# Convergence of Implicit and Explicit Schemes for an Asymptotically Nonexpansive Mapping inq-Uniformly Smooth and Strictly Convex Banach Spaces

Authors: Meng Wen; Changsong Hu; Zhiyu Wu
Journal: Journal of Applied Mathematics
 (2012)

---

## Abstract

We introduce a new iterative scheme with Meir-Keeler contractions for an asymptotically nonexpansive mapping inq-uniformly smooth and strictly convex Banach spaces. We also proved the strong convergence theorems of implicit and explicit schemes. The results obtained in this paper extend and improve many recent ones announced by many others.


---

## Body

## 1. Introduction

LetE be a real Banach space. With J:E→2E*, we denote the normalized duality mapping given by
(1.1)J(x)={f∈E*:〈x,f〉=‖x‖2,‖f‖=‖x‖},
where 〈·,·〉 denotes the generalized duality pairing and E* the dual space of E. In the sequel we will donate single-valued duality mappings by j. Given q>1, by Jq we will denote the generalized duality mapping given by
(1.2)Jq(x)={f∈E*:〈x,f〉=‖x‖q,‖f‖=‖x‖q-1}.We recall that the following relation holds:(1.3)Jq(x)=‖x‖q-2J(x),
for x≠0.We recall that the modulus of smoothness ofE is the function ρE:[0,∞)→[0,∞) defined by
(1.4)ρE(t):=sup⁡{12(‖x+y‖+‖x-y‖)-1:‖x‖≤1,‖y‖≤t}.E is said to be uniformly smooth if limt→0(ρE(t)/t)=0.Letq>1. E is said to be q-uniformly smooth if there exists a constant c>0 such that ρE(t)≤ctq. Examples of such spaces are Hilbert spaces and Lp (or lp).We note that aq-uniformly smooth Banach space is uniformly smooth. This implies that its norm uniformly Fréchet differentiable (see [1]).IfE is uniformly smooth, then the normalized duality map j is single-valued and norm to norm uniformly continuous.LetE be a real Banach space and C is a nonempty closed convex subset of E. A mapping T:C→C is said to be asymptotically nonexpansive if there exists a sequence {hn}⊂[0,∞) with limn→∞hn=0 such that
(1.5)‖Tnx-Tny‖≤(1+hn)‖x-y‖,x,y∈C,n≥1,
and F(T) denotes the set of fixed points of the mapping T; that is, F(T)={x∈C:Tx=x}. For asymptotically nonexpansive self-map T, it is well known that F(T) is closed and convex (see e.g., [2]).Theorem 1.1 (Banach [3]).
Let(X,d) be a complete metric space and let f be a contraction on X; that is, there exists r∈(0,1) such that d(f(x),f(y))≤rd(x,y) for all x,y∈X. Then f has a unique fixed point.Theorem 1.2 (Meir and Keeler [4]).
Let(X,d) be a complete metric space and let ϕ be a Meir-Keeler contraction (MKC) on X, that is, for every ε>0, there exists δ>0 such that d(x,y)<ε+δ implies d(ϕ(x),ϕ(y))<ε for all x,y∈X. Then ϕ has a unique fixed point.This theorem is one of generalizations of Theorem1.1, because contractions are Meir-Keeler contractions.We recall that, given aq-uniformly smooth and strictly convex Banach space E with a generalized duality map Jq:E→E* and C a subset of E, a mapping F:C→C is called(1)
k′-Lipschitzian, if there exists a constant k′>0 such that
(1.6)‖Fx-Fy‖≤k′‖x-y‖holds for everyx and y∈C;(2)
η-strongly monotone, if there exists a constant η>0 such that
(1.7)〈Fx-Fy,jq(x-y)〉≥η‖x-y‖qholds for everyx,y∈C and jq(x-y)∈Jq(x-y).In2010, Ali and Ugwunnadi [5] introduced and considered the following iterative scheme:
(1.8)x0∈H,xn+1=βnxn+(1-βn)yn,yn=(I-αnA)Ti(n+1)p(n+1)xn+αnγf(xn),∀n≥1,
where T1,T2,…,TN a family of asymptotically nonexpansive self-mappings of H with sequences {1+kp(n)i(n)}, such that kp(n)i(n)→0 as n→∞ and f:H→H are a contraction mapping with coefficient α∈(0,1). Let A be a strongly positive-bounded linear operator with coefficient γ¯>0, and 0<γ<γ¯/α. They proved the strong convergence of the implicit and explicit schemes for a common fixed point of the family T1,T2,…,TN, which solves the variational inequality 〈(A-γf)x¯,x¯-x〉≤0,forallx∈⋂i=1N‍Fix(Ti).Motivated and inspired by the results of Ali and Ugwunnadi [5], we introduced an iterative scheme as follows. for x1=x∈C,
(1.9)xn+1=βnxn+(1-βn)yn,yn=(I-μαnF)Tnxn+αnγϕ(xn),∀n≥1,
where T is an asymptotically nonexpansive self-mapping of C with sequences {1+hn}, such that hn→0 as n→∞ and ϕ:C→C are a Meir-Keeler contraction (MKC, for short). Let F is a k′-Lipschitzian and η-strongly monotone operator with 0<μ<min{(qη/Cq(k′)q)1/(q-1),1}. We will prove the strong convergence of the implicit and explicit schemes for a fixed point of T, which solves the variational inequality 〈(γϕ-μF)p,Jq(z-p)〉≤0, for z∈F(T). Our results improve and extend the results of Ali and Ugwunnadi [5] for an asymptotically nonexpansive mapping in the following aspects:(i)
Hilbert space is replaced by aq-uniformly smooth and strictly convex Banach space;(ii)
contractive mapping is replaced by a MKC;(iii)
Theorems3.1 and 4.1 extend the results of Ali and Ugwunnadi [5] from a strongly positive-bounded linear operator A to a k′-Lipschitzian and η-strongly monotone operator F.


## 2. Preliminaries

In order to prove our main results, we need the following lemmas.Lemma 2.1 (see [6]).
Letq>1 and E be a q-uniformly smooth Banach space, then there exists a constant Cq>0 such that
(2.1)‖x+y‖q≤‖x‖q+q〈y,jq(x)〉+Cq‖y‖q,∀x,y∈E.Lemma 2.2 (see [7, Lemma 2.3]).
Letϕ be a MKC on a convex subset C of a Banach space E. Then for each ε>0, there exists r∈(0,1) such that
(2.2)‖x-y‖≥εimplies‖ϕx-ϕy‖≤r‖x-y‖∀x,y∈C.Lemma 2.3 (see [8]).
Let{xn} and {zn} be bounded sequences in a Banach space E and {γn} be a sequence in [0,1] which satisfies the following condition:
(2.3)0<lim⁡inf⁡n→∞γn≤lim⁡sup⁡n→∞γn<1.
Suppose that xn+1=γnxn+(1-γn)zn,n≥0, and limsupn→∞(∥zn+1-zn∥-∥xn+1-xn∥)≤0. Then limn→∞∥zn-xn∥=0.Lemma 2.4 (see [9, 10]).
Let{sn} be a sequence of nonnegative real numbers satisfying
(2.4)sn+1≤(1-λn)sn+λnδn+γn,n≥0,
where {λn}, {δn} and {γn} satisfy the following conditions: (i){λn}⊂[0,1] and ∑n=0∞‍λn=∞, (ii)limsupn→∞δn≤0 or ∑n=0∞‍λnδn<∞,(iii)γn≥0(n≥0),∑n=0∞‍γn<∞. Then limn→∞sn=0.Lemma 2.5 (see [11]).
LetC be a nonempty closed convex subset of a uniformly convex Banach space E and T:C→E is an asymptotically nonexpansive mapping with F(T)≠∅. Then the mapping I-T is demiclosed at zero, that is, xn⇀x and ∥xn-Txn∥→0, then x=Tx.Lemma 2.6.
LetF be a k′-Lipschitzian and η-strongly monotone operator on a q-uniformly smooth Banach space E with k′>0,η>0,0<t<1 and 0<μ<min{(qη/Cq(k′)q)1/(q-1),1}. Then S=(I-tμF):E→E is a contraction with contractive coefficient 1-tτ and τ=(qμη-Cqμq(k′)q)/q.Proof.
From (2.1), we have
(2.5)‖Sx-Sy‖q=‖x-y-tμ(Fx-Fy)‖q≤‖x-y‖q+q〈-tμ(Fx-Fy),Jq(x-y)〉+Cq‖-tμ(Fx-Fy)‖q≤‖x-y‖q-tqμη‖x-y‖q+tCqμq(k′)q‖x-y‖q=[1-t(qμη-Cqμq(k′)q)]‖x-y‖q≤[1-tqμη-Cqμq(k′)qq]q‖x-y‖q=(1-tτ)q‖x-y‖q,
where τ=(qμη-Cqμq(k′)q)/q, and
(2.6)‖Sx-Sy‖≤(1-tτ)‖x-y‖.
Hence S is a contraction with contractive coefficient 1-tτ.Lemma 2.7 (see [5, Lemma 2.9]).
LetT:E→E be a uniformly Lipschitzian with a Lipschitzian constant L≥1, that is, there exists a constant L≥1 such that
(2.7)‖Tnx-Tny‖≤L‖x-y‖,∀x,y∈E.Lemma 2.8 (see, e.g., Mitrinović [12, page 63]).
Letq>1. Then the following inequality holds:
(2.8)ab≤1qaq+q-1qbq/(q-1),
for arbitrary positive real numbers a,b.


## 3. Main Result

Theorem 3.1.
LetE be a q-uniformly smooth and strictly convex Banach space, and C a nonempty closed convex subset of E such that C±C⊂C and have a weakly sequentially continuous duality mapping Jq from E to E*. Let T:C→C be an asymptotically nonexpansive mapping with sequences {1+hn}, such that hn→0 as n→∞ and F*:=F(T)≠∅. Let D be a bounded subset of C such that supx∈D∥Tn+1x-Tnx∥→0. Let F be a k′-Lipschitzian and η-strongly monotone operator on C with 0<μ<min{(qη/Cq(k′)q)1/(q-1),1}, and ϕ be a MKC on C with 0<γ<(qμη-Cqμq(k′)q)/q=τ. Let {αn} be a sequence in (0,1) satisfying the following conditions:(A1)
limn→∞αn=0;(A2)
limn→∞(hn/αn)=0.Let {xn} be defined by
(3.1)xn=αnγϕ(xn)+(I-αnμF)Tnxn.
Then,{xn} converges to a fixed point say p in F* which solves the variational inequality
(3.2)〈(μF-γϕ)p,Jq(p-z)〉≤0,∀z∈F*.Proof.
Letp∈F*. Since αn→0 and hn/αn→0 as n→∞, then (1-αnτ)(hn/αn)→0 as n→∞, so ∃N0∈N such that for all n≥N0, αn<(k′)-1 and (1-αnτ)(hn/αn)<(1/2)(τ-γ). Thus, for n≥N0(3.3)‖xn-p‖q=〈αnγϕ(xn)+(I-αnμF)Tnxn-p,Jq(xn-p)〉=αn〈γϕ(xn)-μFp,Jq(xn-p)〉+〈(I-αnμF)Tnxn-(I-αnμF)p,Jq(xn-p)〉=αn〈γϕ(xn)-γϕ(p),Jq(xn-p)〉+αn〈γϕ(p)-μFp,Jq(xn-p)〉+〈(I-αnμF)Tnxn-(I-αnμF)p,Jq(xn-p)〉≤αnγ‖xn-p‖q+(1-αnτ)(1+hn)‖xn-p‖q+αn〈γϕ(p)-μFp,Jq(xn-p)〉=[1-αn(τ-γ)+(1-αnτ)hn]‖xn-p‖q+αn〈γϕ(p)-μFp,Jq(xn-p)〉≤〈γϕ(p)-μFp,Jq(xn-p)〉(τ-γ)-(1-αnτ)(hn/αn)≤〈γϕ(p)-μFp,Jq(xn-p)〉(τ-γ)-(1/2)(τ-γ)≤2‖γϕ(p)-μFp‖‖xn-p‖q-1τ-γ.
Therefore,
(3.4)‖xn-p‖≤2‖γϕ(p)-μFp‖τ-γ.
Thus, {xn} is bounded and therefore {ϕ(xn)} and {μFTnxn} are also bounded. Also from (3.1), we have
(3.5)‖xn-Tnxn‖=αn‖γϕ(xn)-μFTnxn‖⟶0asn⟶∞.
From (3.5) and ∥Tn+1xn-Tnxn∥→0, we obtain
(3.6)‖Tn+1xn-xn‖≤‖Tn+1xn-Tnxn‖+‖Tnxn-xn‖⟶0,‖Tn+1xn-Txn‖≤(1+h1)‖Tnxn-xn‖⟶0,
Thus,
(3.7)‖Txn-xn‖≤‖Txn-Tn+1xn‖+‖Tn+1xn-xn‖⟶0.
Since {xn} is bounded, now assume that p is a weak limit point of {xn} and a subsequence {xnj} of {xn} converges weakly to p. Then, by Lemma 2.5 and (3.7), we have that p is a fixed point of T, hence p∈F*.
Next we observe that the solution of the variational inequality (3.2) in F* is unique. Assume that q~,p∈F* are solutions of the inequality (3.2), without loss of generality, we may assume that there is a number ε such that ∥p-q~∥≥ε. Then by Lemma 2.2, there is a number r such that ∥ϕp-ϕq~∥≤r∥p-q~∥. From (3.2), we know
(3.8)〈(μF-γϕ)p,Jq(p-q~)〉≤0,(3.9)〈(μF-γϕ)q~,Jq(q~-p)〉≤0.
Adding (3.8) and (3.9), we have
(3.10)〈(μF-γϕ)p-(μF-γϕ)q~,Jq(p-q~)〉≤0.
Noticing that
(3.11)〈(μF-γϕ)p-(μF-γϕ)q~,Jq(p-q~)〉=〈μFp-μFq~,Jq(p-q~)〉-〈γϕp-γϕq~,Jq(p-q~)〉≥μη‖p-q~‖q-γ‖ϕp-ϕq~‖‖p-q~‖q-1≥μη‖p-q~‖q-γr‖p-q~‖q≥(μη-γr)‖p-q~‖q≥(μη-γr)εq>0.
Therefore p=q~. That is, p∈F* is the unique solution of (3.2).
Finally, we show thatxn→p as n→∞. From (3.3), we get
(3.12)‖xn-p‖q≤αn〈γϕ(p)-μFp,Jq(xn-p)〉αn(τ-γ)-(1-αnτ)hn=〈γϕ(p)-μFp,Jq(xn-p)〉(τ-γ)-(1-αnτ)(hn/αn),
and in particular
(3.13)‖xnj-p‖q≤〈γϕ(p)-μFp,Jq(xnj-p)〉(τ-γ)-(1-αnjτ)(hnj/αnj).
Since xnj⇀p, from the above inequality and Jq is a weakly sequentially continuous duality mapping, we have xnj→p as j→∞. Next, we show that p solves the variational inequality (3.2). Indeed, from the relation
(3.14)xn=αnγϕ(xn)+(I-αnμF)Tnxn,
we get
(3.15)(μF-γϕ)xn=-1αn[(I-Tn)xn-αnμFxn+αnμFTnxn].
So, for any z∈F*(3.16)〈(μF-γϕ)xn,Jq(xn-z)〉=-1αn〈(I-Tn)xn-αnμFxn+αnμFTnxn,Jq(xn-z)〉=-1αn〈(I-Tn)xn-(I-Tn)z,Jq(xn-z)〉+〈(μF-μFTn)xn,Jq(xn-z)〉≤-1αn‖xn-z‖q+1αn(1+hn)‖xn-z‖q+〈(μF-μFTn)xn,Jq(xn-z)〉≤hnαn‖xn-z‖q+〈(μF-μFTn)xn,Jq(xn-z)〉.
Now replacing n in (3.16) with nj and letting j→∞, using (μF-μFTn)xnj→(μF-μFTn)p=0 for p∈F*, and the fact that xnj→p as j→∞, we obtain 〈(μF-γϕ)p,Jq(p-z)〉≤0,∀z∈F*. This implies that p∈F* is a solution of the variational inequality (3.2). Every weak limit of {xn} say p belongs to F*. Furthermore, p is a strong limit of {xn} that solves the variational inequality (3.2). As this solution is unique we get that xn→p as n→∞. This completes the proof.Corollary 3.2.
LetE be a q-uniformly smooth and strictly convex Banach space, and let C be a nonempty closed convex subset of E such that C±C⊂C and have a weakly sequentially continuous duality mapping Jq from E to E*. Let T:C→C be a nonexpansive mapping. Let {αn} be a sequence in (0, 1) satisfying limn→∞αn=0. Let ϕ and F be as in Theorem 3.1. For T, let {xn} be defined by
(3.17)xn=αnγϕ(xn)+(I-αnμF)Txn.
Then, {xn} converges to a fixed point say p in F* which solves the variational inequality (3.2).


## 4. Explicit Algorithm

Theorem 4.1.
LetE be a q-uniformly smooth and strictly convex Banach space, and let C be a nonempty closed convex subset of E such that C±C⊂C and have a weakly sequentially continuous duality mapping Jq from E to E*. Let T:C→C be an asymptotically nonexpansive mapping with sequences {1+hn}, such that hn→0 as n→∞ and F*:=F(T)≠∅. Let D be a bounded subset of C such that supx∈D∥Tn+1x-Tnx∥→0. Let F be a k′-Lipschitzian and η-strongly monotone operator on C with 0<μ<min{(qη/Cq(k′)q)1/(q-1),1}, and ϕ be a MKC on C with 0<γ<(qμη-Cqμq(k′)q)/q=τ. Let {αn}, {βn} be sequences in (0,1) satisfying the following conditions:(B1)
limn→∞αn=0;(B2)
limn→∞(hn/αn)=0;(B3)
∑n=1∞‍αn=∞;(B4)
0<liminfn→∞βn≤limsupn→∞βn<1.Then, {xn} defined by (1.9) converges strongly to a fixed point say p in F* which solves the variational inequality (3.2).Proof.
Sinceαn→0 and hn/αn→0 as n→∞, (1-αnτ)(hn/αn)→0 as n→∞. Thus, ∃N0∈N such that (1-αnτ)(hn/αn)<(1/2)(τ-γ) and αn<(k′)-1, foralln≥N0. For any point p∈F* and n≥N0,
(4.1)‖yn-p‖=‖αn(γϕ(xn)-μFp)+(I-αnμF)Tnxn-(I-αnμF)p‖≤αnγ‖xn-p‖+αn‖γϕ(p)-μFp‖+(1-αnτ)(1+hn)‖xn-p‖=[1-αn(τ-γ)+(1-αnτ)hn]‖xn-p‖+αn‖γϕ(p)-μFp‖.
But
(4.2)‖xn+1-p‖≤βn‖xn-p‖+(1-βn)‖yn-p‖.
Therefore,
(4.3)‖xn+1-p‖≤[βn+(1-βn)[1-αn(τ-γ)+(1-αnτ)hn]]‖xn-p‖+αn(1-βn)‖γϕ(p)-μFp‖=[1-αn(1-βn)((τ-γ)-(1-αnτ)hnαn)]‖xn-p‖+αn(1-βn)‖γϕ(p)-μFp‖≤[1-αn(1-βn)[12(τ-γ)]]‖xn-p‖+(αn(1-βn)/2)(τ-γ)(1/2)(τ-γ)‖γϕ(p)-μFp‖≤max⁡{‖xn-p‖,2‖γϕ(p)-μFp‖τ-γ}.
By induction, we have
(4.4)‖xn-p‖≤max⁡{‖xN0-p‖,2‖γϕ(p)-μFp‖τ-γ},n≥N0.
Next we show that
(4.5)lim⁡n→∞‖xn+1-xn‖=0.
From (1.9),
(4.6)yn+1-yn=αn+1γϕ(xn+1)+(I-αn+1μF)Tn+1xn+1-αnγϕ(xn)+(I-αnμF)Tnxn.
Therefore,
(4.7)‖yn+1-yn‖=‖(I-αnμF)Tn+1xn-(I-αnμF)Tnxnαn+1γ(ϕ(xn+1)-ϕ(xn))+(αn+1-αn)γϕ(xn)+(I-αn+1μF)Tn+1xn+1-(I-αn+1μF)Tn+1xn+(I-αn+1μF)Tn+1xn-(I-αnμF)Tn+1xn+(I-αnμF)Tn+1xn-(I-αnμF)Tnxn‖.
Hence,(4.8)‖yn+1-yn‖≤αn+1γ‖xn+1-xn‖+|αn+1-αn|γ‖ϕ(xn)‖+(1-αn+1τ)(1+hn+1)‖xn+1-xn‖+|αn+1-αn|‖μF‖‖Tn+1xn‖+(1-αnτ)‖Tn+1xn-Tnxn‖,lim⁡sup⁡n→∞⁡(‖yn+1-yn‖-‖xn+1-xn‖)≤0,
and by Lemma 2.3(4.9)lim⁡n→∞‖yn-xn‖=0.
Thus, from (1.9),
(4.10)‖xn+1-xn‖=(1-βn)‖yn-xn‖⟶0asn⟶∞.
Next, we show that(4.11)lim⁡n→∞‖xn-Txn‖=0.
Since
(4.12)‖xn-Tnxn‖≤‖xn+1-xn‖+‖xn+1-Tnxn‖=‖xn+1-xn‖+‖βnxn+(1-βn)yn-Tnxn‖≤‖xn+1-xn‖+βn‖xn-Tnxn‖+(1-βn)αn(‖γϕ(xn)‖+‖μFTnxn‖).
Thus,
(4.13)‖xn-Tnxn‖≤11-βn‖xn+1-xn‖+αn(‖γϕ(xn)‖+‖μFTnxn‖).
Hence,
(4.14)lim⁡n→∞‖xn-Tnxn‖=0.
Since T is Lipschitz with constant L and for any positive number n≥1, we have
(4.15)‖xn-Txn‖≤‖xn-Tnxn‖+‖Tnxn-Tn+1xn‖+‖Tn+1xn-Txn‖≤‖xn-Tnxn‖+‖Tnxn-Tn+1xn‖+L‖Tnxn-xn‖⟶0.
Therefore,
(4.16)lim⁡n→∞‖xn-Txn‖=0.
Next we show that
(4.17)lim⁡sup⁡n→∞〈γϕ(p)-μFp,Jq(xn-p)〉≤0,
where p∈F* is the unique solution of inequality (3.2). Let {xnj} be a subsequence of {xn} such that
(4.18)lim⁡sup⁡n→∞〈γϕ(p)-μFp,Jq(xn-p)〉=lim⁡j→∞〈γϕ(p)-μFp,Jq(xnj-p)〉.
Since {xn} is bounded, we may also assume that there exists some z∈C such that xnj⇀z. From (4.11) it follows that
(4.19)xnj-Txnj⟶0asj⟶∞.
By Lemma 2.5, the weak limit z∈C of {xnj} is a fixed point of the mapping T, so this implies that z∈F*. Hence by Theorem 3.1 and Jq is a weakly sequentially continuous duality mapping, we have
(4.20)lim⁡sup⁡n→∞〈γϕ(p)-μFp,Jq(xn-p)〉=〈γϕ(p)-μFp,Jq(z-p)〉≤0.
Finally, we show that ∥xn-p∥→0. By contradiction, there is a number ε0 such that
(4.21)lim⁡sup⁡n→∞‖xn-p‖≥ε0.
Case1. Fixed ε1(ε1<ε0), if for some n≥N∈ℕ such that ∥xn-p∥≥ε0-ε1, and for the other n≥N∈ℕ such that ∥xn-p∥<ε0-ε1.
Let(4.22)Mn=2q〈γϕp-μFp,Jq(yn-p)〉(ε0-ε1)q.
From (4.20), we know limsupn→∞Mn≤0. Hence, there is a number N, when n>N, we have Mn≤τ-γ. We extract a number n0≥N satisfying ∥xn0-p∥<ε0-ε1, then we estimate ∥xn0+1-p∥(4.23)‖yn0-p‖q=‖αn0γϕ(xn0)+(I-μαn0F)Tn0xn0-p‖q=〈(I-μαn0F)Tn0xn0-(I-μαn0F)p,Jq(yn0-p)〉+αn0〈γϕ(xn0)-γϕ(p),Jq(yn0-p)〉+αn0〈γϕ(p)-μFp,Jq(yn0-p)〉≤(1-αn0τ)(1+hn0)‖xn0-p‖‖yn0-p‖q-1+αn0γ‖ϕ(xn0)-ϕ(p)‖‖yn0-p‖q-1+αn0〈γϕ(p)-μFp,Jq(yn0-p)〉<[1-αn0(τ-γ)+(1-αn0τ)hn0](ε0-ε1)‖yn0-p‖q-1+αn0〈γϕ(p)-μFp,Jq(yn0-p)〉=[1-αn0[(τ-γ)-(1-αn0τ)hn0αn0]](ε0-ε1)‖yn0-p‖q-1+αn0〈γϕ(p)-μFp,Jq(yn0-p)〉≤[1-αn02(τ-γ)](ε0-ε1)‖yn0-p‖q-1+αn0〈γϕ(p)-μFp,Jq(yn0-p)〉≤[1-αn02(τ-γ)]1q(ε0-ε1)q+q-1q‖yn0-p‖q+αn0〈γϕ(p)-μFp,Jq(yn0-p)〉≤[1-αn02(τ-γ)](ε0-ε1)q+qαn0〈γϕ(p)-μFp,Jq(yn0-p)〉.
But
(4.24)‖xn0+1-p‖q≤βn0‖xn0-p‖q+(1-βn0)‖yn0-p‖q<βn0(ε0-ε1)q+(1-βn0)‖yn0-p‖q.
Therefore,
(4.25)‖xn0+1-p‖q<βn0(ε0-ε1)q+(1-βn0)[1-αn02(τ-γ)](ε0-ε1)q+qαn0(1-βn0)〈γϕ(p)-μFp,Jq(yn0-p)〉=[1-12αn0(1-βn0)(τ-γ)](ε0-ε1)q+qαn0(1-βn0)〈γϕ(p)-μFp,Jq(yn0-p)〉=[1-12αn0(1-βn0)((τ-γ)-Mn)](ε0-ε1)q<(ε0-ε1)q.
Hence, we have
(4.26)‖xn0+1-p‖<ε0-ε1.
In the same way, we can get
(4.27)‖xn-p‖<ε0-ε1,∀n≥n0.
It contradicts the limsupn→∞∥xn-p∥≥ε0.
Case2. Fixed ε1(ε1<ε0), if ∥xn-p∥≥ε0-ε1, for all n≥N∈ℕ, from Lemma 2.2, there is a number r(0<r<1) such that
(4.28)‖ϕ(xn)-ϕ(p)‖≤r‖xn-p‖,n≥N.
It follows (1.9) that
(4.29)‖yn-p‖q=‖αnγϕ(xn)+(I-μαnF)Tnxn-p‖q=〈(I-μαnF)Tnxn-(I-μαnF)p,Jq(yn-p)〉+αn〈γϕ(xn)-γϕ(p),Jq(yn-p)〉+αn〈γϕ(p)-μFp,Jq(yn-p)〉≤(1-αnτ)(1+hn)‖xn-p‖‖yn-p‖q-1+αnγ‖ϕ(xn)-ϕ(p)‖‖yn-p‖q-1+αn〈γϕ(p)-μFp,Jq(yn-p)〉<[1-αn(τ-γr)+(1-αnτ)hn]‖xn-p‖‖yn-p‖q-1+αn〈γϕ(p)-μFp,Jq(yn-p)〉=[1-αn[(τ-γr)-(1-αnτ)hnαn]]‖xn-p‖‖yn-p‖q-1+αn〈γϕ(p)-μFp,Jq(yn-p)〉≤[1-αn2(τ-γr)]‖xn-p‖‖yn-p‖q-1+αn〈γϕ(p)-μFp,Jq(yn-p)〉≤[1-αn2(τ-γr)]1q‖xn-p‖q+q-1q‖yn-p‖q+αn〈γϕ(p)-μFp,Jq(yn-p)〉≤[1-αn2(τ-γr)]‖xn-p‖q+qαn〈γϕ(p)-μFp,Jq(yn-p)〉.
Therefore,
(4.30)‖xn+1-p‖q≤βn‖xn-p‖q+(1-βn)‖yn-p‖q<βn‖xn-p‖q+(1-βn)[1-αn2(τ-γr)]‖xn-p‖q+(1-βn)αnq〈γϕ(p)-μFp,Jq(yn-p)〉=[1-12αn(1-βn)(τ-γr)]‖xn-p‖q+(1-βn)αnq〈γϕ(p)-μFp,Jq(yn-p)〉.
By Lemma 2.4, we have that ∥xn-p∥→0 as n→∞. It contradicts the ∥xn-p∥≥ε0-ε1. This completes the proof.The following corollary follows from Theorem4.1.Corollary 4.2.
LetE be a q-uniformly smooth and strictly convex Banach space, and let C be a nonempty closed convex subset of E such that C±C⊂C and have a weakly sequentially continuous duality mapping Jq from E to E*. Let T,F*,F,ϕ and {αn} be as in Corollary 3.2. Let {xn} be defined by
(4.31)xn+1=βnxn+(1-βn)yn,yn=(I-μαnF)Txn+αnγϕ(xn),∀n≥0,
Then, {xn} converges strongly to a fixed point say p in F* which solves the variational inequality (3.2).



---

*Source: 474031-2012-08-08.xml*