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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
d
fn(x,y)x, fn(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(fnx, fny) < ε. Ifxandyare not proximal, they are said to bedistal.
Definition1.2(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.
ForS⊆X, δ(S)and ¯S denote the diameter and the closure ofS, respectively. For A, B⊆X, δ(A, B)=sup{d(x, y)|x ∈A, y ∈B}. Following Furi and Vignoli [2], f is said to bedensifyingif for every bounded subsetAofXwith α(A) >0, we have α(f (A)) < α(A), whereα(A)denotes the measure of noncompactness in the sense
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 )= {fnx| 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 α
O x0, f
=max α
f O
x0, f , α
x0
=α
f O
x0, 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
d
x0, f x0
> d
fn1x0, fn1+1x0
> d
fn2x0, fn2+1x0
>···> d
fnix0, fni+1x0
>··· (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
d
fkx0, fk+1x0
≥d
fni−1x0, fni−1+1x0
> d
fnix0, fni+1x0
. (2.3)
SinceO(x0, f )is compact andfis continuous, we may (by selecting a subsequence, if necessary) assume thatfnix0→uandfni+1x0→f ufor someu∈Xasi→ ∞. Thus d(u, f u)=limi→∞d(fnix0, fni+1x0). 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
d
fku, fk+1u
=lim
i→∞d
fni+kx0, fni+k+1x0
≥lim
i→∞d
fni+kx0, fni+k+1x0
=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 thatfnx0≠fn+1x0, for alln∈N0. Choose{nj}j∈N0⊂Nsuch thatnj< nj+1and
d
fnjx0, fnj+1x0
<min 1
j, d
fnj−1x0, fnj−1+1x0
(2.5)
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{fnjix0}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)=d
fn(w,b)w, fn(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
d
fpx, fqy
< δ
∪h∈CISfO(z, h):z∈O(x, y, f )
(2.7) for allx, y∈Xwithfpx≠fqy. Then, we have the following:
(i) f has a unique fixed pointv∈Xsuch thatfnx→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)=max
α
x, f x, . . . , fr−1x , α
fr
O(x, f )
=α frA
. (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¯ A¯isf-invariant. SetD= ∩n∈Nfn(A). It is well known that¯ Dis a nonempty compact subset of ¯Aandf (D)=D. Hencefp(D)=D, fq(D)=D. We now assert thatDis a singleton. Otherwise there exist two distinct pointsu, v∈Dsuch thatδ(D)=d(u, v).
Sincefp(D)=D,fq(D)=D, there existx, y∈Dsuch thatfpx=u, fqy=v and fpx≠fqy. Obviously,δ({∪h∈CISfO(z, h):z∈O(x, y, f )})⊆D. By (2.7) we have
0< δ(D)=d
fpx, 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) < d
fpb, fqv
< δ
∪h∈CISfO(z, h):z∈O(b, v, f )
≤d(b, v), (2.10) which is impossible. Hence v is a unique fixed point of f. Sincefnx, v ∈fn(A),¯ d(fnx, v)≤d(fn(A))¯ →0 asn→ ∞. That is, fnx→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(fnx, v) < ε/3 forn > k.
Consequently,
d
fix, fjx
≤d fix, v
+d fjx, v
<2ε
3 (2.11)
for alli, j > k. It follows that δ
O
fnx, f
=sup d
fix, fjx
:i, j≥n
≤2ε
3 < 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∈N0fn(C)
=max α
∪rn=0−1fn(C) , α
fr
∪n∈N0fn(C)
=α fr
∪n∈N0fn(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∈N0fn(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∈fnY. That is,fn(Y )⊆B(v, ε). Hencefn(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].
Finally we conclude that (v) holds. For anyh∈CISf, it follows from (i) and the def- inition of CISf thath({v})⊆ {v}. That is,hv=v. Hencev is a fixed point of CISf. Note thatf∈CISf. By (i),vis the only fixed point of CISf. This completes the proof.
Remark2.6. By takingp=qinTheorem 2.5, we get the result which improves [4, Theorem 2.1].
Theorem2.7. 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 thatfrandgs are densifying and
d
fpx, gqy
< δ
∪h∈CISfO(a, h):a∈O(x, f )
∪
∪t∈CISgO(b, t):b∈O(y, g) (2.14) for allx, y∈Xwithfpx≠gqy. Then, we have the following:
(i) fandghave a unique common fixed pointv∈Xsuch thatfnx→v,gnx→v, for everyx∈X;
(ii) bothfandghave diminishing orbital diameters;
(iii) for every nonempty compact f-invariant and g-invariant subset Y of X,
∩n∈N0fn(Y )= {v}and∩n∈N0gn(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 thatfp(x)= u,gq(y)=w. Obviously,{∪h∈CISfO(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)=d
fpx, 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)=d
fpu, gqv
< 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.
Remark2.8. By takingf=gandp=qinTheorem 2.7, we obtain the result which generalizes Theorem 2.1 of Liu [4].
Acknowledgment. This work was supported by Korea Research Foundation Grant KRF-2001-005-D00002.
References
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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, Chinju660-701, Korea
E-mail address:[email protected]
Yong Soo Kim: Department of Mathematics, Gyeongsang National University, Chinju660-701, Korea
