NONEXPANSIVE SEMIGROUPS IN BANACH SPACES

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NONEXPANSIVE SEMIGROUPS IN BANACH SPACES

SACHIKO ATSUSHIBA AND WATARU TAKAHASHI Received 24 February 2005

We introduce an implicit iterative process for a nonexpansive semigroup and then we prove a weak convergence theorem for the nonexpansive semigroup in a uniformly convex Banach space which satisfies Opial’s condition. Further, we discuss the strong convergence of the implicit iterative process.

1. Introduction

LetCbe a closed convex subset of a Hilbert space and letT be a nonexpansive mapping fromCinto itself. For eacht∈(0, 1), the contraction mappingTtofCinto itself defined by

T_tx=tu+ (1−t)Tx (1.1)

for everyx∈C, has a unique fixed pointxt, whereuis an element ofC. Browder [4]

proved that{xt}converges strongly to a fixed point ofTast→0 in a Hilbert space. Moti- vated by Browder’s theorem [4], Takahahi and Ueda [20] proved the strong convergence of the following iterative process in a uniformly convex Banach space with a uniformly Gˆateaux diﬀerentiable norm (see also [14]):

x_k=1 kx+

1−1

k

Tx_k (1.2)

for everyk=1, 2, 3,..., wherex∈C. On the other hand, Xu and Ori [21] studied the following implicit iterative process for finite nonexpansive mappingsT1,T2,...,T_r in a Hilbert space:x0=x∈Cand

x_n=α_nx_n−1+1−α_nT_nx_n (1.3) for everyn=1, 2,..., where{α_n}is a sequence in (0, 1) andT_n=T_n+r. And they proved a weak convergence of the iterative process defined by (1.3) in a Hilbert space. Sun et al.

[17] studied the iterations defined by (1.3) and proved the strong convergence of the iterations in a uniformly convex Banach space, requiring one mappingT_iin the family to be semi compact.

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In this paper, using the idea of [17,21], we introduce an implicit iterative process for a nonexpansive semigroup and then prove a weak convergence theorem for the nonexpansive semigroup in a uniformly convex Banach space which satisfies Opial’s condition. Further, we discuss the strong convergence of the implicit iterative process (see also [1,2,7,12,13]).

2. Preliminaries and notations

Throughout this paper, we denote byNandZ⁺the set of all positive integers and the set of all nonnegative integers, respectively. LetEbe a real Banach space. We denote byB_r the set{x∈E:x ≤r}. A Banach spaceEis said to bestrictly convexifx+y/2<1 for eachx,y∈B1withx=y, and it is said to beuniformly convexif for eachε >0, there existsδ >0 such thatx+y/2≤1−δ for eachx,y∈B1 withx−y ≥ε. It is well- known that a uniformly convex Banach space is reflexive and strictly convex (see [19]).

LetCbe a closed subset of a Banach space and letTbe a mapping fromCinto itself. We denote byF(T) andF_ε(T) forε >0, the sets{x∈C:x=Tx}and{x∈C:x−Tx ≤ε}, respectively.

A mappingTofCinto itself is said to becompactifTis continuous and maps bounded sets into relatively compact sets. A mappingT ofCinto itself is said to bedemicompact atξ∈Cif for any bounded sequence{yn}inCsuch that yn−T yn→ξasn→ ∞, there exists a subsequence{ynk}of{yn}andy∈Csuch thatynk→yask→ ∞andy−T y=ξ.

In particular, a continuous mappingT isdemicompact at0 if for any bounded sequence {yn}inCsuch thatyn−T yn→0 asn→ ∞, there exists a subsequence{ynk}of{yn}and y∈Csuch thatynk→yask→ ∞.Tis also said to besemicompactifTis continuous and demicompact at0 (e.g., see [21]).T is said to bedemicompact onCifT is demicompact for each y∈C. IfT is compact onC, thenT is demicompact onC. For examples of demicompact mappings, see [1,2,12,13]. We also denote byIthe identity mapping. A mappingT ofCinto itself is said to benonexpansiveifTx−T y ≤ x−yfor every x,y∈C. We denote byN(C) the set of all nonexpansive mappings fromCinto itself. We know from [5] that ifCis a nonempty closed convex subset of a strictly convex Banach space, thenF(T) is convex for eachT∈N(C) withF(T)= ∅. The following are crucial to prove our results (see [5,6]).

Proposition2.1 (Browder). LetCbe a nonempty bounded closed convex subset of a uni- formly convex Banach space and letT be a nonexpansive mapping fromCinto itself. Let {xn}be a sequence inCsuch that it converges weakly to an elementxofCand{xn−Txn} converges strongly to0. Thenxis a fixed point ofT.

Proposition 2.2 (Bruck). Let E be a uniformly convex Banach space and let C be a nonempty closed convex subset ofE. For anyε >0, there existsδ >0such that for any non- expansive mappingTofCinto itself withF(T)= ∅,

coFδ(T)⊂Fε(T). (2.1)

LetE^∗be the dual space of a Banach spaceE. The value ofx^∗∈E^∗atx∈Ewill be denoted byx,x^∗. We say that a Banach spaceEsatisfiesOpial’s condition[11] if for each

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sequence{x_n}inEwhich converges weakly tox,

nlim→∞

xn−x<lim

n→∞

xn−y (2.2)

for eachy∈Ewithy=x. Since if the duality mappingx→ {x^∗∈E^∗:x,x^∗ = x²= x^∗²}fromEintoE^∗is single-valued and weakly sequentially continuous, thenEsat- isfies Opial’s condition. Each Hilbert space and the sequence spaces^p with 1< p <∞ satisfy Opial’s condition (see [8,11]). Though anL^p-space with p=2 does not usually satisfy Opial’s condition, each separable Banach space can be equivalently renormed so that it satisfies Opial’s condition (see [11,22]).

LetSbe a semigroup. LetB(S) be the Banach space of all bounded real-valued functions onSwith supremum norm. Fors∈Sandf ∈B(S), we define an elementlsf inB(S) by (l_sf)(t)=f(st) for eacht∈S. LetXbe a subspace ofB(S) containing 1. An elementµ inX^∗is said to be ameanonXifµ =µ(1)=1. We often writeµt(f(t)) instead ofµ(f) forµ∈X^∗andf ∈X. LetXbels-invariant, that is,ls(X)⊂Xfor eachs∈S. A meanµon Xis said to beleft invariantifµ(l_sf)=µ(f) for eachs∈Sand f ∈X. A sequence{µ_n} of means onXis said to bestrongly left regularifµn−l_s^∗µn →0 for eachs∈S, where l^∗_s is the adjoint operator ofls. In the case whenSis commutative, a strongly left regular sequence is said to bestrongly regular[9,10]. LetEbe a Banach space, letXbe a subspace ofB(S) containing 1 and letµbe a mean onX. Let f be a mapping fromSintoEsuch that{f(t) :t∈S}is contained in a weakly compact convex subset ofEand the mapping t→ f(t),x^∗is inXfor eachx^∗∈E^∗. We know from [9,18] that there exists a unique elementx0∈Esuch thatx0,x^∗ =µ_tf(t),x^∗for allx^∗∈E^∗. Following [9], we denote suchx0by f(t)dµ(t). LetCbe a nonempty closed convex subset of a Banach spaceE.

A family᏿= {T(t) :t∈S}is said to be anonexpansive semigrouponCif it satisfies the following:

(1) for eacht∈S,T(t) is a nonexpansive mapping fromCinto itself;

(2)T(ts)=T(t)T(s) for eacht,s∈S.

We denote byF(᏿) the set of common fixed points of᏿, that is,_t_∈_SF(T(t)). Let᏿= {T(t) :t∈S}be a nonexpansive semigroup onCsuch that for eachx∈C,{T(t)x:t∈S} is contained in a weakly compact convex subset ofC. LetXbe a subspace ofB(S) with 1∈Xsuch that the mappingt→ T(t)x,x^∗is inXfor eachx∈Candx^∗∈E^∗, and let µbe a mean onX. Following [15], we also writeTµxinstead ofT(t)x dµ(t) forx∈C.

We remark thatTµis nonexpansive on C andTµx=xfor eachx∈F(᏿); for more details, see [19].

We writexn→x(or lim_n_→∞xn=x) to indicate that the sequence{xn}of vectors converges strongly tox. Similarly, we writexnx(or w-lim_n_→∞xn=x) will symbolize weak convergence. For any elementzand any setA, we denote the distance betweenzandAby d(z,A)=inf{z−y:y∈A}.

3. Weak convergence theorem

Throughout the rest of this paper, we assume thatSis a semigroup. LetCbe a nonempty weakly compact convex subset of a Banach space E and let ᏿= {T(s) :s∈S} be

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a nonexpansive semigroup ofC. We consider the following iterative procedure (see [21]):

x0=x∈Cand

xn=αnxn−1+1−αn

Tµnxn (3.1)

for everyn∈N, where{α_n}is a sequence in (0, 1).

Lemma3.1. LetCbe a nonempty weakly compact convex subset of a Banach spaceEand let᏿= {T(t) :t∈S}be a nonexpansive semigroup onCsuch thatF(᏿)= ∅. LetXbe a subspace ofB(S)with1∈Xsuch that the functiont→ T(t)x,x^∗is inXfor eachx∈Cand x^∗∈E^∗. Let{µn}be a sequence of means onSand let{αn}be a sequence of real numbers such that0< α_n<1for everyn∈N. Letx∈Cand let{x_n}be the sequence defined byx0=x and

Tµnxn (3.2)

for everyn∈N. Then,x_n+1−w ≤ x_n−wandlim_n_→∞x_n−wexists for eachw∈ F(᏿).

Proof. Letw∈F(᏿). By the definition of{xn}, we obtain that x_n−w=α_nx_n₋1−w+1−α_nT_µ_nx_n−w

≤α_nx_n−1−w+1−α_nT_µnx_n−w

≤α_nx_n−1−w+1−α_nx_n−w

(3.3)

and hence

αnxn−w≤αnxn−1−w. (3.4) It follows fromαn=0 that{xn−w}is a nonincreasing sequence. Hence, it follows that

lim_n_→∞x_n−wexists.

The following lemma was proved by Shioji and Takahashi [16] (see also [3,9]).

Lemma3.2 (Shioji and Takahashi). LetCbe a nonempty closed convex subset of a uniformly convex Banach spaceEand let᏿= {T(t) :t∈S}be a nonexpansive semigroup onC. LetX be a subspace ofB(S)with1∈Xsuch that it isls-invariant for eachs∈S, and the function t→ T(t)x,x^∗is inXfor eachx∈Candx^∗∈E^∗. Let{µ_n}be a sequence of means onS which is strongly left regular. For eachr >0andt∈S,

lim

n→∞ sup

y∈C∩Br

Tµny−T(t)Tµny=0. (3.5)

The following lemma is crucial in the proofs of the main theorems.

Lemma3.3. LetCbe a nonempty closed convex subset of a uniformly convex Banach space Eand let᏿= {T(t) :t∈S}be a nonexpansive semigroup onCsuch thatF(᏿)= ∅. LetX be a subspace ofB(S)with1∈Xsuch that it isl_s-invariant for eachs∈S, and the function t→ T(t)x,x^∗is inXfor eachx∈Candx^∗∈E^∗. Let{µ_n}be a sequence of means onS

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which is strongly left regular and let{α_n}be a sequence of real numbers such that0< α_n<1 for everyn∈Nand^∞_n₌₁(1−α_n)= ∞. Letx∈Cand let{x_n}be the sequence defined by x0=xand

x_n=α_nx_n−1+1−α_nT_µ_nx_n (3.6) for everyn∈N. Then, for eacht∈S,

nlim→∞xn−T(t)xn=0. (3.7) Proof. Forx∈Candw∈F(᏿), putr= x−wand setD= {u∈E:u−w ≤r} ∩C.

Then,Dis a nonempty bounded closed convex subset ofCwhich isT(s)-invariant for eachs∈Sand containsx0=x. So, without loss of generality, we may assume thatCis bounded. Fixε >0,t∈Sand set M0=sup{z:z∈C}. Then, fromProposition 2.2, there existsδ >0 such that

coF_δT(t)⊂F_εT(t). (3.8)

FromLemma 3.2there existsl∈Nsuch that

Tµiy−T(t)Tµiy< δ (3.9)

for everyi≥landy∈C. We have, for eachk∈N, x_l+k=α_l+kx_l+k−1+1−α_l+k

T_µl+kx_l+k

=α_l+k α_l+k−1x_l+k−2+1−α_l+k−1

T_µl+k−1x_l+k−1

+1−α_l+k

T_µl+kx_l+k

...

= _l+k

i=l

αi

xl−1+

l+k−1 j=l

_l+k

i=j+1

αi

1−αj

Tµjxj

+1−αl+k

Tµl+kxl+k.

(3.10)

Put

yk= 1 1−_l+k

i=lαi

_l₊_k₋₁

j=l

_l₊_k

i=j+1

αi

1−αj

Tµjxj

+1−αl+k Tµl+kxl+k

. (3.11)

From

l+k−1 j=l

_l₊_k

i=j+1

αi

1−αj

+1−αl+k

=1−

l+k

i=l

αi, (3.12)

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we obtainy_k∈co({T_µ_ix_i}ⁱ_i⁼₌^l_l⁺^k) and xl+k=

_l₊_k

i=l

αi

xl−1+

1−^l

+k

i=l

αi

yk. (3.13)

From (3.9), we know that for everyk∈N,T_µ_ix_i∈F_δ(T(t)) fori=l,l+ 1,...,l+k. So, it follows from (3.8) thaty_k∈coF_δ(T(t))⊂F_ε(T(t)) for everyk∈N. We know from Abel- Dini theorem that^∞_i₌_l(1−αi)= ∞implies^∞_i₌_lαi=0. Then, there existsm∈Nsuch that^l_i⁺₌^k_lαi< ε/(2M0) for everyk≥m. From (3.13), we obtain

xl+k−yk= _l₊_k

i=l

αi

xl−1−yk< ε

2M0·2M0=ε (3.14) for everyk≥m. Hence,

T(t)xl+k−xl+k≤T(t)xl+k−T(t)yk+T(t)yk−yk+yk−xl+k

≤2x_l+k−y_k+T(t)y_k−y_k≤2ε+ε=3ε (3.15) for everyk≥m. Sinceε >0 is arbitrary, we get lim_n_→∞T(t)xn−xn =0 for eacht∈S.

Now, we prove a weak convergence theorem for a nonexpansive semigroup in a Banach space.

Theorem3.4. LetCbe a nonempty closed convex subset of a uniformly convex Banach space Ewhich satisfies Opial’s condition and let᏿= {T(t) :t∈S}be a nonexpansive semigroup onCsuch thatF(᏿)= ∅. LetXbe a subspace ofB(S)with1∈Xsuch that it isl_s-invariant for eachs∈S, and the functiont→ T(t)x,x^∗ is inX for eachx∈Candx^∗∈E^∗. Let {µn}be a sequence of means onSwhich is strongly left regular and let{αn}be a sequence of real numbers such that0< α_n<1for everyn∈Nand^∞_n₌1(1−α_n)= ∞. Letx∈Cand let {xn}be the sequence defined byx0=xand

Tµnxn (3.16)

for everyn∈N. Then,{xn}converges weakly to an element ofF(᏿).

Proof. SinceEis reflexive and{xn}is bounded,{xn}must contain a subsequence of{xn} which converges weakly to a point inC. Let{xni}and{xnj}be two subsequences of{xn} which converge weakly toyandz, respectively. FromLemma 3.3andProposition 2.1, we knowy,z∈F(᏿). We will showy=z. Supposey=z. Then fromLemma 3.1and Opial’s condition, we have

nlim→∞xn−y=lim

i→∞xni−y<lim

i→∞xni−z

=lim

n→∞x_n−z=lim

j→∞x_n_j−z

<lim

j→∞xnj−y=lim

j→∞xn−y.

(3.17)

This is a contradiction. Hence{x_n}converges weakly to an element ofF(᏿).

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4. Strong convergence theorems

In this section, we discuss the strong convergence of the iterates defined by (3.1). Now, we can prove a strong convergence theorem for a nonexpansive semigroup in a Banach space (see also [2]).

Theorem4.1. LetCbe a nonempty closed convex subset of a uniformly convex Banach space Eand let᏿= {T(t) :t∈S}be a nonexpansive semigroup onCsuch thatF(᏿)= ∅. LetX be a subspace ofB(S)with1∈Xsuch that it isl_s-invariant for eachs∈S, and the function t→ T(t)x,x^∗is inXfor eachx∈Candx^∗∈E^∗. Let{µn}be a sequence of means onS which is strongly left regular and let{αn}be a sequence of real numbers such that0< αn<1 for everyn∈Nand^∞_n₌₁(1−α_n)= ∞. Letx∈Cand let{x_n}be the sequence defined by x0=xand

Tµnxn (4.1)

for everyn∈N. If there exists someT(s)∈᏿which is semicompact, then{xn}converges strongly to an element ofF(᏿).

Proof. Since the nonexpansive mappingT(s) is semicompact, there exist a subsequence {x_nj}of{x_n}andy∈Csuch thatx_nj→yas j→ ∞. ByLemma 3.3, we have that

0=lim

j→∞xnj−T(t)xnj=y−T(t)y (4.2) for eacht∈Sand hencey∈F(᏿). Then, it follows fromLemma 3.1that

nlim→∞xn−y=lim

j→∞xnj−y=0. (4.3)

Therefore,{x_n}converges strongly to an element ofF(᏿).

Next, we give a necessary and suﬃcient condition for the strong convergence of the iterates.

Theorem4.2. Let Cbe a nonempty weakly compact convex subset of a Banach spaceE and let᏿= {T(t) :t∈S}be a nonexpansive semigroup onCsuch thatF(᏿)= ∅. LetX be a subspace ofB(S)with1∈Xsuch that the functiont→ T(t)x,x^∗is inX for each x∈Candx^∗∈E^∗. Let{µ_n}be a sequence of means onSand let{α_n}be a sequence of real numbers such that0< αn<1for everyn∈N. Letx∈Cand let{xn}be the sequence defined byx0=xand

x_n=α_nx_n−1+1−α_nT_µnx_n (4.4) for everyn∈N. Then,{x_n}converges strongly to a common fixed point of᏿if and only if lim_n_→∞d(x_n,F(᏿))=0.

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Proof. The necessity is obvious. So, we will prove the suﬃciency. Assume lim

n→∞dxn,F(᏿)=0. (4.5)

ByLemma 3.1, we have

xn+1−w≤xn−w (4.6)

for eachw∈F(᏿). Taking the infimum overw∈F(᏿),

dxn+1,F(᏿)≤dxn,F(᏿) (4.7) and hence the sequence{d(xn,F(᏿))}is nonincreasing. So, from lim_n_→∞d(xn,F(᏿))=0, we obtain that

nlim→∞dxn,F(᏿)=0. (4.8)

We will show that{x_n}is a Cauchy sequence. Letε >0. There exists a positive integerN such that for eachn≥N,d(x_n,F(᏿))< ε/2. For anyl,k≥Nandw∈F(᏿), we obtain

xl−w≤xN−w, xk−w≤xN−w (4.9)

byLemma 3.1. So, we obtainxl−xk ≤ xl−w+w−xk ≤2xN−wand hence x_l−x_k≤2 inf x_N−y:y∈F(᏿)=2dx_N,F(᏿)< ε (4.10)

for everyl,k≥N. This implies that{xn}is a Cauchy sequence. SinceCis a closed subset ofE,{xn}converges strongly toz0∈C. Further, sinceF(᏿) is a closed subset ofC, (4.8) implies thatz0∈F(᏿). Thus, we have that{x_n}converges strongly to a common fixed

point of᏿.

Theorem4.3. LetCbe a nonempty closed convex subset of a uniformly convex Banach space Eand let᏿= {T(t) :t∈S}be a nonexpansive semigroup onCsuch thatF(᏿)= ∅. LetX be a subspace ofB(S)with1∈Xsuch that it isl_s-invariant for eachs∈S, and the function t→ T(t)x,x^∗is inXfor eachx∈Candx^∗∈E^∗. Let{µn}be a sequence of means onS which is strongly left regular and let{αn}be a sequence of real numbers such that0< αn<1 for everyn∈Nand^∞_n₌1(1−α_n)= ∞. Assume that there exists∈Sandk >0such that

I−T(s)z≥kdz,F(᏿) (4.11)

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for everyz∈C. Letx∈Cand let{x_n}be the sequence defined byx0=xand xn=αnxn−1+1−αn

Tµnxn (4.12)

for everyn∈N. Then,{x_n}converges strongly to an element ofF(᏿).

Proof. FromLemma 3.3, we obtain that(I−T(s))x_n →0 asn→0. Then, it follows from (4.11) that

nlim→∞kdx_n,F(᏿)=0 (4.13)

for somek >0. Therefore, we can conclude that{x_n}converges strongly to an element of

F(᏿) byTheorem 4.2.

5. Deduced theorems from main results

Throughout this section, we assume thatCis a nonempty closed convex subset of a uniformly convex Banach spaceE,xis an element ofC, and{α_n}is a sequence of real numbers such that 0< αn<1 for eachn∈Nand^∞_n₌₁(1−αn)= ∞. As direct consequences of Theorems3.4and4.1, we can show some convergence theorems.

Theorem5.1. LetTbe a nonexpansive mapping fromCinto itself such thatF(T)= ∅. Let {x_n}be the sequence defined byx0=xand

x_n=α_nx_n−1+1−α_n 1 n+ 1

n i=0

Tⁱx_n (5.1)

for everyn∈N. IfEsatisfies Opial’s condition, then{x_n}converges weakly to a fixed point ofT, and ifTis semicompact, then{xn}converges strongly to a fixed point ofT.

Theorem5.2. LetTbe as inTheorem 5.1. Let{sn}be a sequence of positive real numbers withsn↑1. Let{xn}be the sequence defined byx0=xand

1−sn

^∞

i=0

sniTⁱxn (5.2)

for everyn∈N. IfEsatisfies Opial’s condition, then{xn}converges weakly to a fixed point ofT, and ifTis semicompact, then{x_n}converges strongly to a fixed point ofT.

Theorem 5.3. LetT be as in Theorem 5.1. Let{q_n,m:n,m∈Z⁺}be a sequence of real numbers such thatq_n,m≥0,^∞_m₌₀q_n,m=1for everyn∈Z⁺andlim_n_→∞^∞_m₌₀|q_n,m+1− qn,m| =0. Let{xn}be the sequence defined byx0=xand

xn=αnxn−1+1−αn^∞

m=0

qn,mT^mxn (5.3)

for everyn∈N. IfEsatisfies Opial’s condition, then{xn}converges weakly to a fixed point ofT, and ifTis semicompact, then{x_n}converges strongly to a fixed point ofT.

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Theorem5.4. LetTandUbe commutative nonexpansive mappings fromCinto itself such thatF(T)∩F(U)= ∅. Let{x_n}be the sequence defined byx0=xand

xn=αnxn−1+1−αn 1 (n+ 1)²

n i,j=0

TⁱU^jxn (5.4)

for everyn∈N. IfEsatisfies Opial’s condition, then{xn}converges weakly to a common fixed point ofTandU, and if eitherTorUis semicompact, then{xn}converges strongly to a common fixed point ofTandU.

LetCbe a closed convex subset of a Banach spaceEand let᏿= {T(t) :t∈[0,∞)} be a family of nonexpansive mappings ofCinto itself. Then,᏿is called a one-parameter nonexpansive semigroup onCif it satisfies the following conditions:T(0)=I,T(t+s)= T(t)T(s) for allt,s∈[0,∞) andT(t)xis continuous int∈[0,∞) for eachx∈C.

Theorem5.5. Let᏿= {T(t) :t∈[0,∞)}be a one-parameter nonexpansive semigroup on Csuch thatF(᏿)= ∅. Let{sn}be a sequence of positive real numbers withsn→ ∞. Let {xn}be the sequence defined byx0=xand

xn=αnxn−1+1−αn1 sn

_s_n

0 T(t)xndt (5.5)

for everyn∈N. IfEsatisfies Opial’s condition, then{x_n}converges weakly to a common fixed point of᏿, and if there exists someT(s)∈᏿which is semicompact, then{xn}converges strongly to a common fixed point of᏿.

Theorem5.6. Let᏿be as inTheorem 5.5. Let{rn}be a sequence of positive real numbers withrn→0. Let{xn}be the sequence defined byx0=xand

x_n=α_nx_n−1+1−α_nr_n _∞

0 e⁻^rⁿ^tT(t)x_ndt (5.6) for everyn∈N. IfEsatisfies Opial’s condition, then{xn}converges weakly to a common fixed point of᏿, and if there exists someT(s)∈᏿which is semicompact, then{x_n}converges strongly to a common fixed point of᏿.

Theorem5.7. Let᏿be as inTheorem 5.5. Let{q_n}be a sequence of continuous functions from[0,∞)into[0,∞)such that₀^∞qn(t)dt=1for everyn∈N,lim_n_→∞qn(t)=0fort≥0 andlim_n_→∞₀^∞|qn(t+s)−qn(t)|dt=0for alls≥0. Let{xn}be the sequence defined by x0=xand

x_n=α_nx_n−1+1−α_n^∞

0 q_n(t)T(t)x_ndt (5.7) for everyn∈N. IfEsatisfies Opial’s condition, then{xn}converges weakly to a common fixed point of᏿, and if there exists someT(s)∈᏿which is semicompact, then{x_n}converges strongly to a common fixed point of᏿.

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Acknowledgments

This research was supported by Grant-in-Aid for Young Scientists (B), the Ministry of Ed- ucation, Culture, Sports, Science and Technology, Japan, and Grant-in-Aid for Scientific Research, Japan Society for the Promotion of Science.

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