# Convergence of Path and Approximation of Common Element of Null Spaces of Countably Infinite Family ofm-Accretive Mappings in Uniformly Convex Banach Spaces

Authors: E. U. Ofoedu; Y. Shehu
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## Abstract

We prove path convergence theorems and introduce
a new iterative sequence for a countably infinite family ofm-accretive mappings and prove strong convergence of the sequence to a common zero of these operators in uniformly convex real Banach space. Consequently, we obtain strong convergence theorems for a countably infinite family of pseudocontractive mappings. Our theorems extend and improve some important results which are announced recently by various authors.


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## Body

## 1. Introduction

LetE be a real Banach space with dual E*. We denote by J the normalized duality mapping from E to 2E* defined by
(1.1)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., [1, 2]). 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
(1.2)〈Ax-Ay,j(x-y)〉≥0.
By the result of Kato [3], (1.2) is equivalent to
(1.3)∥x-y∥≤∥x-y+s(Ax-Ay)∥,∀s>0.
If E is a Hilbert space, accretive operators are also called monotone. An operator 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), for all r>0, where cl(D(A)) denotes the closure of the domain of A. It is easy to see that every m-accretive operator satisfies the range condition. An operator A is said to be maximal accretive if it is accretive and the inclusion G(A)⊆G(B) implies G(A)=G(B), where G(A) denotes the graph of A and B is an accretive operator.A mappingT:D(T)⊆E→R(T)⊆E is said to be nonexpansive if
(1.4)∥Tx-Ty∥≤∥x-y∥,∀x,y∈D(T).
It is not difficult to deduce from (1.3) that a mapping A is accretive if and only if its resolvent Jr:=(I+rA)-1, for all r>0, is nonexpansive and single valued on the range of (I+rA). Thus, in particular, JA=J1:=(I+A)-1 is nonexpansive and single valued on the range of (I+A). Furthermore, F(JA):=A-1(0):={x∈D(A):Ax=0}. For more details see, for example, [4, 5].Closely related to the class of accretive operators is the class of pseudocontractive maps. An operatorT with domain D(T) in E and range R(T) in E is called pseudocontractive if A:=I-T is accretive. The importance of these operators in application is well known (see, e.g., [6–9] and the references contained therein).It is well known that the class of pseudocontractive mappings properly contains the class of nonexpansive mappings (see, e.g., [4]). Construction of fixed points of nonexpansive mappings is an important subject in nonlinear operator theory and its applications, in particular, in image recovery and signal processing (see, e.g., [10]).Iterative approximation of fixed points and zeros of nonlinear operators have been studied extensively by many authors to solve nonlinear operator equations as well as variational inequality problems (see, e.g., [11–15]). The iterative scheme
(1.5)x0∈E,xn+1=Jrnxn,n≥0,
(where Jrn is the resolvent of an m-accretive operator A) for example, has been extensively studied over the past forty years or so for construction of zeros of accretive operators (see, e.g., [16–20]).Kim and Xu [21] introduced a modification of Mann iterative scheme in a reflexive Banach space having weakly continuous duality mapping for finding a zero of an m-accretive operator A as follows:
(1.6)x0=u∈E,xn+1=αnu+(1-αn)Jrnxn,n≥0.
They proved that the sequence {xn}n=0∞ generated by (1.6) converges to a zero of m-accretive operator A under the following conditions:(i)
limn→∞αn=0,(ii)
∑n=0∞αn=∞ (equivalently, ∏n=0∞(1-αn)=0),(iii)
∑n=0∞|αn+1-αn|<∞; rn≥ϵ for some ϵ>0 and for all n≥0,(iv)
∑n=1∞|1-rn-1/rn|<∞,(v)
rn≥ϵ for some ϵ>0 and for all n≥0 and ∑n=1∞|rn-rn-1|<∞.In 2007, Qin and Su [22] also considered the following iterative scheme in either a uniformly smooth Banach space or a reflexive Banach space having a weakly sequentially continuous duality mapping:
(1.7)x0=u∈C,yn=βnxn+(1-βn)Jrnxn,xn+1=αnu+(1-αn)ynn≥0,
where {αn}n=0∞ and {βn}n=0∞ are sequences in (0,1). They proved that the sequence {xn}n=0∞ generated by (1.7) converges strongly to a zero of m-accretive operator A provided that {αn}n=0∞ and {βn}n=0∞ satisfy conditions (i), (ii), and (iii), and {rn}n=0∞ satisfies condition (v).Chen and Zhu [23] considered the following viscosity iterative scheme for resolvent Jrn of m-accretive mapping A:
(1.8)x0∈C,xn+1=αnf(xn)+(1-αn)Jrnxn,n≥0,
where f is a contraction mapping defined on C. Under the assumption that {rn}n=0∞ satisfies condition (v), Chen and Zhu [23] proved in a reflexive Banach space having weakly sequentially continuous duality mapping that the sequence {xn}n=0∞ generated by (1.8) converges strongly to a zero of A, which solves a certain variational inequality.Recently, Jung [24] introduced the following viscosity iterative method:
(1.9)x0∈C,yn=βnxn+(1-βn)Jrnxn,xn+1=αnf(xn)+(1-αn)ynn≥0.
Under certain appropriate conditions on the parameters {αn}n=0∞, {βn}n=0∞, {rn}n=0∞ and the sequence {xn}n=0∞; Jung [24] established strong convergence of the sequence {xn}n=0∞ generated by (1.9) to a zero of A, which is a unique solution of a certain variational inequality problem, in either a reflexive Banach space having a weakly sequentially continuous duality mapping or a reflexive Banach space having a uniformly Gâteaux differentiable norm such that every weakly compact convex subset of E has the fixed point property for nonexpansive mappings.In [5], Zegeye and Shahzad proved the following theorem.Theorem 1 ZS.
LetE be a strictly convex reflexive real Banach space which has uniformly Gâteaux differentiable norm and let K be a nonempty closed convex subset of E. Assume that every nonempty closed convex and bounded subset of E has the fixed point property for nonexpansive mappings. Let Ai:K→E, i=1,2,…,r be a finite family of m-accretive mappings with ⋂i=1rAi-1(0)≠∅. For given u,x1∈K, let {xn}n≥1 be generated by the algorithm
(1.10)xn+1=θnu+(1-θn)Srxn,∀n≥1,
where Sr=a0I+a1JA1+⋯+arJAr, with JAi=(I+Ai)-1, 0<ai<1, i=1,…,r, ∑i=1rai=1, and {θn}n≥1 is a sequence in (0,1) satisfying the following conditions:(i)
limn→∞θn=0;(ii)
∑n=1∞θn=∞;(iii)
∑n=1∞|θn-θn-1|<∞ or limn→∞(|θn-θn-1|/θn)=0.Then, {xn}n≥1 converges strongly to a common solution of the equation Aix=0 for i=1,2,…,r.Motivated by the results of the authors mentioned above, it is our purpose in this paper to prove new path convergnce theorems and introduce a new iteration process for a countably infinite family ofm-accretive mappings and prove strong convergence of the sequence to a common zero of these operators in uniformly convex real Banach spaces. As a result, we obtain strong convergence theorems for a countably infinite family of pseudocontractive mappings. Our theorems extend and improve some important results which are announced recently by various authors.


## 2. Preliminaries

LetE be a real normed linear space. 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∥t
exists for each x,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. Furthermore, E is said to be uniformly smooth if the limit exists uniformly for (x,y)∈S×S.LetE be a real normed linear space. The modulus of convexity of E is the function δE:[0,2]→[0,1] defined by
(2.2)δE(ϵ)=inf{1-∥x+y∥2:∥x∥=∥y∥=1,ϵ=∥x-y∥}.
The space E is said to be uniformly convex if and only if δE(ϵ)>0 for all ϵ∈(0,2]. E is said to be strictly convex if for all x,y∈E such that ∥x∥=∥y∥=1 and for all λ∈(0,1) we have ∥λx+(1-λ)y∥<1. It is well known that every uniformly convex Banach space is strictly convex.A mappingT with domain D(T) and range R(T) in E is said to be demiclosed at p if whenever {xn} is a sequence in D(T) such that xn⇀x∈D(T) and Txn→p, then Tx=p.A mappingT:D(T)⊆E→E is said to be demicompact at h if for any bounded sequence {xn} in D(T) such that (xn-Txn)→h as n→∞, there exists a subsequence say {xnj} of {xn} and x*∈D(T) such that {xnj} converges strongly to x* and x*-Tx*=h.We need the following lemmas in the sequel.Lemma 2.1.
LetE be a real normed space, then
(2.3)∥x+y∥2≤∥x∥2+2〈y,j(x+y)〉,
for all x,y∈E and for all j(x+y)∈J(x+y).Lemma 2.2 (Lemma  3 of Bruck [25]).
LetK be a nonempty closed convex subset of a strictly convex real Banach space E. Let {Ti}i=1∞ be a sequence of nonexpansive mappings from K to E such that ⋂i=1∞F(Ti)≠∅. Let {λi}i=1∞ be a sequence of positive numbers such that ∑i=1∞λi=1, then a mapping G on K defined by Gx:=∑i=1∞λiTix for all x∈K is well defined, nonexpansive, and F(G)=⋂i=1∞F(Ti).Lemma 2.3 (Xu [26]).
Let{an} be a sequence of nonnegative real numbers satisfying the following relation:
(2.4)an+1≤(1-αn)an+αnσn,n≥1,
where {αn}n=1∞⊂[0,1] and {σn}n=1∞ is a sequence in ℝ satisfying (i) ∑αn=∞; (ii) lim supσn≤0. Then, an→0 as n→∞.Lemma 2.4 (Suzuki [27]).
Let{xn} and {yn} be bounded sequences in a Banach space X and let {βn} be a sequence in [0,1] with 0<lim infn→∞βn≤lim supn→∞βn<1. Suppose that xn+1=(1-βn)yn+βnxn for all integers n≥1 and lim supn→∞(∥yn+1-yn∥-∥xn+1-xn∥)≤0. Then, limn→∞∥yn-xn∥=0.Lemma 2.5 (Cioranescu [28]).
LetA be a continuous accretive operator defined on a real Banach space E with D(A)=E. Then, A is m-accretive.Lemma 2.6 (C. E. Chidume and C. O. Chidume [29]).
LetK be a nonempty closed convex subset of a real Banach space E. For arbitrary r>0, let Br(0):={x∈E:∥x∥≤1}. Then, there exists a continuous strictly increasing function g:[0,∞)→[0,∞), g(0)=0 such that for every x,y∈Br(0) and for p∈(1,∞), the following inequality holds:
(2.5)4.2pg(12∥x+y∥)≤(p.2p-4)∥x∥p+p.2p〈y,jp(x)〉+4∥y∥p.


## 3. Path Convergence Theorems

We begin with the following lemma.Lemma 3.1.
LetK be a nonempty closed convex subset of a reflexive strictly convex Banach space E. Let T:K→K be a nonexpansive mapping. Let {xn}n=1∞, a bounded sequence in K, be an approximate fixed point sequence of T, that is limn→∞∥xn-Txn∥=0. Let φ(x)=μn∥xn-x∥2, for all x∈K and let Γ={x∈K∩B:φ(x)=minz∈Kφ(z)}, where B is any bounded closed convex nonempty subset of E such that xn∈B for all n∈ℕ. Then T has a fixed point in Γ, provided that F(T)≠∅.Proof.
SinceE is a reflexive Banach space, then Γ is a bounded closed convex nonempty subset of E. Since limn→∞∥xn-Txn∥=0, we have that for all x∈Γ,
(3.1)φ(Tx)=μn∥xn-Tx∥2≤μn(∥xn-Txn∥+∥Txn-Tx∥)2≤μn∥xn-x∥2=φ(x).
Hence, Tx∈Γ, for all x∈Γ, that is, Γ is invariant under T. Let x*∈F(T). Then since every closed convex nonempty subset of a reflexive strictly convex Banach space is a Chebyshev set (see, e.g., [30, Corollary  5.1.19]), there exists a unique u*∈Γ such that
(3.2)∥x*-u*∥=infz∈Γ∥x*-z∥,
but x*=Tx* and Tu*∈Γ. Thus,
(3.3)∥x*-Tu*∥=∥Tx*-Tu*∥≤∥x*-u*∥.
So, Tu*=u*. Hence, F(T)∩Γ≠∅. This completes the proof.Proposition 3.2.
LetK be a nonempty closed convex subset of a real Banach space E. Let Ai:K→E, i=1,2,…, be a countably infinite family of m-accretive mappings and define JAi:=(I+Ai)-1, i=1,2,…. Let {αn}n=1∞, {σi,n}n=1∞, i=1,2,… be sequences in (0,1) such that ∑i=1∞σi,n=(1-αn). Fix δ∈[γ1,γ2], for some γ1,γ2∈(0,1). For arbitrary fixed u∈K, define a map Tn:K→K by
(3.4)Tnx=αnu+∑i=1∞σi,n((1-δ)x+δJAix),∀x∈K.
Then, Tn is a strict contraction on K.Proof.
Letx,y∈K, then
(3.5)∥Tnx-Tny∥=∥∑i=1∞σi,n((1-δ)(x-y)+δ(JAix-JAiy))∥≤∑i=1∞σi,n((1-δ)∥x-y∥+δ∥JAix-JAiy∥)≤(1-αn)∥x-y∥.Thus, for eachn∈ℕ, there is a unique zn∈K satisfying
(3.6)zn=αnu+∑i=1∞σi,n((1-δ)zn+δJAizn).Lemma 3.3.
LetK be a nonempty closed convex subset of a real Banach space E. For each i≥1, let Ai:K→E be a countably infinite family of m-accretive mappings. For n∈ℕ, let {zn} be a sequence satisfying (3.6) and assume ⋂i=1∞Ai-1(0)≠∅. Then, {zn} is bounded.Proof.
Letx*∈⋂i=1∞Ai-1(0)=⋂i=1∞F(JAi). Then, using (3.6), we obtain
(3.7)∥zn-x*∥2=〈αn(u-x*)+∑i=1∞σi,n((1-δ)zn+δJAizn-x*),j(zn-x*)〉≤αn〈u-x*,j(zn-x*)〉+∑i=1∞σi,n∥zn-x*∥2=αn〈u-x*,j(zn-x*)〉+(1-αn)∥zn-x*∥2,
which implies that ∥zn-x*∥≤∥u-x*∥. Thus, {zn} is bounded.Lemma 3.4.
LetK be a nonempty closed convex subset of a uniformly convex real Banach space E. For each i≥1, let Ai:K→E be a countably infinite family of m-accretive mappings such that ⋂i=1∞Ai-1(0)≠∅. Let {αn} be a sequence in (0,1) such that limn→∞(αn/σi,n)=0, for all i≥1, ∑i=1∞σi,n=(1-αn). Let {zn} be a sequence satisfying (3.6). Then, limn→∞∥zn-JAizn∥=0, for all i≥1. Furthermore, if {λi}i=1∞ is a sequence in (0,1) such that ∑i=1∞λi=1; limn→∞∑i=1∞|σi,n-λi|=0 and define G:=(1-δ)I+δT, where T:=∑i=1∞λiJAi, then limn→∞∥zn-Gzn∥=0.Proof.
We start by showing thatlimn→∞∥zn-JAizn∥=0, for all i≥1. For this, let Si:=(1-δ)I+δJAi, where I is the identity operator on K. Since {zn}n=1∞ is bounded, then for each i≥1 and x*∈⋂i=1∞Ai-1(0), we have the following using (2.5):
(3.8)4.2pg(12∥Sizn-zn∥)=4.2pg(12∥Sizn-x*+x*-zn∥)≤(p.2p-4)∥x*-zn∥p+p.2p〈Sizn-x*,jp(x*-zn)〉+4∥Sizn-x*∥p≤(p.2p-4)∥x*-zn∥p+p.2p〈Sizn-zn+zn-x*,jp(x*-zn)〉+4∥Sizn-x*∥p=(p.2p-4)∥x*-zn∥p+p.2p〈Sizn-zn,jp(x*-zn)〉-p.2p〈x*-zn,jp(x*-zn)〉+4∥Sizn-x*∥p≤p.2p〈zn-Sizn,jp(zn-x*)〉.
Hence,
(3.9)4pg(12∥Sizn-zn∥)≤〈zn-Sizn,jp(zn-x*)〉,
and so,
(3.10)4p∑i=1∞σi,ng(12∥Sizn-zn∥)≤4p∑i=1∞σi,n〈zn-Sizn,jp(zn-x*)〉.
Using (3.6), we have
(3.11)〈zn-x*,jp(zn-x*)〉=αn〈u-x*,jp(zn-x*)〉+∑i=1∞σi,n〈Sizn-zn+zn-x*,jp(zn-x*)〉=αn〈u-x*,jp(zn-x*)〉+∑i=1∞σi,n〈Sizn-zn,jp(zn-x*)〉+(1-αn)〈zn-x*,jp(zn-x*)〉,
which implies
(3.12)∑i=1∞σi,n〈zn-Sizn,jp(zn-x*)〉=αn〈u-zn,jp(zn-x*)〉.
Using this and (3.10), we get
(3.13)4p∑i=1∞σi,ng(12∥Sizn-zn∥)≤αn〈u-zn,jp(zn-x*)〉.
Since {zn} is bounded, we have that
(3.14)∑i=1∞σi,ng(12∥Sizn-zn∥)≤αnM,
for some constant M>0. This yields
(3.15)g(12∥Sizn-zn∥)≤αnσi,nM.
Thus, since g is continuous, strictly increasing, g(0)=0, and limn→∞(αn/σi,n)=0, for all i≥1, we have
(3.16)2g(12limn→∞∥Sizn-zn∥)=0.
So, limn→∞∥Sizn-zn∥=0, for all i≥1, but
(3.17)∥Sizn-zn∥=∥(1-δ)zn+δJAizn-zn∥=∥δ(JAizn-zn)∥=δ∥JAizn-zn∥.
Thus,
(3.18)limn→∞∥JAizn-zn∥=0,∀i≥1.
Next, we show that limn→∞∥zn-Gzn∥=0. Observe that
(3.19)zn-Gzn=αnu+∑i=1∞(σi,n-λi)[(1-δ)zn+δJAizn].
So,
(3.20)∥zn-Gzn∥≤αn∥u∥+M∑i=1∞|σi,n-λi|
for some M>0. Hence,
(3.21)limn→∞∥zn-Gzn∥=0.
This completes the proof.Theorem 3.5.
LetK be a nonempty closed convex subset of a uniformly convex real Banach space E with uniformly Gâteaux differentiable norm. Let Ai:K→E, i=1,2,…, be a countably infinite family of m-accretive mappings such that ⋂i=1∞Ai-1(0)≠∅. Let {zn} be a sequence satisfying (3.6). Let {λi}i=1∞ be a sequence in (0,1) such that ∑i=1∞λi=1 and limn→∞∑i=1∞|σi,n-λi|=0. Let G:=(1-δ)I+δT, where T:=∑i=1∞λiJAi. Then, {zn} converges strongly to an element in ⋂i=1∞Ai-1(0).Proof.
Observe that by Lemma2.2, T:=∑i=1∞λiJAi is well defined, nonexpansive, and F(T)=⋂i=1∞F(JAi)=⋂i=1∞Ai-1(0). Furthermore, it is easy to see that G is nonexpansive and that F(G)=F(T)=⋂i=1∞Ai-1(0). Now, since {zn} is bounded and limn→∞∥Gzn-zn∥=0, we have by Lemma 3.1 that there exists a unique z* in the set Ω*:={x∈K∩B*:μn∥zn-x∥2=miny∈K∥zn-y∥} such that Gz*=z*, where B* is a bounded closed convex nonempty subset of E such that u,zn∈B* for all n∈ℕ. Thus, z*∈F(G)=⋂i=1∞Ai-1(0). Let t∈(0,1), then by convexity of K∩B*, we have that (1-t)z*+tu∈K∩B*. Thus, μn∥zn-z*∥2≤μn∥zn-((1-t)z*+tu)∥2=μn∥zn-z*-t(u-z*)∥2. Moreover, we have, by Lemma 2.1 that
(3.22)∥zn-z*-t(u-z*)∥2≤∥zn-z*∥2-2t〈u-z*,j(zn-z*-t(u-z*))〉.
This implies that μn〈u-z*,j(zn-z*-t(u-z*))〉≤0. Furthermore, since E has uniformly Gâteaux differentiable norm, we obtain that
(3.23)limt→0(〈u-z*,j(zn-z*)〉-〈u-z*,j(zn-z*-t(u-z*))〉)=0.
Thus, given ϵ>0, there exists δϵ>0 such that for all t∈(0,δϵ) and for all n∈ℕ,
(3.24)〈u-z*,j(zn-z*)〉<ϵ+〈u-z*,j(zn-z*-t(u-z*))〉.
Taking Banach limit on both sides of this inequality, we obtain
(3.25)μn〈u-z*,j(zn-z*)〉≤ϵ;
and since ϵ>0 is arbitrary, we have that
(3.26)μn〈u-z*,j(zn-z*)〉≤0.
Now, using (3.6), we have that
(3.27)∥zn-z*∥2=〈αn(u-z*)+∑i=1∞σi,n(((1-δ)zn+δJAizn)-z*),j(zn-z*)〉≤αn〈u-z*,j(zn-z*)〉+(1-αn)∥zn-z*∥2.
So,
(3.28)∥zn-z*∥2≤〈u-z*,j(zn-z*)〉.
Again, taking Banach limit, we obtain
(3.29)μn∥zn-z*∥2≤μn〈u-z*,j(zn-z*)〉≤0,
so that μn∥zn-z*∥2=0. Hence, there exists a subsequence {znl}l=1∞ of {zn}n=1∞ such that liml→∞znl=z*. We now show that {zn}n=1∞ actually converges to z*. Suppose there is another subsequence {znk}k=1∞ of {zn}n=1∞ such that limk→∞znk=u*. Then, since limn→∞∥JAizn-zn∥=0 and JAi is continuous for all i∈ℕ, we have that u*∈⋂i=1∞Ai-1(0). Claim 1 (u*=z*).
Suppose for contradiction thatu*≠z*, then ∥u*-z*∥>0, but using (3.6), we have that
(3.30)∥znl-u*∥2=〈αnl(u-u*)+∑l=1∞σi,nl(((1-δ)znl+δJAiznl)-u*),j(znl-u*)〉=αnl〈u-z*,j(znl-u*)〉+αnl〈z*-znl,j(znl-u*)〉+αnl∥znl-u*∥2+∑i=1∞σi,nl〈(1-δ)znl+δJAiznl-u*,j(znl-u*)〉≤αnl〈u-z*,j(znl-u*)〉+αnl〈z*-znl,j(znl-u*)〉+αnl∥znl-u*∥2+(1-δ)(1-αnl)∥znl-u*∥2+δ(1-αnl)∥znl-u*∥2=αnl〈u-z*,j(znl-u*)〉+αnl〈z*-znl,j(znl-u*)〉+∥znl-u*∥2.
Thus,
(3.31)〈u-z*,j(u*-znl)〉≤∥znl-u*∥∥znl-z*∥.
Using the fact that {zn}n=1∞ is bounded and that E has a uniformly Gâteaux differentiable norm, we obtain from (3.31) that
(3.32)〈u-z*,j(u*-z*)〉≤0.
Similarly, we also obtain that 〈u-u*,j(z*-u*)〉≤0 or rather
(3.33)〈u*-u,j(u*-z*)〉≤0.
Adding (3.32) and (3.33), we have that ∥z*-u*∥≤0, a contradiction. Thus, z*=u*. Hence, {zn}n=1∞ converges strongly to z*∈⋂i=1∞Ai-1(0). This completes the proof.Theorem 3.6.
LetK be a nonempty closed convex subset of a uniformly convex real Banach space E with uniformly Gâteaux differentiable norm. Let Ai:K→E, i=1,2,…, be a countably infinite family of m-accretive mappings such that ⋂i=1∞Ai-1(0)≠∅. Let {zn} be a sequence satisfying (3.6). If at least one of the mappings JAi is demicompact, then {zn} converges strongly to an element of ⋂i=1∞Ai-1(0).Proof.
For fixeds∈ℕ, let JAs be demicompact. Since limn→∞∥zn-JAszn∥=0, there exists a subsequence say {znk} of {zn} that converges strongly to some point z*∈K. By continuity of JAi and the fact that limk→∞∥znk-JAiznk∥=0, i=1,2,…, we have that z*∈⋂i=1∞Ai-1(0). Assuming that there is another subsequence {znj}j=1∞ of {zn} that converges strongly to a point say, q*, then following the argument of the last part of proof of Theorem 3.5, we obtain that {zn}n=1∞ converges strongly to z*∈⋂i=1∞Ai-1(0). This completes the proof.Theorem 3.7.
LetK be a nonempty closed convex subset of a uniformly convex real Banach space E with uniformly Gâteaux differentiable norm and also admits weakly sequential continuous duality mapping. Let Ai:K→E, i=1,2,…, be a countably infinite family of m-accretive mappings such that ⋂i=1∞Ai-1(0)≠∅. Let {zn} be a sequence satisfying (3.6). Then, {zn} converges strongly to an element of ⋂i=1∞Ai-1(0).Proof.
Since{zn} is bounded, there exists a subsequence say {znk} of {zn} that converges weakly to some point z*∈K. Using the demiclosedness property of (I-JAi) at 0 for each i≥1 (since JAi is nonexpansive for each i∈ℕ, see, e.g., [31]) and the fact that limk→∞∥znk-JAiznk∥=0, we get that z*∈⋂i=1∞Ai-1(0). We also observe from (3.6) that
(3.34)∥znk-z*∥2=〈αnku+∑i=1∞σi,nk((1-δ)znk+δJAiznk)-z*,j(znk-z*)〉≤αnk〈u-z*,j(znk-z*)〉+∑i=1∞σi,nk∥znk-z*∥2=αnk〈u-z*,j(znk-z*)〉+(1-αnk)∥znk-z*∥2.
This implies that ∥znk-z*∥2≤〈u-z*,j(znk-z*)〉. Using the fact that j is weakly sequential continuous, then we have from the last inequality that {znk} converges strongly to z*. Then following the argument of the last part of proof of Theorem 3.5, we obtain that {zn}n=1∞ converges strongly to z*∈⋂i=1∞Ai-1(0). This completes the proof.


## 4. Convergence Theorems for Countably Infinite Family ofm-Accretive Mappings

For the rest of this paper,{αn}n=1∞ and {σi,n}n=1∞ are in (0,1) satisfying the following additional conditions:(i)
limn→∞αn=0,(ii)
∑n=1∞αn=∞,(iii)
limn→∞∑i=1∞|σi,n+1-σi,n|=0.Theorem 4.1.
LetK be a nonempty closed convex subset of a uniformly convex real Banach space E with uniformly Gâteaux differentiable norm. Let Ai:K→E, i=1,2,…, be a countably infinite family of m-accretive mappings such that ⋂i=1∞Ai-1(0)≠∅. For fixed δ∈[γ1,γ2], for some γ1,γ2∈(0,1), u∈K, let {xn}n=1∞ be generated by
(4.1)x1∈K,xn+1=αnu+∑i=1∞σi,n((1-δ)xn+δJAixn),n≥1,
then limn→∞∑i=1∞σi,n((1-δ)xn+δJAixn-xn)=0. Furthermore, if {αn}n≥1 is such that
(4.2)limn→∞∑i=1∞σi,n((1-δ)xn+δJAixn-xn)αn=0,
then {xn}n=1∞ converges strongly to a common zero of {Ai}i=1∞.Proof.
Using mathematical induction, it is easy to see that forx*∈⋂i=1∞Ai-1(0) fixed
(4.3)∥xn-x*∥≤max{∥u-x*∥,∥x1-x*∥},∀n≥1.
Hence, {xn}n=1∞ is bounded and so {JAixn}n=1∞ is also bounded.
Now, define the sequences{βn}n=1∞ and {yn}n=1∞ by βn:=(1-δ)αn+δ and yn:=(xn+1-xn+βnxn)/βn. Then,
(4.4)yn=αnu+δ∑i≥1σi,n((1-δ)xn+δJAixn)βn.
Observe that {yn}n=1∞ is bounded and that
(4.5)∥yn+1-yn∥-∥xn+1-xn∥≤|αn+11-βn+1-αn1-βn|∥u∥+|δ(1-αn+1)βn+1-1|∥xn+1-xn∥+δMβn+1βn∑i=1∞|σi,n+1-σi,n|+δMβn+1βn|βn+1-βn|,
for some M>0. Thus,
(4.6)lim supn→∞(∥yn+1-yn∥-∥xn+1-xn∥)≤0.
Hence, by Lemma 2.4, we have limn→∞∥yn-xn∥=0. Consequently, we have
(4.7)limn→∞∥xn+1-xn∥=limn→∞βn∥yn-xn∥=0.
From (4.1), we have that
(4.8)xn+1-xn=αn(u-xn)+∑i=1∞σi,n(((1-δ)xn+δJAixn)-xn),
which implies that ∥∑i=1∞σi,n((1-δ)xn+δJAixn-xn)∥≤∥xn+1-xn∥+αn∥u-xn∥ and thus
(4.9)limn→∞∥∑i=1∞σi,n(((1-δ)xn+δJAixn)-xn)∥=0.
Let {zn}n=1∞ be a sequence satisfying (3.6). Then, by Theorem 3.5, zn→z*∈⋂i=1∞Ai-1(0). Using Lemma 2.1, we have that
(4.10)∥zn-xn∥2≤∥∑i=1∞σi,n(((1-δ)zn+δJAizn)-((1-δ)xn+δJAixn)+((1-δ)xn+δJAixn)-xn)∥2+2αn〈u-xn,j(zn-xn)〉≤((1-αn)∥zn-xn∥+∥∑i=1∞σi,n(((1-δ)xn+δJAixn)-xn)∥)2+2αn〈u-zn,j(zn-xn)〉.
This implies that
(4.11)〈u-zn,j(xn-zn)〉≤αn2∥zn-xn∥2+(1-αn)∥zn-xn∥·∥∑i=1∞σi,n(((1-δ)xn+δJAixn)-xn)∥αn+∥∑i=1∞σi,n(((1-δ)xn+δJAixn)-xn)∥22αn
and hence,
(4.12)lim supn→∞〈u-zn,j(xn-zn)〉≤0.
Moreover, we have that
(4.13)〈u-zn,j(xn-zn)〉=〈u-z*,j(xn-z*)〉+〈u-z*,j(xn-zn)-j(xn-z*)〉+〈z*-zn,j(xn-zn)〉.
Using the boundedness of {xn}n=1∞, we have
(4.14)〈z*-zn,j(xn-zn)〉→0,asn→∞.
Also, since j is norm-to-weak* uniformly continuous on bounded subsets of E, we have that
(4.15)〈u-z*,j(xn-zn)-j(xn-z*)〉→0,asn→∞.
From (4.12) and (4.13), we obtain that
(4.16)lim supn→∞〈u-z*,j(xn-z*)〉≤0.
Finally, using Lemma 2.1 on (4.1), we get
(4.17)∥xn+1-z*∥2≤∥∑i=1∞σi,n+1(((1-δ)xn+δJAixn)-z*)∥2+2αn〈u-z*,j(xn+1-z*)〉≤(1-αn)∥xn-z*∥2+2αn〈u-z,j(xn+1-z*)〉.
Using (4.16) and Lemma 2.3 in (4.17), we get that {xn}n=1∞ converges strongly to common zero of the family {Ai}i=1∞ of m-accretive operators.Remark 4.2.
IfK is replaced with E in Theorems 3.5, 3.6, 3.7, and 4.1, then by Lemma 2.5, the assumption that Ai is m-accretive for each i≥1 could be replaced with Ai is continuous for each i≥1.Hence, we have the following theorem.Theorem 4.3.
LetE be a uniformly convex real Banach space with uniformly Gâteaux differentiable norm. Let Ai:E→E, i=1,2,…, be a countably infinite family of continuous accretive operators such that ⋂i=1∞Ai-1(0)≠∅. For arbitrary but fixed δ∈(0,1), u∈K, let {xn}n=1∞ be generated by x1∈K,
(4.18)xn+1=αnu+∑i=1∞σi,n((1-δ)xn+δJAixn),n≥1,
then, {xn}n=1∞ converges strongly to a common zero of {Ai}i=1∞.Proof.
By Lemma2.5, we have that Ai is m-accretive for each i≥1. Then, the rest of the proof follows from Theorem 4.1.We also have the following theorems.Theorem 4.4.
LetK, E, Ai and {xn}n≥1 be as an Theorem 4.1. Suppose that at least one of JAi is demicompact, then {xn}n≥1 converges strongly to a common zero of Ai, i=1,2,….Proof.
The proof follows as in the proof of Theorem4.1 but using Theorem 3.6.Theorem 4.5.
LetK, E, Ai and {xn}n≥1 be as an Theorem 4.1. Suppose that, in addition, E admits weakly sequential continuous duality mapping, then {xn}n≥1 converges strongly to a common zero of Ai, i=1,2,….Proof.
The proof follows as in the proof of Theorem4.1 but using Theorem 3.7.


## 5. Convergence Theorems for Countably Infinite Family of Pseudocontractive Mappings

Theorem 5.1.
LetK be a nonempty closed convex subset of a uniformly convex real Banach space E with uniformly Gâteaux differentiable norm. Let Ti:K→E, i=1,2,…, be a countably infinite family of pseudocontractive mappings such that for each i≥1, (I-Ti) is m-accretive and ⋂i=1∞F(Ti)≠∅. Let JTi=(2I-Ti)-1, i≥1. For fixed δ∈[γ1,γ2], for some γ1,γ2∈(0,1) and u∈K, let {xn}n=1∞ be generated by x1∈K:
(5.1)xn+1=αnu+∑i=1∞σi,n((1-δ)xn+δJTixn),n≥1,
then {xn}n=1∞ converges strongly to a common fixed point of {Ti}i=1∞.Proof.
PutAi:=(I-Ti), i≥1. It is then obvious that Ai-1(0)=F(Ti), for all i∈ℕ and hence ⋂i=1∞Ai-1(0)=⋂i=1∞F(Ti). Furthermore, Ai is m-accretive for each i≥1. Thus, we obtain the conclusion from Theorem 4.1 with JAi in the definition of zn replaced with JTi.Theorem 5.2.
LetE be a uniformly convex real Banach space with uniformly Gâteaux differentiable norm. For each i≥1, let Ti:E→E be a countably infinite family of continuous pseudocontractive mappings such that for each i≥1 and ⋂i=1∞F(Ti)≠∅. Let JTi=(2I-Ti)-1, i≥1. For arbitrary but fixed δ∈[γ1,γ2], for some γ1,γ2∈(0,1) and u∈K, let {xn}n=1∞ be generated by x1∈K:
(5.2)xn+1=αnu+∑i=1∞σi,n((1-δ)xn+δJTixn),n≥1,
then {xn}n=1∞ converges strongly to a common fixed point of {Ti}i=1∞.Remark 5.3.
Theorems similar to Theorems4.4 and 4.5 could also be obtained for countably infinite family of pseudocontractive mappings.Remark 5.4.
Prototypes for our iteration parameters are(5.3)αn=1n+1,σi,n=n2i(n+1),λi=12i.Remark 5.5.
The addition of bounded error terms in any of our recursion formulas leads to no further generalization.Remark 5.6.
Iff:K→K is a contraction map and we replace u by f(xn) in the recursion formulas of our theorems, we obtain what some authors now call viscosity iteration process. We observe that all our theorems in this paper carry over trivially to the so-called viscosity process. One simply replaces u by f(xn), repeats the argument of this paper, using the fact that f is a contraction map.



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*Source: 910487-2009-11-11.xml*