# On characterizations of fixed points

## Full text

(1)

### ON CHARACTERIZATIONS OF FIXED POINTS

ZEQING LIU, LILI ZHANG, and SHIN MIN KANG

(Received 16 January 2001 and in revised form 14 March 2001)

Abstract.We give some necessary and suﬃcient conditions for the existence of ﬁxed points of a family of self mappings of a metric space and we establish an equivalent condi-tion for the existence of ﬁxed points of a continuous compact mapping of a metric space.

2000 Mathematics Subject Classiﬁcation. 54H25.

1. Introduction. Jungck [2] ﬁrst gave a necessary and suﬃcient condition for the existence of ﬁxed points of a continuous self mapping of a complete metric space. Afterwards, Park [8], Leader [4], and Khan and Fisher [3] established a few theorems similar to that of Jungck. Janos [1] and Park [9] proved ﬁxed point theorems for com-pact self mappings of a metric space. Recently, Liu [5] established criteria for the existence of ﬁxed points of a family of self mappings of a metric space. The aim of this paper is to oﬀer some characterizations for the existence of ﬁxed points of a family of self mappings and a continuous compact mappings of metric spaces, respectively. We also establish a ﬁxed point theorem for two compact mappings, which extends properly the results of Janos [1] and Park [9].

Let ωand N denote the sets of nonnegative and positive integers, respectively. Suppose that(X, d)is a metric space. Forx, y∈X, deﬁne

Cf =g|g:X →Xandf g=gf,

Hf =g|g:X →Xandg∩n∈ωfnX⊆ ∩n∈ωfnX,

Hf(x)=hx|h∈Hf, Hf(x, y)=Hf(x)∪Hf(y),

O(x, f )=fnx|n∈ω, O(x, y, f )=O(x, f )∪O(y, f ).

(1.1)

Obviously,Cf ⊆Hf. LetΦbe a family of self mappings ofX. A pointx∈Xis said to be aﬁxed pointofΦiff x=xfor allf∈Φ. LetF:X×X→[0,+∞)be continuous andF (x, y)=0 if and only ifx=y. ForA, B⊂X, deﬁne

δ(A, B)=supF (x, y)|x∈A, y∈B (1.2)

(2)

In order to prove our main results, we need the following lemmas.

Lemma1.1(see [6]). Letfbe a continuous compact self mapping of a metric space (X, d). IfA= ∩n∈ωfnX, then

(a1) Ais compact, (a2) f A=A= ∅,

(a3) d(fnX)d(A)asn→ ∞, (a4) {fn|nω} ⊆Hf.

Lemma1.2(see [7]). Letf be a continuous self mappings of a metric space(X, d) with the following properties:

(a5) fhas a unique ﬁxed pointwinX,

(a6) for everyx∈X, the sequence of iterations{fnx}

n=0converges tow,

(a7) there exists an open neighborhoodUofwwith the property that given any open setVcontainingw, there existsk∈Nsuch thatn≥kimpliesfnUV. Then for eachα∈ (0,1), there exists a metric d∈ M(X)relative to which f is a contraction with Lipschitz constantα.

2. Main results

Theorem2.1. LetΦbe a family of self mappings of a metric space(X, d). Then the following statements are equivalent:

(b1) Φhas a ﬁxed point;

(b2) there existm, n∈Nand continuous compact self mappingsf,gof(X, d)such that eitherΦ⊆Cf orΦ⊆Cgand

Ffmx, gny< δHf(x), Hg(y), (2.1)

for allx, y∈Xwithfmx=gny;

(b3) there existm, n∈Nand continuous self mappingsf,gof(X, d)such thatf g is compact,f∈Cg,Φ∈Cf g, and

Ffmx, gny< δHf g(x, y), (2.2)

for allx, y∈Xwithfmx=gny,

(b4) there exists a continuous compact self mapping of(X, d)withΦ⊆Cf such that

F (f x, f y) <max

F (x, y), F (x, f x), F (y, f y),F (x, f x)F (y, f y) F (x, y) ,

F (f x, f y)F (x, f x) F (x, y) ,

F (x, f y)F (f x, y) F (x, y)

,

(2.3)

for allx, y∈Xwithx=y.

Moreover, if (b2) holds, thenf,g, andΦhave a unique common ﬁxed point; if (b3) holds, thenf gandΦhave a unique common ﬁxed point; if (b4) holds, thenf andΦ

(3)

Proof. Let (b1) hold andwbe a ﬁxed point ofΦ. Deﬁnef , g:XXbyf x=gx= wfor allx∈X. It is easy to show that (b2), (b3), and (b4) hold.

Assume that (b2) holds. LetA= ∩n∈ωfnX,B= ∩n∈ωgnX. Sincef andgare con-tinuous compact self mappings of(X, d), it follows from (a1) and (a2) thatAandB are compact andf A=A,gB =B. Consequently,fmA=A, gnB=B. Suppose that δ(A, B) >0. Then there exista∈A,b∈Bwithδ(A, B)=F (a, b)becauseFis continu-ous andA×Bis compact. SincefmA=A,gnB=B, there existxA,yBsuch that fmx=a,gny=b. In view of (2.1), we have

δ(A, B)=F (a, b)=Ffmx, gny< δHf(x), Hg(y)< δ(A, B), (2.4)

which is impossible, and henceδ(A, B)=0. That is,A=B= {w}for somew∈Xso f w=gw=w. Ifvis another ﬁxed point of f, thenv∈ ∩n∈ωfnX= {w}, that is,

v=w. Hencewis the only ﬁxed point off. Similarly,wis also the only ﬁxed point ofg.

Without loss of generality, we assume thatΦ⊆Cf. It follows from Cf ⊆Hf that hA⊆Afor allh∈Φ. That is,hw=wfor allh∈Φ. Thuswis the only common ﬁxed point off,g, andΦ. Therefore (b1) holds.

Assume that (b3) holds. PutA= ∩n∈ω(f g)nX. ThenAis compact andf gA=A. Sincef is continuous andf∈Cg, we infer that

f A=f∩n∈ω(f g)nX⊆ ∩n∈ω(f g)nf X⊆ ∩n∈ω(f g)nX=A. (2.5)

Similarly, we have

gA⊆A. (2.6)

It follows fromf gA=A, (2.5), and (2.6) that

f A⊆A=f gA⊆f A. (2.7)

That is, f A=A. Similarly, we have gA=A. Suppose thatδ(A) >0. BecauseF is continuous andAis compact, then there exista, b∈Asuch thatδ(A)=F (a, b). Since fmA=gnA=A, there existx, yAwithfmx=a,gny=b. Using (2.2), we have

δ(A)=F (a, b)=fmx, gny< δHF G(x, y)δ(A), (2.8)

which is a contradiction. Henceδ(A)=0. That is,A= {w} for somew∈X. This implies thatf w=gw=f gw=w. As in the proof of above, we can prove thatwis the only ﬁxed point off g, andwis the unique common ﬁxed point off gandΦ. So (b1) holds.

Assume that (b4) holds. As above we infer thatA= ∩n∈ωfnXis compact andf A=

(4)

virtue of (2.3), we get

φ(f w)=F (f w, f f w)

<max

F (w, f w), F (w, f w), F (f w, f f w),F (w, f w)F (f w, f f w) F (w, f w) ,

F (f w, f f w)F (w, f w) F (w, f w) ,

F (w, f f w)F (f w, f w) F (w, f w)

=F (w, f w)

=φ(w).

(2.9)

This is a contradiction to the deﬁnition ofw. Sow is a ﬁxed point off. Iff has a second distinct ﬁxed pointv, by (2.3), we obtain that

F (w, v)=F (f w, f v)

<max

F (w, v), F (w, w), F (v, v),F (w, w)F (v, v) F (w, v) ,

F (w, v)F (w, w) F (w, v) ,

F (w, v)F (w, v) F (w, v)

=F (w, v),

(2.10)

which is a contradiction. Therefore,wis the only ﬁxed point off. It is a simple matter to show thatwis the unique common ﬁxed point off andΦ. Thus (b1) holds. This completes the proof.

Next, we give a theorem about the equivalent condition for the existence of ﬁxed points of a continuous compact self mapping on a metric space.

Theorem2.2. Letsbe a continuous compact self mapping of a metric space(X, d).

Thens has a ﬁxed point if and only if there exists a continuous self mappingf ofX such thatf∈Csand

F (f x, f y) <max

F (sx, sy), F (sx, f x), F (sy, f y),F (sx, f x)F (sy, f y) F (sx, sy) ,

F (f x, f y)F (sx, f x) F (sx, sy) ,

F (sx, f y)F (f x, sy) F (sx, sy)

,

(2.11)

for allx, y∈Xwithsx=sy. Indeed,f andshave a unique common ﬁxed point.

Proof. To see that the stated conditions is necessary, suppose thatshas a ﬁxed pointw∈X. Deﬁnef:X→Xbyf x=wfor allX∈X. Thenf sx=w=sw=sf x for allx∈X, that is,f∈Cs. Clearly, (2.11) holds.

On the other hand, suppose that there exists a continuous self mappingfofXsuch thatf∈Cs and (2.11) holds. LetA= ∩n∈ωSnX. FromLemma 1.1, we infer thatAis compact andsA=A. Sincefis continuous ands∈Cf, we have

(5)

Deﬁne the function φ(x)by φ(x)=F (sx, f x)for all x∈A. It is clear that φ(x) is continuous on A and attains its minimum value at somew ∈A. We claim that sw=f w. If not, from (2.12), there existsp∈Asatisfyingf w=sp. Using (2.11), we conclude that

φ(p)=F (sp, f p)=F (f w, f p)

<max

F (sw, sp), F (sw, f w), F (sp, f p),F (sw, f w)F (sp, f p) F (sw, sp) ,

F (f w, f p)F (sw, f w) F (sw, sp) ,

F (sw, f p)F (f w, sp) F (sw, sp)

=maxF (sw, f w), F (sp, f p)

=F (sw, f w)

=φ(w),

(2.13)

which is a contradiction to the choice ofw. Sosw=f w. By virtue off∈Cs, we have

f sw=sf w=ssw. (2.14)

Now suppose thatssw=sw. By (2.11) and (2.14), we get

F (ssw, sw)=F (f sw, f w)

<max

F (ssw, sw), F (ssw, f sw), F (sw, f w),F (ssw, f sw)F (sw, f w) F (ssw, sw) ,

F (f sw, f w)F (ssw, f sw) F (ssw, sw) ,

F (ssw, f w)F (f sw, sw) F (ssw, sw)

=F (ssw, sw),

(2.15)

which is a contradiction. Thusssw=sw, that is,swis a ﬁxed point ofs. Therefore, the set M of ﬁxed points of s is not empty. Now s is continuous, soM is closed. SinceM ⊆Aand Ais compact, M is compact. Moreover, since f and s commute, f (M)⊆M. Note also that (2.11) restricted toM reduces to (2.3). We can therefore applyTheorem 2.1(b4) tofM:M→Mto obtain a unique common ﬁxed pointuoff ands inM. SinceM contains all the ﬁxed points ofs, uis a unique common ﬁxed pointf ands. This completes the proof.

Theorem2.3. Letf,gbe continuous compact self mappings of a metric space(X, d)

satisfying(2.1). Thenfandghave a unique ﬁxed point, respectively, and furthermore, for anyα∈(0,1), there exist metrics dand d∈M(X)relative to whichf andg satisfy, respectively,

d(f x, f y)≤αd(x, y), d(gx, gy)≤αd(x, y), (2.16)

for allx, y∈X.

Proof. LetA= ∩nωfnX,B= ∩nωgnX, andU=X. As in the proof ofTheorem

(6)

d(fnX), d(gnX)0 asn→ ∞. ThusfnXandgnXsqueeze into any neighborhood of w. That is (a7) is fulﬁlled. ThusTheorem 2.3follows fromLemma 1.2. This completes the proof.

Corollary2.4. Letfbe a continuous compact self mapping of a metric space(X, d)

satisfying

F (f x, f y) < δHf(x), Hf(y), (2.17)

for allx, y∈Xwithx=y. Thenf has a unique ﬁxed point.

Furthermore, for anyα∈(0,1), there exists a metricd∈M(X)relative to whichf satisﬁes

d(f x, f y)≤αd(x, y), (2.18)

for allx, y∈X.

The following simple example reveals that Corollary 2.4 extends properly Theo-rem 1.1 of Janos [1] and Theorem 1 of Park [9].

Example2.5. LetX= {0,2,4,6,9}with the usual metric. Deﬁne a mappingf:X X byf0=f4=f6=6,f2=0, andf9=2. Then f is a continuous compact self mapping ofX. It is easy to check that

d(f x, f y)≤6<9=δHf(x), Hf(y), (2.19)

for allx, y∈Xwithx=y. So the conditions ofCorollary 2.4are satisﬁed. But Theo-rem 1.1 of Janos [1] and Theorem 1 of Park [9] are not applicable since

d(f2, f4)=6>2=12d(2, f2)+d(4, f4),

d(f2, f4)=6=δO(2,4, f ).

(2.20)

Acknowledgement. The authors express their thank to the referee for his help-ful suggestions, and the third author was supported by Korea Research Foundation Grant (KRF-99-005-D00003).

References

[1] L. Janos,On mappings contractive in the sense of Kannan, Proc. Amer. Math. Soc.61(1976), no. 1, 171–175.MR 54#13886. Zbl 342.54024.

[2] G. Jungck,Commuting mappings and ﬁxed points, Amer. Math. Monthly83(1976), no. 4, 261–263.MR 53#4031. Zbl 321.54025.

[3] M. S. Khan and B. Fisher,Some ﬁxed point theorems for commuting mappings, Math. Nachr. 106(1982), 323–326.MR 83k:54053. Zbl 501.54031.

[4] S. Leader,Uniformly contractive ﬁxed points in compact metric spaces, Proc. Amer. Math. Soc.86(1982), no. 1, 153–158.MR 83j:54041. Zbl 507.54040.

[5] Z. Liu,Families of mappings and ﬁxed points, Publ. Math. Debrecen47 (1995), no. 1-2, 161–166.CMP 1 362 279. Zbl 854.54039.

[6] ,On compact mappings of metric spaces, Indian J. Math.37(1995), no. 1, 31–36. CMP 1 366 982. Zbl 839.54031.

(7)

[8] S. Park,Fixed points off-contractive maps, Rocky Mountain J. Math.8(1978), no. 4, 743– 750.MR 80d:54059. Zbl 398.54030.

[9] ,On general contractive-type conditions, J. Korean Math. Soc.17(1980), no. 1, 131– 140.MR 82e:54055. Zbl 448.54048.

Zeqing Liu: Department of Mathematics, Liaoning Normal University, Dalian, Liaoning116029, China