International Journal of Pure and Applied Mathematics
————————————————————————–
Volume 51 No. 2 2009, 239-247
FIXED POINT THEOREMS FOR TWO CLASSES
OF MAPPINGS IN COMPACT METRIC SPACES
Shin Min Kang1 § , Pingping Zheng2 , Zeqing Liu3 , Chahn Yong Jung4
1 Department
of Mathematics
The Research Institute of Natural Science
Gyeongsang National University
Jinju, 660-701, KOREA
e-mail: smkang@nongae.gsnu.ac.kr
2,3 Department of Mathematics
Liaoning Normal University
P.O. Box 200, Dalian, Liaoning, 116029, P.R. CHINA
2 e-mail: pingpingzheng@live.cn
3 e-mail: zeqingliu@sina.com.cn
4 Department of Business Administration
Gyeongsang National University
Jinju, 660-701, KOREA
e-mail: bb5734@nongae.gsnu.ac.kr
Abstract: In this paper sufficient conditions of the existence of fixed points
for two classes of nonlinear mappings are established in compact metric spaces,
and sufficient and necessary conditions of a continuous mapping to possesses
fixed points are given in compact metric spaces.
AMS Subject Classification: 54H25
Key Words: fixed point, continuous mapping, commuting mappings, compact subset, compact metric space
1. Introduction and Preliminaries
Let f be a self mapping of a compact metric space (X, d) and ω denote the
set of all nonnegative integers. In 1988, Jungck [2] defined Cf = {h : h : X →
Received:
January 10, 2009
§ Correspondence
author
c 2009 Academic Publications
240
S.M. Kang, P. Zheng, Z. Liu, C.Y. Jung
X and f h = hf } and proved the following result.
Theorem 1.1. (see [2]) Let f and g be continuous commuting self mappings of a compact metric space (X, d). If f x =
6 gy implies that
d(f x, gy) > d(hx, hy)
(1.1)
for some h ∈ Cf ∩ Cg , then at least one of f or g has a fixed point in X.
Liu [4] extended Jungck’s result and proved the following result.
Theorem 1.2. (see [4]) Let f and g be continuous self mappings of a
n
n
∞
compact metric space (X, d) and f (∩∞
n=1 (gf ) X) = ∩n=1 (gf ) X. If f and g
satisfy that
d(f x, gy) > inf{d(x, f x), d(y, f y), d(x, gx),
d(y, gy), d(hx, hy) : h ∈ Cf ∩ Cg } (1.2)
for all x, y ∈ X with f x 6= gy, then at least one of f or g has a fixed point in
X.
Inspired by the results in [1]-[6], in this paper we introduce and study two
classes of nonlinear mappings f and g as follows:
d(f x, gy) + min{d(f x, y), d(x, gy)}
n
> inf d(x, f x), d(y, f y), d(x, gx), d(y, gy), d(hx, hy),
d(x, f x)d(y, f y) d(x, f x)d(x, gx) d(x, f x)d(y, gy)
,
,
,
d(f x, gy)
d(f x, gy)
d(f x, gy)
d(x, f x)d(hx, hy) d(y, f y)d(x, gx) d(y, f y)d(y, gy)
,
,
,
d(f x, gy)
d(f x, gy)
d(f x, gy)
d(y, f y)d(hx, hy) d(x, gx)d(y, gy) d(x, gx)d(hx, hy)
,
,
,
d(f x, gy)
d(f x, gy)
d(f x, gy)
d(y, gy)d(hx, hy) d(y, gy)[1 + d(y, f y)]
,
,
d(f x, gy)
1 + d(x, f x)
d(hx, hy)[1 + d(y, f y)] d(x, f x)[1 + d(x, y)]
,
,
1 + d(x, f x)
1 + d(y, gy)
o
d(y, f y)[1 + d(x, y)] d(hx, hy)[1 + d(x, y)]
(1.3)
,
: h ∈ Cf ∩ Cg
1 + d(y, gy)
1 + d(y, gy)
and
d(f x, gy) + min{d(f m+1 x, f m y), d(gm x, gm+1 y)}
n
> inf d(hx, f hx), d(hy, f hy), d(hx, ghx),
FIXED POINT THEOREMS FOR TWO CLASSES...
241
d(f p x, f p y)d(gq x, gq y)
,
d(f x, gy)
o
d(hx, f hx)d(kx, gkx) d(hy, f hy)d(ky, gky)
(1.4)
,
: p, q ∈ ω, h, k ∈ Cf ∩ Cg
d(f x, gy)
d(f x, gy)
for all x, y ∈ X with f x 6= gy and some m ∈ ω. Under certain conditions, we
establish some fixed point theorems for the classes of mappings (1.3) and (1.4)
in compact metric spaces. Our results extend and improve Theorems 1.1 and
1.2.
d(hy, ghy), d(hx, hy),
In order to obtain our results, we need the following result, which is due to
Jungck [1] and Leader [3].
Lemma 1.1. (see [1], [3]) Let f be a continuous self mapping of a compact
n
metric space (X, d). If B = ∩∞
n=1 f X, then
(a) B is compact;
(b) B = f B 6= ∅;
(c) hB ⊆ B for h ∈ Cf .
2. Main Results
Now we show the existence of fixed points for mappings (1.3) and (1.4) in
compact metric spaces.
Theorem 2.1. Let f and g be continuous self mappings of a compact
n
n
∞
metric space (X, d) and f (∩∞
n=1 (gf ) X) = ∩n=1 (gf ) X. If f and g satisfy
(1.3) for all x, y ∈ X with f x 6= gy, then at least one of f or g has a fixed point
in X.
n
Proof. Let A = ∩∞
n=1 (gf ) X. Using Lemma 1.1 and f A = A, we conclude
that A is compact and A = gf A = gA 6= ∅. It follows that
n
∞
n
hA ⊆ ∩∞
n=1 h(gf ) X = ∩n=1 (gf ) hX
n
⊆ ∩∞
n=1 (gf ) X = A,
∀h ∈ Cf ∩ Cg . (2.1)
Since f and g are continuous and A is compact, it follows that there exist
a, b ∈ A such that
d(a, f a) = inf{d(x, f x) : x ∈ A},
(2.2)
d(b, gb) = inf{d(x, gx) : x ∈ A}.
Next we consider two possible cases below.
242
S.M. Kang, P. Zheng, Z. Liu, C.Y. Jung
Case 1. Assume that
d(a, f a) ≤ d(b, gb).
(2.3)
It follows from gA = A that there exists some c ∈ A such that gc = a. Suppose
that a 6= f a, that is, f gc 6= gc. In view of (2.1)∼(2.3), we deduce that
d(f gc, gc) + min{d(f gc, c), d(gc, gc)}
n
> inf d(gc, f gc), d(c, f c), d(gc, ggc), d(c, gc), d(hgc, hc),
d(gc, f gc)d(c, f c) d(gc, f gc)d(gc, ggc) d(gc, f gc)d(c, gc)
,
,
,
d(f gc, gc)
d(f gc, gc)
d(f gc, gc)
d(gc, f gc)d(hgc, hc) d(c, f c)d(gc, ggc) d(c, f c)d(c, gc)
,
,
,
d(f gc, gc)
d(f gc, gc)
d(f gc, gc)
d(c, f c)d(hgc, hc) d(gc, ggc)d(c, gc) d(gc, ggc)d(hgc, hc)
,
,
,
d(f gc, gc)
d(f gc, gc)
d(f gc, gc)
d(c, gc)d(hgc, hc) d(c, gc)[1 + d(c, f c)]
,
,
d(f gc, gc)
1 + d(gc, f gc)
d(hgc, hc)[1 + d(c, f c)] d(gc, f gc)[1 + d(gc, c)]
,
,
1 + d(gc, f gc)
1 + d(c, gc)
o
d(c, f c)[1 + d(gc, c)] d(hgc, hc)[1 + d(gc, c)]
,
: h ∈ Cf ∩ Cg
1 + d(c, gc)
1 + d(c, gc)
n
= inf d(a, f a), d(c, f c), d(a, ga), d(c, gc), d(ghc, hc),
d(a, f a)d(c, f c) d(a, f a)d(a, ga) d(a, f a)d(c, gc)
,
,
,
d(f a, a)
d(f a, a)
d(f a, a)
d(a, f a)d(ghc, hc) d(c, f c)d(a, ga) d(c, f c)d(c, gc)
,
,
,
d(f a, a)
d(f a, a)
d(f a, a)
d(c, f c)d(ghc, hc) d(a, ga)d(c, gc) d(a, ga)d(ghc, hc)
,
,
,
d(f a, a)
d(f a, a)
d(f a, a)
d(c, gc)d(ghc, hc) d(c, gc)[1 + d(c, f c)]
,
,
d(f a, a)
1 + d(a, f a)
d(ghc, hc)[1 + d(c, f c)] d(a, f a)[1 + d(gc, c)]
,
,
1 + d(a, f a)
1 + d(c, gc)
o
d(c, f c)[1 + d(gc, c)] d(ghc, hc)[1 + d(gc, c)]
,
: h ∈ Cf ∩ Cg
1 + d(c, gc)
1 + d(c, gc)
≥ inf{d(a, f a), d(b, gb), d(ghc, hc) : h ∈ Cf ∩ Cg } ≥ d(a, f a),
FIXED POINT THEOREMS FOR TWO CLASSES...
243
which implies that
d(a, f a) = d(f gc, gc) > d(a, f a),
which is a contradiction. Hence a = f a.
Case 2. Assume that
d(a, f a) > d(b, gb).
(2.4)
It follows from f A = A that there exists some c ∈ A such that f c = b.
Suppose that b 6= gb, that is, gf c 6= f c. In light of (2.1), (2.2) and (2.4), we
arrive at
d(f c, gf c) + min{d(f c, f c), d(c, gf c)}
n
> inf d(c, f c), d(f c, f f c), d(c, gc), d(f c, gf c), d(hc, hf c),
d(c, f c)d(f c, f f c) d(c, f c)d(c, gc) d(c, f c)d(f c, gf c)
,
,
,
d(f c, gf c)
d(f c, gf c)
d(f c, gf c)
d(c, f c)d(hc, hf c) d(f c, f f c)d(c, gc) d(f c, f f c)d(f c, gf c)
,
,
,
d(f c, gf c)
d(f c, gf c)
d(f c, gf c)
d(f c, f f c)d(hc, hf c) d(c, gc)d(f c, gf c) d(c, gc)d(hc, hf c)
,
,
,
d(f c, gf c)
d(f c, gf c)
d(f c, gf c)
d(f c, gf c)d(hc, hf c) d(f c, gf c)[1 + d(f c, f f c)]
,
,
d(f c, gf c)
1 + d(c, f c)
d(hc, hf c)[1 + d(f c, f f c)] d(c, f c)[1 + d(c, f c)]
,
,
1 + d(c, f c)
1 + d(f c, gf c)
o
d(f c, f f c)[1 + d(c, f c)] d(hc, hf c)[1 + d(c, f c)]
,
: h ∈ Cf ∩ Cg
1 + d(f c, gf c)
1 + d(f c, gf c)
n
= inf d(c, f c), d(b, f b), d(c, gc), d(b, gb), d(hc, f hc),
d(c, f c)d(b, f b) d(c, f c)d(c, gc) d(c, f c)d(b, gb)
,
,
,
d(b, gb)
d(b, gb)
d(b, gb)
d(c, f c)d(hc, f hc) d(b, f b)d(c, gc) d(b, f b)d(b, gb)
,
,
,
d(b, gb)
d(b, gb)
d(b, gb)
d(b, f b)d(hc, f hc) d(c, gc)d(b, gb) d(c, gc)d(hc, f hc)
,
,
,
d(b, gb)
d(b, gb)
d(b, gb)
d(b, gb)d(hc, f hc) d(b, gb)[1 + d(b, f b)]
,
,
d(b, gb)
1 + d(c, f c)
d(hc, f hc)[1 + d(b, f b)] d(c, f c)[1 + d(c, f c)]
,
,
1 + d(c, f c)
1 + d(b, gb)
244
S.M. Kang, P. Zheng, Z. Liu, C.Y. Jung
o
d(b, f b)[1 + d(c, f c)] d(hc, f hc)[1 + d(c, f c)]
,
: h ∈ Cf ∩ Cg
1 + d(b, gb)
1 + d(b, gb)
≥ inf{d(a, f a), d(b, gb), d(f hc, hc) : h ∈ Cf ∩ Cg } ≥ d(b, gb),
which gives that
d(b, f b) = d(f c, gf c) > d(b, gb),
which is absurd. Hence b = gb. This completes the proof.
Remark 2.1. Theorem 2.1 generalizes the corresponding results in [2] and
[4].
Theorem 2.2. Let f and g be continuous commuting self mappings of a
compact metric space (X, d). Assume that there exists m ∈ ω satisfying (1.4)
for all x, y ∈ X with f x 6= gy. Then at least one of f or g has a fixed point in
X.
n
Proof. Let A = ∩∞
n=1 (f g) X. It follows from Lemma 1.1 that A is a
compact subset of X, f gA = A 6= ∅ and hA ⊆ A for all h ∈ Cf ∩ Cg . Note that
n
∞
n
f A = f ∩∞
n=1 (f g) X ⊆ ∩n=1 f (f g) X
n
n
∞
= ∩∞
n=1 (f g) f X ⊆ ∩n=1 (f g) X = A.
(2.5)
Thus (2.5) ensures that
A = f gA ⊆ f A ⊆ A,
which implies that A = f A. Similarly we conclude also that A = gA. As in the
proof of Theorem 2.1, we deduce easily that (2.1) and (2.2) hold. Assume that
(2.3) is satisfied. It follows from gA = A that there exists some c ∈ A such that
gc = a. Suppose that a 6= f a, that is, f gc 6= gc. In view of (2.1)∼(2.3) and
(1.4), we get that
d(f gc, gc) + min{d(f m+1 gc, f m c), d(gm gc, gm+1 c)}
n
> inf d(hgc, f hgc), d(hc, f hc), d(hgc, ghgc), d(hc, ghc), d(hgc, hc),
d(f p gc, f p c)d(gq gc, gq c) d(hgc, f hgc)d(kgc, gkgc)
,
,
d(f gc, gc)
d(f gc, gc)
o
d(hc, f hc)d(kc, gkc)
: p, q ∈ ω, h, k ∈ Cf ∩ Cg
d(f gc, gc)
n
≥ inf d(ha, f ha), d(hc, f hc), d(ha, gha), d(hc, ghc), d(ghc, hc),
o
d(gf p c, f p c), d(ha, f ha), d(hc, f hc) : p, q ∈ ω, h, k ∈ Cf ∩ Cg ≥ d(f a, a),
FIXED POINT THEOREMS FOR TWO CLASSES...
245
which yields that
d(a, f a) = d(f gc, gc) > d(a, f a),
which is impossible. Hence a = f a.
Similarly, if (2.4) holds, we also infer that b is a fixed point of g. This
completes the proof.
It follows from Theorems 2.1 and 2.2 that
Corollary 2.1. Let f and g be continuous self mappings of a compact
n
∞
n
metric space (X, d) and f (∩∞
n=1 (gf ) X) = ∩n=1 (gf ) X. If f and g satisfy that
n
d(f x, gy) > inf d(x, f x), d(y, f y), d(x, gx), d(y, gy), d(hx, hy),
d(x, f x)d(y, f y) d(x, f x)d(x, gx) d(x, f x)d(y, gy)
,
,
,
d(f x, gy)
d(f x, gy)
d(f x, gy)
d(x, f x)d(hy, hy) d(y, f y)d(x, gx) d(y, f y)d(y, gy)
,
,
,
d(f x, gy)
d(f x, gy)
d(f x, gy)
d(y, f y)d(hx, hy) d(x, gx)d(y, gy) d(x, gx)d(hx, hy)
,
,
,
d(f x, gy)
d(f x, gy)
d(f x, gy)
d(y, gy)d(hx, hy) d(y, gy)[1 + d(y, f y)]
,
,
d(f x, gy)
1 + d(x, f x)
d(hx, hy)[1 + d(y, f y)] d(x, f x)[1 + d(x, y)]
,
,
1 + d(x, f x)
1 + d(y, gy)
o
d(y, f y)[1 + d(x, y)] d(hx, hy)[1 + d(x, y)]
,
: h ∈ Cf ∩ Cg
1 + d(y, gy)
1 + d(y, gy)
for all x, y ∈ X with f x 6= gy, then at least one of f or g has a fixed point in
X.
Corollary 2.2. Let f and g be continuous commuting self mappings of a
compact metric space (X, d). Assume that there exists m ∈ ω satisfying
d(f x, gy) + min{d(f m+1 x, f m y), d(gm x, gm+1 y)}
n
> inf d(f n+1 x, f n x), d(f n+1 y, f n y), d(gn+1 x, gn x), d(gn+1 y, gn y),
d(f n x, f n+1 x)d(gp x, gp+1 x)
,
d(f x, gy)
d(f n y, f n+1 y)d(gp y, gp+1 y) d(f n x, f n y)d(gp x, gp y)
,
:
d(f x, gy)
d(f x, gy)
d(hx, hy),
o
n, p ∈ ω, h ∈ Cf ∩ Cg .
246
S.M. Kang, P. Zheng, Z. Liu, C.Y. Jung
Then at least one of f and g has a fixed point in X.
Theorem 2.3. Let f be a continuous self mapping of a compact metric
space (X, d). Then the following statements are equivalent:
(a) f has a fixed point in X;
(b) f x 6= f y implies d(f x, f y) > d(hx, hy) for some h ∈ Cf ;
(c) f x 6= f y implies
n
d(f x, f y) + min{d(f x, y), d(x, f y)} > inf d(x, f x), d(y, f y), d(hx, hy),
d(x, f x)d(y, f y) d2 (x, f x) d(x, f x)d(hx, hy)
,
,
,
d(f x, f y)
d(f x, f y)
d(f x, f y)
d2 (y, f y) d(y, f y)d(hx, hy) d(y, f y)[1 + d(y, f y)]
,
,
,
d(f x, f y)
d(f x, f y)
1 + d(x, f x)
d(hx, hy)[1 + d(y, f y)] d(x, f x)[1 + d(x, y)]
,
,
1 + d(x, f x)
1 + d(y, f y)
o
d(y, f y)[1 + d(x, y)] d(hx, hy)[1 + d(x, y)]
,
: h ∈ Cf ∩ Cg .
1 + d(y, f y)
1 + d(y, f y)
Proof. It follows from [1] that (a) and (b) are equivalent. It is clear that
(b) implies (c). Taking g = f in Theorem 2.1, we conclude that (c) implies (a).
This completes the proof.
Ackowledgements
This work was supported by the Science Research Foundation of Educational
Department of Liaoning Province (2008352).
References
[1] G. Jungck, Commuting mappings and fixed points, Amer. Math. Monthly,
83 (1976), 261-263.
[2] G. Jungck, Common fixed poins for commuting and compatible maps on
compacta, Proc. Amer. Math. Soc., 103 (1988), 977-983.
[3] S. Leader, Uniformly contractive fixed point in compact metric space, Proc.
Amer. Math. Soc., 86, No. 1 (1982), 153-158.
FIXED POINT THEOREMS FOR TWO CLASSES...
247
[4] Z. Liu, A fixed point theorem in compact metric spaces, Bull. Malaysian
Math. Soc., 15 (1992), 69-71.
[5] Z. Liu, A note on fixed point in compact metric spaces, Indian J. Math.,
34 (1992), 173-176.
[6] Z. Liu, C. Feng, S.M. Kang, Y.S. Kim, Some fixed points for expansive
mappings and families of mappings, East Asian Math., 18 (2002), 127136.
248