On coincidence and common fixed points of nearly densifying mappings

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ON COINCIDENCE AND COMMON FIXED POINTS OF NEARLY

DENSIFYING MAPPINGS

ZEQING LIU and JEONG SHEOK UME

(Received 7 February 2000)

Abstract.Coincidence and common ﬁxed point theorems for certain new classes of nearly densifying mappings are established. Our results extend, improve, and unify a lot of pre-viously known theorems.

Keywords and phrases. Complete metric space, coincidence point, common ﬁxed point, nearly densifying mapping, attractor, left reversible semigroup, F-diminishing orbital diameter.

2000 Mathematics Subject Classiﬁcation. Primary 54H25.

1. Introduction. Furi and Vignoli [4] established first the existence of fixed point for densifying mappings. Afterwards Chatterjee [1], Diviccaro, Khan and Sessa [2], Fisher and Khan [3], Iseki [6, 7], Jain and Jain [8], Janos, Ko, and Tan [9], Khan [10], Khan and Fisher [11], Khan and Liu [12], Khan and Rao [13], Khan [14], Liu [17, 18, 19, 20, 21, 16], Pande [24, 25], Rao [22], Ray and Fisher [26], Sastry and Naidu [27], Sharma [28], Sharma and Srivastava [29] and others obtained fixed and coincidence point theo-rems for densifying and nearly densifying mappings, respectively. Huang, Huang, and Jeng [5] proved a common fixed point theorem for a left reversible semigroup, which consists of a number of continuous self-mappings in compact metric spaces.

The purpose of this paper is to establish coincidence and common ﬁxed point theo-rems for certain new classes of nearly densifying mappings in complete metric spaces. In Section 2, we introduce notation, terminology and prove a lemma, which plays an important role in the paper. In Section 3, we obtain some common ﬁxed point orems for families of mappings. In Section 4, we give general coincidence point the-orems for two pairs of mappings. Our results extend, improve, and unify the corre-sponding results of Chatterjee [1], Diviccaro, Khan, and Sessa [2], Huang, Huang, and Jeng [5], Janos, Ko, and Tan [9], Khan [10], Khan and Liu [12], Khan and Rao [13]. Liu [17, 18, 19], Rao [22], Sharma and Srivastava [29] and others.

2. Preliminaries. Recall that a semigroupGis said to be left reversible if for any

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Throughout this paper,(X,d)denotes a metric space,N,R+_{, and}_R_{denote the sets} of positive integers, nonnegative real numbers, and real numbers, respectively, and

ω=N{0}. Deﬁne

=F|F:X×X →R+_and_F(x,y)₌_{0 if and only if}_x₌_y_, 1= {F|F∈ andF is upper semicontinuous inX×X}, 2= {F|F∈ andF is lower semicontinuous inX×X}.

(2.1)

LetGbe a family of self-mappings inX. A subsetYofXis calledG-invariant ifgY⊆Y

for allg∈G. Let

NCIG= {Y|Y is nonempty compactG-invariant subset of X},

CISG=g|g:X →XandgY⊆Y , ∀Y∈NCIG, (2.2)

andG∗ _{be the semigroup generated by}_G_{under composition. Clearly,}_CIS_G_⊇_G∗_⊇ {gn_:_n_∈_ω}_{for any}_g_∈_G_{. For}_A,B_⊆_{X, x,y}_∈_{X, f}_∈_G,_and_F_∈_{, deﬁne}

δF(A,B)=supF(a,b):a∈A,b∈B, δF(A)=δF(A,A), δF(x,A)=δF{x},A, δF(x,y)=δF{x},{y}, Of(x)=fnx:n∈ω, Of(x,y)=Of(x)

Of(y), (2.3) Cf =h|h:X →X,f h=hf, G∗x= {x}gx:g∈G∗,

CISf=CIS{f}, NCIf=NCI{f}.

¯

A denotes the closure of A. f is said to have diminishing orbital diameter if limn→∞δd(Of(fnx)) < δd(Of(x))for allx∈Xwithδd(Of(x)) >0. fis called con-tractive with respect todifd(f x,f y) < d(x,y)for all distinctx,y∈X.

Deﬁnition2.1. LetGbe a semigroup of self-mappings on a metric space(X,d)

and F ∈ . G is said to haveF-diminishing orbital diameter, if for anyx ∈X with

δF(Gx) >0 there iss∈Gsuch thatδF(Gsx) < δF(Gx).

Deﬁnition2.2(see [15]). LetAbe a bounded subset of a metric space(X,d). Then

α(A), the measure of noncompactness ofA, is the inﬁmum of allε >0 such thatA

admits a ﬁnite covering consisting of subsets with diameters less thanε. The following properties ofαare well known.

Lemma2.3. Let(X,d)be a metric space andA,Bbe bounded subsets ofX. Then

αAB=maxα(A),α(B); (2.4)

α(A)=0⇐⇒Ais pre-compact, i.e., A is totally bounded; (2.5)

α(A)=αA¯. (2.6)

Deﬁnition2.4(see [4]). A continuous self-mapping f in a metric space (X,d)

is said to be densifying ifα(f (A)) < α(A)for every bounded subset A of X with

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Deﬁnition2.5(see [27]). A self-mappingf in a metric space(X,d)is said to be nearly densifying ifα(f (A)) < α(A)for every bounded andf-invariant subsetAof

Xwithα(A) >0.

Obviously, each densifying mapping is nearly densifying, but the converse is false. Deﬁnition2.6(see [23]). LetXbe a topological space,fa self-mapping inX, and

Ma nonempty subset ofX. Mis an attractor for compact sets underf if (i) Mis compact andf M⊆M,

(ii) given any compact setC⊆Xand any open neighborhoodUofM, there exists

k∈Nsuch thatfn_C_⊆_U_{for all}_n_≥_k_.

LetGbe a left reversible semigroup. We deﬁne a relation≥onGbya≥b if and only ifa∈bG{b}. It is easy to check that(G,≥)is a directed set.

Lemma2.7. LetGbe a left reversible semigroup of continuous self-mappings in a compact metric space(X,d), A= f∈Gf x,andF∈ 1. Then

lim

f∈GδF(f x)=δF(A);

A∈NCIG and f A=A, ∀f∈G.

(2.7)

Proof. Note thatf X ⊆gX for allf ,g∈G with f ≥g. Thus{δF(f X)}f∈G is a bounded decreasing net inR. Obviously, limf∈Gδf(f X)exists inRand

δF(A)≤lim

f∈GδF(f x). (2.8)

We now prove thatf Xis a compact subset ofXfor eachf∈G. Letxbe inXand

xnn∈N⊆Xwith limn→∞f xn=x. The compactness ofXensures that there exists a subsequence{xnk}k∈Nof{xn}n∈Nsuch that it converges to some pointt∈X. Since fis continuous, sox=f t∈f X. Thereforef Xis closed. That is,f Xis compact. This means thatAis compact.

We next prove that

δF(A)≥lim

f∈GδF(f X). (2.9)

Let f ∈ G. Since F is upper semicontinuous and f X×f X is compact, there exist

xf,yf ∈f X with F(xf,yf)=δF(f X). From the compactness ofX we can choose two subnets{xfk}and{yfk}of{xf}and{yf}, respectively, such thatxfk→xand

yfk→y for somex,y∈X. For everyg∈Gandfk≥g,we get thatxfk, yfk∈gX. By virtue of closedness of gX, we infer thatx,y∈gX. This means thatx,y∈A. Consequently,

lim

f∈GδF(f X)=flim∈GF

xf,yf=lim k F

xfk,yfk

=F(x,y)≤δF(A). (2.10)

Thus (i) follows from (2.8) and (2.9).

Letn∈Nandf1,f2,...,fn∈G. It follows from the left reversibility ofGthat there existg1,g2,...,gn∈G with f1g1=f2g2= ··· =fngn =h for some h∈G. Hence

n

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We ﬁnally prove thatf A=A for allf ∈G. Letf ∈G and x∈A. For anyg∈G

there exista,b∈Gwithf a=gb. Note thatx∈A⊆aX. Thus there isy∈Xwith

x=ay. It follows thatf x=f ay=gby∈gX. This implies thatf A⊆ g∈GgX=A forf∈G. For the reverse inclusion, letf ,g∈Gandy∈A. It follows fromy∈f gX

that there existsxg∈gXwithf xg=y. The compactnessXensures that there exists a convergent subnetxgk

of xg such that xgk →x for somex ∈X. Therefore

y=f x. For anyh,g∈Gwithg≥h, we obtain thathX is closed andxg belongs to hX. Thus the limit pointxofxglies inhX. That is,x∈A. Note thaty=f x∈f A. Therefore,A⊆f Aforf∈G. This completes the proof.

Remark2.8. Lemma 2.7 generalizes Lemma 2.3 of Huang, Huang, and Jeng [5].

3. Common ﬁxed point theorems for nearly densifyingmappings

Theorem3.1. LetGandHbe ﬁnite families of continuous and nearly densifying self-mappings in a complete metric space(X,d). If there existg∈G∗_{, h}_∈_H∗_{, F}_∈ 1, x0,y0∈Xsuch that

F(gx,hy) < δF  

s∈CISG

sG∗_x, t∈CISH

tH∗_y 

_, _∀x,y_∈_X_with_gx_≠_hy_; _(3.1)

G∗_x0,H∗_y0_{are bounded and}_G∗_,H∗_{are left reversible.Then the following statements}

hold:

(i) GandHhave a unique common ﬁxed pointw∈X, andwis also the only ﬁxed point ofGandH, respectively;

(ii) lims∈G∗Fsx0,w=lim_t∈H∗Fty0,w=lim_s∈G∗δ_FsG∗x0=lim_t∈H∗δ_FtH∗y0

=0;

(iii) for anyC∈NCIG∗ and anyD∈NCI_H∗, _s∈G∗sC= _t∈H∗tD= {w}.

Proof. LetA= s∈G∗sG∗x0andB= _t∈H∗tH∗y0. SinceG∗x0=x0_s∈GsG∗x0 andGis ﬁnite, so

αG∗_x0₌_max_α_x0_,α_sG∗_x0_:_s_∈_G₌_max_α_sG∗_x0_:_s_∈_G_. _(3.2)

Note that each s ∈ G is nearly densifying. Thus, αG∗_x0 ₌ _{0. It follows from} Lemma 2.3 thatG∗_x0_{is pre-compact. Completeness of}_(X,d)_{ensures that}_G∗_x0_is compact. Since everys∈G∗_{is continuous,}_sG∗_x0_⊆_sG∗_x0_⊆_G∗_x0_{. By Lemma 2.7 we} immediately conclude thatA∈NCIG∗andf A=Afor allf∈G∗. Similarly,B∈NCI_H∗ andf B=Bfor allf∈H∗_.

We assert thatδF(A,B)=0. OtherwiseδF(A,B) >0. SinceFis upper semicontinuous andA×Bis compact, we can easily choosea∈Aandb∈BwithF(a,b)=δF(A,B). Therefore, there existx∈Aand y∈B such thata=gx and b=hy. Using (3.1), we have

F(a,b)=F(gx,hy) < δF  

s∈CISG

sG∗_x, t∈CISH

tH∗_y  

≤δF(A,B)=F(a,b),

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which is a contradiction. Consequently,A=B=asingleton, say,{w}for somew∈X. Thusw=f wfor allf∈GH. That is,GandHhave a common ﬁxed pointw∈X. IfGhas another ﬁxed pointv∈Xandv≠w, by (3.1) we infer that

F(v,w)=F(gv,hw) < δF  

s∈CISG

sG∗_v, t∈CISH

tH∗_w 

₌_F(v,w), _(3.4)

which is absurd. HenceGhas a unique ﬁxed pointw. Similarly, we conclude thatH

has also a unique ﬁxed pointw. It follows from Lemma 2.7 that

lim s∈G∗δF

sG∗_x0₌_δ_F_(A)₌_δ_F_{w}₌₀₌_δ_F_(B)₌ _lim t∈H∗δF

tH∗_y0_. _(3.5)

Note thatsx0∈sG∗_{x0, ty0}_∈_tH∗_y0_and_w_∈_sG∗_x0 _tH∗_y0_{for all}_s_∈_G∗_{, t}_∈_H∗_. Thus Theorem 3.1(ii) follows immediately from (3.5).

LetC∈NCIG∗ andY = _s∈G∗sC. Lemma 2.7 ensures thatY ∈NCI_G∗ andf Y=Y for all f ∈ G∗_{. Suppose that} _δ_F_{(Y ,w) >} _{0. Then there exists} _x _∈ _Y _{such that} F(gx,w)=δF(Y ,w). In view of (3.1) and Theorem 3.1(i). We obtain thatF(gx,w) < δF(s∈CISGsG∗x,w)≤δF(Y ,w), which is impossible. HenceδF(y,w)=0. That is,

Y = s∈G∗sC= {w}. Similarly, we obtain that _t∈H∗tD= {w}ifD∈NCIH∗. This completes the proof.

Theorem3.2. LetGandHbe ﬁnite families of continuous and nearly densifying self-mappings in a complete bounded metric space(X,d)satisfying (3.1). Assume that

G∗_{, H}∗_{are near commutative. Then Theorem 3.1(i), (iii), and the following statements}

hold:

(i)

lim

s∈G∗F(sx,w)=_t∈Hlim∗F(ty,w)=_s∈Glim∗δF

sG∗_x

= lim t∈H∗δF

tH∗_y₌₀_, _∀x,y_∈_X_; (3.6)

(ii) G∗_and_H∗_have_F_{-diminishing orbital diameter.}

Proof. Letx,ybe inX. PutA= s∈G∗sG∗xandB= _t∈H∗tH∗y. As in the proof of Theorem 3.1, we conclude thatA∈NCIG∗, f A=Afor allf∈G∗ andB∈NCI_H∗,

gB=Bfor allg∈H∗_{. It follows from Theorem 3.1(ii) that}

A= s∈G∗

sA= {w} =B= t∈H∗

tB. (3.7)

Thus (3.6) follows from Lemma 2.7 and the deﬁnitions ofG∗_x_and_H∗_y_.

Givens,t∈G∗_{. Since}_G∗_{is commutative, there is}_g_∈_G∗_with_ts₌_sg_{. This means that}

δFG∗sx=δF{sx}tsx:t∈G∗≤δFsG∗x≤δFsG∗x. (3.8)

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We now state without proof analogues of Theorems 3.1 and 3.2.

Theorem3.3. LetG be a ﬁnite family of continuous and nearly densifying self-mappings in a complete metric space(X,d). If there existg,h∈G∗_{, F}_{∈ 1, x0}_∈_X

such that

F(gx,hy) < δF  

s∈CISG

sG∗_x_sG∗_y 

_, _∀x,y_∈_X_with_gx_≠_hy_; _(3.9)

G∗_x0_{is bounded and}_G∗_{is left reversible.Then the following statements hold:} (i) Ghas a unique common ﬁxed pointw∈X,and

lim s∈G∗F

sx0,w= lim s∈G∗δF

sG∗_x0₌_0; _(3.10)

(ii) for anyC∈NCIG∗, _s∈G∗sC= {w}.

Theorem3.4. Letfandgbe continuous self-mappings in a complete metric space

(X,d). Assume that there existi,j,p,q∈N, F∈ 1, x0,y0∈Xsuch that

(i) F(fp_x,gq_y)<δ

Fs∈CISfsOf(x),

t∈CISgtOg(y)

,∀x,y∈Xwithfp_x_≠_gq_y_; (ii) fi_and_gj_{are nearly densifying;}

(iii) Of(x0)andOg(y0)are bounded.

Then the following statements hold:

(1) fandghave a unique common ﬁxed pointw∈X, andwis also the only ﬁxed point off andg, respectively;

(2) limn→∞F(fnx0,w)=limn→∞F(gny0,w)=limn→∞δF(fnOf(x0))= limn→∞δF(gnOg(y0))=0;

(3) for anyC∈NCIf and anyD∈NCIg,

n∈N

fn_C₌ n∈N

gn_D_{= {w}.} _(3.11)

Proof. SetA= n∈NfnOf(x0)andB= n∈NgnOg(y0). In view of Theorem 3.4(ii), (iii) and

αOfx0=maxαfkx0: 0≤k≤i−1,αfiOfx0, (3.12)

we conclude easily thatA∈NCIf andf A=A. Similarly,B∈NCIg andgB=B. The rest of the proof is the same as that of Theorem 3.1. This completes the proof.

Remark3.5. Theorem 3.4 extends Theorems 3 and 4 of Liu [19], the theorem of Sharma and Srivastava [29]. Akin to Theorem 3.4, we have the following.

Theorem3.6. Letf be continuous self-mapping in a complete metric space(X,d). Assume that there existi,p,q∈N, F∈ 1, x0∈Xsuch that

(i) F(fp_x,fq_{y) < δ}

Fs∈CISfsOf(x,y)

,∀x,y∈Xwithfp_x_≠_fq_y_; (ii) fi_{is nearly densifying;}

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Then the following statements hold:

(1) fhas a unique ﬁxed pointw∈X, andlimn→∞F(fnx0,w)=limn→∞δFfnOf(x0) =0;

(2) for anyC∈NCIf, n∈NfnC= {w}.

Remark3.7. Theorem 4 of Khan [10] and Theorem 4 of Rao [22] are special cases of Theorem 3.6.

Theorem3.8. Letf and g be continuous self-mappings in a complete bounded metric space(X,d). Assume that there existi,j,p,q∈Nsatisfying Theorem 3.4(ii) and

dfp_x,gq_y_{< δ} d

  

s∈CISf

sOf(x),

t∈CISg

tOg(y)   ,

∀x,y∈Xwithfp_x_≠_gq_y.

(3.13)

Then Theorem 3.4(1) and (3.11) and the following statements hold:

(i) limn→∞d(fnx,w)=limn→∞d(gny,w)=limn→∞δd(fnOf(x))= limn→∞δd(gnOg(y))=0,∀x,y∈X;

(ii) there exist bounded complete metricsd1,d2onXwhich are equivalent todsuch thatf ,gare contractive with respect tod1andd2, respectively;

(iii) CISf and CISghave a unique common ﬁxed pointw∈X, andwis also the only

ﬁxed point of CISf and CISg, respectively; (iv) f andghave diminishing orbital diameter.

Proof. It follows from Theorem 3.4 that Theorem 3.4(1), (3.11), and Theorem 3.8(i) hold. By the deﬁnitions ofCISf and CISg, we conclude easily that Theorem 3.8(iii) holds. Sincefn_O_f_(x)₌ _O_f_(fn_x) _and _gn_O_g_(y)₌_O_g_(gn_y)_{, so Theorem 3.6(iv) is} satisﬁed. Now we prove that Theorem 3.8(ii) holds. Assume thatBbe any nonempty compact subset ofX. Using Lemma 2.3, we have

α n∈ωf n_B =max   α  i−1

n=0 fn_B

 _,α

 ∞

n=i fn_B

     =α  ∞ n=i fn_B

 ₌_α

fi n∈ωf n_B . (3.14)

Thusn∈ωfnBis totally bounded becausefiis nearly densifying. SetC=n∈ωfnB. Since f is continuous and X is complete, we infer that C is compact and f C ⊆ fn∈ωfnB ⊆ C. Hence (3.11) ensures that n∈ωfnC = {w}. This means that δd(fnC)↓0 asn→ ∞. For each open neighborhoodUofw, there exists an open ball B(w,ε)= {x : x ∈X and d(x,w) < ε}with B(w,ε)⊆ U. Note thatδd(fnC) ↓0 asn→ ∞. Thus there existsk∈Nsuch thatδd(fnC) < εfor alln≥k. Givenx∈fnC and n ≥k, we obtain that d(x,w) ≤ δd(fnC) < ε. Consequently, fnB ⊆ fnC ⊆ B(w,ε)⊆Ufor alln≥k. This shows that{w}is an attractor for compact sets underf. Thus Theorem 3.8(ii) follows from theorem of [9] and Remark 1 of [9]. This completes the proof.

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Theorem3.9. Letf be a continuous self-mapping in a complete bounded metric space(X,d). Assume that there existi,p,q∈Nsatisfying Theorem 3.6(ii) and

dfp_x,fq_y_{< δ} d

  

s∈CISf

sOf(x,y)  

, ∀x,y∈Xwithfp_x_≠_fq_y. _(3.15)

Then Theorem 3.6(2) and the following statements hold:

(i) f has a unique ﬁxed pointw∈X, and has diminishing orbital diameter and

lim n→∞d(f

n_x,w)₌_lim n→∞δd(f

n_O

f(x))=0,∀x∈X;

(ii) there exists a bounded complete metricd1onXwhich is equivalent todsuch thatf is contractive with respect tod1;

(iii) CISf has a unique common ﬁxed pointw∈X.

Remark3.10. Theorem 3.8 generalizes Theorem 4 of [2] and Theorem 4 of [22]. Theorem 3.9 extends and improves Theorem 3 of [1], Corollary 2 of [9], Theorem 3.1 of [17], and Theorems 1 and 2 of [18]

4. Coincidence point theorems for two pairs of nearly densifyingmappings Theorem4.1. Letf , g, s, andtbe a continuous and nearly densifying mappings from a complete metric space(X,d)into itself satisfying

f gt=f tg=tf g and gst=sgt=stg. (4.1)

LetG= {f ,g,s,t}. Assume that there existF1,F2∈ andx0∈Xsuch that

F1orF2∈ 2; (4.2)

F1(f x,gy) <max

F2(sx,ty),F2(sx,f x),F1(ty,gy),

minF2(sx,gy),F1(f x,ty),

F2(sx,ty)2 F1(f x,gy) ,

F2(sx,f x)2 F1(f x,gy) ,

F1(ty,gy)2 F1(f x,gy) , F2(sx,ty)F1(f x,ty)

F1(f x,gy) ,

F2(sx,f x)F1(f x,ty) F1(f x,gy) , F1(ty,gyF1(f x,ty)

F1(f x,gy) ,

F2(sx,gy)F1(f x,ty) F1(f x,gy) ,

F2(sx,f x)2 F2(sx,ty) ,

F2(sx,f x)F1(ty,gy) F2(sx,ty) , F2(sx,f x)F1(f x,gy)

F2(sx,ty) ,

F2(sx,f x)F1(f x,ty) F2(sx,ty) , F1(ty,gy)F1(f x,ty)

F2(sx,ty) ,

F2(sx,gy)F1(sx,ty) F2(sx,ty)

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for allx,y∈Xwithsx≠ty, f x≠gy;

F2(gx,f y) <max

F1(tx,sy),F1(tx,gx),F2(sy,f y),

minF1(gx,sy),F2(tx,f y),

F1(tx,sy)2 F2(gx,f y) ,

F1(tx,gx)2 F2(gx,f y) ,

F2(sy,f y)2 F2(gx,f y) ,

F1(tx,sy)F1(gx,sy) F2(gx,f y) , F1(tx,gx)F1(gx,sy)

F2(gx,f y) ,

F2(sy,f y)F1(gx,sy) F2(gx,f y) ,

F2(tx,f y)F1(gx,sy) F2(gx,f y) ,

F1(tx,gx)2 F1(tx,sy) ,

F1(tx,gx)F2(sy,f y) F1(tx,sy) , F1(tx,gx)F2(gx,f y)

F1(tx,sy) ,

F1(tx,gx)F1(gx,sy) F2(tx,sy) , F2(sy,f y)F1(gx,sy)

F1(tx,sy) ,

F1(gx,sy)F2(tx,f y) F1(tx,sy)

(4.4)

for allx,y∈Xwithgx≠f y, tx≠sy;

G∗_x0_{is bounded and}_G∗_{is left reversible}_. _(4.5)

Thenf andsorgandthave a coincidence point inX.

Proof. PutA=G∗_x0_{. It follows that} _A_{= {x0}}_{f A}_gA_sA_tA_{. This yields} that

α(A)=maxα(f A),α(gA),α(sA),α(tA). (4.6) It is evident to see that α(A)=0. Thus ¯A is compact by completeness of X. Set

B h∈G∗hA¯. Lemma 2.7 ensures thatf B=gB=sB=tB=B≠∅andBis compact. LetF1be in2. Deﬁner :B→R+ _{by putting}_{r (x)}₌_F1(tx,gx)_{. Since}_r _{is a lower} semi-continuous function on the compact setB, so there existsb∈Bwith

r (b)=F1(tb,gb)=inf

x∈BF1(tx,gx). (4.7) Suppose that neitherfandsnorgandthave a coincidence point. Then

tf gc≠gf gc, tstc≠gstc, stgc≠f tgc, (4.8) whereb=stc∈B. In view of (4.1), (4.3), (4.4), (4.7) and (4.8), we have

r (f gc)=F1(tf gc,gf gc)=F1(f tgc,gf gc)

<max

F2(stgc,tf gc),F2(stgc,f tgc),F1(tf gc,gf gc),

minF2(stgc,gf gc),F1(f tgc,tf gc)[F2(stgc,tf gc)]2 F1(f tgc,gf gc) , [F2(stgc,f tgc)]2

F1(f tgc,gf gc) ,

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F2(stgc,tf gc)F1(f tgc,tf gc) F1(f tgc,gf gc) ,

F2(stgc,f tgc)F1(f tgc,tf gc) F1(f tgc,gf gc) , F1(tf gc,gf gc)F1(f tgc,tf gc)

F1(f tgc,gf gc) ,

F2(stgc,gf gc)F1(f tgc,tf gc) F1(f tgc,gf gc) ,

[F2(stgc,f tgc)]2 F2(stgc,tf gc) ,

F2(stgc,f tgc)F1(tf gc,gf gc) F2(stgc,tf gc) , F2(stgc,f tgc)F1(f tgc,gf gc)

F2(stgc,tf gc) ,

F2(stgc,f tgc)F1(f tgc,tf gc) F2(stgc,tf gc) , F1(tf gc,gf gc)F1(f tgc,tf gc)

F2(stgc,tf gc) ,

F2(stgc,gf gc)F1(f tgc,tf gc) F2(stgc,tf gc)

=maxF2(gstc,f gtc),F2(gstc,f gtc),r (f gc),0,[F2(gstc,f gtc)]_{r (f gc)} 2, [F2(gstc,f gtc)]2

r (f gc) ,r (f gc),0,0,0,0,F2(gstc,f gtc), r (f gc),r (f gc),0,0,0

=max

F2(gstc,f gtc),[F2(gstc,f gtc)] 2 r (f gc)

=F2(gstc,f gtc)

<max

F1(tstc,sgtc),F1(tstc,gstc),F2(sgtc,f gtc)

minF1(gstc,sgtc),F2(tstc,f gtc),[F1(tstc,sgtc)]_{F2(gstc,f gtc)}2, [F1(tstc,gstc)]2

F2(gstc,f gtc) ,

[F2(sgtc,f gtc)]2 F2(gstc,f gtc) , F1(tstc,sgtc)F1(gstc,sgtc)

F2(gstc,f gtc) ,

F1(tstc,gstc)F1(gstc,sgtc) F2(gstc,f gtc) , F2(sgtc,f gtc)F1(gstc,sgtc)

F2(gstc,f gtc) ,

F2(tstc,f gtc)F1(gstc,sgtc) F2(gstc,f gtc) , [F1(tstc,gstc)]2

F1(tstc,sgtc) ,

F1(tstc,gstc)F2(sgtc,f gtc) F1(tstc,sgtc) , F1(tstc,gstc)F2(gstc,f gtc)

F1(tstc,sgtc) ,

F1(tstc,gstc)F1(gstc,sgtc) F1(tstc,sgtc) , F2(sgtc,f gtc)F1(gstc,sgtc)

F1(tstc,sgtc) ,

F1(gstc,sgtc)F2(tstc,f gtc) F1(tstc,sgtc)

=max

r (b),r (b),F2(gstc,f gtc),0,_{F2(gstc,f gtc)}[r (b)]2 ,_{F2(gstc,f gtc)}[r (b)]2 , F2(gstc,f gtc),0,0,0,0,r (b),

F2(gstc,f gtc),F2(gstc,f gtc),0,0,0

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which implies that

r (b)≤r (f gc) < r (b), (4.10)

which is a contradiction. Hencefandsorgandtmust have a coincidence point. The argument is similar forF2∈ 2. This completes the proof.

Theorem4.2. Letf ,g,s, andtbe continuous and nearly densifying mappings from a complete metric space (X,d) into itself satisfying f ,g ∈ Cs Ct. Let G = {f ,g,s,t} andH = {s,t}. Assume that there exist F1,F2∈ and x0∈X such that (4.2), (4.3), (4.4), and the following statement hold:

G∗_x0_{is bounded and}_H∗_{is left reversible}_. _(4.11)

Thenf andsorgandthave a coincidence point inX.

Proof. PutA=G∗_x0_and_B₌

h∈H∗hA¯. As in the proof of Theorem 4.1, we infer thatBis nonempty compact subset of ¯AandsB=tB=B⊇f BgB. The remaining part of the proof is as in Theorem 4.1. This completes the proof.

Remark4.3. Theorem 3.1 of [12] and Theorem 3.1 of [13] are special cases of Theorem 4.2.

Remark4.4. The following example reveals that f ,g,s, andt in Theorems 4.1 and 4.2 do not necessarily have a coincidence point and that if eitherf ands org

andthave a coincidence point, then the coincidence point may not be unique. Example4.5. Let X = {1,3,6}with the usual metric d and F1=F2=d. Deﬁne

f ,g,s,t:X→Xbyf1=g3=g6=1, f3=f6=g1=3 ands=t=iX—the identity mapping onX. TakeG= {f ,g,s,t}and H= {s,t}. Clearly,g2₌_f₌_f2_{, g}₌_{f g}₌ gf=g3_{, G}₌_G∗_{, H}₌_H∗_{, and}_G∗_and_H∗_{are left reversible. It is easy to verify that}

d(f x,gy)=2<3=d(sx,ty) (4.12)

for allx,y∈Xwithsx≠ty,f x≠gy,and

d(gx,f y)=2<3=d(tx,sy) (4.13)

for allx,y∈Xwithtx≠sy,gx≠f y. Thus the conditions of Theorems 4.1 and 4.2 are satisﬁed. However,fandshave two coincidence points 1 and 3, whilef ,g,s, andt

have none.

Theorem4.6. Letf ,g,s, andtbe continuous and nearly densifying mappings from a complete metric space(X,d)into itself satisfyingf ,g,s∈Ctandg∈Cs. Assume that

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Proof. Sincef ,g,s∈Ct, g∈Cs, andf b=gb=sb=tb, we havet2b=tf b= f tb=tgb=gtb=tsb=stb. Suppose thatt2_b_≠_tb_{. From (4.3) and (4.4) we}

con-clude that

F1t2_b,tb₌_{F1(f tb,gb)}

<max

F2(stb,tb),F2(stb,f tb),F1(tb,gb),

minF2(stb,gb),F1(f tb,tb),[F2(stb,tb)]2 F1(f tb,gb) , [F2(stb,f tb)]2

F1(f tb,gb) ,

[F1(tb,gb)]2 F1(f tb,gb) ,

F2(stb,tb)F1(f tb,tb) F1(f tb,gb) , F2(stb,f tb)F1(f tb,tb)

F1(f tb,gb) ,

F1(tb,gb)F1(f tb,tb) F1(f tb,gb) , F2(stb,gb)F1(f tb,tb)

F1(f tb,gb) ,

[F2(stb,f tb)]2 F2(stb,tb) ,

F2(stb,f tb)F1(tb,gb) F2(stb,tb) , F2(stb,f tb)F1(f tb,gb)

F2(stb,tb) ,

F2(stb,f tb)F1(f tb,gb) F2(stb,tb) , F1(tb,gb)F1(f tb,tb) F2(stb,tb) , F2(stb,gb)F1(f tb,tb) F2(stb,tb)

=maxF2t2_b,tb_,F2t2b,tb2 F1t2_b,tb ,F1

t2_b,tb =F2t2_b,tb₌_{F2(gtb,f b)}

<max

F1t2_b,sb_,F1_t2_b,gtb_{,F2(sb,f b),}

minF1gtb,sb,F2t2_{b,f b}_,

F1t2_b,sb2 F2(gtb,f b) ,

F1t2_b,gtb2 F2(gtb,f b) ,

F2(sb,f b)2 F2(gtb,f b) ,

F1t2_b,sb_F1(gtb,sb) F2(gtb,f b) , F1t2_b,gtb_F1(gtb,sb)

F2(gtb,f b) ,

F2(sb,f b)F1(gtb,sb) F2(gtb,f b) , F2t2_{b,f b}_F1(gtb,sb)

F2(gtb,f b) ,

F1t2_b,gtb2 F1t2_b,sb , F1t2_b,gtb_{F2(sb,f b)}

F1t2_b,sb ,

F1t2_b,gtb_{F2(gtb,f b)} F1t2_b,sb , F1t2_b,gtb_F1(gtb,sb)

F1t2_b,sb ,

F2(sb,f b)F1(gtb,sb) F1t2_b,sb , F1(gtb,sb)F2t2_{b,f b}

F1t2_b,sb

=maxF1t2_b,tb_,

F1t2_b,tb2 F2t2_b,tb ,F2

t2_b,tb

=F1t2_b,tb_, _(4.14)

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common ﬁxed point off ,g,s, and t. The uniqueness of a common ﬁxed point fol-lows from (4.3) and (4.4). This completes the proof.

Remark4.7. Theorem 4.6 extends Theorem 3.2 of [12] and Theorem 3.2 of [13]. Theorem4.8. Letf ,g,s, andtbe continuous and nearly densifying mappings from a complete metric space(X,d)into itself andG= {f ,g,s,t}. Suppose that there exist

F∈ 2andx0∈Xsuch that (4.5) and the following hold:

F(f x,gy) >infF(f z,sz),F(gz,tz):z∈G∗_x_G∗_y_,

∀x,y∈Xwithf x≠gy. (4.15)

Thenf andsorgandthave a coincidence point in X.

Proof. DeﬁneA=G∗_x0 _and_B₌

h∈G∗hA¯. As in the proof of Theorem 4.1, we infer thatBis compact,hB=B≠∅for allh∈G∗_{, and there are}_a,b_∈_B_{such that}

F(f a,sa)=infF(f x,sx):x∈B, F(gb,tb)=infF(gx,tx):x∈B. (4.16)

Without loss of generality, we assume that

F(f a,sa)≤F(gb,tb). (4.17)

Sincef ,g,s, andt∈G∗_{, it follows that} _{f B}₌_gB ₌_sB₌_tB ₌_B_{. Thus there exist} v,w∈Bwitha=gvandsa=gw. We claim thatf a=sa. If not, thenf gv≠gw. By virtue of (4.15), (4.16), and (4.17), we have

F(f a,sa)=F(f gv,gw)

>infF(f z,sz),F(gz,tz):z∈G∗_gv_G∗_y ≥infF(f z,sz),F(gz,tz):z∈B

=F(f a,sa),

(4.18)

which is a contradiction. Hencef a=sa. This completes the proof.

Theorem4.9. Letf andgbe continuous and nearly densifying mappings from a complete metric space(X,d)into itself andG= {f ,g}. Suppose that there existF∈ 2

andx0∈Xsatisfying (4.5) and

F(f x,gy) >infF(f z,z),F(gz,z),F(hx,hy):z∈G∗_x_G∗_{y, h}_∈_C f

Cg

G∗_, ∀x,y∈Xwithf x≠gy.

(4.19)

Thenf orghas a ﬁxed point inX.

Proof. It may be completed following the proof of Theorem 4.8.

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Zeqing Liu: Department Of Mathematics, Liaoning Normal University, Dalian, Liaoning,116029, China