Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 750403, 9 pages
doi:10.1155/2012/750403
Research Article
A Unique Common Triple Fixed Point Theorem for
Hybrid Pair of Maps
K. P. R. Rao,1 G. N. V. Kishore,2 and Kenan Tas3
1
Department of Mathematics, Acharya Nagarjuna University, Nagarjuna Nagar, Guntur 522510, India
Department of Mathematics, Swarnandhra Institute of Engineering and Technology, Seetharampuram,
Narspur 534 280, India
3
Department of Mathematics and Computer Science, Cankaya University, 06810 Ankara, Turkey
2
Correspondence should be addressed to Kenan Tas, kenan@cankaya.edu.tr
Received 25 June 2012; Accepted 29 August 2012
Academic Editor: Nikolaos Papageorgiou
Copyright q 2012 K. P. R. Rao et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We obtain a unique common triple fixed point theorem for hybrid pair of mappings in metric
spaces. Our result extends the recent results of B. Samet and C. Vetro 2011. We also introduced a
suitable example supporting our result.
1. Introduction
The study of fixed points for multivalued contraction mappings using the Hausdorff metric
was initiated by Nadler 1.
Let X, d be a metric space. We denote CBX the family of all nonempty closed and
bounded subsets of X and CLX the set of all nonempty closed subsets of X. For A, B ∈
CBX and x ∈ X, we denote Dx, A inf{dx, a : a ∈ A}. Let H be the Hausdorff metric
induced by the metric d on X, that is,
HA, B max sup dx, B, sup d y, A ,
x∈A
1.1
y∈B
for every A, B ∈ CBX.
It is clear that for A, B ∈ CBX and a ∈ A, we have da, B ≤ HA, B.
Definition 1.1. An element x ∈ X is said to be a fixed point of a set-valued mapping T : X →
CBX if and only if x ∈ T x.
2
Abstract and Applied Analysis
In 1969, Nadler 1 extended the famous Banach contraction principle 2 from singlevalued mapping to multivalued mapping and proved the following fixed point theorem for
the multivalued contraction.
Theorem 1.2 see, Nadler 1. Let X, d be a complete metric space and let T be a mapping from
X into CBX. Assume that there exists c ∈ 0, 1 such that
H T x, T y ≤ cd x, y ,
1.2
for all x, y ∈ X. Then, T has a fixed point.
Lemma 1.3 see, Nadler 1. Let A, B ∈ CBX and α > 1. Then for every a ∈ A, there exists
b ∈ B such that da, b ≤ αHA, B.
Lemma 1.4 see, Nadler 1. Let α > 0. If A, B ∈ CBX with HA, B ≤ α, then for each a ∈ A,
there exists b ∈ B such that da, b ≤ α.
Lemma 1.5 see, Nadler 1. Let {An } be a sequence in CBX with limn → ∞ HAn , A 0, for
A ∈ CBX. If xn ∈ An and limn → ∞ dxn , x 0, then x ∈ A.
The existence of fixed points for various multivalued contractive mappings has been
studied by many authors under different conditions. For details, we refer the reader to 1, 3–
11 and the references therein.
The concept of coupled fixed point for multivalued mapping was introduced by Samet
and Vetro 12, and later several authors, namely, Hussain and Alotaibi 13, Aydi et al. 14,
and Abbas et al. 15, proved coupled coincidence point theorems in partially ordered metric
spaces.
Definition 1.6 see, Samet and Vetro 12. Let F : X × X → CLX be a given mapping. We
say that x, y ∈ X × X is a coupled fixed point of F if and only if
x ∈ F x, y ,
y ∈ F y, x .
1.3
Definition 1.7 see, Hussain and Alotaibi 13. Let the mappings F : X × X → CBX and
g : X → X be given. An element x, y ∈ X × X is called
1 a coupled coincidence point of a pair {F, g} if gx ∈ Fx, y and gy ∈ Fy, x;
2 a coupled common fixed point of a pair {F, g} if x gx ∈ Fx, y and y gy ∈
Fy, x.
Berinde and Borcut 16 introduced the concept of triple fixed points and obtained a
tripled fixed point theorem for single valued map.
Now we give the following.
Definition 1.8. Let X be a nonempty set, T : X × X × X → 2X collection of all nonempty
subsets of X. f : X → X.
i The point x, y, z ∈ X × X × X is called a tripled fixed point of T if
x ∈ T x, y, z ,
y ∈ T y, x, y ,
z ∈ T z, y, x .
1.4
Abstract and Applied Analysis
3
ii The point x, y, z ∈ X × X × X is called a tripled coincident point of T and f if
fx ∈ T x, y, z ,
fy ∈ T y, x, y ,
fz ∈ T z, y, x .
1.5
iii The point x, y, z ∈ X × X × X is called a tripled common fixed point of T and f if
x fx ∈ T x, y, z ,
y fy ∈ T y, x, y ,
z fz ∈ T z, y, x .
1.6
Definition 1.9. Let T : X × X × X → 2X be a multivalued map and f be a self map on X. The
Hybrid pair {T, f} is called w-compatible if fT x, y, z ⊆ T fx, fy, fz whenever x, y, z
is a tripled coincidence point of T and f.
2. Main Results
Theorem 2.1. Let X, d be a metric space and let T : X × X × X → CBX and f : X → X
mappings satisfying
2.1.1 HT x, y, z, T u, v, w ≤ jdfx, fy kdfy, fv ldfz, fw, for all x, y, z,
u, v, w ∈ X and j, k, l ∈ 0, 1 with j k l ≤ h < 1, where h is a fixed number,
2.1.2 T X × X × X ⊆ fX and fX is a complete subspace of X.
Then the maps T and f have a tripled coincidence point.
Further, T and f have a tripled common fixed point if one of the following conditions holds.
2.1.3 (a) {T, f} is w-compatible, there exist u, v, w ∈ X such that limn → ∞ f n x u,
limn → ∞ f n y v and limn → ∞ f n z w, whenever x, y, z is a tripled coincidence point
of {T, f} and f is continuous at u, v, w.
(b) There exist u, v, w ∈ X such that limn → ∞ f n u x, limn → ∞ f n v y and
limn → ∞ f n w z whenever x, y, z is a tripled coincidence point of {T, f} and f is
continuous at x, y, and z.
Proof. Let x0 , y0 , z0 ∈ X. From 2.1.2, there exist sequences {xn }, {yn }, and {zn } in X such
that fxn1 ∈ T xn , yn , zn , fyn1 ∈ T yn , xn , yn and fzn1 ∈ T zn , yn , xn , n 0, 1, 2, 3, . . ..
For simplification, denote
dnx d fxn−1 , fxn ,
y
dn d fyn−1 , fyn ,
dnz d fzn−1 , fzn .
2.1
4
Abstract and Applied Analysis
From 2.1.1, we obtain
d2x d fx1 , fx2
≤ H T x0 , y0 , z0 , T x1 , y1 , z1 h
≤ jd fx0 , fx1 kd fy0 , fy1 ld fz0 , fz1 h
i
y
jd1x kd1 ld1z h,
y
d2 d fy1 , fy2
≤ H T y0 , x0 , y0 , T y1 , x1 , y1 h
≤ jd fy0 , fy1 kd fx0 , fx1 ld fy0 , fy1 h
y
kd1x j l d1 h,
d2z d fz1 , fz2
≤ H T z0 , y0 , x0 , T z1 , y1 , x1 h
≤ jd fz0 , fz1 kd fy0 , fy1 ld fx0 , fx1 h
ii
iii
y
ld1x kd1 jd1z h,
d3x d fx2 , fx3
≤ H T x1 , y1 , z1 , T x2 , y2 , z2 h2
≤ jd fx1 , fx2 kd fy1 , fy2 ld fz1 , fz2 h2
y
jd2x kd2 ld2z h2
y
y
≤ j jd1x kd1 ld1z h k kd1x j l d1 h
iv
y
l ld1x kd1 jd1z h h2
y j 2 k2 l2 d1x 2jk 2lk d1 2jl d1z h2 j k l h
y j 2 k2 l2 d1x 2jk 2lk d1 2jl d1z 2h2 ,
y
d3 d fy2 , fy3
≤ H T y1 , x1 , y1 , T y2 , x2 , y2 h2
≤ jd fy1 , fy2 kd fx1 , fx2 ld fy1 , fy2 h2
y
kd2x j l d2 h2
y
y
≤ k jd1x kd1 ld1z h j l kd1x j l d1 h h2
y
2
2jk lk d1x j l k2 d1 kld1z j k l h h2
y
2
≤ 2jk lk d1x j l k2 d1 kld1z 2h2 ,
v
Abstract and Applied Analysis
d3z d fz2 , fz3
≤ H T z1 , y1 , x1 , T z2 , y2 , x2 h2
≤ jd fz1 , fz2 kd fy1 , fy2 ld fx1 , fx2 h2
y
5
y
jd2z kd2 ld2x h2 ld2x kd2 jd2z h2
y
y
≤ l jd1x kd1 ld1z h k kd1x j l d1 h
vi
y
j ld1x kd1 jd1z h h2
y 2jl k2 d1x 2 jk lk d1 j 2 l2 d1z j k l h h2
y ≤ 2jl k2 d1x 2 jk lk d1 j 2 l2 d1z 2h2 .
Let A j
k l
k jl 0
l k j
denoted by
a
1 b1 c1
d1 e1 f1
g1 b1 h1
.
Clearly, a1 b1 c1 d1 e1 f1 g1 b1 h1 j k l ≤ h < 1.
Then,
⎤
⎡
⎤
⎡
2jl
j 2 k2 l2 2jk 2lk
a2 b2 c2
⎥
⎢
2
A2 ⎣ 2jk lk
j l k2 kl ⎦ denote A2 by⎣d2 e2 f2 ⎦.
2
g2 b2 h2
2jk 2lk j 2 l2
2jl k
2.2
It is clear that a2 b2 c2 d2 e2 f2 g2 b2 h2 j k l2 ≤ h2 < 1.
Now we prove by induction that
⎤
an bn cn
An ⎣dn en fn ⎦,
gn bn hn
2.3
n
an bn cn dn en fn gn bn hn j k l ≤ hn < 1.
2.4
⎡
where
Equation 2.3 is true for n 1, 2.
Assume that 2.3 is true for some n. Consider
⎡
An1
⎤⎡
⎤
an bn cn
j k l
An · A ⎣dn en fn ⎦⎣k j l 0⎦
gn bn hn
l k j
⎤
⎡
jan kbn lcn kan j l bn kcn lan jcn
⎥
⎢
⎣jdn ken lfn kdn j l en kfn ldn jfn ⎦.
jgn kbn lhn kgn j l bn khn lgn jhn
2.5
6
Abstract and Applied Analysis
We have
n1
≤ hn1 < 1.
an1 bn1 cn1 j k l an bn cn j k l
2.6
Similarly, we have
n1
dn1 en1 fn1 gn1 bn1 hn1 j k l
≤ hn1 < 1.
2.7
Thus 2.3 is true for all ve integer values of n.
Now from i–vi and continuing this process, we get
⎡ x⎤
⎡ x ⎤
dn1
⎡
⎡ n⎤
⎤ d1
⎥
⎥
⎢
an bn cn ⎢
nh
⎢ y⎥
⎢ y ⎥
⎢d ⎥ ≤ ⎣dn en fn ⎦⎢d ⎥ ⎣nhn ⎦,
⎢ 1⎥
⎢ n1 ⎥
⎦
⎣
gn bn hn ⎣ ⎦
nhn
z
z
dn1
d1
2.8
for all n 1, 2, 3, . . .. That is,
y
x
dn1
≤ an d1x bn d1 cn d1z nhn ,
y
y
dn1 ≤ dn d1x en d1 fn d1z nhn ,
y
z
dn1
≤ gn d1x bn d1 hn d1z nhn ,
∀n 1, 2, 3, . . . .
For m > n, we have
d fxm , fxn ≤ d fxm , fxm−1 d fxm−1 , fxm−2
· · · d fxn2 , fxn1 d fxn1 , fxn
x
x
x
x
dm
dm−1
· · · dn2
dn1
y
≤ am−1 d1x bm−1 d1 cm−1 d1z m − 1hm−1
y
am−2 d1x bm−2 d1 cm−2 d1z m − 2hm−2
y
· · · an1 d1x bn1 d1 cn1 d1z n 1hn1
y
an d1x bn d1 cn d1z nhn
2.9
Abstract and Applied Analysis
7
≤ am−1 am−2 · · · an1 an d1x
y
bm−1 bm−2 · · · bn1 bn d1
cm−1 cm−2 · · · cn1 cn d1z
m − 1hm−1 m − 2hm−2 · · · n 1hn1 nhn
m−1
y
≤ hm−1 hm−2 · · · hn1 hn d1x d1 d1z jhj
jn
≤
m−1
hn x
y
d1 d1 d1z jhj −→ 0 as n −→ ∞,
1−h
jn
because 0 ≤ h < 1.
2.10
Hence {fxn } is a Cauchy. Similarly, we can show that {fyn } and {fzn } are Cauchy.
Suppose fX is complete, the sequences {fxn }, {fyn }, and {fzn } are convergent to
some α, β, γ in fX, respectively. There exist x, y, z ∈ X such that α fx, β fy, and γ fz.
Now, we have
d T x, y, z , α ≤ d T x, y, z , fxn1 d fxn1 , α
≤ H T x, y, z , T xn , yn , zn d fxn1 , α
2.11
≤ jd fx, fxn kd fy, fyn ld fz, fzn d fxn1 , α
jd α, fxn kd β, fyn ld γ, fzn d fxn1 , α .
Letting n → ∞, we get dT x, y, z, α ≤ 0 so that α ∈ T x, y, z. That is, fx ∈ T x, y, z.
Similarly, we can show that fy ∈ T y, x, y and fz ∈ T z, y, x. Thus x, y, z is a tripled
coincidence point of T and f. Suppose 2.1.3 a holds.
Since x, y, z is a tripled coincidence point of T and f, there exist u, v, w ∈ X such that
limn → ∞ f n x u, limn → ∞ f n y v and limn → ∞ f n z w.
Since f is continuous at u, v and w, we have fu u, fv v and fw w.
Since fx ∈ T x, y, z, we have f 2 x ∈ fT x, y, z ⊆ T fx, fy, fz.
Since fy ∈ T y, x, y, we have f 2 y ∈ fT y, x, y ⊆ T fy, fx, fy.
Since fz ∈ T z, y, x, we have f 2 z ∈ fT z, y, x ⊆ T fz, fy, fx.
Then fx, fy, fz is tripled coincidence point of T and f.
Similarly, we can show that f n x, f n y, f n z is a tripled coincidence point of T and f.
Also it is clear that
f n x ∈ T f n−1 x, f n−1 y, f n−1 z ,
f n y ∈ T f n−1 y, f n−1 x, f n−1 y ,
f n z ∈ T f n−1 z, f n−1 y, f n−1 x .
2.12
8
Abstract and Applied Analysis
From 2.1.1, we have
d fu, T u, v, w ≤ d fu, f n x d f n x, T u, v, w
≤ d fu, f n x H T f n−1 x, f n−1 y, f n−1 z , T u, v, w
2.13
≤ d fu, f n x jd f n x, fu kd f n y, fv ld f n z, fw .
Letting n → ∞, we obtain
d fu, T u, v, w ≤ 0,
2.14
fu ∈ T u, v, w.
2.15
which implies that
Thus u fu ∈ T u, v, w. Similarly, we can show that v fv ∈ T v, u, v and w fw ∈
T w, v, u. Thus u, v, w is a tripled common fixed point of T and f. Suppose 2.1.3 b
holds.
Since x, y, z is a tripled coincidence point of {T, f}, there exist u, v, w ∈ X such that
limn → ∞ f n u x, limn → ∞ f n v y and limn → ∞ f n w z.
Since f is continuous at x, y and z, we have fx x, fy y and fz z. Thus x fx ∈
T x, y, z, y fy ∈ T y, x, y and z fz ∈ T z, y, x. Hence x, y, z is a tripled common
fixed point of {T, f}.
The following example illustrates Theorem 2.1.
Example 2.2. Let X 0, 1, T : X × X × X → CBX and f : X → X defined as T x, y, z 0, 1/8 sin x 1/4 sin y 1/3 sin z and fx 7/8x. Then
1
1
1
sin x sin y sin z
H T x, y, z , T u, v, w 8
4
3
1
1
1
−
sin u sin v sin w 8
4
3
≤
1
1
|sin x − sin u| sin y − sin v
8
4
1
|sin z − sin w|
3
1
1
1
≤ |x − u| y − v |z − w|
8
4
3
2 7
7
7 8 7
7 1 7
x − u y − v z − w
7 8
8
7 8
8
21 8
8
2 8 1 d fx, fu d fy, fv d fz, fw .
7
7
21
2.16
Abstract and Applied Analysis
9
It is clear that all conditions of Theorem 2.1 are satisfied and 0, 0, 0 is the tripled common
fixed point of T and f.
The following example shows that T and f have no tripled common fixed point if
2.1.3 a or 2.1.3 b is not satisfied.
Example 2.3. Let X 0, 4, T x, y, z 1.5, 2 and fx 2−1/2x. Then 0, 1/2, 1 is a tripled
coincidence point of T and f. Clearly T and f have no tripled common fixed point.
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