFix(T)(0) which is the minimum norm fixed point of T.Corollary 3.8.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T:C→C a nonexpansive mapping with Fix(T)≠∅. Let β and δ be two constants in (0,1) satisfying β+δ<1. For any x0∈C, let {xn} be the sequence defined by the following explicit manner:
(3.38)xn+1=(βλ+δ)xn+β(1-λ)Txn+(1-β-δ)PC[(1-αn)xn],n≥0,
where αn∈(0,1) satisfies the following conditions:  (C1)
limn→∞αn=0;(C2)
∑n=0∞‍αn=∞. Then the sequence {xn} converges strongly to PFix(T)(0) which is the minimum norm fixed point of  T.


## 4.  Conclusion

Finding fixed points of nonlinear mappings (especially, nonexpansive mappings) has received vast investigations due to its extensive applications in a variety of applied areas of inverse problem, partial differential equations, image recovery and signal processing. It is wellknown that strictly pseudocontractive mappings have more powerful applications than nonexpansive mappings in solving inverse problems. In this paper, we devote to construct the methods for computing the fixed points of strictly pseudocontractive mappings. Two iterative methods have been presented. Especially, we can use these two methods to find the minimum norm fixed point of strictly pseudocontractive mappings. The ideas contained in the present paper can help us to solve the minimum norm problems in the applied science.



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