# Path Convergence and Approximation of Common Zeroes of a Finite Family ofm-Accretive Mappings in Banach Spaces

Authors: Yekini Shehu; Jerry N. Ezeora
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

LetE be a real Banach space which is uniformly smooth and uniformly convex. Let K be a
nonempty, closed, and convex sunny nonexpansive retract of E, where Q is the sunny nonexpansive retraction. If E admits weakly sequentially continuous duality mapping j, path convergence is proved for a nonexpansive
mapping T:K→K. As an application, we prove strong convergence theorem for common zeroes of a finite
family of m-accretive mappings of K to E. As a consequence, an iterative scheme is constructed to converge to
a common fixed point (assuming existence) of a finite family of pseudocontractive mappings from K to E under certain mild conditions.


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## Body

## 1. Introduction

LetE be a real Banach space with dual E* and K a nonempty, closed and convex subset of E. A mapping T:K→K is said to be nonexpansive if for all x,y∈K, we have
(1.1)∥Tx-Ty∥≤∥x-y∥.
A point x∈K is called a fixed point of T if Tx=x. The fixed points set of T is the set F(T)∶={x∈K:Tx=x}.Construction of fixed points of nonexpansive mappings is an important subject in nonlinear mapping theory and its applications; in particular, in image recovery and signal processing (see, e.g., [1–3]). Many authors have worked extensively on the approximation of fixed points of nonexpansive mappings. For example, the reader can consult the recent monographs of Berinde [4] and Chidume [5].We denote byJ the normalized duality mapping from E to 2E* defined by
(1.2)Jx∶={f*∈E*:〈x,f*〉=∥x∥2=∥f*∥2},
where 〈·,·〉 denotes the generalized duality pairing between members of E and E*. It is well known that if E* is strictly convex then J is single valued (see, e.g., [5, 6]). In the sequel, we will denote the single-valued normalized duality mapping by j.A mappingA:D(A)⊆E→E is called accretive if, for all x,y∈D(A), there exists j(x-y)∈J(x-y) such that
(1.3)〈Ax-Ay,j(x-y)〉≥0.
By the results of Kato [7], (1.3) is equivalent to
(1.4)∥x-y∥≤∥x-y+s(Ax-Ay)∥,∀s>0.
If E is a Hilbert space, accretive mappings are also called monotone. A mapping A is called m-accretive if it is accretive and R(I+rA), range of (I+rA), is E for all r>0; and A is said to satisfy the range condition if cl(D(A))⊆R(I+rA),forallr>0, where cl(D(A)) denotes the closure of the domain of A. A is said to be maximal accretive if it is accretive and the inclusion G(A)⊂G(B), where G(A) is a graph of A, with B accretive, implies G(A)=G(B). It is known (see e.g., [8]) that every maximal accretive mapping is m-accretive and the converse holds if E is a Hilbert space. Interest in accretive mappings stems mainly from their firm connection with equations of evolution. It is known (see, e.g., [9]) that many physically significant problems can be modelled by initial-value problems of the following form:
(1.5)u′(t)+Au(t)=0,u(0)=u0,
where A is an accretive mapping in an appropriate Banach space. Typical examples where such evolution equations occur can be found in the heat, wave, or Schrödinger equations. One of the fundamental results in the theory of accretive mappings, due to Browder [10], states that if A is locally Lipschitzian and accretive, then A is m-accretive. This result was subsequently generalized by Martin [11] to the continuous accretive mappings. If in (1.5), u(t) is independent of t, then (1.5) reduces to
(1.6)Au=0,
whose solutions correspond to the equilibrium points of the system (1.5). Consequently, considerable research effects have been devoted, especially within the past 30 years or so, to iterative methods for approximating these equilibrium points.Closely related to the class of accretive mappings is the class of pseudocontractive mappings. A mappingT with domain D(T) in E and range R(T) in E is called pseudocontractive if A∶=I-T is accretive. It is then clear that any zero of A is a fixed point of T. Consequently, the study of approximating fixed points of pseudocontractive mappings, which correspond to equilibrium points of the system (1.5), became a flourishing area of research for numerous mathematicians (see, e.g., [12–14] and the references therein).It is not difficult to deduce from (1.4) that the mapping A is accretive if and only if (I+rA)-1,forallr>0, is nonexpansive on the range of (I+rA). Thus, in particular, JA∶=(I+A)-1 is nonexpansive and single valued on the range of (I+A). Furthermore, F(JA)∶=N(A)∶={x∈D(A):Ax=0}. It is well known that every nonexpansive mapping is pseudocontractive and the converse does not, however, hold.Very recently, Yao  et al. [15] proved path convergence for a nonexpansive mapping in a real Hilbert space. In particular, they proved the following theorem.Theorem 1.1 (Yao et al. [15]).
LetK be a nonempty, closed, and convex subset of a real Hilbert space H. Let T:K→K be a nonexpansive mapping with F(T)≠∅. For t∈(0,1), let the net {xt} be generated by xt=TPK[(1-t)xt], then as t→0, the net {xt} converges strongly to a fixed point of T.Furthermore, they applied Theorem1.1 to prove the following theorem.Theorem 1.2 (Yao et al. [15]).
LetK be a nonempty, closed, and convex subset of a real Hilbert space H. Let T:K→K be a nonexpansive mapping such that F(T)≠∅. Let {αn}n=1∞ and {βn}n=1∞ be two real sequences in (0,1). For an arbitrary x1∈K, let the sequence {xn}n=1∞ be generated iteratively by
(1.7)yn=PK[(1-αn)xn],xn+1=(1-βn)xn+βnTyn,n≥1.
Suppose that the following conditions are satisfied:(a)
limαn=0 and ∑n=1∞αn=∞,(b)
0<liminfn→∞βn≤limsupn→∞βn<1,
then the sequence{xn}n=1∞ generated by (1.7) converges strongly to a fixed point of T.Motivated by the results of Yao  et al. [15], we proved  path convergence for a nonexpansive mapping in a uniformly smooth real Banach space which is also uniformly convex and E admits weakly sequentially continuous duality mapping j. As an application, a strong convergence is proved for common zeroes of a finite family of m-accretive mappings of K to E. As a consequence, an iterative scheme is constructed to converge to a common fixed point (assuming existence) of a finite family of pseudocontractive mappings from K to E under certain mild conditions.


## 2. Preliminaries

LetE be a real Banach space and let S∶={x∈E:∥x∥=1}. E is said to have a Gâteaux differentiable norm (and E is called smooth) if the limit (2.1)limt→0∥x+ty∥-∥x∥texists for eachx,y∈S; E is said to have a uniformly Gâteaux differentiable norm if for each y∈S the limit is attained uniformly for x∈S. Further, E is said to be uniformly smooth if the limit exists uniformly for (x,y)∈S×S. The modulus of smoothness of E is defined by (2.2)ρE(τ)∶=sup{∥x+y∥+∥x-y∥2-1:∥x∥=1,∥y∥=τ}.E is equivalently said to be smooth if ρE(τ)>0, for any τ>0.Let dimE≥2. The modulus of convexity of E is the function δE:(0,2]→[0,1] defined by (2.3)δE(ϵ)∶=inf{1-∥x+y2∥:∥x∥=∥y∥=1;ϵ=∥x-y∥}.E is uniformly convex if for any ϵ∈(0,2], there exists a δ=δ(ϵ)>0 such that if x,y∈E with ∥x∥≤1,∥y∥≤1 and ∥x-y∥≥ϵ, then ∥(1/2)(x+y)∥≤1-δ. Equivalently, E is uniformly convex if and only if δE(ϵ)>0 for all ϵ∈(0,2]. E is called strictly convex if for all x,y∈E,x≠y,∥x∥=∥y∥=1, we have ∥λx+(1-λ)y∥<1,forallλ∈(0,1). It is known that every uniformly convex Banach space is reflexive.LetK⊂E be closed and convex and Q be a mapping of E onto K. Then Q is said to be sunny if Q(Q(x)+t(x-Q(x)))=Q(x) for all x∈E and t≥0. A mapping Q of E into E is said to be a retraction if Q2=Q. If a mapping Q is a retraction, then Q(z)=(z) for every z∈R(Q), where R(Q) is the range ofQ. A subset K of E is said to be a sunny nonexpansive retract of E if there exists a sunny nonexpansive retraction of E onto K and it is said to be a nonexpansive retract of E if there exists a nonexpansive retraction of E onto K. If E=H, the metric projectionPK is a sunny nonexpansive retraction from H to any closed and convex subset of H. But this is not true in a general Banach spaces. We note that if E is smooth and Q is retraction of K onto F(T), then Q is sunny and nonexpansive if and only if for each x∈K and z∈F(T) we have 〈Qx-x,J(Qx-z)〉≤0, (see [16–18] for more details).A mappingT with domain D(T) and range R(T) in E is said to be demiclosed at p if whenever {xn}n=1∞ is a sequence in D(T) such that xn⇀x∈D(T) and Txn→p then Tx=p.Suppose thatJ is single valued. Then, J is said to be weakly sequentially continuous if for each {xn}n=1∞⊂E which converges weakly to x implies J(xn) converges in weak* to J(x).We need the following lemmas in the sequel.Lemma 2.1 (Browder [19], Goebel and Kirk [20]).
LetE be a real uniformly convex Banach space and let K be a nonempty, closed, and convex subset of E and T:K→K is a nonexpansive mapping such that F(T)≠∅, then, I-T is demiclosed at zero.Lemma 2.2 (Suzuki [21]).
Let{xn}n=1∞ and {yn}n=1∞ be bounded sequences in a Banach space X and let {βn}n=1∞ be a sequence in [0,1] with 0<liminfn→∞βn≤limsupn→∞βn<1. Suppose that xn+1=(1-βn)yn+βnxn  for all integers n≥1 and limsupn→∞(∥yn+1-yn∥-∥xn+1-xn∥)≤0, then limn→∞∥yn-xn∥=0.Lemma 2.3 (Chidume [5], Reich [22]).
LetE be a uniformly real smooth Banach space, then there exists a nondecreasing continuous function β:[0,∞)→[0,∞) with limt→0+β(s)=0 and β(cs)≤cβ(s) for c≥1 such that for all x,y∈E, the following inequality holds:
(2.4)∥x+y∥2≤∥x∥2+2〈y,j(x)〉+max{∥x∥,1}∥y∥β(∥y∥).Lemma 2.4 (Xu [23]).
Let{an} be a sequence of nonnegative real numbers which satisfies the following relation:
(2.5)an+1≤(1-αn)an+αnσn,n≥1,
where {an}n=1∞⊂[0,1] and {σn}n=1∞ is a sequence in ℝ satisfying the following: (i)
∑αn=∞,(ii)
limsupσn≤0, then, an→0 as n→∞.Lemma 2.5 (Cioranescu [8]).
LetA be a continuous accretive mapping defined on a real Banach space E with D(A)=E, then A is m-accretive.Lemma 2.6 (Zegeye and Shahzad [24]).
LetK be a nonempty, closed, and convex subset of a real strictly convex Banach space E. For each r=1,2,…,N let Ar:K→E be an m-accretive mapping such that ⋂r=1NN(Ai)≠∅. Let a0,a1,a2,…,aN be real numbers in (0,1) such that ∑i=0Nai=1, and let SN∶=a0I+a1JA1+a2JA2+⋯+aNJAN, with JAr∶=(I+Ar)-1, then SN is nonexpansive and F(SN)=⋂r=1NN(Ar).


## 3. Path Convergence Theorem

LetK be a nonempty, closed, and convex sunny nonexpansive retract of a uniformly smooth Banach space E which is also uniformly convex where QK is the sunny nonexpansive retraction of E onto K. Let T:K→K be nonexpansive. For each t∈(0,1), we define the mapping Tt:K→K by
(3.1)Ttx∶=TQK[(1-t)x].
We will show that Tt is a contraction.From (3.1), we have(3.2)∥Ttx-Tty∥≤∥QK(1-t)x-QK(1-t)y∥≤(1-t)∥x-y∥,
which implies that Tt is a contraction. Therefore, by the Banach contraction mapping principle, there exists a unique fixed point zt of Tt in K. That is,(3.3)zt=TQK[(1-t)zt].Next, we prove that{zt} is bounded. Let x*∈F(T), then using (3.3), we have (3.4)∥zt-x*∥=∥TQK(1-t)zt-TQKx*∥≤∥QK(1-t)zt-QKx*∥≤∥(1-t)zt-tx*+tx*-x*∥=∥(1-t)(zt-x*)-tx*∥≤(1-t)∥zt-x*∥+t∥x*∥.Thus,∥zt-x*∥≤∥x*∥. This implies that {zt} is bounded.We next show that∥zt-Tzt∥→0 as t→0, as follows:(3.5)∥zt-Tzt∥=∥TQK(1-t)zt-TQKzt∥≤∥QK(1-t)zt-QKzt∥≤∥(1-t)zt-zt∥≤t∥zt∥→0,(sincet→0).Next, we show that{zt} is relatively norm compact as t→0. Let {tn} be a sequence in (0,1) such that tn→0 as n→∞. Put zn∶=ztn. From (3.5), we obtain that(3.6)∥zn-Tzn∥→0.Remark 3.1.
Letx*∈F(T) and r1>0 be sufficiently large such that zt∈Br1(x*)¯∩K for each t∈(0,1), where Br1(x*)¯∶={z∈E:∥z-x*∥≤r1}. For the next theorem, we define A∶=max{1,2r1} and assume that the function β from Lemma 2.3 satisfies the following condition: β(s)≤s/A.Theorem 3.2.
LetE be a real Banach space which is uniformly smooth and uniformly convex and let K be a nonempty, closed, and convex sunny nonexpansive retract of E, where QK is the sunny nonexpansive retraction of E onto K. Let T:K→K be a nonexpansive mapping with F(T)≠∅. For each t∈(0,1), let {zt} be generated by (3.3), then as t→0,{zt} converges strongly to a fixed point of T if E admits weak sequential continuous duality mapping j.Proof.
From (3.3), we get for u∈F(T),
(3.7)∥zt-u∥2=∥TQK[(1-t)zt]-TQKu∥2≤∥(1-t)zt-u∥2=∥zt-u-tzt∥2≤∥zt-u∥2-2t〈zt,j(zt-u)〉+max{∥zt-u∥,1}t∥zt∥β(t∥zt∥)≤∥zt-u∥2-2t〈zt,j(zt-u)〉+max{2r1,1}t∥zt∥β(t∥zt∥)≤∥zt-u∥2-2t〈zt,j(zt-u)〉+t2∥zt∥2=∥zt-u∥2-2t〈zt-u,j(zt-u)〉-2t〈u,j(zt-u)〉+t2∥zt∥2≤∥zt-u∥2-2t∥zt-u∥2-2t〈u,j(zt-u)〉+t2∥zt∥2.
This implies that
(3.8)∥zt-u∥2≤〈u,j(u-zt)〉+t2∥zt∥2.
In particular,
(3.9)∥zn-u∥2≤〈u,j(u-zn)〉+tn2∥zn∥2.
Since {zn} is bounded, without loss of generality, we can assume that {zn} converges weakly to z*. Using the demiclosedness property of (I-T) at zero and the fact that ∥zn-Tzn∥→0 as n→∞, we obtain that z*∈F(T). Therefore, we can substitute z* for u in (3.9) to obtain
(3.10)∥zn-z*∥2≤〈z*,j(z*-zn)〉+tn2∥zn∥2.
Using the fact that j is weakly sequentially continuous, we have from the last inequality that {zn} converges strongly to z*. We now show that {zt} actually converges to z*. Suppose that {ztm} converges strongly to x*. Put zm∶=ztm, then since ∥zm-Tzm∥→0 as m→∞ and I-T is demiclosed at zero, we have that x*∈F(T).
Claim 3 (z*=x*).
Suppose in contradiction thatx*≠z*. Using (3.3), we obtain using similar argument as above that
(3.11)∥zm-z*∥2≤〈z*,j(z*-zm)〉+tm2∥zm∥2.
Thus,
(3.12)∥x*-z*∥2≤〈z*,j(z*-x*)〉.
Interchanging x* and z*, we obtain
(3.13)∥z*-x*∥2≤〈x*,j(x*-z*)〉.
Adding (3.12) and (3.13) yields
(3.14)2∥x*-z*∥2≤∥x*-z*∥2
and implies that x*=z*. This completes the proof.Corollary 3.3.
LetE∶=lp,1<p<∞ and let K be a nonempty, closed, and convex sunny nonexpansive retract of E, where QK is the sunny nonexpansive retraction of E onto K. Let T:K→K be a nonexpansive mapping with F(T)≠∅. For each t∈(0,1), let {zt} be generated by (3.3) then as t→0,{zt} converges strongly to a fixed point of T.


## 4. Iterative Methods and Convergence Theorems

Theorem 4.1.
LetE be a real Banach space which is uniformly smooth and uniformly convex, and let K be a nonempty, closed, and convex sunny nonexpansive retract of E, where QK is the sunny nonexpansive retraction of E onto K. For each r=1,2,…,N, let Ar:K→E be an m-accretive mapping such that ⋂r=1NN(Ar)≠∅. Let {αn}n=1∞ and {βn}n=1∞ be two real sequences in (0,1). For an arbitrary x1∈K, let the sequence {xn}n=1∞ be generated iteratively by
(4.1)yn=QK[(1-αn)xn],xn+1=(1-βn)xn+βnSNyn,n≥1,
where SN∶=a0I+a1JA1+a2JA2+⋯+aNJAN, with JAr∶=(I+Ar)-1,r=1,2,…,N for 0<ai<1,i=0,1,2,…,N,∑i=0Nai=1. Suppose that the following conditions are satisfied:(a)
limn→∞αn=0 and ∑n=1∞αn=∞,(b)
0<liminfn→∞βn≤limsupn→∞βn<1,then the sequence {xn}n=1∞ converges strongly to a common zero of {Ar}r=1N if E admits weakly sequentially continuous duality mapping j.Proof.
By Lemma2.6, SN is nonexpansive and F(SN)=⋂r=1NN(Ar). Now, we first show that the sequence {xn}n=1∞ is bounded. Let x*∈⋂r=1NN(Ar)=F(SN), we have from (4.1) that
(4.2)∥xn+1-x*∥=∥(1-βn)(xn-x*)+βn(SNyn-x*)∥≤(1-βn)∥xn-x*∥+βn∥SNyn-x*∥≤(1-βn)∥xn-x*∥+βn[(1-αn)∥xn-x*∥+αn∥x*∥]=(1-αnβn)∥xn-x*∥+αnβn∥x*∥≤max{∥xn-x*∥,∥x*∥}⋮≤max{∥x1-x*∥,∥x*∥}.
Hence, {xn}n=1∞ is bounded and {SNxn} is also bounded. Set un=SNyn,n≥1. It follows that
(4.3)∥un+1-un∥=∥SNyn+1-SNyn∥≤∥yn+1-yn∥≤∥(1-αn+1)xn+1-(1-αn)xn∥≤∥xn+1-xn∥+αn+1∥xn+1∥+αn∥xn∥.
Hence, limsupn→∞(∥un+1-un∥-∥xn+1-xn∥)≤0. This together with Lemma 2.2 implies that limn→∞∥un-xn∥=0. Thus,
(4.4)limn→∞∥xn+1-xn∥=limn→∞βn∥xn-un∥=0,∥xn-SNxn∥≤∥xn-xn+1∥+∥xn+1-SNxn∥≤∥xn-xn+1∥+(1-βn)∥xn-SNxn∥+βn∥SNyn-SNxn∥≤∥xn-xn+1∥+(1-βn)∥xn-SNxn∥+βn∥yn-xn∥=∥xn-xn+1∥+(1-βn)∥xn-SNxn∥+βn∥QK[(1-αn)xn]-QKxn∥≤∥xn-xn+1∥+(1-βn)∥xn-SNxn∥+βn∥(1-αn)xn-xn∥=∥xn-xn+1∥+(1-βn)∥xn-SNxn∥+αnβn∥xn∥≤∥xn-xn+1∥+(1-βn)∥xn-SNxn∥+αn∥xn∥,sinceβn∈(0,1),
that is,
(4.5)∥xn-SNxn∥≤1βn{∥xn-xn+1∥+αn∥xn∥}→0.
Let {zt} be defined by (3.3) for T=SN, then from Theorem 3.2,  zt→x*∈F(SN)=⋂r=1NN(Ar) as t→0 (This is guaranteed because E admits weakly sequentially continuous duality mapping). Next, we show that
(4.6)limsupn→∞〈x*,j(x*-xn)〉≤0.
Now, since {xn}n=1∞ and {zt} are bounded, there exist r1>0,r2>0 such that zt∈Br1(x*)¯∩K for each t∈(0,1) and xn∈Br2(x*)¯∩K for any n≥1. Let ρ=r1+r2, A=max{2ρ,1}, and β(s)≤s/A. Hence, by Lemma 2.3, we have
(4.7)∥zt-xn∥2=∥zt-SNxn+SNxn-xn∥2≤∥zt-SNxn∥2+2〈SNxn-xn,j(zt-SNxn)〉+max{∥zt-SNxn∥,1}∥SNxn-xn∥β(∥SNxn-xn∥)≤∥zt-SNxn∥2+2〈SNxn-xn,j(zt-SNxn)〉+max{∥zt-x*∥+∥xn-x*∥,1}∥SNxn-xn∥β(∥SNxn-xn∥)≤∥zt-SNxn∥2+2〈SNxn-xn,j(zt-SNxn)〉+max{2ρ,1}∥SNxn-xn∥β(∥SNxn-xn∥)≤∥zt-SNxn∥2+2〈SNxn-xn,j(zt-SNxn)〉+∥SNxn-xn∥2≤∥zt-SNxn∥2+2∥SNxn-xn∥∥zt-SNxn∥+∥SNxn-xn∥2≤∥zt-SNxn∥2+M∥SNxn-xn∥≤∥SNQK(1-t)zt-SNxn∥2+M∥SNxn-xn∥≤∥QK(1-t)zt-xn∥2+M∥SNxn-xn∥=∥(1-t)zt-xn∥2+M∥SNxn-xn∥=∥zt-xn-tzt∥2+M∥SNxn-xn∥≤∥zt-xn∥2-2t〈zt,j(zt-xn)〉+max{∥zt-xn∥,1}t∥zt∥β(t∥zt∥)+M∥Txn-xn∥≤∥zt-xn∥2-2t〈zt,j(zt-xn)〉+max{∥zt-x*∥+∥xn-x*∥,1}t∥zt∥β(t∥zt∥)+M∥Txn-xn∥≤∥zt-xn∥2-2t〈zt,j(zt-xn)〉+max{2ρ,1}t∥zt∥β(t∥zt∥)+M∥Txn-xn∥≤∥zt-xn∥2-2t〈zt,j(zt-xn)〉+t2∥zt∥2+M∥Txn-xn∥≤∥zt-xn∥2-2t〈zt,j(zt-xn)〉+t2M1+M∥Txn-xn∥
for some M>0 and M1>0. Thus, 〈zt,j(zt-xn)〉≤M1t/2+(M/2t)∥SNxn-xn∥. Therefore,
(4.8)limt→0limsupn→∞〈zt,j(zt-xn)〉≤0.
Moreover,
(4.9)〈-zt,j(xn-zt)〉=〈-x*,j(xn-x*)〉+〈-x*,j(xn-zt)〉-〈-x*,j(xn-x*)〉+〈x*-zt,j(xn-zt)〉=〈-x*,j(xn-x*)〉+〈-x*,j(xn-zt)-j(xn-x*)〉+〈x*-zt,j(xn-zt)〉.
Since {xn}n=1∞ is bounded, we have that 〈x*-zt,j(xn-zt)〉→0 as t→0 and since j is norm-to-weak* uniformly continuous on bounded sets, we have 〈-x*,j(xn-zt)-j(xn-x*)〉→0 as t→0. Using (4.8) and (4.9), we obtain
(4.10)limsupn→∞〈-x*,j(xn-x*)〉≤0.
From (4.1), we have
(4.11)∥yn-x*∥=∥SNQK(1-αn)xn-x*∥2≤∥(1-αn)xn-x*∥2=∥xn-x*-αnx*∥2≤∥xn-x*∥2-2αn〈xn,j(xn-x*)〉+max{1,2ρ}αn∥xn∥β(αn∥xn∥)≤∥xn-x*∥2-2αn〈xn,j(xn-x*)〉+αn2∥xn∥2=∥xn-x*∥2+2αn〈xn-x*+x*,j(x*-xn)〉+αn2∥xn∥2=∥xn-x*∥2+2αn〈x*,j(x*-xn)〉-2αn〈x*-xn,j(x*-xn)〉+αn2∥xn∥2=∥xn-x*∥2+2αn〈x*,j(x*-xn)〉+αn2∥xn∥2-2αn∥xn-x*∥2=(1-αn)∥xn-x*∥2-αn∥xn-x*∥2+2αn〈x*,j(x*-xn)〉+αn2∥xn∥2≤(1-αn)∥xn-x*∥2+2αn〈x*,j(x*-xn)〉+αn2∥xn∥2.
Also, from (4.1), we obtain
(4.12)∥xn+1-x*∥2≤(1-βn)∥xn-x*∥2+βn∥yn-x*∥2≤(1-βn)∥xn-x*∥2+βn[(1-αn)∥xn-x*∥2+2αn〈x*,j(x*-xn)〉+αn2∥xn∥2]≤(1-αnβn)∥xn-x*∥2+2αnβn〈x*,j(x*-xn)〉+αn2βnM2=(1-αnβn)∥xn-x*∥2+αnβn[2〈x*,j(x*-xn)〉+αnβnM2],
where M2∶=supn≥1∥xn∥2. Using Lemma 2.4, we get that {xn}n=1∞ converges strongly to x*∈F(SN)=⋂r=1NN(Ar). This completes the proof.If in Theorem4.1, we consider K=E, the condition that each A_r,r=1,2,…,N is m-accretive may be replaced with continuity of each Ar. Thus, we have this theorem.Theorem 4.2.
LetE be a real Banach space which is uniformly smooth and also uniformly convex. For each r=1,2,…,N, let Ar:E→E be a continuous accretive mapping such that ⋂r=1NN(Ar)≠∅. Let {αn}n=1∞ and {βn}n=1∞ be two real sequences in (0,1). For an arbitrary x1∈E, let the sequence {xn}n=1∞ be generated iteratively by
(4.13)yn=(1-αn)xn,xn+1=(1-βn)xn+βnSNyn,n≥1,
where SN∶=a0I+a1JA1+a2JA2+⋯+aNJAN, with JAr∶=(I+Ar)-1,r=1,2,…,N for 0<ai<1,i=0,1,2,…,N,∑i=0Nai=1. Suppose that the following conditions are satisfied:(a)
limn→∞αn=0 and ∑n=1∞αn=∞,(b)
0<liminfn→∞βn≤limsupn→∞βn<1,then the sequence {xn}n=1∞ converges strongly to a common zero of {Ar}r=1N if E admits weakly sequentially continuous duality mapping j.Proof.
TakeQK=I in Theorem 4.1. By Lemma 2.5, we have that Ar is m-accretive for each r=1,2,…,N. Then, the result follows from Theorem 4.1.The following theorems give strong convergence to a common fixed point of a finite family of pseudocontractive mappings.Theorem 4.3.
LetE be a real Banach space which is uniformly smooth and uniformly convex, and let K be a nonempty, closed, and convex sunny nonexpansive retract of E, where QK is the sunny nonexpansive retraction of E onto K. For each r=1,2,…,N, let Tr:K→E be a pseudocontractive mapping such that (I-Tr) is m-accretive on K with ⋂r=1NF(Tr)≠∅. Let {αn}n=1∞ and {βn}n=1∞ be two real sequences in (0,1) and JTr∶=(2I-Tr)-1 for each r=1,2,…,N. For an arbitrary x1∈K let sequence {xn}n=1∞ be generated iteratively by
(4.14)yn=QK[(1-αn)xn],xn+1=(1-βn)xn+βnSNyn,n≥1,
where SN∶=a0I+a1JT1+a2JT2+⋯+aNJTN for 0<ai<1,i=0,1,2,…,N,∑i=0Nai=1. Suppose that the following conditions are satisfied:(a)
limn→∞αn=0 and ∑n=1∞αn=∞,(b)
0<liminfn→∞βn≤limsupn→∞βn<1,then the sequence {xn}n=1∞ converges strongly to a common fixed point of {Tr}r=1N if E admits weakly sequentially continuous duality mapping j.Proof.
LetAr∶=(I-Tr) for each r=1,2,…,N. Then, clearly, F(Tr)=N(Ar) and hence ⋂r=1NN(Ar)=⋂r=1NF(Tr)≠∅. Furthermore, each Ar for r=1,2,…,N is m-accretive. The result follows from Theorem 4.1.Theorem 4.4.
LetE be a real Banach space which is uniformly smooth and uniformly convex. For each r=1,2,…,N, let Tr:K→E be a continuous pseudocontractive mapping on E such that ⋂r=1NF(Tr)≠∅. Let {αn}n=1∞ and {βn}n=1∞ be two real sequences in (0,1) and JTr∶=(2I-Tr)-1 for each r=1,2,…,N. For arbitrary x1∈K let sequence {xn}n=1∞ be generated iteratively by
(4.15)yn=(1-αn)xn,xn+1=(1-βn)xn+βnSNyn,n≥1,
where SN∶=a0I+a1JT1+a2JT2+⋯+aNJTN for 0<ai<1,i=0,1,2,…,N,∑i=0Nai=1. Suppose that the following conditions are satisfied:(a)
limn→∞αn=0 and ∑n=1∞αn=∞,(b)
0<liminfn→∞βn≤limsupn→∞βn<1,then the sequence {xn}n=1∞ converges strongly to a common fixed point of {Tr}r=1N if E admits weakly sequentially continuous duality mapping j.Proof.
The proof follows from Theorem4.2.



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*Source: 285376-2010-08-24.xml*