# Strong Convergence Theorems for  a Common Fixed Point of a Finite Family of  Pseudocontractive Mappings

Authors: O. A. Daman; H. Zegeye
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

It is our purpose, in this paper, to prove strong convergence of Halpern-Ishikawa iteration method to a common fixed point of finite family of Lipschitz pseudocontractive mappings. There is no compactness assumption imposed either onC or on T. The results obtained in this paper improve most of the results that have been proved for this class of nonlinear mappings.


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## Body

## 1. Introduction

LetC be a nonempty subset of a real Hilbert space H. The mapping T:C→H is called Lipschitz if there exists L≥0 such that
(1.1)∥Tx-Ty∥≤L∥x-y∥,∀x,y∈C.IfL=1, then T is called  nonexpansive, and if L<1, then T is called a contraction. It follows from (1.1) that every contraction mapping is nonexpansive and every nonexpansive mapping is Lipschitz.A mappingT:C→H is called α-strictly pseudocontractive [1] if for all x,y∈C there exists α∈[0,1) such that
(1.2)〈x-y,Tx-Ty〉≤∥x-y∥2-α∥(I-T)x-(I-T)y∥2.A mappingT is called pseudocontractive if
(1.3)〈x-y,Tx-Ty〉≤∥x-y∥2,∀x,y∈C.We note that (1.2) and (1.3) can be equivalently written as
(1.4)∥Tx-Ty∥2≤∥x-y∥2+α∥(I-T)x-(I-T)y∥2,∀x,y∈C.(1.5)∥Tx-Ty∥2≤∥x-y∥2+∥(I-T)x-(I-T)y∥2,∀x,y∈C,respectively.We observe from (1.4) and (1.5) that every nonexpansive mapping is α-strict pseudocontractive mapping and every α-strict pseudocontractive mapping is pseudocontractive mapping, and hence class of pseudocontractive mappings is a more general class of mappings. Furthermore, pseudocontractive mappings are related with the important class of nonlinear monotone mappings, where a mapping A with domain D(A) and range R(A) in H is called monotone if the inequality
(1.6)〈x-y,Ax-Ay〉≥0,
holds for every x,y∈D(A). We note that T is pseudocontractive if and only if A:=I-T is monotone, and hence a fixed point of T, F(T):={x∈D(T):Tx=x} is a zero of A, N(A):={x∈D(A):Ax=0}. It is now well known (see, e.g., [2]) that if A is monotone, then the solutions of the equation Ax=0 correspond to the equilibrium points of some evolution systems. Consequently, many researchers have made efforts to obtain iterative methods for approximating fixed points of T, when T is pseudocontractive (see, e.g., [3–10] and the references contained therein).LetC be a closed subset of a Hilbert space H, and let T:C→C be a contraction. Then the Picard iteration method given by
(1.7)x0∈C,xn+1=Txn,n≥1,
converges to the unique fixed point of T. However, this Picard iteration method may not always converge to a fixed point of T, when T is nonexpansive mapping. We can take, for example, T to be the anticlockwise rotation of the unit disk in ℝ2 (with the Euclidean norm) about the origin of coordinate of an angle, say, θ.The scheme that has been used to approximate fixed points of nonexpansive mappings is theMann iteration method[5] given by
(1.8)x0∈C,xn+1=(1-αn)xn+αnTxn,n≥0,
where {αn} is a real sequence in the interval (0,1) satisfying certain conditions. But it is worth mentioning that the Mann iteration process does not always converge strongly to a fixed point of nonexpansive mapping T. One has to impose compactness assumption on C (e.g., C is compact) or on T (e.g., T is semicompact) to get strong convergence of Mann iteration method to a fixed point of nonexpansive self-map T (see, e.g., [11, 12]).We also note that efforts to approximate a fixed point of a Lipschitz pseudocontractive mapping defined even on a compact convex subset of a Hilbert space by Mann iteration method proved abortive. One can see an example of a Lipschitz pseudocontractive self-map of acompact convex subset of a Hilbert space with a unique fixed point for which no Mann sequence converges by Chidume and Mutangadura [13]. This leads now to our next concern.Can we construct an iterative sequence for approximating fixed point of the Lipschitz pseudocontractive mappings?In 1974, Ishikawa [14] introduced an iteration process which converges to a fixed point of Lipschitz pseudocontractive self-map T of C, when C is compact. In fact, he proved the following theorem.Theorem I.
IfC is a compact convex subset of a Hilbert space H,T:C↦C is a Lipschitz pseudocontractive mapping and x0 is any point of C, then the sequence {xn}n≥0 converges strongly to a fixed point of T, where {xn} is defined iteratively for each integer n≥0 by
(1.9)xn+1=(1-αn)xn+αnTyn,yn=(1-βn)xn+βnTxn,
here {αn},{βn} are sequences of positive numbers satisfying the conditions
(1.10)(i)0≤αn≤βn<1,(ii)limn→∞βn=0,(iii)∑n≥0‍αnβn=∞.We observe that Theorem I imposes compactness assumption onC, and it is still an open problem whether or not scheme (1.9), known as the Ishikawa iterative method, can be used to approximate fixed points of Lipschitz pseudocontractive mappings without compactness assumption onCor onT.In order to obtain a strong convergence theorem for pseudocontractive mappings without the compactness assumption, Zhou [15] established the hybrid Ishikawa algorithm for Lipschitz pseudocontractive mappings as follows:
(1.11)yn=(1-αn)xn+αnTxn,zn=(1-βn)xn+βnTyn,Cn={z∈C:∥zn-z∥2≤∥xn-z∥2-αnβn(1-2αn-L2αn2)∥xn-Txn∥2},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0,n≥1.He proved that the sequence{xn} defined by (1.11) converges strongly to PF(T)x0, where PC is the metric projection from H into C.Recently, several authors (see, e.g., [16–18]) also used the hybrid Mann and hybrid Ishikawa algorithm methods to obtain strong convergence to a fixed point of Lipschitz pseudocontractive mappings. But it is worth mentioning that the hybrid schemes are not easy to compute. They involve computation of Cn and Qn for each n≥1.Another iteration scheme was introduced and studied by Chidume and Zegeye [19] with which they approximated fixed point of Lipschitz pseudocontractive mapping in a more general real Banach space.LetK be a convex nonempty subset of real Banach space E, and let T:K→K be a mapping. From arbitrary x1∈K, define {xn}n≥1 by
(1.12)xn+1=(1-λn)xn+λnTxn-λnθn(xn-x1),n∈ℕ,
where {λn}n≥1 and {θn}n≥1 are real sequences in (0,1) satisfying the following conditions: (i)limn→∞θn=0; (ii)λn=o(θn); (iii)∑n=1∞‍λnθn=∞; (iv)limn→∞((θn-1/θn-1)/λnθn)=0, λn(1+θn)<1. Examples of real sequences which satisfy these conditions are λn=1/(n+1)a and θn=1/(n+1)b,  where 0<b<a and a+b<1. They proved the following theorem.Theorem CZ.
LetC be a nonempty closed convex subset of a reflexive real Banach space E with a uniformly Gâteaux differentiable norm. Let T:C→C be a Lipschitz pseudocontractive mapping with Lipschitz constant L>0 and F(T)≠∅. Suppose every closed convex and bounded subset of K has the fixed point property for nonexpansive self-mappings. Let a sequence {xn}n≥1 be generated iteratively by (1.12). Then {xn}n≥1 converges strongly to a fixed point of T.Theorem CZ solves the open problem of approximating fixed point of Lipschitz pseudocontractive mappings that has been in the air for many years. However, it is still an open problem whether or not this scheme can be used to approximate a common fixed point of a family of Lipschitz pseudocontractive mappings. Moreover, we observe that the conditions on the real sequences{θn} and {λn} excluded the natural choice, θn=1/(n+1) and λn=1/(n+1).Our concern now is the following:can we construct an iterative sequence for a common fixed point of a family of Lipschitz pseudocontractive mappings?For a sequence{αn} of real numbers in [0,1] and an arbitrary u∈C, let the sequence {xn} in C be iteratively defined by x0∈C:(1.13)xn+1=αnu+(1-αn)Txn,n≥1.
The recursion formula (1.13) known as Halpern scheme was first introduced in 1967 by Halpern [20] in the framework of Hilbert spaces. He proved that {xn} convergs strongly to a fixed point of nonexpansive self-mapping T of C.Recently, considerable research efforts have been devoted to developing iterative methods for approximating a common fixed point of a family of several nonlinear mappings (see, e.g., [4, 21, 22]). In 1996, Bauschke [3] introduced the following Halpern-type iterative process for approximating a common fixed point for a finite family of N nonexpansive self-mappings. In fact, he proved the following theorem.Theorem B.
Let C be a nonempty closed convex subset of a Hilbert spaceH, and let  T1,T2,…,TN be a finite family of nonexpansive mappings of C into itself with F:=F(T1TN⋯T2)=⋯=F(TN-2⋯T1TN)≠∅. Let {αn} be a real sequence in [0,1] which satisfies certain mild conditions. Given points, x0∈C, let {xn} be generated by
(1.14)xn+1=αn+1u+(1-αn+1)Tn+1xn,n≥0,
where Tn=Tn(modN). Then {xn} converges strongly to PFu, where PFu:H→F is the metric projection.But it is worth mentioning that it is still an open problem whether or not this scheme can be used to approximate a common fixed points of Lipschitz pseudocontractive mappings?In 2008, Zhou [22] studied weak convergence of an implicit scheme to a common fixed point of finite family of pseudocontractive mappings. More precisely, he proved the following theorem.Theorem Z.
LetE be a real uniformly convex Banach space with a Frêchet differentiable norm. Let C be a closed convex subset of E, and let {Ti}i=1r be a finite family of Lipschitzian pseudocontractive self-mappings of C such that F:=∩i=1rF(Ti)≠∅. Let {xn} be defined by
(1.15)xn=αnxn-1+(1-αn)Tnxn,n≥1,
where Tn=Tn(modr). If {αn} is chosen so that αn∈(0,1) with limsupn→∞αn<1, then {xn} converges weakly to a common fixed point of the family {Ti}i=1r.Here, we remark that the scheme in Theorem Z isimplicit, and the convergence is weak convergence.More recently, Zegeye et al. [23] proved the following strong convergence of Ishikawa iterative process for a common fixed point of finite family of Lipschitz pseudocontractive mappings.Theorem ZSA (see [23]).
LetC be a nonempty, closed and convex subset of a real Hilbert space H. Let Ti:C→C,i=1,2,…,N, be a finite family of Lipschitz pseudocontractive mappings with Lipschitzian constants Li, for i=1,2,…,N, respectively. Assume that the interior of F:=∩i=1nF(Ti) is nonempty. Let {xn} be a sequence generated from an arbitrary x0∈E by
(1.16)yn=(1-βn)xn+βnTnxn,xn+1=(1-αn)xn+αnTnyn,n≥1,
where Tn:=Tn(modN) and {αn},{βn}⊂(0,1) satisfying certain appropriate conditions. Then, {xn} converges strongly to a common fixed point of {T1,T2,…,TN}.From Theorem ZSA, we observe that the assumption that the interior ofF(T)  is nonempty is  severe restriction.Motivated by Halpern [20] and Zegeye et al. [23], it is our purpose, in this paper, to prove strong convergence of  Halpern-Ishikawa  algorithm (3.3) to a common fixed point of a finite family of Lipschitz pseudocontractive mappings. No compactness assumption is imposed either on one of the mappings or on C. The assumption that interior of F(T)   is nonempty is dispensed with. Moreover, computation of closed and convex set Cn for each n≥1 is not required. The results obtained in this paper improve and extend the results of Theorems I and ZSA, Zhou [15], Yao et al. [17], and Tang et al. [16].


## 2. Preliminaries

In what follows we will make use of the following lemmas.Lemma 2.1.
LetH be a real Hilbert space. Then for any given x,y∈E, the following inequality holds:
(2.1)∥x+y∥2≤∥x∥2+2〈y,x+y〉.Lemma 2.2 (see [24]).
LetC be a convex subset of a real Hilbert space H. Let x∈H. Then x0=PCx if and only if
(2.2)〈z-x0,x-x0〉≤0,∀z∈C.Lemma 2.3 (see [25]).
Let{an} be a sequence of nonnegative real numbers satisfying the following relation:
(2.3)an+1≤(1-βn)an+βnδn,n≥n0,
where {βn}⊂(0,1) and {δn}⊂R satisfying the following conditions: limn→∞βn=0,∑n=1∞‍βn=∞, and limsupn→∞δn≤0. Then, limn→∞an=0.Lemma 2.4 (see [18]).
LetH be a real Hilbert space, let  C  be a closed convex subset of H, and let  T:C→C be a continuous pseudocontractive mapping; then(i)
F(T) is closed convex subset of C;(ii)
(I-T) is demiclosed at zero; that is,  if {xn} is a sequence in C such that xn⇀x and Txn-xn→0, as n→∞, then x=T(x).Lemma 2.5 (see [26]).
Let{an} be sequences of real numbers such that there exists a subsequence {ni} of {n} such that ani<ani+1 for all i∈N. Then there exists a nondecreasing sequence {mk}⊂N such that mk→∞,and the following properties are satisfied by all (sufficiently large) numbers k∈N:
(2.4)amk≤amk+1,ak≤amk+1.
In fact, mk=max{j≤k:aj<aj+1}.Lemma 2.6 (see [27]).
LetH be a real Hilbert space. Then for all xi∈H and αi∈[0,1] for i=1,2,…,n such that α1+α2+⋯+αn=1 the following equality holds:
(2.5)∥α0x0+α1x1+⋯+αnxn∥2=∑i=0n‍αi∥xi∥2-∑0≤i,j≤n‍αiαj∥xi-xj∥2.


## 3. Main Result

We now prove the following lemma and theorems.Lemma 3.1.
LetC be a nonempty convex subset of a real Hilbert space H. Let Ti:C→C,i=1,2,…,N, be a finite family of Lipschitz pseudocontractive mappings with constants Li, respectively. Let S=θ1T1+θ2T2+⋯+θNTN, where θ1+θ2+⋯+θN=1. Then S is Lipschitz pseudocontractive mapping on C.Proof.
Letx,y∈C. Then
(3.1)〈Sx-Sy,x-y〉=θ1〈T1x-T1y,x-y〉+θ2〈T2x-T2y,x-y〉+⋯+θN〈TNx-TNy,x-y〉≤θ1∥x-y∥2+θ2∥x-y∥2+⋯+θN∥x-y∥2=∥x-y∥2.
HenceS is pseudocontractive. Moreover, since
(3.2)∥Sx-Sy∥=∥(θ1T1+θ2T2+⋯+θNTN)x-(θ1T1+θ2T2+⋯+θNTN)y∥≤θ1∥T1x-T1y∥+θ2∥T2x-T2y∥+⋯+θN∥TNx-TNy∥≤L∥x-y∥,
where L:=max{Li:i=1,2,…,N}, we get that S is L-Lipschitz. The proof is complete.Let{Ti:i=1,2,…,N} be a finite family of pseudocontractive mappings. The family is said to satisfy condition(H) if 〈Tix-x,Tjx-x〉≥0, for i,j∈{1,2,…,N}.Theorem 3.2.
LetC be a nonempty, closed and convex subset of a real Hilbert space H. Let Ti:C→C,i=1,2,…,N be a finite family of Lipschitz pseudocontractive mappings with Lipschitz constants Li, respectively, satisfying condition (H). Assume that F:=⋂i=1N‍F(Ti) is nonempty. Let a sequence {xn} be a sequence generated from an arbitrary x1=w∈C by
(3.3)yn=(1-βn)xn+βnSnxn,xn+1=αnw+(1-αn)(γnSnyn+(1-γn)xn),
where Sn:=θn,1T1+θn,2T2+⋯+θn,NTN,  for {θn,i}⊆[a,b]⊂(0,1) such that θn,1+θn,2+⋯+θn,N=1, for all n≥1 and {αn},{βn},{γn}⊂(0,1) satisfying the following conditions: (i) 0≤αn≤c<1,  for all n≥1 such that limn→∞αn=0 and ∑‍αn=∞; (ii) 0<α≤γn≤βn≤β<1/[(1+L2)+1], for all  n≥1, for L:=max{Li:i=1,2,…,N}. Then, {xn} converges strongly to a common fixed point of {T1,T2,…TN} nearest to x1=w.Proof.
Letp=PFw. Then from (3.3), Lemma 2.6, (1.5), and Lemma 3.1 we have the following:
(3.4)∥xn+1-p∥2=∥αnw+(1-αn)(γnSnyn+(1-γn)xn)-p∥2≤αn∥w-p∥2+(1-αn)∥γnSnyn+(1-γn)xn-p∥2=αn∥w-p∥2+(1-αn)[γn∥Snyn-p∥2+(1-γn)∥xn-p∥2-γn(1-γn)∥Snyn-xn∥2]=αn∥w-p∥2+(1-αn)γn∥Snyn-p∥2≤αn∥w-p∥2+(1-αn)γn+(∥yn-p∥2+∥yn-Snyn∥2)+(1-αn)(1-γn)∥xn-p∥2-γn(1-γn)(1-αn)∥Snyn-xn∥2=αn∥w-p∥2+(1-αn)γn∥yn-p∥2+(1-αn)γn∥yn-Snyn∥2+(1-αn)(1-γn)∥xn-p∥2-γn(1-γn)(1-αn)∥Snyn-xn∥2+(1-αn)(1-γn)∥xn-p∥2-γn(1-αn)(1-γn)∥Snyn-xn∥2.In addition, we have that(3.5)∥yn-Snyn∥2=∥(1-βn)(xn-Snyn)+βn(Snxn-Snyn)∥2=(1-βn)∥xn-Snyn∥2+βn∥Snxn-Snyn∥2-βn(1-βn)∥xn-Snxn∥2≤(1-βn)∥xn-Snyn∥2+βnL2∥xn-yn∥2-βn(1-βn)∥xn-Snxn∥2=(1-βn)∥xn-Snyn∥2+βn3L2∥xn-Snxn∥2-βn(1-βn)∥xn-Snxn∥2=(1-βn)∥xn-Snyn∥2+βn(L2βn2+βn-1)∥xn-Snxn∥2,(3.6)||yn-p||2=∥(1-βn)(xn-p)+βn(Snxn-p)∥2=(1-βn)∥xn-p∥2+βn∥Snxn-p∥2-βn(1-βn)∥xn-Snxn∥2≤(1-βn)∥xn-p∥2+βn[∥xn-p∥2+∥xn-Snxn∥2]-βn(1-βn)∥xn-Snxn∥2=∥xn-p∥2+βn2∥xn-Snxn∥2.
Substituting (3.5) and (3.6) into (3.4) we obtain that
(3.7)∥xn+1-p∥2≤αn∥w-p∥2+(1-αn)γn[∥xn-p∥2+βn2∥xn-Snxn∥2]+(1-αn)γn[(1-βn)∥xn-Snyn∥2+βn(L2βn2+βn-1)×∥xn-Snxn∥2]+(1-αn)(1-γn)∥xn-p∥2-γn(1-γn)(1-αn)∥Snyn-xn∥2+(1-αn)γnβn2∥xn-Snxn∥2+(1-αn)γnβn(L2βn2+βn-1)×∥xn-Snxn∥2+[(1-αn)(1-βn)γn-(1-αn)(1-γn)γn]×∥xn-Snyn∥2=αn∥w-p∥2+(1-αn)∥xn-p∥2-(1-αn)γnβn(1-2βn-L2βn2)×∥xn-Snxn∥2+(1-αn)γn(γn-βn)∥xn-Snyn∥2=αn∥w-p∥2+[(1-αn)γn+(1-αn)(1-γn)]∥xn-p∥2.
Since from (ii), we have that (γn-βn)≤0 and 1-2βn-L2βn2≥1-2β-L2β2>0 for all n≥1, (3.7) implies that
(3.8)∥xn+1-p∥2≤αn∥w-p∥2+(1-αn)∥xn-p∥2-(1-αn)γnβn(1-2β-L2β2)∥xn-Snxn∥2≤αn∥w-p∥2+(1-αn)∥xn-p∥2.
Thus, by induction,
(3.9)∥xn+1-p∥2≤max{∥x1-p∥2,∥w-p∥2},∀n≥1,
which implies that {xn} and hence {yn} are bounded.Furthermore, from (3.3), Lemma 2.1, and following the methods used in (3.7) we get that
(3.10)∥xn+1-p∥2=∥αn(w-p)+(1-αn)[γnSnyn+(1-γn)xn-p]∥2≤(1-αn)∥γnSnyn+(1-γn)xn-p∥2+2αn〈w-p,xn+1-p〉=(1-αn)γn∥Snyn-p∥2+(1-αn)(1-γn)∥xn-p∥2-γn(1-γn)(1-αn)∥Snyn-xn∥2+2αn〈w-p,xn+1-p〉≤(1-αn)γn[∥yn-p∥2+∥yn-Snyn∥2]+(1-αn)(1-γn)∥xn-p∥2-γn(1-γn)(1-αn)∥Snyn-xn∥2+2αn〈w-p,xn+1-p〉=(1-αn)γn∥yn-p∥2+(1-αn)γn∥yn-Snyn∥2+(1-αn)(1-γn)∥xn-p∥2-γn(1-γn)(1-αn)∥Snyn-xn∥2≤(1-αn)γn[∥xn-p∥2+βn2∥xn-Snxn∥2]+(1-αn)γn×[(1-βn)∥xn-Snyn∥2+βn(L2βn2+βn-1)∥xn-Snxn∥2]+(1-αn)(1-γn)∥xn-p∥2-γn(1-γn)(1-αn)∥Snyn-xn∥2+2αn〈w-p,xn+1-p〉≤(1-αn)∥xn-p∥2+2αn〈w-p,xn+1-p〉-(1-αn)γnβn(1-2βn-βn2L2)∥Snyn-xn∥2≤(1-αn)∥xn-p∥2+2αn〈w-p,xn+1-p〉-(1-c)α2(1-2β-β2L2)∥Snxn-xn∥2+2αn〈w-p,xn+1-p〉.On the other hand, using Lemma2.6 and condition (H), we get that
(3.11)||xn-Snxn||2=∥xn-(θn,1T1+θn,2T2+⋯+θn,NTN)xn∥2=∥θn,1(xn-T1xn)+θn,2(xn-T2xn)+⋯+θn,N(xn-TNxn)∥2=θn,1∥xn-T1xn∥2+θn,2∥xn-T2xn∥2+⋯+θn,N∥xn-TNxn∥2-∑1≤i,j≤N‍θn,iθn,j∥Tixn-Tjxn∥2-∑1≤i,j≤N,i≠j‍θn,iθn,j[∥Tixn-xn∥2+∥xn-Tjxn∥2]=θn,1[1-θn,2-θn,3-⋯-θn,N]∥xn-T1xn∥2+θn,2[1-θn,1-θn,3-θn,4-⋯-θn,N]∥xn-T2xn∥2…+θn,N[1-θn,1-θn,2-θn,4-⋯-θn,N]∥xn-TNxn∥2≥θn,1∥xn-T1xn∥2+θn,2∥xn-T2xn∥2+⋯+θn,N∥xn-TNxn∥2.
Thus, substituting (3.11) into (3.10) we obtain that(3.12)∥xn+1-p∥2≤(1-αn)∥xn-p∥2+2αn〈w-p,xn+1-p〉-(1-c)α2(1-2β-β2L2)×[θn,1(1-θn,2-θn,3-⋯-θn,N)∥xn-T1xn∥2+θn,2(1-θn,1-θn,3-θn,4-⋯-θn,N)∥xn-T2xn∥2…+θn,N(1-θn,1-θn,2-θn,4-⋯-θn,N-1)∥xn-TNxn∥2](3.13)≤(1-αn)∥xn-p∥2+2αn〈w-p,xn+1-p〉.
Now, we consider the following two cases.Case  1.Suppose that there exists n0∈N such that {||xn-p||} is nonincreasing. Then, we get that {||xn-p||)} is convergent. Thus, from (3.12) and the fact that αn→0, as n→∞, we have that
(3.14)xn-Tixn→0,asn→∞,
for each i=1,2,…,N. Let zn=γnSnyn+(1-γn)xn. Then from (3.3) we obtain that
(3.15)xn+1-zn=αn(w-zn)→0,asn→∞.Furthermore, from (3.3) and (3.14) we get that
(3.16)∥yn-xn∥=∥βn(Snxn-xn)∥≤∥Snxn-xn∥≤θn,1∥T1xn-xn∥+θn,2∥T2xn-xn∥+⋯+θn,N∥TNxn-xn∥→0,
as n→∞, and hence (3.16) and the fact that Sn is L-Lipschitz imply that
(3.17)∥zn-xn∥=∥γn(Snyn-xn)∥=∥γn(Snyn-Snxn)+γn(Snxn-xn)∥≤γnL∥yn-xn∥+γn∥Snxn-xn∥→0,asn→∞.
Now, (3.15) and (3.17) imply that
(3.18)xn+1-xn→0,asn→∞.Moreover, since{xn} is bounded and E is reflexive, we choose a subsequence {xni+1} of {xn} such that xni+1⇀z and limsupn→∞〈w-p,xn+1-p〉=limi→∞〈w-p,xni+1-p〉. This implies from (3.18) that xni⇀z. Then, from (3.14) and Lemma 2.4 we have that z∈F(Ti), for each i=1,2,…,N. Hence, z∈∩i=1NF(Ti). Therefore, by Lemma 2.2, we immediately obtain that
(3.19)limsupn→∞〈w-p,xn+1-p〉=limi→∞〈w-p,xni+1-p〉=〈w-p,z-p〉≤0.
Then, since from (3.13) we have that
(3.20)∥xn+1-p∥2≤(1-αn)∥xn-p∥2+2αn〈w-p,xn+1-p〉.It follows from (3.20), (3.19), and Lemma 2.3 that ||xn-p||→0, as n→∞. Consequently, xn→p.Case  2.Suppose that there exists a subsequence {ni} of {n} such that
(3.21)∥xni-p∥<∥xni+1-p∥,
for all i∈N. Then, by Lemma 2.5, there exists a nondecreasing sequence {mk}⊂N such that mk→∞, ||xmk-p||≤||xmk+1-p|| and ||xk-p||≤||xmk+1-p|| for all k∈N. Now, from (3.12) and the fact that αn→0, we get that xmk-Tixmk→0, as k→∞, for each i=1,2,…,N. Thus, as in Case 1, we obtain that xmk+1-xmk→0 and that(3.22)limsupk→∞〈w-p,xmk+1-p〉≤0.Now, from (3.13) we have that
(3.23)∥xmk+1-p∥2≤(1-αmk)∥xmk-p∥2+2αmk〈w-p,xmk+1-p〉,
and hence, since ∥xmk-p∥2≤∥xmk+1-p∥2, (3.23) implies that
(3.24)αmk∥xmk-p∥2≤∥xmk-p∥2-∥xmk+1-p∥2+2αmk〈w-p,xmk+1-p〉≤2αmk〈w-p,xmk+1-p〉.
But noting that αmk>0, we obtain that
(3.25)∥xmk-p∥2≤2〈w-p,xmk+1-p〉.Then, from (3.22) we get that ||xmk-p||→0, as k→∞. This together with (3.23) gives that ||xmk+1-p||→0, as k→∞. But ||xk-p||≤||xmk+1-p||, for all k∈N; thus we obtain that xk→p. Therefore, from the previous two cases, we can conclude that {xn} converges strongly to an element of F,  and the proof is complete.If, in Theorem3.2, we consider single Lipschitz pseudocontractive mapping, then the assumption of condition (H) is not required. In fact, we have the following corollary.Corollary 3.3.
LetC be a nonempty, closed and convex subset of a real Hilbert space H. Let T:C→C be a Lipschitz pseudocontractive mapping with Lipschitz constants L. Assume that F(T) is nonempty. Let a sequence {xn} be a sequence generated from an arbitrary x1=w∈C by
(3.26)yn=(1-βn)xn+βnTxn,xn+1=αnw+(1-αn)(γnTyn+(1-γn)xn),
where {αn},{βn}⊂(0,1) satisfying the following conditions: (i)0<αn≤c<1, for all n≥1 such that limn→∞αn=0 and ∑‍αn=∞; (ii)  0<α≤γn≤βn≤β<1/[(1+L2)+1], for all  n≥1. Then, {xn} converges strongly to a fixed point of T nearest to x1=w.Proof.
PuttingSn:=T in (3.3) the scheme reduces to scheme (3.26), and following the method of proof of Theorem 3.2 we get that (see, (3.10))
(3.27)∥xn+1-p∥2≤(1-αn)∥xn-p∥2+2αn〈w-p,xn+1-p〉-(1-αn)γnβn(1-2β-β2L2)∥Txn-xn∥2≤(1-αn)∥xn-p∥2+2αn〈w-p,xn+1-p〉-(1-c)γnβn(1-2β-β2L2)∥Txn-xn∥2≤(1-αn)∥xn-p∥2+2αn〈w-p,xn+1-p〉.
Now, considering cases as in the proof of Theorem3.2 we obtain the required result.
We now state and prove a convergence theorem for a common zero of finite family of monotone mappings.Corollary 3.4.
LetH be a real Hilbert space. Let Ai:H→H,i=1,2,…,N be a finite family of Lipschitz monotone mappings with Lipschitz constants Li, respectively, satisfying 〈Aix,Ajx〉≥0, for all i,j∈{1,2,…,N}.
Assume thatF:=⋂i=1N‍N(Ai) is nonempty. Let a sequence {xn} be generated from an arbitrary x1∈H by
(3.28)yn=xn-βnAnxn,xn+1=αnw+(1-αn)(xn-γnAnyn),
where An:=θn,1A1+θn,2A2+⋯+θn,NAN, for {θn,i}⊆[a,b]⊂(0,1) such that θn,1+θn,2+⋯+θn,N=1, for all n≥1 and {αn},{βn},{γn}⊂(0,1) satisfying the following conditions: (i)  0<αn≤c<1, for all n≥0 such that limn→∞αn=0 and ∑‍αn=∞; (ii) 0<α≤γn≤βn≤β<1/[(1+L2)+1], for all n≥1, for L:=max{(1+Li):i=1,2,…,N}. Then, {xn} converges strongly to a common zero point of {A1,A2,…AN} nearest to x1=w.Proof .
LetTix:=(I-Ai)x, for i=1,2,…,N. Then we get that every Ti for all i∈{1,2,…,N} is Lipschitz pseudocontractive mapping with Lipschitz constants L′i:=(1+Li) and ∩i=1NF(Ti)=∩i=1N(Ai)≠∅. Moreover, when An is replaced with (I-Tn), for each i∈{1,2,…,N}, we get that scheme (3.28) reduces to scheme (3.3), and hence the conclusion follows from Theorem 3.2.
If, in Corollary3.4 we consider a single Lipschitz monotone mapping, then we obtain the following corollary.Corollary 3.5.
LetH be a real Hilbert space. Let A:H→H be Lipschitz monotone mappings with Lipschitz constant L. Assume that N(A) is nonempty. Let a sequence {xn} be generated from an arbitrary x1∈H by
(3.29)yn=xn-βnAxn,xn+1=αnw+(1-αn)(xn-γnAyn),
where {αn},{βn},{γn}⊂(0,1) satisfying the following conditions: (i) 0<αn≤c<1, for all n≥1 such that limn→∞αn=0 and ∑‍αn=∞; (ii) 0<α≤γn≤βn≤β<1/[(1+L2)+1], foralln≥1. Then, {xn} converges strongly to a zero point of A nearest to x1=w.We now give examples of Lipschitz pseudocontractive mappings satisfying condition(H). Let X:=ℝ and C:=[-24,3]⊂R. Let T1,T2:=C→C be defined by
(3.30)T1x:={x,x∈[-24,0),x-x3,x∈[0,3],T2x:={x,x∈[-24,2),3x-x2,x∈[2,3].Then we observe thatF(T1)=[-24,0],and F(T2)=[-24,2], and hence common fixed point of T1 and T2 is [-24,0] which is nonempty. Now, we show that T1 and T2 are pseudocontractive mappings. But, since
(3.31)A1x:=(I-T1)x={0,x∈[-24,0),x3,x∈[0,3],A2x:=(I-T2)x={0,x∈[-24,2),-2x+x2,x∈[2,3]
are monotone, we have that T1 and T2 are pseudocontractive mappings.Now, we show thatT1 and T2 are Lipschitzian mappings. First, we show that T1 is Lipschitzian with constant L=28. Let C1=[-24,0), C2=[0,3]. If x,y∈C1, then we have that
(3.32)|T1x-T1y|=|x-y|≤28|x-y|.
If x,y∈C2,then we have that
(3.33)|T1x-T1y|=|x-x3-(y-y3)|≤|x-y|+|x3-y3|=|x-y|+|x-y||x2+xy+y2|=(1+|x2+xy+y2|)|x-y|≤28|x-y|.Ifx∈C1 and y∈C2, then we get that
(3.34)|T1x-T1y|=|x-(y-y3)|≤|x-y|+|y3|=|x-y|+y2|y|≤|x-y|+y2|y-x|=(y2+1)|x-y|≤10|x-y|≤28|x-y|.
Therefore, from (3.32), (3.33), and (3.34), we obtain that T1 is Lipschitz.Next, we show thatT2 is Lipschitz with constant L=9.LetD1=[-24,2), D2=[2,3]. If x,y∈D1, then we have that
(3.35)|T2x-T2y|=|x-y|≤9|x-y|.
If x,y∈D2,then we get that
(3.36)|T2x-T2y|=|3x-x2-(3y-y2)|≤3|x-y|+|x2-y2|=3|x-y|+|x-y||x+y|≤(3+|x+y|)|x-y|≤9|x-y|.
If x∈D1 and y∈D2 then we have that
(3.37)|T2x-T2y|=|x-(3y-y2)|≤|x-y|+|y2-2y|
and for x∈[0,2) (3.37) implies that
(3.38)|T2x-T2y|≤|x-y|+|y2-2y-(x2-2x)|≤|x-y|+2|x-y|+|x-y||x+y|≤(3+|x+y|)|x-y|≤9|x-y|.
For x∈[-24,0) inequality (3.37) gives that
(3.39)|T2x-T2y|≤|x-y|+|y-2||y|,≤|x-y|+|y-2||y-x|=(1+|y-2|)|x-y|≤2|x-y|≤9|x-y|.Therefore, from (3.35), (3.36), and (3.39) we obtain that T2 is Lipschitz. Furthermore, we show that T1 and T2 satisfy condition (H). If x∈D1,then we have that 〈T1x-x,T2x-x〉=0, and if x∈D2 we get that 〈T1x-x,T2x-x〉=〈x-x3-x,(3x-x2)-x〉=〈-x3,2x-x2〉=-x3(2x-x2)≥0. Therefore, T1 and T1 satisfy property (H).Remark 3.6.
Theorem3.2 provides convergence sequence to a common fixed point of finite family of Lipschitzian pseudocontractive mappings whereas Corollary 3.4 provides convergence sequence to a common zero of finite family of monotone mappings in Hilbert spaces. No compactness assumption is imposed either on T or C. This provides affirmative answer to the question raised.Remark 3.7.
Theorem3.2 improves Theorem I, Theorem 3.1 of Zhou [15], Theorem 3.1 of Yao  et al. [17], and Theorem 3.1 of Tang  et al. [16] in the sense that either our convergence does not require compactness of T or computation of Cn+1 from Cn for each n≥1.Remark 3.8.
Theorem3.2 improves Theorems I and ZSA in the sense that our convergence is for a fixed point of a finite family of Lipschitz pseudocontractive mappings. The condition that interior of F(T) is nonempty is dispensed with.



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*Source: 405315-2012-09-20.xml*