APPROXIMATING CURVES FOR NONEXPANSIVE MAPPINGS ANDACCRETIVE OPERATORS

YISHENG SONG∗

COLLEGE OF MATHEMATICS AND INFORMATION SCIENCE,HENAN NORMAL UNIVERSITY, P.R. CHINA, 453007.

Abstract. In this paper, under the hypotheses that E is a reflexive Banach space which has aweakly continuous duality mapping Jϕ with gauge function ϕ, we investigate several alternativeviscosity approximation methods for finding a common element of the set of fixed points of infinitenon-expansive mappings, and prove two strong convergence theorems.

Key Words and Phrases: Alternative viscosity approximation methods, non-expansive mappings,m−accretive operator, weakly continuous duality mapping.2000 AMS Subject Classification: 49J40; 47J20; 47J25; 65J15; 41A50; 47H10; 54H25.

1. Introduction

Let E be a Banach space and E∗ be its dual space. Let K be a nonempty closed convex subsetof E and T : K → K be a mapping. T is said to be non-expansive if ‖Tx − Ty‖ ≤ ‖x − y‖ for allx, y ∈ K. The fixed point set of T is denoted by Fix(T ) := {x ∈ K;Tx = x}. We write xn ⇀ x

(respectively xn∗⇀ x) to indicate that the sequence xn weakly (respectively weak∗) converges to x;

as usual xn → x will symbolize strong convergence.In nonlinear analysis, a common approach to solving a problem with multiple solutions is to replace

it by a family of perturbed problems admitting a unique solution, and to obtain a particular originalsolution as the limit of these perturbed solutions as the perturbation vanishes. This principle arisesfor instance in minimization problems (Tikhonov regularization [3]), in partial differential equations(viscosity solutions [33, Section 33.11]), in monotone inclusions [33, Section 32.18], in variationalinequalities [5], in evolution equations, and in fixed point theory (approximating curves [6, 28]);further examples will be found in [4, 9, 33] and the references therein. One classical way to researchfixed point of nonexpansive mappings is to use contractions to approximate a non-expansive mapping(Halpern [13] and Browder [7] and Reich [18] and Xu[29]). More precisely, take t ∈ (0, 1) and definea contraction Tt : K → K by

Ttx = tu + (1− t)Tx, x ∈ K,

where u ∈ K is a fixed point. Banachs Contraction Mapping Principle guarantees that Tt has a uniquefixed point xt in K. It is unclear, in general, what is the behavior of xt as t → 0, even if T has a fixedpoint. However, in the case of T having a fixed point, Halpern [13](u = 0) and Browder [7] provedthat if E is a Hilbert space, then xt does converges strongly to the fixed point of T that is nearest to u,respectively. Reich [18] extended Halpern’s and Browder’s result to the setting of Banach spaces andproved that if E is a uniformly smooth Banach space, then xt converges strongly to a fixed point of Tand the limit defines the (unique) sunny non-expansive retraction from K onto Fix(T ). Recently, Xu[29] and Song-Chen [21] independently showed that the above result holds in a Banach space whichhas a weakly continuous duality mapping. In 2000, for T : K → K a nonexpansive mapping withF (T ) 6= ∅, and f : K → K a fixed contractive mapping, Moudafi[15] introduced the following viscosityiterative process {xn}:

xn+1 = αnf(xn) + (1− αn)Txn,

* Corresponding author Email: songyisheng123@yahoo.com.cn.1

2 YISHENG SONG

and prove that {xn} converges to a fixed point p of T in a Hilbert space. Xu [31] extended Moudafi’sresults to a uniformly smooth Banach space. (Other similar results see [22, 21, 23, 27, 24, 20].) Re-cently, in the framework of Hilbert space, Patrick Combettes-Hirstoaga [9] (also see Hirstoaga [14] andSong [20]) introduced a new iterative scheme (1.1) for a infinite family of non-expansive mappings,and showed several strongly convergent theorems. The following theorem is main one of them.

Theorem CH [9, Theorem 2.3] Let {Tt}t∈(0,1) and {St}t∈(0,1) be families of non-expansive oper-ators from Hilbert space H into itself with domain H, let f : H → H be a contraction, and supposethat F = ∩t∈(0,1)Fix(Tt) 6= ∅. Then there exists a unique point x∗ ∈ F such that x∗ = PF (fx∗). Nowset

(∀t ∈ (0, 1)) xt = Tt(xt + t(fStxt − xt)). (1.1)Then {xt}t∈(0,1) is uniquely defined. In addition, if {xt}t∈(0,1) is τ−focused with respect to {Tt}t∈(0,1)

(see Definition 3.1), F ⊂ ⋂t∈(0,1)

Fix(St), and, for every x ∈ H and every sequence {tn} in ]0, 1[ such

that limn→∞

tn = 0,

xtn→ x ∈ F and xtn

− Ttnxtn

→ 0 implies Stnxtn

→ x, (1.2)then xt → x∗ as t → 0.

The goal of this paper is to analyze the properties of the alternative viscosity approximating curves(1.3) and (1.4) for finding a common element of the set of fixed points of infinite non-expansivemappings in a Banach space which has a weakly continuous duality mapping.

zt = Tr(t)(tf(Sr(t)zt) + (1− t)zt), (1.3)

xn+1 = Trn(αnf(Srn

xn) + (1− αn)Trnxn), n ≥ 1. (1.4)

Then, we prove several strong convergence theorems which improves and develops the results ofCombettes-Hirstoaga [9], Hirstoaga [14], Song [20] and Xu [29, 31] and many other literatures whichisn’t mentioned. In particular, our main results not only generalize the main results of Combettes-Hirstoaga [9] and Hirstoaga [14] from a Hilbert space to a Banach space, but also extend the iterate(1.1) to an explicit scheme. Using this result, we obtain several corollaries which complement anddevelop the corresponding ones in [1, 8, 2, 29, 32].

2. Preliminaries

By a gauge function we mean a continuous strictly increasing function ϕ : [0,∞) → [0,∞) suchthat ϕ(0) = 0 and lim

r→∞ϕ(r) = ∞. The mapping J : E → 2E∗ defined by

Jϕ(x) = {f ∈ E∗; 〈x, f〉 = ‖x‖‖f‖, ‖f‖ = ϕ(‖x‖)},∀x ∈ E},is called the duality mapping with gauge function ϕ. In particular, the duality mapping with gaugefunction ϕ(t) = t, denoted by J , is referred to as the normalized duality mapping. Browder [6] initiatedthe study of certain classes of nonlinear operators by means of the duality mapping Jϕ. FollowingBrowder [6], we say that a Banach space E has a weakly continuous duality mapping if there exists agauge ϕ for which the duality map Jϕ is single-valued and weak-weak∗ sequentially continuous (thatis, if {xn} is a sequence in E weakly convergent to a point x, then the sequence Jϕ(xn) convergesweak∗ to Jϕ(x)). It is known that lp(1 < p < ∞) has a weakly continuous duality map with gaugeϕ(t) = tp−1. Set

Φ(t) =∫ t

0

ϕ(τ)dτ, t ≥ 0.

ThenJϕ(x) = ∂Φ(‖x‖), x ∈ E,

where ∂ denotes the subdifferential in the sense of convex analysis.

ALTERNATIVE VISCOSITY APPROXIMATING CURVES 3

The first part of the following lemma is an immediate consequence of the subdifferential inequalityand the proof of the second part can be found in [10, Theorem 5]; see also [11].

Lemma 2.1 Assume that E has a weakly continuous duality mapping Jϕ with gauge ϕ.(i) For all x, y ∈ E, there holds the inequality

Φ(‖x + y‖) ≤ Φ(‖x‖) + 〈y, Jϕ(x + y)〉.(ii) Assume a sequence {xn} in E is weakly convergent to a point x. Then there holds the identity

lim supn→∞

Φ(‖xn − y‖) = lim supn→∞

Φ(‖xn − x‖) + Φ(‖y − x‖)

for all x, y ∈ E. In particular, E satisfies Opial’s property; that is, if {xn} is a sequence weaklyconvergent to x, then there holds the inequality

lim supn→∞

‖xn − x‖ < lim supn→∞

‖xn − y‖, y ∈ E, y 6= x.

Recall that an operator T with domain D(T ) and range R(T ) in E is said to be(i) non-expansive if ‖Tx− Ty‖ ≤ ‖x− y‖ for all x, y ∈ D(T );(ii) contractive if ‖Tx− Ty‖ ≤ β‖x− y‖ for all x, y ∈ D(T ) and some β ∈ [0, 1).(iii) Let A : E → 2E be a set-valued operator. The set grA = {(x, u) ∈ E2;u ∈ Ax} is the graph of

A, the inverse A−1 of A is the set-valued operator with graph {(u, x) ∈ E2;u ∈ Ax}. Moreover, A isaccretive if for all (x, u) ∈ grA, (y, v) ∈ grA, there is a j(x−y) ∈ J(x−y) such that 〈u−v, j(x−y)〉 ≥ 0.

(iv) Let A : E → 2E∗ be a set-valued operator. The set grA = {(x, u) ∈ E × E∗;u ∈ Ax} is thegraph of A. A is said to be monotone if 〈x− y, u− v〉 ≥ 0 for all (x, u) ∈ grA, (y, v) ∈ grA. It is wellknown that a monotone operator coincides with a accretive operator whenever E is a Hilbert space.

An accretive operator A is m−accretive if R(I + rA) = E for all r > 0. Denote by A−1(0) the zeroset of A, i.e., A−1(0) = {x ∈ D(A); 0 ∈ Ax}. The resolvent and Yosida approximation of A (r > 0)are respectively defined by Jr = (I + rA)−1 and Ar = I−Jr

r . It is known that Fix(Jr) = A−1(0)for all r > 0 and Jr is single-valued and is a non-expansive mapping from E to D(A) which will beassumed convex. For more details see [28, Capter 4,5]. We also need the demiclosedness principle fornon-expansive mappings(see [12]).

Lemma 2.2 ([12, Lemma 4]). Let E be a Banach space satisfying Opials condition and let K be anonempty closed convex subset of E. Let T : K → K be a non-expansive mapping. Then (I − T ) isdemiclosed at zero, i.e., if {xn} is a sequence in K which converges weakly to x and if the sequence{xn − Txn} converges strongly to zero, then x− Tx = 0.

If C is a nonempty convex subset of a Banach space E and D is a nonempty subset of C, then amapping P : C → D is called a retraction if P is continuous with F (P ) = D. A mapping P : C → Dis called sunny if

P (Px + t(x− Px)) = Px, ∀x ∈ C

whenever Px + t(x−Px) ∈ C and t > 0. A subset D of C is said to be a sunny non-expansive retractof C if there exists a sunny non-expansive retraction of C onto D, for more details, see [17, 19, 28].The following Lemma is well known [28, 19, 29, 32].

Lemma 2.3 Let C be nonempty convex subset of a smooth Banach space E, ∅ 6= D ⊂ C, J : E →E∗ the normalized duality mapping of E, and P : C → D a retraction. Then P is both sunny andnon-expansive if and only if there holds the inequality:

〈x− Px, J(y − Px)〉 ≤ 0 for all x ∈ C and y ∈ D. (2.1)

Hence there is at most one sunny non-expansive retraction from C onto D.Note that the inequality (2.1) is equivalent to the inequality (2.2)

〈x− Px, Jϕ(y − Px)〉 ≤ 0 for all x ∈ C and y ∈ D, (2.2)

4 YISHENG SONG

where ϕ is an arbitrary gauge. This is because there holds the relation (also see[29, 32])

Jϕ(x) =ϕ(‖x‖)‖x‖ J(x),∀x 6= 0.

Lemma 2.4([30, Lemma 2.5]) Let {an} be a sequence of nonnegative real numbers satisfying theproperty

an+1 ≤ (1− γn)an + γnβn, n ≥ 0,

where {γn} ⊂ (0, 1) and {βn} real number sequence such that

(i)∞∑

n=0γn = ∞; (ii) lim sup

n→∞βn ≤ 0.

Then {an} converges to zero, as n →∞.

3. Strong convergence of iteration scheme (1.3)

Definition 3.1 ([9]) Let {Tr(t)} (r(t) ∈ (0,+∞)) be a family of operators from E to E with domainE and let {zt}t∈(0,1) be a family in E. Then {zt}t∈(0,1) is τ−focused with respect to {Tr(t)} if, forevery x ∈ E and every sequence {tn} in (0, 1) such that tn → 0,

ztn⇀ x and ztn

− Tr(tn)ztn→ 0 implies Tr(t)x = x. (3.1)

In the sequel, we assume that E be reflexive Banach space which has a weakly sequentially contin-uous duality mapping Jϕ with gauge function ϕ.

Example 3.2 Let T : E → E be a non-expansive operator such that Fix(T ) 6= ∅, let r(t) = kt ∈(0, 1) such that inft∈(0,1) kt > 0, set Tr(t) = I + kt(T − I), and take {zt}t∈(0,1) in E. Then {zt}t∈(0,1)

is τ−focused with respect to Tr(t).

Proof. Suppose that tn → 0, ztn ⇀ x and ztn −Tr(tn)ztn → 0. Then, since inft∈(0,1) kt > 0, we obtainztn − Tztn → 0 and Lemma 2.1 and 2.2 yields x = Tx = Tr(t)x. ¤

Example 3.3Let A : E → E be a m−accretive operator such that A−1(0) 6= ∅, let rt ∈ (0,+∞)such that rt ≥ ε > 0, set Tr(t) = Jrt = (I + rtA)−1, resolvent of A, and take {zt}t∈(0,1) in E. Then{zt}t∈(0,1) is τ−focused with respect to Tr(t).

Proof. Suppose that tn → 0, ztn⇀ x and ztn

− Tr(tn)ztn→ 0. Then, since rt ≥ ε > 0, we obtain

Tr(tn)ztn = Jrtnztn ⇀ x and Artn

ztn =ztn

− Jrtnztn

rtn

→ 0.

Taking the limit as tn → 0 in the relation (Jrtnztn

, Artnztn

) ∈ grA, we get (x, 0) ∈ grA by the weaklysequentially continuity of the duality mapping Jϕ and the relation Jϕ(x) = ϕ(‖x‖)

‖x‖ J(x),∀x 6= 0. Thatis, x ∈ A−1(0) = Fix(Jrt

) = Fix(Tr(t)). ¤

Now, we present the concept of uniformly asymptotically regular semigroups of nonexpansiveoperator[8, 1, 2]. Let {Tr; r > 0} be a non-expansive operator semigroups on a Banach space E.A (one-parameter) non-expansive semigroups is a family {Tr; r > 0} of self-mappings of E such that

(i) T (0)x = x for x ∈ K;(ii) Tt+sx = TtTsx for t, s > 0 and x ∈ D(T );(iii) lim

r→0Trx = x for x ∈ D(T );

(iv) for each t > 0, Tr is non-expansive, that is,

‖Trx− Try‖ ≤ ‖x− y‖,∀x, y ∈ D(Tr).

ALTERNATIVE VISCOSITY APPROXIMATING CURVES 5

Then {Tr : r > 0} is said to be uniformly asymptotically regular (in short, u.a.r.) on E if for all h ≥ 0and any bounded subset K of E,

limr→∞

supx∈K

‖Th(Trx)− Trx‖ = 0.

Example 3.4 Let {Tr; r > 0} be an u.a.r. non-expansive operator semigroups on E such thatF := ∩r>0Fix(Tr) 6= ∅, let r(t) ∈ (0,+∞) such that r(t) → ∞ as t → 0, and take the boundedsequence {zt}t∈(0,1) in E. Then {zt} is τ−focused with respect to Tr.

Proof. Suppose that tn → 0, ztn⇀ x and ztn

− Tr(tn)ztn→ 0. Then, since r(tn) → ∞ and {Tr} is

u.a.r. non-expansive semigroups, then for all h > 0,

limn→∞

‖Th(Tr(tn)ztn)− Tr(tn)ztn

‖ ≤ limn→∞

supx∈C

‖Th(Tr(tn)x)− Tr(tn)x‖ = 0,

where C is any bounded subset of E containing {ztn}. Hence,

‖ztn − Thztn‖ ≤ ‖ztn − Tr(tn)ztn‖+ ‖Tr(tn)ztn− Th(Tr(tn)ztn

)‖+ ‖Th(Tr(tn)ztn)− Thztn

‖≤ 2‖ztn − Tr(tn)ztn‖+ ‖Th(Tr(tn)ztn)− Tr(tn)ztn‖ → 0.

That is, for all h > 0,lim

n→∞‖ztn

− Thztn‖ = 0,

and Lemma 2.1 and 2.2 yields x = Thx. Since h is arbitrary, then x ∈ F . ¤

Theorem 3.5 Let {Tr(t)} and {Sr(t)} be two families of non-expansive operators from E intoitself with domain E and ∅ 6= F = ∩r(t)∈(0,∞)Fix(Tr(t)) ⊂ ∩r(t)∈(0,∞)Fix(Sr(t)), and f : E → E acontractive mapping with contractive coefficient β ∈ (0, 1). Suppose that

(∀t ∈ (0, 1)) zt = Tr(t)(zt + t(fSr(t)zt − zt)). (3.2)

Then {zt}t∈(0,1) is well defined. In addition, if {zt}t∈(0,1) is τ−focused with respect to {Tr(t)}, thenzt → x∗ as t → 0, where x∗ is the unique point such that x∗ = PF (fx∗) and PF is the unique sunnynon-expansive retraction from E to F .

Proof. For each fixed t ∈ (0, 1), let At be defined by At := Tr(t)((1 − t)I + tf(Sr(t))). Then for allx, y ∈ E,

‖Atx−Aty‖ =‖Tr(t)((1− t)x + tf(Sr(t)x))− Tr(t)((1− t)y + tf(Sr(t)y))‖≤(1− t)‖x− y‖+ t‖f(Sr(t)x)− f(Sr(t)y)‖≤(1− t(1− β))‖x− y‖.

Namely, At is a contractive mapping in E. Thus, Banach’s Contraction Mapping Principle guaranteesthat At has a unique fixed point zt in E, i.e., {zt} satisfies (3.2) is uniquely defined for each t ∈ (0, 1).

The remainder of the proof given below employs the similar idea as the proof in [21, Theorem 2.2](also see [9, Theorem 2.3] and [11, Theorem 2.2]). Take p ∈ F to deduce that, for t ∈ (0, 1),

‖zt − p‖ =‖Tr(t)((1− t)zt + tf(Sr(t)zt))− p‖ ≤ (1− t)‖zt − p‖+ t‖f(Sr(t)zt)− p‖≤(1− t)‖zt − p‖+ t‖f(Sr(t)zt)− f(Sr(t)p)‖+ t‖f(p)− p‖≤(1− t(1− β))‖zt − p‖+ t‖f(p)− p‖.

Thus

‖zt − p‖ ≤ ‖f(p)− p‖1− β

.

Consequently, {zt} is bounded and, since

‖f(Sr(t)zt)− zt‖ ≤ ‖f(Sr(t)zt)− f(Sr(t)p)‖+ ‖f(p)− zt‖,

6 YISHENG SONG

we obtain M = supt∈(0,1)

‖f(Sr(t)zt) − zt‖ < +∞. Now set yt = zt + t(fSr(t)zt − zt). Then (3.2) yields

xt = Tr(t)yt and

‖zt − Tr(t)zt‖ =‖Tr(t)((1− t)zt + tf(Sr(t)zt))− Tr(t)zt‖≤t‖f(Sr(t)zt)− p‖ ≤ tM → 0(t → 0),

that is,

limt→0

‖zt − Tr(t)zt‖ = 0. (3.3)

Likewise,

limt→0

‖yt − zt‖ = limt→0

‖yt − Tr(t)yt‖ = 0. (3.4)

Assume tn → 0+ as n → ∞ and {ztn} is bounded. Since E is reflexive, we may assume thatztn

⇀ x∗ for some x∗ ∈ E. Since {zt}t∈(0,1) is τ−focused with respect to {Tr(t)} and (3.3) implieslimn→∞ ‖ztn − Tr(tn)ztn‖ = 0, we gain by Definition 3.1, x∗ ∈ F.

Finally we prove that {zt} converges strongly to x∗ ∈ F . Let {tn} be a sequence in (0, 1) such thattn → 0 and ztn

⇀ x∗ as n → ∞. Then the argument above shows that x∗ ∈ F . We next show thatztn

→ x∗. As a matter of fact, since Jϕ is weakly continuous, we have by Lemma 2.1,

Φ(‖ztn − x∗‖) = Φ(‖Tr(tn)((1− tn)ztn + tnf(Sr(tn)ztn))− x∗‖)≤Φ(‖(1− tn)ztn

+ tnf(Sr(tn)ztn)− x∗‖) = Φ(‖ytn

− x∗‖)=Φ(‖(1− tn)(ztn

− x∗) + tn(f(Sr(tn)ztn)− f(Sr(tn)x

∗)) + tn(f(x∗)− x∗‖)≤Φ((1− tn)‖ztn

− x∗‖+ tn‖f(Sr(tn)ztn)− f(Sr(tn)x

∗‖) + tn〈f(x∗)− x∗, Jϕ(ytn− x∗)〉

≤(1− tn(1− β))Φ(‖ztn − x∗‖) + tn〈f(x∗)− x∗, Jϕ(ytn − x∗)〉.This implies that

(1− β)Φ(‖ztn− x∗‖) ≤ 〈f(x∗)− x∗, Jϕ(ytn

− x∗)〉. (3.5)

Now observing that ytn⇀ x∗ from (3.4) implies Jϕ(ytn

− x∗) ∗⇀ 0, we conclude from the inequality

(3.5) that

Φ(‖ztn− x∗‖) → 0.

Hence ztn → x∗; furthermore, ytn → x∗.We finally prove that the entire net {zt} converges strongly. Towards this end, we use [16, Propo-

sition 2.1.31(e)] (that is, if every subnet of the entire net {zt} in a topological space E has a subnetconverging to x∗ ∈ E, then zt → x∗). Now we have proved that every subnet of the entire net {zt} inE has a subnet converging to some x∗ ∈ E as t → 0. Subsequently, we shall show that every clusterpoint of {zt} equal to x∗ as t → 0.

Suppose that there exists another subnet {zsn} ⊂ {zt} such that zsn → z, as sn → 0. Then wealso have z ∈ F . Next we show z = x∗. Indeed, for any fixed p ∈ F , it is easy to see that (notingzt = Tr(t)yt)

Φ(‖yt − p‖) =Φ(‖zt − p + t(f(Sr(t)zt)− zt)‖)≤Φ(‖Tr(t)yt − Tr(t)p‖) + t〈f(Sr(t)zt)− zt, Jϕ(yt − p)〉≤Φ(‖yt − p‖) + t〈f(Sr(t)zt)− zt, Jϕ(yt − p)〉.

This implies for t ∈ (0, 1) and p ∈ F ,

〈f(Sr(t)zt)− zt, Jϕ(p− yt)〉 ≤ 0.

In particular,

〈f(Sr(tn)ztn)− ztn

, Jϕ(p− ytn)〉 ≤ 0 and 〈f(Sr(sn)zsn

)− zsn, Jϕ(p− ysn

)〉 ≤ 0. (3.6)

ALTERNATIVE VISCOSITY APPROXIMATING CURVES 7

Furthermore,|〈f(Sr(t)zt)− zt, Jϕ(p− yt)〉 − 〈f(x∗)− x∗, Jϕ(p− x∗)〉|= |〈f(Sr(t)zt)− zt − (f(x∗)− x∗), Jϕ(p− yt)〉+ 〈f(x∗)− x∗, Jϕ(p− yt)− Jϕ(p− x∗)〉|≤ ‖(f(Sr(t)zt)− f(x∗)) + (x∗ − zt)‖ϕ(‖yt − p‖) + |〈f(x∗)− x∗, Jϕ(p− yt)− Jϕ(p− x∗)〉|≤ (1 + β)L‖x∗ − zt‖+ |〈f(x∗)− x∗, Jϕ(p− yt)− Jϕ(p− x∗)〉|,

where L is a constant such that L ≥ ϕ(‖yt − p‖). Since the duality mapping Jϕ is weakly continuousand ztn , ytn → x∗, therefore as n →∞, we obtain

|〈f(Sr(tn)ztn)− ztn

, Jϕ(z − ytn)〉 − 〈f(x∗)− x∗, Jϕ(z − x∗)〉| → 0.

In (3.6), p is replaced by z, we gain

〈f(x∗)− x∗, Jϕ(z − x∗)〉 = limn→∞

〈f(Sr(tn)ztn)− ztn , Jϕ(z − ytn)〉 ≤ 0.

Likewise, substituting p with x∗ in (3.6), we also have

〈f(z)− z, Jϕ(x∗ − z)〉 ≤ 0.

Adding up gets

(1− β)‖x∗ − z‖ϕ(‖x∗ − z‖) ≤ 〈(x∗ − z)− (f(x∗)− f(z)), Jϕ(x∗ − z)〉 ≤ 0.

Hence z = x∗ and {zt} converges strongly to x∗. Moreover, x∗ is the unique solution the followingvariational inequality:

〈f(x∗)− x∗, Jϕ(p− x∗)〉 ≤ 0.

Applying Lemma 2.3 and (2.2), we obtain that x∗ = PF (fx∗) and PF is sunny non-expansive retractionfrom E to F . ¤

Corollary 3.6 Let T be a non-expansive operator from E into itself with domain E and Fix(T ) 6= ∅,and f : E → E a contractive mapping with contractive coefficient β ∈ (0, 1). Suppose that kt ∈ (0, 1)such that inft∈(0,1) kt > 0. If {zt}t∈(0,1) and {yt}t∈(0,1) are given respectively by

(∀t ∈ (0, 1)) zt = (I + kt(T − I))(zt + t(fzt − zt)) (3.7)

and(∀t ∈ (0, 1)) yt = (I + kt(T − I))(yt + t(f(I + kt(T − I)yt)− yt)). (3.8)

Then both {zt} and {yt} are well defined, and both zt and and yt strongly converge to x∗ as t → 0, wherex∗ is the unique point such that x∗ = PFix(T )(fx∗) and PFix(T ) is the unique sunny non-expansiveretraction from E to Fix(T ).

Proof. Let Tr(t) = I + kt(T − I) and Sr(t) ≡ I (or Sr(t) ≡ Tr(t)), (3.2) turn into (3.7) (or (3.8)). Itfollows from Theorem 3.5 and Example 3.2 that the conclusion is reached. ¤

Analogously, we also obtain the following results.Corollary 3.7 Let A be a m−accretive operator such that C = A−1(0) 6= ∅, let rt ∈ (0,+∞) such

that rt ≥ ε > 0, set Jrt= (I + rtA)−1, resolvent of A, and f : E → E a contractive mapping with

contractive coefficient β ∈ (0, 1). Suppose that {zt}t∈(0,1) and {yt}t∈(0,1) are given respectively by

(∀t ∈ (0, 1)) zt = Jrt(zt + t(fzt − zt)) (3.9)

and(∀t ∈ (0, 1)) yt = Jrt

(yt + t(f(Jrtyt)− yt)). (3.10)

Then both {zt} and {yt} are well defined, and both zt and and yt strongly converge to x∗ as t → 0,where x∗ is the unique point such that x∗ = PC(fx∗) and PC is the unique sunny non-expansiveretraction from E to C.

Corollary 3.8 Let {Tr; r > 0} be an u.a.r. non-expansive operator semigroups on E such thatF := ∩r>0Fix(Tr) 6= ∅, let r(t) ∈ (0,+∞) such that r(t) →∞ as t → 0, and f : E → E a contractive

8 YISHENG SONG

mapping with contractive coefficient β ∈ (0, 1). Suppose that {zt}t∈(0,1) and {yt}t∈(0,1) are givenrespectively by

(∀t ∈ (0, 1)) zt = Trt(zt + t(fzt − zt)) (3.11)

and

(∀t ∈ (0, 1)) yt = Trt(yt + t(f(Trt

yt)− yt)). (3.12)

Then both {zt} and {yt} are well defined, and both zt and and yt strongly converge to x∗ as t → 0,where x∗ is the unique point such that x∗ = PF (fx∗) and PF is the unique sunny non-expansiveretraction from E to F .

Remark 3.9 (i) Theorem 3.5 generalizes Combettes-Hirstoaga’s result [9] to a Banach space andCorollary 3.6 is an improvement of [9, Corollary 2.8].

(ii) It can be seen from the proof of Theorem 3.5 that the condition:

xtn→ x ∈ F and xtn

− Ttnxtn

→ 0 implies Stnxtn

→ x

is not necessary in [9, Theorem 2.3] which contains in the hypothesis: {zt}t∈(0,1) is τ−focused withrespect to {Tr(t)} and t → 0.

(iii)If E is a Hilbert space, then Corollary 3.7 for a monotone operator A also holds. Corollary 3.6and Corollary 3.7 also reinforce and develop [21, Theorem 2.2] and [29, Theorem 3.1], respectively.

(iv) Corollary 3.8 is an important evolution and complementarity of [8, Theorem 3.1] and [32,Theorem 3.3].

(v) In Theorem 3.5 and Corollary 3.6, 3.7, 3.8, if f(x) ≡ u ∈ E, ∀x ∈ E and let PF u = limt→0

zt, then

PF is a sunny non-expansive retraction from E to F by Lemma 2.3. Moreover, F = ∩r>0Fix(Tr) (orC = A−1(0)) is a sunny non-expansive retract of E.

(vi) The conclusion of Theorem 3.5 should hold in a reflexive and strictly convex Banach space witha uniformly Gaeaux differentiable norm or in a uniformly smooth Banach space. The proof should bewith the help of the property of Banach Limits (see [18, 22, 23, 24, 25, 26, 28, 29, 31]).

4. Strong convergence of explicit iteration scheme

In this section, let E be a reflexive Banach space which has a weakly sequentially continuousduality mapping Jϕ with gauge function ϕ. We study the following alternative iterative method fortwo families of non-expansive operators {Tr} and {Sr}:

xn+1 = Trn(αnf(Srn

xn) + (1− αn)xn), n ≥ 1. (4.1)

Condition 4.1.(i) ∅ 6= F = ∩r∈(0,∞)Fix(Tr) ⊂ ∩r∈(0,∞)Fix(Sr);

(ii) limn→∞

αn = 0 and∞∑

n=1αn = ∞;

(iii) For every subsequence {xnk} ⊂ {xn},

{xnk

⇀ x

xn+1 − Trnxn → 0

implies x ∈ F.

Example 4.2 Let A : E → E be a m−accretive operator such that A−1(0) 6= ∅, let rn ∈ (0,+∞)such that rn → ∞, set Trn = Jrn = (I + rnA)−1, resolvent of A, and take a bounded sequence {xn}in E. Then {xn} satisfies Condition 4.1 (iii).

Proof. For any fixed subsequence {xnk} ⊂ {xn}, suppose that

xnk⇀ x(k →∞) and xn+1 − Trn

xn → 0(n →∞).

ALTERNATIVE VISCOSITY APPROXIMATING CURVES 9

Then, since rn →∞ and the boundedness of {xn} tegother with the boundedness of {Jrnxn} by the

non-expansivity of Jr, we obtain

Trnk−1xnk−1 = Jrnk−1xnk−1 ⇀ x and Arnk−1xnk−1 =xnk−1 − Jrnk−1xnk−1

rnk−1→ 0.

Taking the limit as n → ∞ in the relation (Jrnk−1xnk−1, Arnk−1xnk−1) ∈ grA, we get (x, 0) ∈grA by the weakly sequentially continuity of the duality mapping Jϕ and the relation Jϕ(x) =ϕ(‖x‖)‖x‖ J(x) for all x 6= 0. That is, x ∈ A−1(0) = F = ∩r∈(0,∞)Fix(Jr) = ∩r∈(0,∞)Fix(Tr). ¤

Example 4.3 Let {Tr; r > 0} be an u.a.r. non-expansive operator semigroups on E such thatF := ∩r>0Fix(Tr) 6= ∅, let rn ∈ (0,+∞) such that rn →∞ as n →∞, and take a bounded sequence{xn} in E. Then {xn}t∈(0,1) satisfies Condition 4.1 (iii).

Proof. For any fixed subsequence {xnk} ⊂ {xn}, suppose that

xnk⇀ x(k →∞) and xn+1 − Trnxn → 0(n →∞).

Then, since rn →∞ and {Tr} is u.a.r. non-expansive semigroups, then for all h > 0,

limn→∞

‖Th(Trnxn)− Trn

xn‖ ≤ limn→∞

supx∈C

‖Th(Trnx)− Trn

x‖ = 0,

where C is any bounded subset of E containing {xn}. Hence,

‖xn+1 − Thxn+1‖ ≤ ‖xn+1 − Trnxn‖+ ‖Trn

xn − Th(Trnxn)‖+ ‖Th(Trn

xn)− Thxn+1‖≤ 2‖xn+1 − Trn

xn‖+ ‖Th(Trnxn)− Trn

xn‖ → 0.

That is, for all h > 0,lim

n→∞‖xn+1 − Thxn+1‖ = 0,

and Lemma 2.1 and 2.2 yields x = Thx. Since h is arbitrary, then x ∈ F . ¤

Theorem 4.4 Let {Tr} and {Sr} be two families of non-expansive operators from E into itself withdomain E, and f : E → E a contractive mapping with contractive coefficient β ∈ (0, 1). Suppose that{xn} is defined by (4.1) and Condition 4.1 is satisfied. If there exists a sunny non-expansive retractionPF from E to F , then xn → x∗ as n →∞, where x∗ is the unique point such that x∗ = PF f(x∗).

Proof. Since PF is a sunny non-expansive retraction from E to F , then PF f is a contractive mappingof E into itself. In fact,

‖PF f(x)− PF f(y)‖ ≤ ‖f(x)− f(y)‖ ≤ β‖x− y‖, for all x, y ∈ E.

So, PF f is a contractive mapping of E into itself. Since E is complete, there exists a unique elementx∗ ∈ E such that x∗ = PF f(x∗). Such a x∗ ∈ E is an element of F and is the unique solution thefollowing variational inequality by Lemma 2.3:

〈f(x∗)− x∗, Jϕ(p− x∗)〉 ≤ 0, ∀p ∈ F. (4.2)

Take u ∈ F . Then from u = Tru = Sru and (4.1), we have

‖xn+1 − u‖ =‖Trn(αnf(Srn

xn) + (1− αn)xn)− u‖≤αn‖f(Srn

xn)− f(Srnu) + fu− u‖+ (1− αn)‖xn − u‖

≤(1− αn(1− β)‖xn − u‖+ αn‖f(u)− u‖

≤max{‖xn − u‖, ‖f(u)− u‖1− β

}...

≤max{‖x0 − u‖, ‖f(u)− u‖1− β

}.

10 YISHENG SONG

This implies that {xn} is bounded. Since ‖Trnxn − u‖ ≤ ‖xn − u‖ and ‖f(Srn

xn)− u‖ ≤ β‖xn − u‖,then both {f(Srn

xn)} and {Trnxn} are also bounded. Henceforth,

‖xn+1 − Trnxn‖ ≤ ‖αnf(Srn

xn) + (1− αn)xn − xn‖ ≤ αn‖f(Srnxn)− xn‖ → 0. (4.3)

Set yn = αnf(Srnxn) + (1− αn)xn, then {yn} is bounded and

‖yn − xn‖ = αn‖f(Srnxn)− xn‖ → 0. (4.4)

We next prove thatlim sup

n→∞〈f(x∗)− x∗, Jϕ(yn − x∗)〉 ≤ 0 (4.5)

Indeed, we can take a subsequence {yni} of {yn} such that

lim supn→∞

〈f(x∗)− x∗, Jϕ(yn − x∗)〉 = limi→∞

〈f(x∗)− x∗, Jϕ(yni− x∗)〉.

We may assume that yni ⇀ p by the reflexivity of E and the boundedness of {yn}. By (4.4) we havexni

⇀ p. It follows from Condition 4.1(iii) together with (4.3) that p ∈ F . From the weak continuityof the duality mapping Jϕ and (4.2), we obtain that

lim supn→∞

〈f(x∗)− x∗, Jϕ(yn − x∗)〉 = limi→∞

〈f(x∗)− x∗, Jϕ(yni − x∗)〉 = 〈f(x∗)− x∗, Jϕ(p− x∗)〉 ≤ 0.

That is, (4.5) holds. Finally to prove that xn → x∗, we apply Lemma 2.1 to get

Φ(‖xn+1 − x∗‖) = Φ(‖Trn(αnf(Srn

xn) + (1− αn)xn)− x∗‖)≤Φ(‖αnf(Srnxn) + (1− αn)xn − x∗‖) = Φ(‖yn − x∗‖)=Φ(‖(1− αn)(xn − x∗) + αn(f(Srn

xn)− f(Srnx∗)) + αn(f(x∗)− x∗)‖)

≤Φ(‖(1− αn)(xn − x∗) + αn(f(Srnxn)− f(Srn

x∗))‖) + αn〈f(x∗)− x∗, Jϕ(yn − x∗)〉≤(1− αn(1− β))Φ(‖xn − x∗‖) + αn〈f(x∗)− x∗, Jϕ(yn − x∗)〉.

Since∞∑

n=1(1− β)αn = ∞ and lim sup

n→∞1

1−β 〈f(x∗)− x∗, Jϕ(yn − x∗)〉 ≤ 0 by (4.5), then an application

of Lemma 2.4 yields that Φ(‖xn − x∗‖) → 0; that is, ‖xn − x∗‖ → 0. ¤

Corollary 4.5 Let A : E → E be a m−accretive operator such that C = A−1(0) 6= ∅, let rn ∈(0,+∞) such that rn →∞, set Trn

= Jrn= (I +rnA)−1, resolvent of A, and f : E → E a contractive

mapping with contractive coefficient β ∈ (0, 1). Suppose that {xn} and {yn} are given respectively by

xn+1 = Jrn(xn + αn(fxn − xn)) (4.6)

andyn+1 = Jrn

(yn + αn(f(Jrnyn)− yn)). (4.7)

If limn→∞

αn = 0 and∞∑

n=1αn = ∞, then both {xn} and {yn} strongly converge to x∗ as n → ∞, where

x∗ is the unique point such that x∗ = PCf(x∗) and PC is the unique sunny non-expansive retractionfrom E to C.

Proof. Since Trn= Jrn

= (I + rnA)−1 is single-valued and non-expansive, then let Srn≡ I (or

Srn ≡ Trn), (4.1) becomes into (4.6) (or (4.7)). Furthermore, from Corollary 3.7, we know thatC = A−1(0) is a sunny non-expansive retract of E. It follows from Theorem 4.4 and Example 4.2 thatthe ultimateness is arrived. ¤

Similarly, using Corollary 3.8, Theorem 4.4 and Example 4.3, we also have the following result.Corollary 4.6 Let {Tr; r > 0} be an u.a.r. non-expansive operator semigroups on E such that

F := ∩r>0Fix(Tr) 6= ∅, let rn ∈ (0,+∞) such that rn →∞ as n →∞, and f : E → E a contractivemapping with contractive coefficient β ∈ (0, 1). Suppose that {xn} and {yn} are given respectively by

xn+1 = Trn(xn + αn(fxn − xn)) (4.8)

ALTERNATIVE VISCOSITY APPROXIMATING CURVES 11

andyn+1 = Trn(yn + αn(f(Trnyn)− yn)). (4.9)

If limn→∞

αn = 0 and∞∑

n=1αn = ∞, then both {xn} and {yn} strongly converge to x∗ as n → ∞, where

x∗ is the unique point such that x∗ = PF (f(x∗)) and PF is the unique sunny non-expansive retractionfrom E to F .

Remark 4.7 (i) In Theorem 4.4, we deals with the explicit alternative viscosity approximationmethods (4.1) which develops Combettes-Hirstoaga’s main result in [9], while in [9], Combettes-Hirstoaga only study the implicit scheme.

(ii) Corollary 4.5 is a significant supplement and development of [29, Theorem 4.1,4.2].(iii) If f(x) ≡ u, then Corollary 4.6 extends [1, Theorem 20] to a Banach space, and which also

gets rid of the condition∞∑

n=0|αn − αn+1| < ∞,

∞∑n=0

|rn − rn+1| < ∞ in [1, Theorem 20]. At the same

time, which is an important evolution and complementarity of [32, Theorem 3.3], [8, Theorem 3.2]and [2, Theorem 3.1], and in particular, the control conditions lim

n→∞αn

αn+1= 1 in [8, Theorem 3.2] and

∞∑n=0

|αn − αn+1| < ∞ in [2, Theorem 3.1] are also removed, respectively.

(iv) In order to get the strong convergence of an explicit algorithm from an implicit one in Theo-rem 4.4, the key step is to consider hypothesis (iii). Thus, it will be very interesting to remove thishypothesis.

Problem. Can the hypothesis (iii) in Theorem 4.4 be removed or be replaced by the other weakerconditions?

References

1. A. Aleyner and Y. Censor, Best approximation to common fixed points of a semigroup of nonexpansive operators,J. Nonlinear Convex Anal. 6 (2005) no. 1 137-151.

2. A. Aleyner and S. Reich, An explicit Construction of sunny nonexpansive retractions in Banach spaces, Fixed Point

Theory and Applications 2005:3 (2005) 295-305.3. H. Attouch, Viscosity solutions of minimization problems, SIAM J. Optim. 6 (1996) 769–806.

4. H. Attouch and R. Cominetti, A dynamical approach to convex minimization coupling approximation with the

steepest descent method, J. Differential Equations 128 (1996) 519–540.5. F. E. Browder, Existence and approximation of solutions of nonlinear variational inequalities, Proc. Nat. Acad.

Sci. U.S.A. 56 (1966) 1080–1086.

6. F.E. Browder, Convergence theorems for sequences of nonlinear operators in Banach spaces, Math. Z. 100 (1967)201-225.

7. F.E. Browder, Fixed point theorems for noncompact mappings in Hilbert space, Proc. Nat. Acad. Sci. U.S.A. 53(1965) 1272-1276.

8. T. D. Benavides, G. Lopez Acedo and H.-K. Xu, Construction of sunny nonexpansive retractions in Banach spaces,

Bull. Austral. Math. Soc. 66 (2002) No. 1. 9-16.9. P. L. Combettes and S. A. Hirstoaga, Approximating Curves for Nonexpansive and Monotone Operators,Journal

of Convex Analysis 13 (2006) 633–646.

10. J.P. Gossez and E.L. Dozo, Some geometric properties related to the fixed point theory for non-expansive mappings,Pacfic J. Math. 40(1972) 565-573.

11. N. M. Gulevich, Fixed points of non-expansive mappings, Journal of Mathematical Sciences, 79(1996) No. 1 755-815.

12. J. Gornicki, Weak convergence theorems for asymptotically non-expansive mappings in uniformly convex Banachspaces, Comment. Math. Univ. Carolin. 30(1989) 249-252.

13. B. Halpern, Fixed points of nonexpansive maps, Bull. Amer. Math. Soc. 73(1967) 957–961.14. S. A. Hirstoaga, Iterative selection methods for common fixed point problems, J. Math. Anal Appl. 324(2006)

1020-1035.

15. A. Moudafi, Viscosity Approximation Methods for Fixed-Points Problems, J. Math. Anal Appl., 241(2000) 46-55.16. R. E. Megginson, An introduction to Banach space theory , 1998 Springer-Verlag New Tork, Inc.

17. S. Reich, Approximating zeros of accretive operators, Proc. Amer. Math. Soc. 51(1975) 381-384.

18. S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal Appl.75(1980) 287-292.

12 YISHENG SONG

19. S. Reich, Asymptotic behavior of contractions in Banach spaces, J. Math. Anal Appl. 44(1973) 57-70.

20. Y. Song, Iterative selection methods for the common fixed point problems in a Banach space, Applied Mathematicsand Computation, 193(2007) 7–17.

21. Y. Song and R. Chen, Viscosity approximation methods for nonexpansive nonself-mappings, J. Math. Anal. Appl.

321(2006) 316–326.22. Y. Song and R. Chen, Strong convergence theorems on an iterative method for a family of finite nonexpansive

mappings, Applied Mathematics and Computation, 180 (2006) 275-287.

23. Y. Song and R. Chen, Iterative approximation to common fixed points of nonexpansive mapping sequences inreflexive Banach spaces, Nonlinear Analysis, 66 (2007) 591–603.

24. Y. Song and S. Xu, Strong convergence theorems for nonexpansive semigroup in Banach spaces, J. Math. Anal.

Appl., 338(2008) 152–161.25. Y. Song and R. Chen, Convergence theorems of iterative algorithms for continuous pseudocontractive mappings,

Nonlinear Anal., 67 (2007) 3058–3063.26. Y. Song and R. Chen, An approximation method for continuous pseudocontractive mappings, Journal of Inequalities

and Applications Volume 2006, Article ID 28950, Pages 1-9 DOI 10.1155/JIA/2006/28950.

27. Y. Song, R. Chen and H. Zhou, Viscosity approximation methods for nonexpansive mapping sequences in BanachspacesNonlinear Analysis, 66(2007) 1016–1024.

28. W. Takahashi, Nonlinear Functional Analysis– Fixed Point Theory and its Applications, Yokohama Publishers inc,

Yokohama, 2000(Japanese).29. H.K. Xu, Strong convergence of an iterative method for nonexpansive and accretive operators, J. Math. Anal. Appl.

314 (2006) 631-643.

30. H.K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. 66 (2002) 240-256.31. H.K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298(2004) 279-291.

32. H.K. Xu, A strong convergence theorem for contraction semigroups in Banach spaces, Bull. Austral. Math. Soc. 72

(2005) 371-379.33. E. Zeidler, Nonlinear Functional Analysis and Its Applications II/B C Nonlinear Monotone Operators, Springer-

Verlag, New York, 1990.