Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 812690, 12 pages
doi:10.1155/2011/812690
Research Article
Tripled Fixed Points of Multivalued
Nonlinear Contraction Mappings in Partially
Ordered Metric Spaces
Mujahid Abbas,1 Hassen Aydi,2 and Erdal Karapınar3
1
Department of Mathematics, Lahore University of Management Sciences, 54792 Lahore, Pakistan
Institut Supérieur d’Informatique et des Technologies de Comminucation de Hammam Sousse,
Université de Sousse, Route GP1, 4011 Hammam Sousse, Tunisia
3
Department of Mathematics, Atilim University, İncek, 06836 Ankara, Turkey
2
Correspondence should be addressed to Hassen Aydi, hassen.aydi@isima.rnu.tn
Received 5 August 2011; Accepted 19 October 2011
Academic Editor: Yong Zhou
Copyright q 2011 Mujahid Abbas 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.
Berinde and Borcut 2011, introduced the concept of tripled fixed point for single mappings in partially ordered metric spaces. Samet and Vetro 2011 established some coupled fixed point theorems for multivalued nonlinear contraction mappings in partially ordered metric spaces. In this paper, we obtain existence of tripled fixed point of multivalued nonlinear contraction mappings in the
framework of partially ordered metric spaces. Also, we give an example.
1. Introduction and Preliminaries
Let X, d be a metric space. Consistent with 1, we denote by CBX the family of all nonempty closed bounded and nonempty closed subsets of X. Let CLX {A ⊂ X : A /
∅
and A A}, where A denotes the closure of A in X. For A, B ∈ CBX, and x ∈ X, set
Dx, A : inf{dx, a : a ∈ A}. We define a Hausdorff metric H on CBX by
HA, B : max sup Da, B, sup Db, A .
a∈A
1.1
b∈B
A point x ∈ K is called a fixed point of T if x ∈ T x.
The study of fixed points for multivalued contractions and nonexpansive maps using
the Hausdorff metric was initiated by Markin 2. Later, an interesting and rich fixed point
2
Abstract and Applied Analysis
theory for such maps was developed. Several authors studied the problem of existence of
fixed point of multivalued mappings satisfying different contractive conditions see, e.g., 3–
10. The theory of multivalued maps has application in control theory, convex optimization,
differential equations, and economics.
Existence of fixed points in ordered metric spaces has been initiated in 2004 by Ran
and Reurings 11, further studied by Nieto and Rodrı́guez-López 12. Samet and Vetro
13 introduced the notion of fixed point of N order in case of single-valued mappings. In
particular for N 3 tripled case, we have the following definition.
Definition 1.1 see, e.g., 13. An element x, y, z ∈ X 3 is called a tripled fixed point of a mapping F : X 3 → X if and only if
x F x, y, z ,
y F y, z, x ,
z F z, x, y .
1.2
Recently, Berinde and Borcut 14 established the existence of tripled fixed point of
single-valued mappings in partially ordered metric spaces. The aim of this paper is to initiate
the study of tripled fixed point of multivalued mappings in the framework of partially ordered metric spaces which in turn extend and strengthen various known results 5, 15.
2. Tripled Fixed Point Results for Multivalued Mappings
First, we introduce the following concepts.
Definition 2.1. An element x, y, z ∈ X 3 is called a tripled fixed point of F : X 3 → CLX if
x ∈ F x, y, z ,
y ∈ F y, z, x ,
z ∈ F z, x, y .
2.1
Definition 2.2. A mapping f : X 3 → R is called lower semicontinuous if, for any sequences
{xn }, {yn }, {zn } in X and x, y, z ∈ X 3 , one has
lim xn , yn , zn x, y, z ⇒ f x, y, z ≤ lim inf xn , yn , zn .
n→∞
n→∞
2.2
Let X, d be a metric space endowed with a partial order and T : X → X. Define
the set Ψ ⊂ X 3 by
Ψ
x, y, z ∈ X 3 : T x T y T z .
2.3
Definition 2.3. A mapping F : X 3 → X is said to have a Ψ-property if
x, y, z ∈ Ψ ⇒ F x, y, z × F y, z, x × F z, y, x ⊂ Ψ.
2.4
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3
We give some examples to illustrate Definition 2.3.
Example 2.4. Let X R be endowed with the usual order ≤, and T : X → X. Define F : X 3 →
CLX by
F x, y, z {x} ∀x, y, z ∈ R.
2.5
Obviously, F has the Ψ-property.
Example 2.5. Let X R be endowed with the usual order ≤, and let T : X → X be defined
by T x expx. Define F : X 3 → CLX by
F x, y, z {x z}
∀x, y, z ∈ R .
2.6
We have Ψ {x, y, z ∈ X 3 , expx ≤ expy ≤ expz}. Moreover, F has the Ψ-property.
Now, we prove the following theorem.
Theorem 2.6. Let X, d be a complete metric space endowed with a partial order and Ψ /
∅; that
is, there exists x0 , y0 , z0 ∈ Ψ. Suppose that F : X 3 → CLX has a Ψ-property such that f : X 3 →
0, ∞ given by
f x, y, z D x, F x, y, z D y, F y, z, x D z, F z, x, y
∀x, y, z ∈ X
2.7
is lower semicontinuous and there exists a function φ : 0, ∞ → M, 1, 0 < M < 1, satisfying
lim sup φr < 1 for each t ∈ 0, ∞.
r → t
2.8
If for any x, y, z ∈ Ψ there exist u ∈ Fx, y, z, v ∈ Fy, z, x, and w ∈ Fz, y, x with
φ f x, y, z dx, u d y, v dz, w ≤ f x, y, z
2.9
fu, v, w ≤ φ f x, y, z dx, u d y, v dz, w ,
2.10
such that
then F has a tripled fixed point.
Proof. By our assumption, φfx, y, z < 1 for each x, y, z ∈ X 3 . Hence, for any x, y, z ∈
X 3 , there exist u ∈ Fx, y, z, v ∈ Fy, z, x, and w ∈ Fz, x, y satisfying
φ f x, y, z dx, u ≤ D x, F x, y, z ,
φ f x, y, z d y, v ≤ D y, F y, z, x ,
φ f x, y, z dz, w ≤ D z, F z, x, y .
2.11
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Abstract and Applied Analysis
Let x0 , y0 , z0 be an arbitrary point in Ψ. By 2.9 and 2.10, we can choose x1 ∈ Fx0 , y0 , z0 ,
y1 ∈ Fy0 , z0 , x0 , and z1 ∈ Fz0 , x0 , y0 satisfying
φ f x0 , y0 , z0 dx0 , x1 d y0 , y1 dz0 , z1 ≤ f x0 , y0 , z0
2.12
f x1 , y1 , z1 ≤ φ f x0 , y0 , z0 dx0 , x1 d y0 , y1 dz0 , z1 .
2.13
such that
By 2.12 and 2.13, we obtain
f x1 , y1 , z1 ≤ φ f x0 , y0 , z0 dx0 , x1 d y0 , y1 dz0 , z1 ≤ φ f x0 , y0 , z0
φ f x0 , y0 , z0 dx0 , x1 d y0 , y1 dz0 , z1 ≤
φ f x0 , y0 , z0 f x0 , y0 , z0 .
2.14
Thus,
f x1 , y1 , z1 ≤ φ f x0 , y0 , z0 f x0 , y0 , z0 .
2.15
Since F has a Ψ-property and x0 , y0 , z0 ∈ Ψ, so we have
F x0 , y0 , z0 × F y0 , z0 , x0 × F z0 , x0 , y0 ⊂ Ψ
2.16
which implies that x1 , y1 , z1 ∈ Ψ.
Again by 2.9 and 2.10, we can choose x2 ∈ Fx1 , y1 , z1 , y2 ∈ Fy1 , z1 , x1 , and
z2 ∈ Fz1 , x1 , y1 satisfying
φ f x1 , y1 , z1 dx1 , x2 d y1 , y2 dz1 , z2 ≤ f x1 , y1 , z1
2.17
f x2 , y2 , z2 ≤ φ f x1 , y1 , z1 dx1 , x2 d y1 , y2 dz1 , z2 .
2.18
such that
Thus, we have
f x2 , y2 , z2 ≤ φ f x1 , y1 , z1 f x1 , y1 , z1
2.19
and x2 , y2 , z2 ∈ Ψ.
Continuing this process, we can choose sequences {xn }, {yn }, {zn } in X such that for
each n ∈ N with xn , yn , zn ∈ Ψ.
Abstract and Applied Analysis
5
xn1 ∈ Fxn , yn , zn , yn1 ∈ Fyn , zn , xn and zn1 ∈ Fzn , xn , yn satisfying
φ f xn , yn , zn dxn , xn1 d yn , yn1 dzn , zn1 ≤ f xn , yn , zn
2.20
such that
f xn1 , yn1 , zn1 ≤ φ f xn , yn , zn dxn , xn1 d yn , yn1 dzn , zn1 .
2.21
Hence, we obtain
f xn1 , yn1 , zn1 ≤ φ f xn , yn , zn f xn , yn , zn
2.22
with xn1 , yn1 , zn1 ∈ Ψ. We claim that fxn , yn , zn → 0 as n → ∞. If fxn , yn , zn 0
for some n ∈ N, then Dxn , Fxn , yn , zn 0 implies that xn ∈ Fxn , yn , zn Fxn , yn ,
zn . Analogously, Dyn , Fyn , zn , xn 0 implies that yn ∈ Fyn , zn , xn and Dzn ,
Fzn , yn , xn 0 implies that zn ∈ Fzn , yn , xn . Hence, xn , yn , zn becomes a tripled fixed
point of F for such n and the result follows. Suppose that fxn , yn , zn > 0 for all n ∈ N.
Using 2.22 and φt < 1, we conclude that {fxn , yn , zn } is a decreasing sequence of
positive real numbers. Thus, there exists a δ ≥ 0 such that
lim f xn , yn , zn δ.
n→∞
2.23
We will show that δ 0. Assume on contrary that δ > 0. Letting n → ∞ in 2.22 and by assumption 2.8, we obtain
δ≤
lim sup
fxn ,yn ,zn → δ
φ f xn , yn , zn δ < δ,
2.24
a contradiction. Hence,
lim f xn , yn , zn 0 .
n→∞
2.25
Now, we prove that {xn }, {yn }, {zn } ⊂ X are Cauchy sequences in X, d. Assume that
α
lim sup
fxn ,yn ,zn → 0
φ f xn , yn , zn .
2.26
By 2.8, we conclude that α < 1. Let k be a real number such that α < k < 1. Thus, there exists
n0 ∈ N such that
φ f xn , yn , zn < k
for each n ≥ n0 .
2.27
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Abstract and Applied Analysis
Using 2.22, we obtain
f xn1 , yn1 , zn1 < kf xn , yn , zn
for each n ≥ n0 .
2.28
By mathematical induction,
f xn1 , yn1 , zn1 < kn1−n0 f xn0 , yn0 , zn0
for each n ≥ n0 .
2.29
Since φt ≥ M > 0 for all t ≥ 0 so 2.20, and 2.29 gives that
kn−n0 dxn , xn1 d yn , yn1 dzn , zn1 < √ f xn0 , yn0 , zn0
M
for each n ≥ n0
2.30
which yields that {xn }, {yn }, {zn } ⊂ X are Cauchy sequences in X. Since X is complete, there
exists a, b, c ∈ X 3 such that
lim xn a,
n→∞
lim yn b,
n→∞
lim zn c.
2.31
n→∞
Finally, we show that a, b, c ∈ X 3 is tripled fixed point of F. As f is lower semicontinuous,
2.25 implies that
0 ≤ fa, b, c Da, Fa, b, c Db, Fb, c, a Dc, Fc, a, b ≤ lim inf f xn , yn , zn 0.
n→∞
2.32
Hence, Da, Fa, b, c Db, Fb, c, a Dc, Fc, a, b 0 gives that a, b, c is a tripled
fixed point of F.
Theorem 2.7. Let X, d be a complete metric space endowed with a partial order and Ψ /
∅; that
is, there exists x0 , y0 , z0 ∈ Ψ. Suppose that F : X 3 → CLX has a Ψ-property such that function
f : X 3 → 0, ∞ defined by
f x, y, z D x, F x, y, z D y, F y, z, x D z, F z, x, y
∀x, y, z ∈ X,
2.33
is lower semicontinuous and there exists a function φ : 0, ∞ → M, 1, 0 < M < 1, satisfying
lim sup φr < 1
r → t
for each t ∈ 0, ∞.
2.34
If for any x, y, z ∈ Ψ there exist u ∈ Fx, y, z, v ∈ Fy, z, x, and w ∈ Fz, y, x satisfying
φΔΔ ≤ D x, f x, y, z D y, f y, z, x D z, f z, x, y
2.35
Abstract and Applied Analysis
7
such that
D u, fu, v, w D v, fv, w, u D w, fw, u, v ≤ φΔΔ,
2.36
where Δ Δx, y, z, u, v, w dx, u dy, v dz, w, then F admits a tripled fixed point.
Proof. By replacing φfx, y, z with φdx, u dy, v dz, w in the proof of
Theorem 2.6, we obtain sequences {xn }, {yn }, {zn } ⊂ X such that for each n ∈ N with
xn , yn , zn ∈ Ψ,
xn1 ∈ F xn , yn , zn ,
yn1 ∈ F yn , zn , xn ,
zn1 ∈ F zn , xn , yn ,
2.37
such that
φΔn Δn ≤ D xn , F xn , yn , zn D yn , F yn , zn , xn D zn , F zn , xn , yn ,
2.38
D xn1 , F xn1 , yn1 , zn1 D yn1 , F yn1 , zn1 , xn1 D zn1 , F zn1 , xn1 , yn1
≤ φΔn D xn , F xn , yn , zn D yn , F yn , zn , xn D zn , F zn , xn , yn
,
2.39
where
Δn Δn
xn , yn , zn , xn1 , yn1 , zn1 dxn , xn1 d yn , yn1 dzn , zn1 .
2.40
Again, following arguments similar to those given in the proof of Theorem 2.6, we deduce
that
D xn , F xn , yn , zn D yn , F yn , zn , xn D zn , F zn , xn , yn
2.41
is a decreasing sequence of real numbers. Thus, there exists a δ > 0 such that
lim D xn , F xn , yn , zn D yn , F yn , zn , xn zn , F zn , xn , yn δ.
n→∞
2.42
Now, we need to prove that {Δn } admits a subsequence converging to certain η for some
η ≥ 0. Since φt ≥ M > 0, using 2.38, we obtain
1 Δn ≤ √ D xn , F xn , yn , zn D yn , F yn , zn , xn D zn , F zn , xn , yn
.
a
2.43
From 2.42 and 2.43, it is clear that the sequence
D xn , F xn , yn , zn D yn , F yn , zn , xn D zn , F zn , xn , yn
2.44
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Abstract and Applied Analysis
is bounded. Therefore, there is some θ ≥ 0 such that
lim inf Δn θ.
2.45
n → ∞
From 2.37, we have xn1 ∈ Fxn , yn , zn , yn1 ∈ Fyn , zn , xn , and zn1 ∈ Fzn , xn , yn ,
Δn ≥ D xn , F xn , yn , zn D yn , F yn , zn , xn D zn , F zn , xn , yn
for each n ≥ 0.
2.46
So comparing 2.42 to 2.45, we get that θ ≥ δ. Now, we shall show that θ δ. If δ 0, then,
by 2.42 and 2.43, we get θ : lim infn → ∞ Δn 0 and consequently θ δ 0. Suppose that
δ > 0. Assume on contrary that θ > δ. From 2.42 and 2.45, there is a positive integer n0
such that
θ−δ
,
D xn , F xn , yn , zn D yn , F yn , zn , xn D zn , F zn , xn , yn < δ 4
δ−
θ−δ
< Δn ,
4
2.47
2.48
for all n ≥ n0 . We combine 2.38, 2.47 to 2.48 to obtain
θ−δ
φΔn δ −
≤ φΔn Δn
4
≤ D xn , F xn , yn , zn D yn , F yn , zn , xn D zn , F zn , xn , yn
<δ
θ−δ
,
4
2.49
for all n ≥ n0 . It follows that
φΔn ≤
θ 3δ
3θ δ
∀n ≥ n0 .
2.50
By 2.39 and 2.50, we have
D xn1 , F xn1 , yn1 , zn1 D yn1 , F yn1 , zn1 , xn1 D zn1 , F zn1 , xn1 , yn1
≤ h D xn , F xn , yn , zn D yn , F yn , zn , xn D zn , F zn , xn , yn
∀n ≥ n0 ,
2.51
Abstract and Applied Analysis
9
where h θ 3δ/3θ δ. Since θ > δ > 0, therefore h < 1, so proceeding by induction and
combining the above inequalities, it follows that
δ ≤ D xn0 k0 , F xn0 k0 , yn0 k0 , zn0 k0 D yn0 k0 , F yn0 k0 , zn0 k0 , xn0 k0
D zn0 k0 , F zn0 k0 , xn0 k0 , yn0 k0
≤ hk0 D xn0 , F xn0 , yn0 , zn0 D yn0 , F yn0 , zn0 , xn0 D zn0 , F zn0 , xn0 , yn0
< δ,
2.52
for a positive integer k0 . Then, we obtain a contradiction, so we must have θ δ.
Now, we shall show that θ 0. Since
θ δ ≤ D xn , F xn , yn , zn D yn , F yn , zn , xn D zn , F zn , xn , yn ≤ Δn ,
2.53
then we rewrite 2.45 as
lim inf Δn θ .
n → ∞
2.54
Hence, there exists a subsequence {Δnk } of {Δn } such that lim infk → ∞ Δnk θ .
By 2.34, we have
lim sup φΔnk < 1.
Δn k → θ 2.55
From 2.39, we obtain
D xnk 1 , F xnk 1 , ynk 1 , znk 1 D ynk 1 , F ynk 1 , znk 1 , xnk 1
D znk 1 , F znk 1 , xnk 1 , ynk 1
≤ φΔnk D xnk , F xnk , ynk , znk D ynk , f ynk , znk , xnk D znk , F znk , xnk , ynk
.
2.56
Taking the limit as k → ∞ and using 2.42, we have
δ lim sup D xnk 1 , F xnk 1 , ynk 1 , znk 1 D ynk 1 , F ynk 1 , znk 1 , xnk 1
k → ∞
D znk 1 , f znk 1 , xnk 1 , ynk 1
≤ lim sup φΔnk k → ∞
10
Abstract and Applied Analysis
lim sup D xnk , F xnk , ynk , znk D ynk , F ynk , znk , xnk D znk , F znk , xnk , ynk
k → ∞
⎛
⎞
⎝lim sup φΔnk ⎠δ.
Δn k → θ 2.57
Assume that δ > 0, then from 2.57 we get that
1 ≤ lim sup φΔnk ,
2.58
a contradiction with respect to 2.55, so δ 0. Now, from 2.39, and 2.42 we have
α lim sup φΔn < 1.
2.59
Δn k → θ Δn → 0
The rest of the proof is similar to the proof of Theorem 2.6, so it is omitted.
We improved and corrected the example of Samet and Vetro 13.
Example 2.8. Let X 0, 1, and let d : X × X → 0, ∞ be the usual metric. Suppose that
T x M for all x ∈ 0, 1 where M is a constant in 0, 1, and F : X 3 → CLX is defined,
for all y, z ∈ X as follow:
⎧ ⎪
15
15
x2
⎪
⎪
if x ∈ 0,
,1 ,
∪
⎨
4
32
32
F x, y, z 2.60
⎪
15
1
15
⎪
⎪
⎩
,
.
if x 96 5
32
Obviously, F has the Ψ-property. Set φ : 0, ∞ → 0, 1:
⎧
2
11
⎪
⎪
t
if
t
∈
0,
,
⎪
⎨ 12
3
φt ⎪
2
10
⎪
⎪
if t ∈
,∞ .
⎩
16
3
Consider the function
⎧
1
⎪
⎪
x y z − x2 y2 z2
⎪
⎪
4
⎪
⎪
⎪
⎪
⎪
43
1
⎪
⎪
⎪x y − x 2 y 2 ⎨
4
160
f x, y, z ⎪
1
86
⎪
⎪
⎪x − x2 ⎪
⎪
4
160
⎪
⎪
⎪
⎪
⎪
⎪
⎩ 129
160
if
if
if
if
2.61
15
15
∪
,1 ,
x, y, z ∈ 0,
32
32
15
15
15
x, y ∈ 0,
∪
, 1 with z ,
32
32
32
15
15
15
x, y ∈ 0,
∪
, 1 with y z ,
32
32
32
15
xyz
32
2.62
Abstract and Applied Analysis
11
which is lower semicontinuous. Thus, for all x, y, z ∈ X with x, y, z /
15/32, there exist u ∈
Fx, y, z {x2 /4}, v ∈ Fy, z, x {y2 /4}, and w ∈ Fz, y, x {z2 /4} such that
Du, Fu, v, w Dv, Fv, w, u Dw, Fw, v, u
x2 x4 y2 y4 z2 z4
−
−
−
4
64
4
64
4
64
y2
y2
x2
x2
z2
z2
1
x
x−
y
y−
z
z−
4
4
4
4
4
4
4
1
≤
4
x2
x
4
1
≤ max
4
10
max
<
12
dx, u y2
y
4
x2
x
4
x2
x−
4
,
y2
y
4
z2
z
4
d y, v z2
z
4
,
,
,
y2
y−
4
z2
z−
4
dz, w
2.63
dx, u d y, v dz, w
dx, u d y, v dz, w
≤ φ dx, u d y, v dz, w dx, u d y, v dz, w .
Hence, for all x, y, z ∈ X with x, y, z /
15/32, the conditions 2.35 and 2.36 are satisfied.
Analogously, one can easily show that conditions 2.35 and 2.36 are satisfied for the cases
x, y ∈ 0, 15/32 ∪ 15/32, 1 with z 15/32 and x ∈ 0, 15/32 ∪ 15/32, 1 and y z 15/32. For the last case, that is, x y z 15/32, we assume that u v w 15/96, so it
follows that
15 2
> .
dx, u d y, v dz, w 16 3
2.64
As a consequence, we get that
φ dx, u d y, v dz, w dx, u d y, v dz, w
10 15 129
<
D x, F x, y, z D y, F y, z, x D z, F z, x, y ,
16 16 160
15 1 15 2
−
Du, Fu, v, w Dv, Fv, w, u Dw, Fw, v, u 3
96 4 96
<
2.65
10 15
φ dx, u d y, v dz, w dx, u d y, v dz, w .
16 16
Thus, we conclude that all the conditions of Theorem 2.7 are satisfied and F admits a tripled
fixed point a 0, 0, 0.
12
Abstract and Applied Analysis
Remark 2.9. If we replace the function φ with the following, we get the results again:
⎧
7
⎪
⎪
⎪
⎨ 12 t
φt ⎪
7
⎪
⎪
⎩
16
3
if t ∈ 0, ,
4
3
if t ∈
,∞ .
4
2.66
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