Fixed point theorems for densifying mappings and compact mappings

PII. S0161171202204093 http://ijmms.hindawi.com © Hindawi Publishing Corp.

FIXED POINT THEOREMS FOR DENSIFYING MAPPINGS

AND COMPACT MAPPINGS

ZEQING LIU, LI WANG, SHIN MIN KANG, and YONG SOO KIM

Received 8 April 2002

The purpose of this note is to establish fixed point theorems for densifying mappings and compact mappings which are contractive in metric spaces and to investigate the existence of fixed points for a family of mappings in bounded metric spaces. The results of this note generalize the results of Bailey (1966) and Liu (1994).

2000 Mathematics Subject Classification: 54H25.

1. Introduction and preliminaries. Letfbe a self-mapping of a metric space(X, d), Ndenote the positive integers set, andN0=N∪{0}. Contractive mappings in metric spaces have been interested for many years, one of those is

dfn(x,y)_{x, f}n(x,y)_{y< d(x, y) (1.1)}

for all distinctx, y∈X, wheren(x, y)∈N. Bailey [1] has investigated the existence of fixed points for the contractive mapping (1.1) in compact metric spaces. Liu [4] first introduced the definition of the family of mappings CISf, and showed fixed point the-orems for CISf. In this note, we prove a few fixed point theorems for densifying and compact mappings which satisfy (1.1) in complete metric spaces and metric spaces, respectively. We also give an example to show that our results are the proper gener-alizations of the result of Bailey [1]. On the other hand, we go on investigating the existence of fixed points for CISf in bounded metric spaces, and our results extend the result of Liu [4].

The following definitions were introduced by Bailey [1] and Nussbaum [5].

Definition1.1(see [1]). Forx, y∈X,xis proximal toyunderfprovided that for eachε >0 there existsn∈Nsuch thatd(fn_{x, f}n_{y) < ε. Ifxandyare not proximal,} they are said to bedistal.

Definition_{1.2(see [5])}. _{A nonempty subsetMofXis said to be anattractor for}

compact sets under f if (1)Mis compact andf (M)⊆M, and (2) given any compact setC⊆Xand any open neighborhoodUofM, there existsk∈Nsuch thatfn_(C)⊆U forn≥k.

of Kuratowski. The mappingf is said to be compact if there exists a compact sub-set Y of X such that f (X)⊆ Y. It is well known that every continuous compact mapping is a densifying mapping. Introduced by Liu [4], CISf = {h: X →X | for every nonempty compact f-invariant subsetAof h(A)⊆A}. LetO(x, f )= {fn_x|

n∈ N0} and O(x, y, f )=O(x, f )∪O(y, f ), for x, y ∈X, where f0 is the iden-tity mapping inX. The mapping f is said to havediminishing orbital diameters if limn→∞δ(O(fnx, f )) < δ(O(x, f ))for allx∈Xwith 0< δ(O(x, f )) <∞.

2. Main results

Theorem2.1. Let(X, d)be a complete metric space andf:X→Xbe a continuous

densifying mapping satisfying (1.1). Suppose thatO(x0, f )is bounded for somex0∈X.

Thenf has a unique fixed point inX.

Proof. _{Sincef is densifying and}

αOx0, f

=maxαfOx0, f

, αx0

=αfOx0, f

, (2.1)

it follows thatα(O(x0, f ))=0. From the completeness of(X, d), we know thatO(x0, f ) is compact inX. We claim thatx0andf x0are proximal underf. Iffnx0=fn+1x0for somen∈N, thenx0andf x0are proximal underf. Iffnx0≠fn+1x0for anyn∈N, from (1.1) we infer that there exists a sequence{ni}i∈N⊂Nsuch that

dx0, f x0> dfn1x0, fn1+1x0

> dfn2_{x0, f}n2+1_x0>···_{> df}ni_{x0, f}ni+1_x0>··· (2.2)

for alli∈N. Suppose that eachniis chosen as the smallest positive integer in order to satisfy (2.2). Then for anyk∈N there existsni∈N such thatni−1≤k < ni. It follows from (2.2) that

dfk_x

0, fk+1x0≥dfni−1x0, fni−1+1x0> dfnix0, fni+1x0. (2.3)

SinceO(x0, f )is compact andfis continuous, we may (by selecting a subsequence, if necessary) assume thatfn_ix

0→uandfni+1x0→f ufor someu∈Xasi→ ∞. Thus

d(u, f u)=limi→∞d(fni_{x0, f}ni+1_{x0). Now assume thatx0andf x0are distal. Then}

there existsε0>0 satisfyingd(fmx0, fm+1x0)≥ε0for allm∈N. It is easy to see thatni+k≤nkfor everyk. In view of (2.3) we get that

dfku, fk+1u=lim i→∞d

fni+k_{x0, f}ni+k+1_x0

≥lim i→∞d

fni+k_{x0, f}ni+k+1_x0

=d(u, f u),

(2.4)

which is a contradiction to (1.1). Hencex0andf x0are proximal underf.

Next, we assert thatfhas a fixed point inX. Without loss of generality, we assume thatfn_x

0≠fn+1x0, for alln∈N0. Choose{nj}j∈N0⊂Nsuch thatnj< nj+1and

dfnj_x

0, fnj+1x0

<min ₁

j, d

fnj−1_x

0, fnj−1+1x0

for all j ∈N, where n0=0. It follows from the compactness ofO(x0, f )and the continuity off that there existsw∈Xand the subsequence{fnji_{x0}i∈N such that} fnjix0→wandfnji+1x0→f w asi→ ∞. Replacingjbyjiin (2.5) and lettingi→ ∞, we deduce thatd(w, f w)≤min{0, d(w, f w)} =0. That is,w=f w.

To prove the uniqueness of the fixed point off, we assume thatfhas another fixed pointb∈Xwithb≠w. From (1.1) we infer that

d(w, b)=dfn(w,b)_{w, f}n(w,b)_{b< d(w, b), (2.6)}

which is impossible. This completes the proof.

The next theorem follows fromTheorem 2.1.

Theorem 2.2. Let(X, d)be a metric space andf :X→X be a continuous and

compact mapping satisfying (1.1). Thenfhas a unique fixed point inX.

Remark _2.3. _{The following example shows that Theorems 2.1 and 2.2 are the} proper generalizations of [1, Corollary 2].

Example2.4. LetX=[0,∞)with the usual metric and definef:X→Xbyf x=

|sinx|for allx∈X. Choosen(x, y)=1 for allx, y∈Xwith 0<|x−y| ≤2 and n(x, y)=2 for allx, y∈Xwith |x−y|>2. It is easy to verify that the conditions of Theorems2.1and2.2are satisfied andf has a unique fixed point 0∈X. But [1, Corollary 2] is not applicable sinceXis not compact.

Theorem2.5. Let(X, d)be a bounded complete metric space and letf:X→Xbe

a continuous mapping. Suppose that there existp, q, r ∈Nsuch thatfr _{is densifying}

and

dfpx, fqy< δ∪h∈CIS_fO(z, h):z∈O(x, y, f ) (2.7)

for allx, y∈Xwithfp_x≠fq_{y. Then, we have the following:}

(i) f has a unique fixed pointv∈Xsuch thatfn_{x→vfor everyx∈X;} (ii) f has diminishing orbital diameters;

(iii) for every nonempty compactf-invariant subsetY ofX,∩n∈N0fn(Y )= {v};

(iv) there exists a bounded complete metricd∗ onX which is equivalent todsuch thatf is contractive with respect tod∗, that is,d∗(f x, f y) < d∗(x, y)for all x, y∈Xwithx≠y;

(v) CISf has a unique fixed pointv∈X.

Proof. _{Letxbe an arbitrary element inXandA}=_{O(x, f ). Then,}

α(A)=maxαx, f x, . . . , fr−1_{x, αf}r_{O(x, f )=αf}r_{A. (2.8)}

Sincefr _{is densifying,Ais precompact. It follows from the completeness of(X, d)} that ¯Ais compact. By the continuity off, we conclude thatf (A)¯ ⊆f (A)⊆_A¯_{. Thus}

¯

Sincefp_(D)=D,fq_{(D)=D, there existx, y∈Dsuch thatf}p_{x=u, f}q_{y=v and}

fp_x≠fq_{y. Obviously,δ({∪}

h∈CISfO(z, h):z∈O(x, y, f )})⊆D. By (2.7) we have

0< δ(D)=dfpx, fqy< δ∪h∈CISfO(z, h):z∈O(x, y, f )

≤δ(D), (2.9)

which is a contradiction. SoDis a singleton, say,D= {v}. Thereforevis a fixed point off.

Next we prove thatvis the unique fixed point off. Otherwiseb (≠v)is another fixed point off. From (2.7) we get that

0=d(b, v) < dfp_{b, f}q_{v< δ∪}

h∈CIS_fO(z, h):z∈O(b, v, f )≤d(b, v), (2.10)

which is impossible. Hence v is a unique fixed point of f. Sincefn_{x, v ∈f}n_(A)¯,

d(fn_{x, v)≤d(f}n_{(A))¯ →0 asn→ ∞. That is, f}n_{x→v as n→ ∞. Since (X, d)is} bounded, for eachx∈X− {v}, we have 0< δ(O(x, f )) <∞. In the light of (i), we get that for arbitraryε >0 there existsk∈Nsuch thatd(fn_{x, v) < ε/3 forn > k.} Consequently,

dfix, fjx≤dfix, v+dfjx, v<2ε

3 (2.11)

for alli, j > k. It follows that

δOfnx, f=supdfix, fjx:i, j≥n≤2₃ε< ε (2.12)

forn > k. This means that limn→∞δ(O(fnx, f ))=0, so f has diminishing orbital

diameters.

Similarly we can show that for every nonempty compactf-invariant subsetY ofX, ∩n∈N0fn(Y )= {v}.

Now, we prove that (iv) holds. LetCbe any nonempty compact subset ofX. Then

α∪n∈N0f

n_(C)=maxα∪r−1 n=0fn(C)

, αfr∪n∈N0f

n_(C)

=αfr_∪ n∈N0f

n_(C). (2.13)

Let Y = ∪n∈N0fn(C). Sincefr is densifying and (X, d)is complete, Y is compact

and f (Y )⊆f (∪n∈N0fn(C))⊆Y. It follows from (iii) that∩n∈N0f

n_{(Y )= {v}. This}

implies thatδ(fn_{(Y ))→0 asn→ ∞. For every open neighborhood U of v, there} exists an open ballB(v, ε)= {x|x∈Xandd(x, v) < ε}such thatB(v, ε)⊆U. Since limn→∞δ(fn(Y ))=0, there existsk∈Nsuch thatδ(fn(Y )) < εforn≥k. It follows

thatd(x, v)≤δ(fn_{(Y )) < εfor allx∈f}n_{Y. That is,f}n_{(Y )⊆B(v, ε). Hencef}n_(C)⊆

fn_{(Y )⊆B(v, ε)⊆Uforn≥k. This shows that{v}is an attractor for compact sets} underf. Thus (iv) follows from [3, Theorem and Remark 1].

Remark2.6. By takingp=qinTheorem 2.5, we get the result which improves [4, Theorem 2.1].

Theorem_2.7. _{Let(X, d)be a bounded complete metric space and letf , g:X}→_X

be continuous and commuting mappings. Suppose that there existr , s, p, q∈Nsuch thatfr_andgs _{are densifying and}

dfp_{x, g}q_{y< δ∪}

h∈CIS_fO(a, h):a∈O(x, f )

∪∪t∈CISgO(b, t):b∈O(y, g)

(2.14) for allx, y∈Xwithfp_x≠gq_{y. Then, we have the following:}

(i) fandghave a unique common fixed pointv∈Xsuch thatfn_x→v,gn_x→v,

for everyx∈X;

(ii) bothfandghave diminishing orbital diameters;

(iii) for every nonempty compact f-invariant and g-invariant subset Y of X, ∩n∈N0f

n_{(Y )= {v}and∩} n∈N0g

n_{(Y )= {v};}

(iv) there exist bounded complete metricsd∗andd∗∗onXwhich are equivalent to dsuch thatf andgare contractive with respect tod∗ andd∗∗, respectively, that isd∗(f x, f y) < d∗(x, y)andd∗∗(gx, gy) < d∗∗(x, y)for allx, y∈X withx≠y;

(v) CISf andCISghave a unique common fixed pointv∈X.

Proof. _{Forx, y}∈_{X, setA}=_{O(x, f ),B}=_{O(y, g),C}=∩_n∈Nfn_(A)¯_,D= ∩_n∈Nfn_(B)¯_. As is the proof ofTheorem 2.5we get that ¯A, ¯Bare nonempty compact andf-invariant, g-invariant subsets ofX with f (C)=C, g(D)=D, respectively. Hencefp_(C)=C,

gq_{(D)=D. Suppose thatδ(C, D) >0. There existu∈C,w∈Dsuch thatδ(C, D)=}

d(u, w) >0. Fromfp_(C)=C,gq_{(D)=D, there existx∈C,y∈Dsuch thatf}p_(x)=

u,gq_{(y)=w. Obviously,{∪}

h∈CIS_fO(a, h):a∈O(x, f )} ⊆Cand{∪t∈CISgO(b, t):b∈ O(y, g)} ⊆D. According to (2.14) we get

0< δ(C, D)=dfpx, gqy

< δ∪h∈CISfO(a, h):a∈O(x, f )

∪∪t∈CISgO(b, t):B∈O(y, g)

≤δ(C, D),

(2.15)

which is a contradiction. Henceδ(C, D)=0, that isC=D= {v}for somev∈X. Then vis a common fixed point offandg.

Now, we prove thatf andghave the only common fixed pointv∈X. Otherwiseu is a second common fixed point offandg. Using (2.14), we have

0< d(u, v)=dfp_{u, g}q_v

< δ∪h∈CISfO(u, h):a∈O(u, f )

∪∪t∈CISgO(v, t):b∈O(v, g)

≤δ(u, v),

(2.16)

which is impossible. The rest of the proof goes in a similar fashion as that ofTheorem 2.5, so we omit it. This completes the proof.

Acknowledgment. This work was supported by Korea Research Foundation

Grant KRF-2001-005-D00002.

References

[1] D. F. Bailey,Some theorems on contractive mappings, J. London Math. Soc.41(1966), 101– 106.

[2] M. Furi and A. Vignoli,A fixed point theorem in complete metric spaces, Boll. Un. Mat. Ital. (4)2(1969), 505–509.

[3] L. Janos, H. M. Ko, and K. K. Tan,Edelstein’s contractivity and attractors, Proc. Amer. Math. Soc.76(1979), no. 2, 339–344.

[4] Z. Liu,Fixed point theorems for condensing and compact maps, Kobe J. Math.11(1994), no. 1, 129–135.

[5] R. D. Nussbaum,Some asymptotic fixed point theorems, Trans. Amer. Math. Soc.171(1972), 349–375.

Zeqing Liu: Department of Mathematics, Liaoning Normal University, P.O. Box₂₀₀, Dalian₁₁₆₀₂₉, Liaoning, China

E-mail address:[email protected]

Li Wang: Department of Mathematics, Liaoning Normal University, P.O. Box₂₀₀, Dalian₁₁₆₀₂₉, Liaoning, China

Shin Min Kang: Department of Mathematics, Gyeongsang National University, Chinju_660-701, Korea

E-mail address:[email protected]