Common fixed point theorems for left reversible and near commutative semigroups and applications

LEFT REVERSIBLE AND NEAR-COMMUTATIVE

SEMIGROUPS AND APPLICATIONS

ZEQING LIU AND SHIN MIN KANG

Received 21 May 2003

We prove some common fixed point theorems for left reversible and near-commutative semigroups in compact and complete metric spaces, respectively. As applications, we get the existence and uniqueness of solutions for a class of nonlinear Volterra integral equa-tions.

1. Introduction

Recently, Y.-Y. Huang and C.-C. Hong [15,16], T.-J. Huang and Y.-Y. Huang [14], and Y.-Y. Huang et al. [17] obtained a few fixed point theorems for left reversible and near-commutative semigroups of contractive self-mappings in compact and complete metric spaces, respectively. These results subsume some theorems in Boyd and Wong [1], Edel-stein [3], and Liu [20].

In this paper, motivated by the results in [14,15,16,17], we establish common fixed point theorems for certain left reversible and near-commutative semigroups of self-mappings in compact and complete metric spaces. As applications, we use our main re-sults to show the existence and uniqueness of solutions of nonlinear Volterra integral equations. Our results generalize, improve, and unify the corresponding results of Fisher [4,5,6,7,8,9,10,11,12], Hegedus and Szilagyi [13], Y.-Y. Huang and C.-C. Hong [16], T.-J. Huang and Y.-Y. Huang [14], Y.-Y. Huang et al. [17], Liu [18,19,20], Ohta and Nikaido [21], Rosenholtz [22], Taskovic [23], and others.

Recall that a semigroup F is said to be left reversible if, for anys,t∈F, there exist u,v∈Fsuch thatsu=tv. It is easy to see that the notion of left reversibility is equivalent to the statement that any two right ideals ofFhave nonempty intersection. A semigroup Fis callednear commutativeif, for anys,t∈F, there existsu∈Fsuch thatst=tu. Clearly, every commutative semigroup is near commutative, and every near-commutative semi-group is left reversible, but the converses are not true.

Throughout this paper, (X,d) denotes a metric space,N,R+_{, andRdenote the sets} of positive integers, nonnegative real numbers, and real numbers, respectively. LetF be a semigroup of self-mappings onX and let f be a self-mapping onX. For A,B⊆X, x,y∈X, define

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δd(A,B)=supd(a,b) :a∈A,b∈B, δd(A)=δd(A,A), δd(x,A)=δd

{x},A, Fx= {x} ∪ {gx:g∈F},

Of(x)=fnx:n∈ {0} ∪N, Of(x,y)=Of(x)∪Of(y), Cf = {h:h:X−→X, f h=h f},

Hf =

h:h:X−→X,h∩n∈N fnx⊆ ∩n∈NfnX, HF=

h:h:X−→X,h∩g∈FgX

⊆ ∩g∈FgX

,

Φ=φ:φ:R+_−→R+_{is upper semicontinuous from the right,} φ(0)=0,φ(t)< tfort >0.

(1.1)

A denotes the closure of A. Clearly, Hf ⊇Cf ⊇ {fn:n∈N} ∪ {iX}, whereiX is the

identity mapping on X. The mapping f is called a closed mapping if, y= f x when-ever{xn}n∈N⊆X such that limn→∞xn=xand limn→∞f xn=y for somex,y∈X. It is

simple to check that the composition of two closed self-mappings in compact metric spaces is closed. The mapping f is called alocal contractionif, for eachx∈X, there is an open setUcontainingxand a real numberm <1 such thatd(f z,f y)≤md(z,y) for all z,y∈U. The mapping f is said to havediminishing orbital diametersif, for everyx∈X withδd(Of(x))>0, there existsn∈Nsuch thatδd(Of(fnx))< δd(Of(x)). Clearly,f has

diminishing orbital diameters if and only if limn→∞δd(Of(fnx))< δd(Of(x)) for allx∈X

withδd(Of(x))>0. The semigroupFis said to havediminishing orbital diametersif, for

anyx∈Xwithδd(Fx)>0, there existsg∈Fsuch thatδd(Fgx)< δd(Fx).

2. Common fixed points for left reversible semigroups in compact metric spaces LetFbe a left reversible semigroup. We define a relation≥onF bya≥bif and only if a∈bF∪ {b}.

It is easy to verify that (F,≥) is a directed set. We need the following lemma for our main theorems.

Lemma2.1. LetFbe a left reversible semigroup of closed self-mappings in a compact metric

space(X,d)and letA= ∩f∈Ff X. Then

(i) limf∈Fδd(f X)=δd(A);

(ii)Ais nonempty, compact and f A=Afor all f ∈F.

Proof. Note that f X⊆gX for all f,g∈F with f ≥g. Thus{δd(f X)}f∈F is a bounded

decreasing net inR. Obviously, limf∈Fδd(f X) exists inRand

δd(A)≤lim

f∈Fδd(f X). (2.1)

We now prove that f X is a compact subset ofX for each f ∈F. Letxbe inX and

{xn}n∈N⊆Xwith limn→∞f xn=x. The compactness ofXensures that there exists a

sub-sequence {xnk}k∈N of{xn}n∈N such that it converges to some pointt∈X. In view of

We next prove that

δd(A)≥lim

f∈Fδd(f X). (2.2)

Given f ∈F, there existxf,yf ∈ f Xwithd(xf,yf)=δd(f X). SinceXis compact, we

can choose two subnets{xfk}and{yfk}of{xf}and{yf}, respectively, such thatxfk→x

and yfk→y for somex,y∈X. For every g∈F and fk≥g, we get that xfk,yfk∈gX.

By virtue of closedness ofgX, we infer thatx,y∈gX. This means thatx,y∈A. Conse-quently,

lim

f∈Fδd(f X)=limf∈Fd

xf,yf

=lim

k d

xfk,yfk

=d(x,y)≤δd(A). (2.3)

Thus (i) follows from (2.1) and (2.2).

Let n∈N and f1,f2,. . .,fn∈F. It follows from the left reversibility of F that there

existg1,g2,. . .,gn∈Fwithf1g1=f2g2= ··· = fngn=hfor someh∈F. Hence,∩ni=1fiX⊇ hX= ∅. The compactness ofXimplies thatA= ∅.

We last prove that f A=Afor all f ∈F. Let f ∈Fandx∈A. For anyg∈F, there exista,b∈F with f a=gb. Note thatx∈A⊆aX. Thus there is y∈X withx=ay. It follows that f x=f ay=gby∈gX. This implies that f A⊆ ∩g∈FgX=Afor f ∈F. For

the reverse inclusion, let f,g∈F and y∈A. It follows from y∈ f gX that there exists xg∈gXwith f xg=y. The compactnessXensures that there exists a convergent subnet {xgk}of{xg}such thatxgk→xfor somex∈X. The closedness of f implies thaty= f x.

For anyh,g∈Fwithg≥h, we obtain thathXis closed and thatxgbelongs tohX. Thus

the limit pointx of{xg}lies inhX. That is,x∈A. Note that y=f x∈ f A. Therefore,

A⊆ f Afor f ∈F. This completes the proof.

Now, we are ready to prove our main theorems.

Theorem2.2. LetFandGbe left reversible semigroups of closed self-mappings in a compact

metric space(X,d). Assume that there exist f ∈F,g∈Gsatisfying

d(f x,g y)< δd

su:u∈Fx,s∈HF

,tv:v∈Gy,t∈HG

(2.4)

for allx,y∈Xwith f x=g y. ThenFandGhave a unique common fixed pointw∈Xand the pointwis also a unique fixed point ofFandG, respectively. Moreover, ifF(resp.,G) is near commutative, thenF(resp.,G) has diminishing orbital diameters.

Proof. LetA= ∩s∈FsX andB= ∩t∈GtX. Ifδd(A,B)>0, then byLemma 2.1there exist a,x∈Aandb,y∈B withδd(A,B)=d(a,b),a= f x, andb=g y. It follows from (2.4)

that

δd(A,B)=d(f x,g y)

< δdsu:u∈Fx,s∈HF,tv:v∈Gy,t∈HG ≤δd(A,B),

(2.5)

which is a contradiction. Therefore,δd(A,B)=0. That is,A=B= {w}for somew∈X.

pointw. Ifvis a fixed point ofF orG, thenv∈ ∩f∈Ff X= {w}orv∈ ∩g∈GgX= {w}.

This means thatv=w. Hence,FandGhave a unique common fixed pointw∈X and the pointwis also a unique fixed point ofFandG, respectively.

Assume that one ofForG, sayF, is near commutative. Letxbe inXwithδd(Fx)>0.

So, for any f,g∈F, there existsh∈Fsuch thatg f =f h. It follows that

δd(F f x)=δd{f x} ∪ {g f x:g∈F}≤δd(f X). (2.6)

It follows from (2.6) andLemma 2.1that

lim

f∈Fδd(F f x)=limf∈Fδd(f X)=0< δd(Fx), (2.7)

which implies thatFhas diminishing orbital diameters. This completes the proof. Using the argument above, we can conclude the following two results.

Theorem 2.3. Let F be a left reversible semigroup of closed self-mappings in a compact

metric space(X,d). Assume that there exist f,g∈Fsatisfying

d(f x,g y)< δd

su:u∈Fx∪F y,s∈HF

(2.8)

for allx,y∈Xwith f x=g y. ThenFhas a unique fixed pointw∈X. Moreover, ifFis near commutative, then it has diminishing orbital diameters.

Theorem 2.4. Let F be a left reversible semigroup of closed self-mappings in a compact

metric space(X,d). Assume that there exists f ∈Fsatisfying

d(f x,f y)< δd

su:u∈Fx∪F y,s∈HF

(2.9)

for allx,y∈Xwith f x=f y. ThenFhas a unique fixed pointw∈X. Moreover, ifFis near commutative, then it has diminishing orbital diameters.

Corollary2.5. Let f be a closed self-mapping of a compact metric space(X,d). Assume

that there existp,q∈Nsuch that

dfpx,fqy< δd

su:u∈Of(x,y), s∈Hf

(2.10)

for allx,y∈Xwith fp_x=fq_{y. Then f has both a unique fixed pointw∈Xand}

diminish-ing orbital diameters.

Proof. TakeF= {fn_{:n∈N}. ThenF is a commutative semigroup. Obviously,Fhas a}

unique fixed pointw∈X if and only if f has a unique fixed pointw∈X. Note that Fx=Of(x) for allx∈Xand thatHF=Hf. ThusCorollary 2.5follows immediately from

Theorem 2.3. This completes the proof.

Theorem2.7. Let(X,d)be a compact and connected metric space and letF be a left re-versible semigroup of self-mappings inXsuch that each f inFis a local contraction. Then Fhas both a unique fixed pointw∈Xand diminishing orbital diameters. Moreover, for any x∈Xand f ∈F, the sequence of iterates{fn_x}

n∈Nconverges tow.

Proof. Letf be inF. We now show thatf has a unique fixed point inX. The compactness ofXensures that there existrf >0 andmf <1 such that

d(f x,f y)≤mfd(x,y) (2.11)

for allx,y∈Xwithd(x,y)< rf. Condition (2.11) ensures that f is continuous. Assume

thatV1,V2,. . .,Vnf are a fixed finite open cover ofXwith sets of diameters less thanrf.

For anyx,y∈X, the connectedness ofXimplies that there is a chain of open sets fromx toy, chosen from the setsV1,V2,. . .,Vnf. This means thatd(x,y)≤nfrf. It follows from

(2.11) thatd(f x,f y)≤mfnfrf. It is easy to check that

dfk_x,fk_y≤m f

k

nfrf (2.12)

for allk∈N. By choosingkso large that (mf)knf<1, we infer thatδd(fkX)< rf. So the

mapping f restricted to the setfk_{X, which maps f}k_{Xto itself, is a contraction. Note that} fk_{Xis closed. By the Banach contraction theorem, the restricted mapping has a unique}

fixed pointwf ∈ fkX. Obviously,wf is a unique fixed point of f inX. In view of (2.12),

we have

δd

fj_X,w f

≤δd

fj_X≤m f

j

nfrf (2.13)

for allj∈N. Therefore,

lim

j→∞δd

fj_X,w f

=0, (2.14)

which implies that both limj→∞fjx=wf and f has diminishing orbital diameters.

Let f andg be inF. Then there existwf,wg∈Xsuch thatwf = f wf,wg=gwg, and

(2.14) and the following equation hold:

lim

j→∞δd

gj_X,w g

=0. (2.15)

Givenj∈N, it follows from the left reversibility ofFthat there areaj,bj∈Fwithfjaj= gj_b

j. From (2.14) and (2.15), we infer that

dwf,wg≤dwf,fjajx+dgjbjx,wg ≤δd

wf,fjX

+δd

gj_X,w g

−→0 asj−→ ∞.

(2.16)

This means thatwf =wg. That is,F has a unique fixed point inX. This completes the

Remark 2.8. Theorem 2.7is a generalization of [22, Theorem 1]. Now we like to give two concrete examples for Theorems2.4and2.7.

Example 2.9. LetX= {0, 2/3} ∪ {1/n:n∈N}with the usual metricd. Define f,g:X→

Xby

f0= f2

3=g0=g 2

3=g1=0, f 1 n=

1

n+ 1, g 1 n+ 1=

1

n+ 2 (2.17)

for alln∈N. Obviously,g f = f2_{, f g=g}2_{,g f1=1/3=0= f g1, (X,d) is a compact} metric space, and f andg are closed. LetF be the semigroup generated by f and g. Now for anya,b∈F, there exista1,. . .,ak,b1,. . .,bn∈ {f,g}witha=a1···ak andb= b1···bn. Hence,ab=(bn)n+k=b(bn)k. ThusFis near commutative, and therefore it is

left reversible also. But it is not commutative. Note that∩s∈FsX= {0}. It is easy to verify

that

d(f x,f y)≤1

2<1=δd

su:u∈Fx∪F y,s∈HF

(2.18)

for allx,y∈Xwithf x=f y. Hence,Fsatisfies the conditions ofTheorem 2.4and clearly 0 is the unique fixed point ofF.

Example 2.10. LetX=[0, 1] with the usual metricd. Define f,g:X→Xby

f x=1

2x, gx= 2

3x (2.19)

for allx∈X. Then (X,d) is a compact and connected metric space, f g=g f, and any one of f andgis a local contraction. LetFbe the semigroup generated by f andg. It is easy to see that every element inFis a local contraction and thatFis commutative. It follows fromTheorem 2.7thatFhas a unique fixed point.

3. Common fixed points for near-commutative semigroups in complete metric spaces

Lemma3.1 (see [15]). Ifφis inΦ, thenlimn→∞φn(t)=0.

Lemma3.2 (see [2]). Ifφ:R+_→R+_{is an upper semicontinuous function withφ(0)=0}

andφ(t)< tfort >0, then there exists a strictly increasing continuous functionψ:R+_→R+ such thatψ(0)=0andφ(t)≤ψ(t)< tfort >0.

Theorem3.3. LetFandGbe near-commutative semigroups of closed self-mappings in a

complete metric space(X,d). Assume that the following conditions are satisfied: (i)for anyx∈X,FxandGxare bounded;

(ii)there existsφ∈Φsuch that, for anyf ∈F,g∈G, there arenf,mg∈Nsatisfying

dfp_x,gq_y≤φδ

d(Fx,Gy) (3.1)

for allx,y∈Xandp≥nf,q≥mg.

ThenFandGhave both a unique common fixed pointw∈Xand diminishing orbital diam-eters. Moreover, for each f ∈F∪Gandx∈X, the sequence of iterates{fn_x}

Proof. We assert that, for any f ∈F,g∈G,x,y∈X, andk≥max{nf,mg},

δd

F fk_x,Ggk_y≤φδ

d(Fx,Gy)

. (3.2)

Takeu∈F fk_xandv∈Ggk_{y. Then there ares∈Fandt∈Gwithu=s f}k_xandv=tgk_y.

The near commutativity ofF andGensures that there exista∈F andb∈Gsuch that s fk_=fk_aandtgk_=gk_{b. It follows from (3.1) that}

d(u,v)=ds fkx,tgky=dfkax,gkby

≤φδd(Fax,Gby)

≤φδd(Fx,Gy)

, (3.3)

which implies that

δdF fkx,Ggky=supd(u,v) :u∈F fkx,v∈Ggky ≤φδd(Fx,Gy)

(3.4)

for anyx,y∈Xandk≥max{nf,mg}. That is, (3.2) holds. For anyx,y∈Xandk∈ {0} ∪

N, putak=δd(F fkmax{nf,mg}x,Ggkmax{nf,mg}y). It follows from (3.2), (i), andLemma 3.1

that

ak≤φ

δd

F f(k−1) max{nf,mg}_x,Gg(k−1) max{nf,mg}_y =φak−1

≤ ··· ≤φk_a

0

=φk_δ

d(Fx,Gy)

−→0 ask−→ ∞.

(3.5)

Let n be in N. Then there exist k,r∈ {0} ∪N and r <max{nf,mg} such that n= kmax{nf,mg}+r. In view of (3.5), we have

δd

F fn_x,Ggn_y≤δ d

F fkmax{nf,mg}_x,Ggkmax{nf,mg}_y ≤φak−1

≤ak−1−→0 asn−→ ∞.

(3.6)

This implies that

maxδd

F fnx,δd

Ggny

≤maxδd

F fnx,gn+1y+δd

gn+1y,F fnx,δd

Ggny,fn+1x+δd

fn+1x,Ggny

≤2δd

F fnx,Ggny−→0 asn−→ ∞.

(3.7)

Therefore,F andG have diminishing orbital diameters. Note that {fn+i_x}

i∈N⊆F fnx and{gn+i_y}

i∈N⊆Ggny. So{fnx}n∈Nand{gny}n∈Nare Cauchy sequences. Since (X,d) is complete,{fn_x}

n∈Nand{gny}n∈Nconverge to some pointsw,b∈X, respectively. Con-sequently,w∈ ∩n∈NF fnxandb∈ ∩n∈NGgny. It follows from (3.6) that

d(w,b)≤δd

That is,w=b. The closedness of f andgimplies thatw= f wandb=gb. Hence, f and g have a common fixed pointw. By arbitrariness of f andg, we conclude thatFandG have a common fixed pointw.

Suppose thatFandGhave also a common fixed pointv∈X. By virtue of (3.1), for any f ∈Fandg∈G, we have

d(w,v)=dfnf_w,gmg_v≤_φδ

d(Fw,Gv)=φd(w,v), (3.9)

which implies thatw=v. This completes the proof.

Theorem3.4. LetFbe near-commutative semigroup of closed self-mappings in a complete

metric space(X,d). Assume that the following conditions are satisfied: (iii)for anyx∈X,Fxis bounded;

(iv)there existsφ∈Φsuch that, for anyf ∈F, there isnf ∈Nsatisfying

dfp_x,fq_x≤φδ d(Fx)

(3.10)

for allx∈Xandp,q≥nf.

ThenF has both a fixed point inXand diminishing orbital diameters. Moreover, for each f ∈Fandx∈X, the sequence of iterates{fn_x}

n∈Nconverges to some fixed point ofF.

Proof. It follows from (3.10) that (3.1) is satisfied forF=G,x=y, f =g, andnf =mg.

ThusTheorem 3.4follows fromTheorem 3.3. This completes the proof.

Theorem3.5. LetFbe near-commutative semigroup of closed self-mappings in a complete

metric space(X,d). Assume that condition (iii) and the following condition (v) hold: (v)there existsφ∈Φsuch that, for anyf ∈F, there isnf ∈Nsatisfying

dfp_x,fq_y≤φδ

d(Fx∪F y) (3.11)

for allx,y∈Xandp,q≥nf.

ThenFhas both a unique common fixed pointw∈Xand diminishing orbital diameters. Moreover, for each f ∈Fandx∈X, the sequence of iterates{fn_x}

n∈Nconverges tow.

Proof. Note that (3.11) implies that both (3.10) is satisfied andF has at most one fixed point inX. ThusTheorem 3.5follows fromTheorem 3.4. This completes the proof. Remark 3.6. It follows fromLemma 3.2thatTheorem 3.5extends [16, Theorem 2.1].

Theorem3.7. LetFandGbe near-commutative semigroups of self-mappings in a complete

metric space(X,d). Assume that condition (i) and the following condition hold: (vi)there existsφ∈Φsuch that, for any f ∈F,g∈Gandx,y∈X,

d(f x,g y)≤φδd(Fx,Gy). (3.12)

ThenFandGhave both a unique common fixed pointw∈Xand diminishing orbital diam-eters. Moreover, for each f ∈F∪Gandx∈X, the sequence of iterates{fn_x}

Proof. Let f andg be inFandG, respectively. As in the proof ofTheorem 3.3, we infer easily that there exists a unique pointw∈Xsuch that

lim

n→∞f

n_x=lim n→∞g

n_{y=w, lim} n→∞δd

F fn_x=lim n→∞δd

Ggn_{y=0 (3.13)}

for allx,y∈X. Thus,FandGhave diminishing orbital diameters and

lim

n→∞f

n_x=lim n→∞g

n_{w=w, lim} n→∞δd

F fn_w=lim n→∞δd

Ggn_{w=0. (3.14)}

Sincew∈(∩n∈NF fnw)∩(∩n∈NGgnw), it follows that

maxδd

F fn_w,w,δ d

Ggn_w,w≤maxδ d

F fn_w,δ d

Ggn_{w (3.15)}

for alln∈N. Using (3.14) and (3.15), we have

lim

n→∞δd

F fnw,w=lim

n→∞δd

Ggnw,w=0. (3.16)

Let>0 be arbitrary. By virtue of (3.14) and (3.16), there existsk∈Nsuch that, for all n≥k,

maxdw,fn+1_w,dw,gn+1_w,δ

dF fnw,w,δdGgnw,w<. (3.17)

For anyh∈G, from (3.12) and (3.17), we immediately conclude that

d(w,hw)≤dw,fn+1_w+dfn+1_w,hw≤+φδ

dF fnw,Gw ≤+φδd

F fnw,w+δd(w,Gw)

≤+φ+δd(w,Gw)

, (3.18)

which implies that

δd(w,Gw)≤+φ

+δd(w,Gw)

. (3.19)

Letting →0 in the above inequality, we obtain that δd(w,Gw)≤φ(δd(w,Gw)). This

means thatδd(w,Gw)=0. That is,Gw= {w}. Similarly,Fw= {w}. Therefore,FandG

have a common fixed pointw. The uniqueness of common fixed point ofFandGfollows

immediately from (3.12). This completes the proof.

From Theorems3.5and3.7, we have the following.

Theorem3.8. LetFbe a near-commutative semigroup of self-mappings in a complete

met-ric space(X,d). Assume that condition (iii) and the following condition (vii) hold: (vii)there existsφ∈Φsuch that, for anyf ∈Fandx,y∈X,

d(f x,f y)≤φδd(Fx∪F y)

. (3.20)

ThenFhas both a unique common fixed pointw∈Xand diminishing orbital diameters. Moreover, for each f ∈Fandx∈X, the sequence of iterates{fn_x}

Corollary3.9. Let f be a self-mapping of a complete metric space(X,d)and satisfy the following:

(viii)for eachx∈X,Of(x)is bounded;

(ix)there existsφ∈Φsuch that, for anyx,y∈X,

d(f x,f y)≤φδd

Of(x,y)

. (3.21)

Then f has both a unique fixed pointw∈Xand diminishing orbital diameters. Moreover, the sequence of iterates{fn_x}

n∈Nconverges towfor eachx∈X. Proof. PutF= {fn_{:n∈N}. Condition (3.21) ensures that}

dfn_x,fn_y≤φδ d

Of

fn−1_x,fn−1_y

≤φδd

Of(x,y)

=φδd(Fx∪F y)

(3.22)

for alln∈Nandx,y∈X. SoCorollary 3.9follows fromTheorem 3.8. This completes

the proof.

Remark 3.10. Theorem 3.8extends, improves, and unifies [4, Theorem 2], [13, Theorem 5], and [19, Theorem 1].

4. Applications

Throughout this section, let (X, · X) be a real Banach space andI=[a,b]⊆R. Define

C(I,X)= {f :f :I−→Xis continuous}, C(I×I×X,X)= {f :f :I×I×X−→Xis continuous},

fC=sup t∈I

_f(t)

X

(4.1)

for all f ∈C(I,X). It is easy to verify that (C(I,X), · C) is a real Banach space also.

Now we investigate the existence problem of common solutions for nonlinear Volterra integral equations of the from

xα(t)=v(t) +λ

t

aKα

t,s,xα(s)

ds, α∈A,t∈I,

yβ(t)=v(t) +λ

t

aMβ

t,s,yβ(s)ds, β∈B,t∈I,

(4.2)

wherev(t)∈C(I,X) is a given function,λ∈Ris an arbitrary parameter,KαandMβare

inC(I×I×X,X),AandBare index sets.

Theorem4.1. LetF= {fα:α∈A}andG= {gβ:β∈B}be near-commutative semigroups

and satisfy the following:

(i)for anyx∈C(I,X),max{δ·X(Fx),δ·X(Gx)}<∞;

(ii)there existsL >0such that, for anyα∈A,β∈B,x,y∈C(I,X), andt,s∈I,

_Kα

t,s,x(s)−Mβ

where

fαx(t)=v(t) +λ

t

aKα

t,s,x(s)ds, α∈A,x∈C(I,X),t∈I,

gβy(t)=v(t) +λ

_t

aMβ

t,s,y(s)ds, β∈B, y∈C(I,X),t∈I.

(4.4)

Then (4.2) have a unique common solutionw∈C(I,X). Moreover, for eachα∈A,β∈B, andx∈C(I,X), the sequences defined by

fαnx(t)=v(t) +λ

_t

aKα

t,s,fαn−1x(s)ds, t∈I,n∈N,

gβ

n

x(t)=v(t) +λ

t

aMβ

t,s,gβ

n−1

x(s)ds, t∈I,n∈N,

(4.5)

converge to the unique solutionwin the norm · C.

Proof. For anyx∈C(I,X), definex∗=supt∈Ie−(1+|λ|)Ltx(t)X. It is easy to show that

e−(1+|λ|)LbxC≤ x∗≤e−(1+|λ|)LaxC. (4.6)

Therefore, the norm · ∗and the sup-norm · Care equivalent to each other.

Obvi-ously, (C(I,X), · ∗) is also a real Banach space. Note that allfαandgβare self-mappings

of (C(I,X), · ∗). In view of (4.3) and (4.5), we infer that, for all α∈A,β∈B, and x,y∈C(I,X),

_{fαx(t)−gβy(t)}

∗

=sup

t∈I

e−(1+|λ|)Lt_|λ|t a

Kαt,s,x(s)−Mβt,s,y(s)ds

X

≤ |λ|sup

t∈I

e−(1+|λ|)Lt

t

a

_Kα

t,s,x(s)−Mβ

t,s,y(s)_Xds

≤ |λ|sup

t∈I

t

ae

(1+|λ|)L(s−t)_e−(1+|λ|)Ls_K α

t,s,x(s)−Mβ

t,s,y(s)_Xds

≤ |λ|sup

t∈I

t

ae

(1+|λ|)L(s−t)_sup

s∈I

e−(1+|λ|)Ls_Lδ

·X(Fx,Gy)ds

≤ |λ|Lsup

t∈I

t

ae

(1+|λ|)L(s−t)_δ

·∗(Fx,Gy)ds

≤ |λ|

1 +|λ|δ·X(Fx,Gy) sup t∈I

1−e(1+|λ|)L(a−t)

= |λ|

1 +|λ| 1−e(1+|λ|)L(a−b)

δ·X(Fx,Gy) =φδ·∗(Fx,Gy)

,

whereφ(t)=(|λ|/(1 +|λ|))(1−e(1+|λ|)L(a−b)_{)t fort∈R}+_{. It follows fromTheorem 3.7} that F and G have a unique common fixed point w∈C(I,X) and the sequences

{(fα)nx}n∈N and{(gβ)nx}n∈N converge tow in the norm · ∗. Therefore, (4.2) have a unique common solution w∈C(I,X) and by (4.6) the sequences{(fα)nx}n∈N and

{(gβ)nx}n∈Nconverge towin the norm · C. This completes the proof.

From Theorems3.8and4.1, we have the following.

Theorem4.2. LetF= {fα:α∈A}be a near-commutative semigroup and satisfy the

fol-lowing:

(iii)for anyx∈C(I,X),δ·X(Fx)<∞;

(iv)there existsL >0such that, for anyα∈A,x,y∈C(I,X)andt,s∈I,

_Kα

t,s,x(s)−Kαt,s,y(s)_X≤Lδ·X(Fx∪F y), (4.8)

where

fαx(t)=v(t) +λ

t

aKα

t,s,x(s)ds, α∈A,x∈C(I,X), t∈I. (4.9)

Then the following equations

xα(t)=v(t) +λ

t

aKα

t,s,x(s)ds, α∈A,t∈I, (4.10)

have a unique common solutionw∈C(I,X). Moreover, for eachα∈Aandx∈C(I,X), the sequence defined by

fα

n

x(t)=v(t) +λ

t

aKα

t,s,fα

n−1

x(s)ds, t∈I,n∈N, (4.11)

converges to the unique solutionwin the norm · C.

ByTheorem 4.2andCorollary 3.9, we have the following.

Corollary4.3. LetKbe inC(I×I×X,X)and satisfy the following:

(v)there existsL >0such that, for anyx,y∈C(I,X)andt,s∈I,

_K

t,s,x(s)−Kt,s,y(s)_X≤Lδ·X

Of(x,y)

, (4.12)

where

f x(t)=v(t) +λ

t

aK

t,s,x(s)ds, x∈C(I,X),t∈I; (4.13)

(vi)for everyx∈C(I,X),δ·X(Of(x))<∞.

Then the following equation

x(t)=v(t) +λ

t

aK

has a unique solutionw∈C(I,X). Moreover, for eachx∈C(I,X), the sequence defined by

fnx(t)=v(t) +λ

t

aK

t,s,fn−1x(s)ds, t∈I,n∈N, (4.15)

converges to the unique solutionwin the norm · C.

Remark 4.4. [23, Theorem 6] is a special case ofCorollary 4.3.

Acknowledgment

This work was supported by the Science Research Foundation of Educational Depart-ment of Liaoning Province (2004C063) and Korea Research Foundation Grant (KRF-2001-005-D00002).

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Zeqing Liu: Department of Mathematics, Liaoning Normal University, P.O. Box 200, Dalian, Liaoning 116029, China

E-mail address:[email protected]

Shin Min Kang: Department of Mathematics and RINS, Gyeongsang National University, Chinju 660-701, Korea