# Minimum-Norm Fixed Point of Pseudocontractive Mappings

Authors: Habtu Zegeye; Naseer Shahzad; Mohammad Ali Alghamdi
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

LetK be a closed convex subset of a real Hilbert space H and let T:K→K be a continuous pseudocontractive mapping. Then for β∈(0,1) and each t∈(0,1), there exists a sequence {yt}⊂K satisfying yt=βPK[(1−t)yt]+(1−β)T(yt) which converges strongly, as t→0+, to the minimum-norm fixed point of T. Moreover, we provide an explicit iteration process which converges strongly to a minimum-norm fixed point of T provided that T is Lipschitz. Applications are also included. Our theorems improve several results in this direction.


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## Body

## 1. Introduction

LetK be a nonempty subset of a real Hilbert space H.  A mapping T:K→H is called Lipschitz if there exists L≥0 such that
(1.1)∥Tx-Ty∥≤L∥x-y∥,∀x,y∈K.
If L∈[0,1), then T is called a contraction; if L=1 then T is called a nonexpansive. It is easy to see from (1.1) that every contraction mapping is nonexpansive, and every nonexpansive mapping is Lipschitz.A mappingT is called strongly pseudocontractive if there exists α∈(0,1) such that inequality
(1.2)〈Tx-Ty,x-y〉≤α∥x-y∥2,
holds for all x,y∈K. T is called pseudocontractive if the inequality
(1.3)〈Tx-Ty,x-y〉≤∥x-y∥2,
holds for all x,y∈K. Note that inequality (1.3) can be equivalently written as
(1.4)∥Tx-Ty∥2≤∥x-y∥2+∥(I-T)x-(I-T)y∥2,∀x,y∈K.It is easy to see that nonexpansive and strongly pseudocontractive mappings are pseudocontractive mappings. However, the converse may not be true (see [1, 2] for details).Interest in pseudocontractive mappings stems mainly from their firm connection with the important class of nonlinearmonotone mappings, where a mapping A with domain D(A) and range R(A) in H is called monotone if the inequality
(1.5)〈Ax-Ay,x-y〉≥0,
holds for every x,y∈D(A). We note that A is monotone if and only if T:=I-A is pseudocontractive, and hence a zero of A, N(A):={x∈D(A):Ax=0} is a fixed point of T, F(T):={x∈D(T):Tx=x}.LetK be a nonempty closed convex subset of a real Hilbert space H and T:K→K a pseudcontractive mapping. Assume that the set of fixed points of T is nonempty. It is known from [3] that F(T) is closed and convex.Let the variational inequality(VI) be given as finding a point x* with the property that
(1.6)x*∈F(T)suchthat〈x*,x-x*〉≥0,∀x∈F(T).
Then, x* is the minimum-norm fixed point of T which exists uniquely and is exactly the (nearest point or metric) projection of the origin onto F(T), that is,  x*=PF(T)(0). We also observe that the minimum-norm fixed point of pseudocontractive T is the minimum-norm solution of a monotone operator equation Ax=0, where A=(I-T).It is quite often to seek the minimum-norm solution of a given nonlinear problem. In an abstract way, we may formulate such problems as finding a pointx* with the property
(1.7)x*∈K,∥x*∥=minx∈K∥x∥.
In other words, x* is the projection of the origin onto K, that is,
(1.8)x*=PK(0).A typical example is the split feasibility problem (SFP), formulated as finding a pointx* with the property that
(1.9)x*∈K,Ax*∈Q,
where K and Q are nonempty closed convex subsets of the infinite-dimension real Hilbert spaces H1 and H2, respectively, and A is bounded linear mapping from H1 to H2. Equation (1.9) models many applied problems arising from image reconstructions and learning theory (see, e.g., [4]). Some works on the finite dimensional setting with relevant projection methods for solving image recovery problems can be found in [5–7]. Defining the proximity function f by
(1.10)f(x):=12∥Ax-PQAx∥2,
we consider the convex optimization problem:
(1.11)minx∈Kf(x):=minx∈K12∥Ax-PQAx∥2.
It is clear that x* is a solution to the split feasibility problem (1.9) if and only if x*∈K and Ax*-PQAx*=0 which is the minimum-norm solution of the minimization problem (1.11).Motivated by the above split feasibility problem, we study the general case of finding the minimum-norm fixed point of a pseudocontractive mappingT:K→K, that is, we find minimum norm fixed point of T which satisfies
(1.12)x*∈F(T)such  that∥x*∥=min{∥x∥:x∈F(T)}.
Let T:K→K be a nonexpansive self-mapping on closed convex subset K of a Banach space E. For a given u∈K and for a given t∈(0,1) define a contraction Tt:K→K by
(1.13)Ttx=(1-t)u+tTx,x∈K.
By Banach contraction principle, it yields a fixed point zt∈K of Tt, that is, zt is the unique solution of the equation:
(1.14)zt=(1-t)u+tTzt.
Browder [8] proved that as t→1, zt converges strongly to a fixed point of T which is closer to u, that is, the nearest point projection of u onto F(T). In 1980, Reich [9] extended the result of Browder to a more general Banach spaces. Furthermore, Takahashi and Ueda [10] and Morales and Jung [11] improved results of Reich [9] to the class of continuous pseudocontractive mappings. For other results on pseudocontractive mappings, we refer to [12–15].We note that the above methods can be used to find the minimum-norm fixed pointx* of T if 0∈K. However, if 0∉K neither Browder's, Reich's, Takahashi and Ueda's, nor Morales and Jung's method works to find minimum-norm fixed point of T.Our concern is now the following:  is it possible to construct a scheme, implicit or explicit, which converges strongly to the minimum-norm fixed point ofT for any closed convex domain K of T?In this direction, Yang et al. [4] introduced an implicit and explicit iteration processes which converge strongly to the minimum-norm fixed point of nonexpansive self-mapping T, in real Hilbert spaces. In fact, they proved the following theorems.Theorem YLY1 (see [4]).
LetK be a nonempty closed convex subset of a real Hilbert space H and T:K→K a nonexpansive mapping with F(T)≠∅. For β∈(0,1) and each t∈(0,1), let yt be defined as the unique solution of fixed point equation:
(1.15)yt=βTyt+(1-β)PK[(1-t)yt],t∈(0,1).
Then the net {yt} converges strongly, as t→0, to the minimum-norm fixed point of T.Theorem YLY2 (see [4]).
LetK be a nonempty closed convex subset of a real Hilbert space H, and let T:K→K be a nonexpansive mapping with F(T)≠∅. For a given x0∈K, define a sequence {xn} iteratively by
(1.16)xn+1=βTxn+(1-β)PK[(1-αn)xn],n≥1,
where β∈(0,1) and αn∈(0,1), satisfying certain conditions. Then the sequence {xn} converges strongly to the minimum-norm fixed point of T.A natural question arises whether the above theorems can be extended to a more general class of pseudocontractive mappings or not.LetK be a closed convex subset a real Hilbert space H and let T:K→K be continuous pseudocontractive mapping.It is our purpose in this paper to prove that forβ∈(0,1) and each t∈(0,1), there exists a sequence {yt}⊂K satisfying yt=βPK[(1-t)yt]+(1-β)T(yt) which converges strongly, as t→0+, to the minimum-norm fixed point of T. Moreover, we provide an explicit iteration process which converges strongly to the minimum-norm fixed point of T provided that T is Lipschitz. Our theorems improve Theorem YLY1 and Theorem YLY2 of Yang et al. [4] and Theorems  3.1, and  3.2 of Cai et al. [16].


## 2. Preliminaries

In what follows, we shall make use of the following lemmas.Lemma 2.1 (see [11]).
LetH be a real Hilbert space. Then, for any given x,y∈H, the following inequality holds:
(2.1)∥x+y∥2≤∥x∥2+2〈y,x+y〉.Lemma 2.2 (see [17]).
LetK be a closed and convex subset of a real Hilbert space H. Let x∈H. Then x0=PKx if and only if
(2.2)〈z-x0,x-x0〉≤0,∀z∈K.Lemma 2.3 (see [18]).
Let{λn}, {αn}, and {γn} be sequences of nonnegative numbers satisfying the conditions: limn→∞αn=0, ∑n=1∞‍αn=∞, and γn/αn→0, as n→∞. Let the recursive inequality:
(2.3)λn+1≤λn-αnψ(λn+1)+γn,n=1,2,…,
be given where ψ:[0,∞)→[0,∞) is a strictly increasing function such that it is positive on (0,∞) and ψ(0)=0. Then λn→0, as n→∞.Lemma 2.4 (see [3]).
LetH be a real Hilbert space, K be a closed convex subset of H and T:K→K be a continuous pseudocontractive mapping, then(i)
F(T) is closed convex subset of K;(ii)
(I-T) is demiclosed at zero, that is, if {xn} is a sequence in K such that xn⇀x and Txn-xn→0, as n→∞, then x=T(x).Lemma 2.5 (see [19]).
LetH be a real Hilbert space. Then for all x,y∈H and α∈[0,1], the following equality holds:
(2.4)∥αx+(1-α)x∥2=α∥x∥2+(1-α)∥y∥2-α(1-α)∥x-y∥2.


## 3. Main Results

Theorem 3.1.
LetK be a nonempty closed and convex subset of a real Hilbert space H. Let T:K→K be a continuous pseudocontractive mapping with F(T)≠∅. Then for β∈(0,1) and each t∈(0,1), there exists a sequence {yt}⊂K satisfying the following condition:
(3.1)yt=βPK[(1-t)yt]+(1-β)T(yt)
and the net {yt} converges strongly, as t→0+, to the minimum-norm fixed point of T.Proof.
Forβ∈(0,1) and each t∈(0,1) let Tt(y):=βPK[(1-t)y]+(1-β)T(y). Then using nonexpansiveness of PK and pseudocontractivity of T, for x,y∈K, we have that
(3.2)〈Ttx-Tty,x-y〉=β〈PK[(1-t)x]-PK[(1-t)y],x-y〉+(1-β)〈T(x)-T(y),x-y〉≤β(1-t)∥x-y∥2+(1-β)∥x-y∥2≤(1-tβ)∥x-y∥2.
This implies that Tt is strongly pseudocontractive on K. Thus, by Corollary 1 of [20] Tt has a unique fixed point, yt, in K. This means that the equation:
(3.3)yt=βPK[(1-t)yt]+(1-β)T(yt)
has a unique solution for each t∈(0,1). Furthermore, since F(T)≠∅, for y*∈F(T), we have that
(3.4)∥yt-y*∥2=〈βPK[(1-t)yt]+(1-β)Tyt-y*,yt-y*〉=β〈PK[(1-t)yt]-PKy*,yt-y*〉+(1-β)〈Tyt-Ty*,yt-y*〉≤β∥(1-t)yt-y*∥·∥yt-y*∥+(1-β)∥yt-y*∥2≤β[(1-t)∥yt-y*∥+t∥y*∥]∥yt-y*∥+(1-β)∥yt-y*∥2,
which implies that
(3.5)∥yt-y*∥≤β(1-t)∥yt-y*∥+βt∥y*∥+(1-β)∥yt-y*∥,
and hence ∥yt-y*∥≤∥y*∥. Therefore, {yt} and hence {Tyt} is bounded.
Furthermore, from (3.3) and using nonexpansiveness of PK we get that
(3.6)∥yt-Tyt∥=∥βPK[(1-t)yt]+(1-β)T(yt)-Tyt∥=β∥PK[(1-t)yt]-PKTyt∥≤β∥(1-t)yt-Tyt∥≤β∥yt-Tyt∥+βt∥yt∥,
which implies that
(3.7)∥yt-Tyt∥≤β(1-β)t∥yt∥→0,ast→0.
Furthermore, from (3.3), convexity of ∥·∥2, (1.4), and (3.7), we get that
(3.8)∥yt-y*∥2=∥(1-β)(Tyt-y*)+β(PK[(1-t)yt]-PKy*)∥2=(1-β)∥Tyt-y*∥2+β∥PK[(1-t)yt]-PKy*∥2≤(1-β)[∥yt-y*∥2+∥Tyt-yt∥2]+β∥(1-t)yt-y*∥2≤(1-β)∥yt-y*∥2+(1-β)∥Tyt-yt∥2+β∥(1-t)yt-y*∥2≤(1-β)∥yt-y*∥2+β2(1-β)t2∥yt∥2+β[∥yt-y*∥2-2t∥yt-y*∥2-2t〈y*,yt-y*〉+t2∥yt∥2].
This implies that
(3.9)∥yt-y*∥2≤〈y*,y*-yt〉+tM,forsomeM>0.
Now, for tn→0, as n→∞, let {yn:=ytn} be a subsequence of {yt} such that yn⇀y'. Then, we have from (3.7) and Lemma 2.4 that y'∈F(T). Furthermore, replacing y* by y' in (3.9) and the fact that yn⇀y' imply that
(3.10)∥yn-y'∥2≤〈y′,y′-yn〉+tnM→0,asn→∞,
which implies that
(3.11)yn→y',asn→∞.
Thus, from (3.9) and (3.11), we have that
(3.12)∥y'-y*∥2≤〈y*,y*-y′〉,asn→∞,
which is equivalent to the inequality:
(3.13)〈y′,y*-y′〉≥0and  hencey′=PF0.
If there is another subsequence {ym} of {yt} such that ym⇀y′′, similar argument gives that y′′=PF0, which implies, by uniqueness of PF0, that y′′=y′. Therefore, the net yt→y′=PF0 which is the minimum-norm of fixed point of T. The proof is complete.We now state and prove a convergence theorem for the minimum-norm zero of a monotone mappingA.Theorem 3.2.
LetH be a real Hilbert space. Let A:H→H be a continuous monotone mapping with N(A)≠∅. Then for β∈(0,1) and each t∈(0,1), there exists a sequence {yt}⊂H satisfying the following condition:
(3.14)yt=β(1-t)yt+(1-β)(I-A)yt,
and the net {yt} converges strongly, as t→0+, to the minimum-norm zero of A.Proof.
LetTx:=(I-A)x. Then, we get that T is continuous pseudocontractive mapping with F(T)=N(A)≠∅. Moreover, since PH is an identity mapping on H, when A is replaced with (I-T) scheme (3.14) reduces to scheme (3.1), and hence the conclusion follows from Theorem 3.1.If in Theorem3.1, we consider {tn},{βn}⊂(0,1) such that tn→0, βn→0 and yn:=ytn, the method of proof of Theorem 3.1 provides the following corollary.Corollary 3.3.
LetK be a nonempty closed and convex subset of a real Hilbert space H. Let T:K→K be continuous pseudocontractive mapping with F(T)≠∅. Then the sequence {yn}⊂K defined by
(3.15)yn=βnPK[(1-tn)yn]+(1-βn)T(yn),
where {tn},{βn}⊂(0,1) such that tn→0, βn→0, as n→∞, converges strongly, as n→∞, to the minimum-norm fixed point of T.The following proposition and lemma play an important role in proving the next theorem.Proposition 3.4.
LetK be a nonempty closed and convex subset of a real Hilbert space H. Let T:K→K be continuous pseudocontractive mapping. Then the sequence {yn} in (3.15) satisfies the following inequality:
(3.16)∥yn-yn-1∥≤|θn-1-θn|θntn[∥yn∥+∥PK[(1-tn)yn-1]∥]+θn-1θn|tn-tn-1|tn∥yn-1∥,
where θn:=βn/(1-βn) for {βn} decreasing sequence.Proof.
If we putθn:=βn/(1-βn),  (3.15) reduces to
(3.17)yn=Tyn+θn(PK[(1-tn)yn]-yn).
Thus, using pseudocontractivity of T and nonexpansiveness of PK we get that
(3.18)∥yn-yn-1∥2=∥Tyn+θn(PK[(1-tn)yn]-yn)-Tyn-1-θn-1(PK[(1-tn-1)yn-1]-yn-1)∥2=∥Tyn-Tyn-1+θn-1yn-1-θnyn+θn-1yn-θn-1yn+θnPK[(1-tn)yn]-θn-1PK[(1-tn-1)yn-1]∥2=〈Tyn-Tyn-1+θn-1(yn-1-yn)+(θn-1-θn)yn,yn-yn-1〉+〈θnPK[(1-tn)yn]-θnPK[(1-tn)yn-1],yn-yn-1〉+〈θnPK[(1-tn)yn-1]-θn-1PK[(1-tn)yn-1],yn-yn-1〉+〈θn-1PK[(1-tn)yn-1]-θn-1PK[(1-tn-1)yn-1],yn-yn-1〉≤∥yn-yn-1∥2-θn-1∥yn-yn-1∥2+(θn-1-θn)∥yn∥×∥yn-yn-1∥+θn(1-tn)∥yn-yn-1∥2+(θn-θn-1)∥PK[(1-tn)yn-1]∥·∥yn-1-yn∥+θn-1|tn-tn-1|·∥yn-1∥∥yn-yn-1∥,
which implies, using the fact that θn is decreasing, that
(3.19)∥yn-yn-1∥≤[1-θn-1+θn(1-tn)]∥yn-yn-1∥+|θn-1-θn|[∥yn∥+∥PK[(1-tn)yn-1]∥]+θn-1|tn-tn-1|·∥yn-1∥≤(1-tnθn)∥yn-yn-1∥+|θn-1-θn|[∥yn∥+∥PK[(1-tn)yn-1]∥]+θn-1|tn-tn-1|·∥yn-1∥,
and hence
(3.20)∥yn-yn-1∥≤|θn-1-θn|θntn[∥yn∥+∥PK[(1-tn)yn-1]∥]+θn-1θn|tn-tn-1|tn∥yn-1∥.
The proof is complete.For the rest of this paper, let{λn}, {θn} (decreasing) and {tn} be real sequences in (0,1] satisfying the following conditions: (i) limn→∞θn=0=limn→∞tn; (ii) λn(1+θn)≤1, ∑‍λnθntn=∞,  limn→∞λn/θntn=0; (iii) limn→∞[θn-1-θn]/λnθn2tn2=0 and limn→∞[tn-1-tn]/λnθntn2=0. Examples of real sequences which satisfy these conditions are λn=1/(n+1)1/2, θn=1/(n+1)1/3 and tn=1/(n+1)1/14.Lemma 3.5.
LetK be a nonempty closed convex subset of a real Hilbert space H. Let T:K→K be a Lipschitz pseudocontractive mapping with Lipschitz constant L≥0 and F(T)≠∅. Let a sequence {xn} be generated from arbitrary x1∈K by
(3.21)xn+1:=(1-λn)xn+λnTxn-λnθn(xn-PK[(1-tn)xn]),
for all positive integers n≥1. Then {xn} is bounded.Proof.
We follow the method of proof of Chidume and Zegeye [21]. Since λn/(θntn)→0, there exists N0>0 such that λn/(θntn)≤d:=1/(2(3+L)2), for all n≥N0. Let x*∈F(T) and r>0 be sufficiently large such that xN0∈Br(x*) and ∥x*∥≤r/(2(4+L)). Now, we show by induction that {xn} belongs to B:=Br(x*)¯ for all integers n≥N0. By construction, we have  xN0∈B. Assume that xn∈B for any n>N0. Then, we prove that xn+1∈B. Suppose xn+1 is not in B. Then ∥xn+1-x*∥>r, and thus from the recursion formula (1.2) and Lemma 2.1 we get that
(3.22)∥xn+1-x*∥2=∥xn-x*-λn((xn-Txn)+θn(xn-PK[(1-tn)xn]))∥2=∥xn-x*∥2-2λn〈(xn-Txn)+θn(xn-PK[(1-tn)xn]),j(xn+1-x*)〉=∥xn-x*∥2-2λnθn〈xn+1-x*,xn+1-x*〉+2λn〈θn(xn+1-xn)-(xn-Txn)+θn(PK[(1-tn)xn]-x*)+(xn+1-Txn+1)-(xn+1-Txn+1),j(xn+1-x*)〉.
Since T is pseudocontractive we have 〈xn+1-Txn+1,j(xn+1-x*)〉≥0. Thus, (3.22) gives
(3.23)∥xn+1-x*∥2≤∥xn-x*∥2-2λnθn∥xn+1-x*∥2+2λn(2+L)∥xn+1-xn∥·∥xn+1-x*∥+2λnθn〈PK[(1-tn)xn]-x*,j(xn+1-x*)〉=∥xn-x*∥2-2λnθn∥xn+1-x*∥2+2λn(2+L)∥xn+1-xn∥·∥xn+1-x*∥+2λnθn〈PK[(1-tn)xn]-PK[(1-tn)xn+1]+PK[(1-tn)xn+1]-PK[(1-tn)x*]+PK[(1-tn)x*]-x*,j(xn+1-x*)〉,
which implies that
(3.24)∥xn+1-x*∥2≤∥xn-x*∥2-2λnθntn∥xn+1-x*∥2+2λn(2+L+(1-tn))∥xn+1-xn∥·∥xn+1-x*∥+2λnθn∥PK[(1-tn)x*]-x*∥·∥xn+1-x*∥=∥xn-x*∥2-2λnθntn∥xn+1-x*∥2+2λn(3+L)[λn∥θn(PK[(1-tn)xn]-PK[(1-tn)x*]+PK[(1-tn)x*]-x*+x*-xn)+Txn-Tx*+x*-xn∥]×∥xn+1-x*∥+2λnθntn∥x*∥·∥xn+1-x*∥≤∥xn-x*∥2-2λnθntn∥xn+1-x*∥2+2λn2(3+L)2∥xn-x*∥·∥xn+1-x*∥+2λnθntn(4+L)∥x*∥∥xn+1-x*∥.
Since ∥xn+1-x*∥>∥xn-x*∥, from (3.24) we get that
(3.25)∥xn+1-x*∥≤λnθntn(3+L)2∥xn-x*∥+(4+L)∥x*∥,
and hence ∥xn+1-x*∥≤r, since xn∈B, ∥x*∥≤r/(2(4+L)) and λn/θntn≤1/2(3+L)2foralln≥N0. But this is a contradiction. Therefore, xn∈B for all positive integers n≥N0, and hence the sequence {xn} is bounded.For the next theorem, let{yn} denotes the sequence defined by yn:=ysn=snTysn+(1-sn)PK[(1-tn)yn],sn=1/(1+θn),foralln≥1, guaranteed by Corollary 3.3 (which reduces to θn(PK[(1-tn)yn]-yn)-(yn-Tyn)=0).Theorem 3.6.
LetK be a nonempty closed convex subset of a real Hilbert space H. Let T:K→K be a Lipschitz pseudocontractive mapping with Lipschitz constant L≥0 and F(T)≠∅. Let a sequence {xn} be generated from arbitrary x1∈K by
(3.26)xn+1:=(1-λn)xn+λnTxn-λnθn(xn-PK[(1-tn)xn]),
for all positive integers n≥1. Then {xn} converges strongly to the minimum-norm fixed point of T, as n→∞.Proof.
By Lemma3.5, we have that the sequence {xn} is bounded. Now, we show that it converges strongly to a minimum-norm fixed point of T. But from (3.26) and Lemma 2.1, we have that
(3.27)∥xn+1-yn∥2≤∥xn-yn∥2-2λnθn〈(xn+1-yn),j(xn+1-yn)〉+2λn〈θn(xn+1-yn)-(xn-Txn)-θn(xn-PK[(1-tn)xn]),j(xn+1-yn)〉=∥xn-yn∥2-2λnθn∥xn+1-yn∥2+2λn〈θn(xn+1-xn)+[θn(PK[(1-tn)yn]-yn)-(yn-Tyn)]-[(xn+1-Txn+1)-(yn-Tyn)]+θn(PK[(1-tn)xn]-PK[(1-tn)yn])+[(xn+1-Txn+1)-(xn-Txn)],j(xn+1-yn)〉.
Observe that by the property of yn and pseudocontractivity of T we have θn(PK[(1-tn)yn]-yn)-(yn-Tyn)=0 (see (3.17)) and 〈(xn+1-Txn+1)-(yn-Tyn),j(xn+1-yn)〉≥0 for all n≥1. Thus, we have from (3.27) that
(3.28)∥xn+1-yn∥2≤∥xn-yn∥2-2λnθn∥xn+1-yn∥2+2λn〈θn(xn+1-xn)+θn(PK[(1-tn)xn]-PK[(1-tn)xn+1]+PK[(1-tn)xn+1]-PK[(1-tn)yn])+(xn+1-Txn+1)-(xn-Txn),j(xn+1-yn)〉≤∥xn-yn∥2-2λnθntn∥xn+1-yn∥2+2λn(3+L)∥xn+1-xn∥·∥xn+1-yn∥.
But by Corollary 3.3, we have that {yn} is bounded. Therefore, there exists M1>0 such that max{(3+L)∥xn+1-yn∥·∥xn-Txn+θn(xn-PK[(1-tn)xn])∥}≤M1. Thus from (3.28), we get that
(3.29)∥xn+1-yn∥2≤∥xn-yn∥2-2λnθntn∥xn+1-yn∥2+2λn2M1.
But using triangle inequality and Proposition 3.4, we have that
(3.30)∥xn-yn∥2≤[∥xn-yn-1∥+∥yn-1-yn∥]2≤∥xn-yn-1∥2+∥yn-1-yn∥M2≤∥xn-yn-1∥2+|θn-1-θn|θntnM3+|tn-tn-1|tnM3,
for some M2,M3>0, and for all n≥N0. Now, substituting (3.30) in (3.29) we obtain that
(3.31)∥xn+1-yn∥2≤∥xn-yn-1∥2-2λnθntn∥xn+1-yn∥2+2λn2M4+θn-1-θnθntnM4+|tn-tn-1|tnM4,
for some constant M4>0. Now, by Lemma 2.3 and the conditions on {λn}, {θn}, and {tn} we get xn+1-yn→0. Consequently, ∥xn-yn∥→0 as n→∞.
Therefore, since by Corollary3.3 we have that yt→y*∈F(T), where y* is with the minimum-norm in F(T), we get that {xn} converges strongly to the minimum-norm of fixed point of T.Corollary 3.7.
LetH be a real Hilbert space. Let A:H→H be a Lipschitz monotone mapping with Lipschitz constant L≥0 and N(A)≠∅. Let a sequence {xn} be generated from arbitrary x1∈H by
(3.32)xn+1=xn-λnAxn+λnθntnxn,
for all positive integers n. Then {xn} converges strongly to the minimum-norm solution of the equation Ax=0.Proof.
LetT:=(I-A). Then T is a Lipschitz pseudocontractive mapping with Lipschitz constant L':=(L+1), and the minimum-norm solution of the equation Ax=0 is the minimum-norm fixed point of T. Moreover, if we replace T  by (I-A) in (3.26), then the equation reduces to (3.32). Thus, the conclusion follows from Theorem 3.6.


## 4.  Applications

For the rest of this paper, letH be a Hilbert space and A:H→H  a bounded linear operator. Consider the convexly constrained linear inverse problem, which has extensively been discussed in the literature (see, e.g., [22]), given by:
(4.1)x∈K,Ax=b,
where K is closed and convex subset of H and b∈H, which is a special case of the SFP problem (1.9). Set
(4.2)φ(x):=12∥Ax-b∥2.
The least-square solution of (4.1) is the least-norm minimizer of the minimization problem (4.2). Let Ω denote the solution set of (4.2). It is known that Ω is nonempty if and only if PA(K)¯(b)∈A(K). In this case, Ω has a unique element with minimum norm which is a least-square solution of (4.1), that is, there exists a unique point x*∈Ω such that
(4.3)∥x*∥=min{∥x∥:x∈Ω}.
We note that φ(x) is a quadratic function with gradient:
(4.4)∇φ(x)=A*(Ax-b),
where A* is adjoint of A. Let γ>0 and x*∈Ω. Thus, x* is the minimum-norm solution of the minimization problem (4.2) if and only if x* a solution of
(4.5)γ∇φ(x)=γA*(Ax-b)=0.
Now, we state applications of our theorems.Theorem 4.1.
Assume that the solution set of convexly constrained linear inverse problem (4.1) with K:=H, a real Hilbert space, is nonempty and that ∇φ is monotone. Then for β∈(0,1) and each t∈(0,1), there exists a sequence {yt}⊂H satisfying the following condition:
(4.6)yt=β(1-t)yt+(1-β)(yt-γA*(Ayt-b)),
where A* is adjoint of A, and the net {yt} converges strongly, as t→0+, to the minimum-norm solution of the split feasibility problem (4.1).Proof.
We note thatφ(x) is continuously differentiable function with gradient:
(4.7)∇φ(x)=A*(Ax-b),
where A* is adjoint of A, which is Lipschitz (see Lemma 8.1 of [5]) and monotone (by hypothesis). Thus, the conclusion follows from Theorem 3.2.Theorem 4.2.
Assume that the solution set of split feasibility problem (4.1) is nonempty and that ∇φ with K:=H, a real Hilbert space, is monotone. Let a sequence {xn} be generated from arbitrary x1∈E by
(4.8)xn+1=xn-λnγA*(Axn-b)+λnθntnxn,
for all positive integers n, where γ>0 and A* is adjoint of A. Then, {xn} converges strongly to the minimum-norm solution of the split feasibility problem (4.1).Remark 4.3.
Theorem3.1 improves Theorem YLY1 and Theorem  3.1 of Cai  et al. [16] to a more general class of pseudocontractive mappings. Moreover, Theorem 3.6 improves Theorem YLY1 and Theorem  3.2 of Cai et al. [16] in the sense that our scheme provides a minimum-norm fixed point of pseudocontractive mapping T.



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*Source: 926017-2012-09-26.xml*