Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2011, Article ID 979586, 10 pages
doi:10.1155/2011/979586
Research Article
Fixed-Point Results for Generalized Contractions
on Ordered Gauge Spaces with Applications
Cristian Chifu and Gabriela Petrus¸el
Faculty of Business, Babes¸-Bolyai University, Horia Street no. 7, 400174 Cluj-Napoca, Romania
Correspondence should be addressed to Cristian Chifu, [email protected]
Received 6 December 2010; Accepted 31 December 2010
Academic Editor: Jen Chih Yao
Copyright q 2011 C. Chifu and G. Petrus¸el. 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.
The p urpose of this paper is to present some fixed-point results for single-valued ϕ-contractions
on ordered and complete gauge space. Our theorems generalize and extend some recent results in
the literature. As an application, existence results for some integral equations on the positive real
axis are given.
1. Introduction
Throughout this paper will denote a nonempty set E endowed with a separating gauge
structure D {d
α
}
α∈Λ
,whereΛ is a directed set see 1 for definitions.Let : {0, 1, 2, }
and
∗
: \{0}.Wealsodenoteby the set of all real numbers and by
:0, ∞.
The set F
f
: {x ∈ E | x fx} denotes the fixed-point set of f.
On the other hand, Ran and Reurings 2 proved the following Banach-Caccioppoli
type principle in ordered metric spaces.
Theorem 1.1 Ran and Reurings 2. Let X be a partially ordered set such that every pair x, y ∈ X
has a lower and an upper bound. Let d be a metric on X such that the metric space X, d is complete.
2 Fixed Point Theory and Applications
Let f : X → X be a continuous and monotone (i.e., either decreasing or increasing) operator. Suppose
that the following two assertions hold:
1 there exists a ∈ 0, 1 such that dfx,fy ≤ a · dx, y,foreachx, y ∈ X with x ≥ y;
2 there exists x
0
∈ X such that x
0
≤ fx
0
or x
0
≥ fx
0
.
Then f has an unique fixed point x
∗
∈ X,thatis,fx
∗
x
∗
,andforeachx ∈ X the sequence
f
opez 3, 7,Nietoetal.5,Petrus¸el and Rus 4, Agarwal et al. 8,O’Regan
and Petrus¸el 6,etc..
2. Preliminaries
Let X be a nonempty set and f : X → X be an operator. Then, f
0
: 1
X
, f
1
: f, ,f
n1
f ◦ f
n
, n ∈ denote the iterate operators of f.LetX be a nonempty set and let sX :
{x
n
n∈N
| x
n
∈ X, n ∈ N}.LetcX ⊂ sX asubsetofsX and Lim : cX → X
an operator. By definition the triple X, cX, Lim is called an L-space Fr
´
echet 9 if the
following conditions are satisfied.
i If x
n
x,foralln ∈ N,thenx
n
∈ cX and Limx
n
i
i∈N
x.
By definition, an element of cX is a convergent sequence, x : Limx
n
n∈N
is the
limit of this sequence and we also write x
n
→ x as n → ∞.
InwhatfollowwedenoteanL-space by X, → .
In this setting, if U ⊂ X × X,thenanoperatorf : X → X is called orbitally U-
continuous see 5 if x ∈ X and f
ni
x → a ∈ X,asi → ∞ and f
ni
x,a ∈ U for
any i ∈
imply f
ni1
x → fa,asi → ∞. In particular, if U X × X,thenf is called
orbitally continuous.
Let X, ≤ be a partially ordered set, that is, X is a nonempty set and ≤ is a reflexive,
transitive, and antisymmetric relation on X.Denote
X
≤
n∈
→ x, y
n
n∈
→ y and x
n
≤ y
n
,foreachn ∈ ⇒ x ≤ y.
If
:E, D is a gauge space, then the convergence structure is given by the family of
gauges D {d
α
}
α∈Λ
.Hence,E, D, ≤ is an ordered L-space, and it will be called an ordered
gauge space, see also 10, 11.
Recall that ϕ :
→
is said to be a comparison function if it is increasing and
ϕ
k
t → 0, as k → ∞. As a consequence, we also have ϕt <t,foreacht>0, ϕ00
and ϕ is right continuous at 0. For example, ϕtat where a ∈ 0, 1, ϕtt/1 t and
ϕtln1 t, t ∈
are comparison functions.
≤
∈ If × f;
iii if x, y ∈ X
≤
and y, z ∈ X
≤
,thenx, z ∈ X
≤
;
iv there exists x
0
∈ X
≤
such that x
0
,fx
0
∈ X
≤
;
4 Fixed Point Theory and Applications
v f is orbitally continuous;
vi there exists a comparison function ϕ :
→
such that, for each α ∈ Λ one has
d
α
≤
. Suppose first that x
0
/
fx
0
. Then, from ii
we obtain
f
x
0
,f
2
x
0
,
f
2
x
0
,f
x
0
,f
n1
x
0
≤ ϕ
n
d
α
x
0
,f
x
0
, for each n ∈
. 3.3
Since ϕ
n
d
d
α
f
n
x
0
,f
n2
x
0
≤ d
α
f
n
x
0
,f
n1
x
0
x
0
,f
n1
x
0
≤ ε.
3.4
Now since f
n
x
0
,f
n2
x
0
∈ X
≤
see iii we have for any n ≥ N that
d
α
f
n
x
x
0
,f
n3
x
0
<ε− ϕ
ε
ϕ
d
α
f
n
x
0
,f
n2
x
0
is a Cauchy sequence in . From the completeness of the gauge space we
have f
n
x
0
n∈
→ x
∗
,asn → ∞.
Let x ∈ E be arbitrarily chosen. T hen;
1 If x, x
0
∈ X
≤
then f
n
x,f
n
x
0
∈ X
≤
and thus, for each α ∈ Λ,wehave
d
α
f
n
x,f
n
0
∈ X
≤
. From t he second relation, a s before, we get, for each α ∈
Λ,thatd
α
f
n
x
0
,f
n
cx, x
0
≤ ϕ
n
d
α
x
0
,cx, x
0
,foreachn ∈ and hence
f
n
cx, x
0
n∈
→ x
fx
∗
.Ifwehavefyy
for some y ∈ E, then from above, we must have f
n
y → x
∗
,soy x
∗
.
If fx
0
x
0
,thenx
0
plays the role of x
∗
.
Remark 3.2. Equivalent representation of condition iv are as follows.
iv’Thereexistsx
0
∈ E such that x
0
≤ fx
0
or x
0
≥ fx
0
iv
a f is orbitally continuous or
b if an increasing sequence x
n
n∈
converges to x in E,thenx
n
≤ x for all n ∈ ;
v there exists a comparison function ϕ :
→
such that
d
α
f
x
,f
y
≤ ϕ
d
α
0
≤ ···f
n
x
0
≤ ···.Hencefromv we obtain d
α
f
n
x
0
,f
n1
x
0
≤
ϕ
n
d
α
x
0
,fx
0
,foreachn ∈ . By a similar approach as in the pr oof of Theorem 3.1 we
obtain
d
α
f
as n → ∞.
By the orbital continuity of the operator f we get that x
∗
∈ F
f
.Ifivb takes place,
then, since f
n
x
0
n∈
→ x
∗
, given any >0thereexistsN
∈
∗
such that for each n ≥ N
we have d
α
f
n
x
0
,x
∗
<. On the other hand, for each n ≥ N
d
α
f
f
n
x
0
,f
x
∗
≤ d
α
x
∗
,f
n1
x
0
y
∗
. There are two possible cases.
a If x
∗
,y
∗
∈ X
≤
,thenwehave0 <d
α
y
∗
,x
∗
d
α
f
n
y
∗
,f
n
x
∗
≤
ϕ
n
d
α
The monotonicity condition implies that f
n
x
∗
and f
n
c
∗
are comparable
as well as f
n
c
∗
and f
n
y
∗
.Hence0 <d
α
y
∗
,x
∗
d
α
f
n
y
∗
,f
,c
∗
ϕ
n
d
α
c
∗
,x
∗
→ 0as
n → ∞, which is again a contradiction. Thus x
∗
y
∗
.
4. Applications
We will apply the above result to nonlinear integral equations on the real axis
x
t
t
0
K
t, s, x
iii there exists a comparison function ϕ :
→
,withϕλt ≤ λϕt for each t ∈
and
any λ ≥ 1,suchthat
|
K
t, s, u
− K
t, s, v
|
≤ ϕ
|
u − v
|
, for each t, s ∈
,u,v∈
n
,u≤ v; 4.2
Fixed Point Theory and Applications 7
iv there exists x
0
∈ C
n
.
Proof. Let E : C0, ∞,
n
and the family of pseudonorms
x
n
: max
t∈0,n
|
x
t
|
e
−τt
, where τ>0. 4.4
Define now d
n
x, y : x − y
n
for x, y ∈ E.
Then D :d
n
n∈
∗
is family of gauges on E.ConsideronE the partial order defined
Ax
t
:
t
0
K
t, s, x
s
ds g
t
,t∈
. 4.6
First observe that from ii A is increasing. Also, for each x, y ∈ E with x ≤ y and for
t ∈ 0,n,wehave
Ax
t
− Ay
ϕ
x
s
− y
s
ds
t
0
ϕ
x
s
− y
−τs
ds
≤ ϕ
d
n
x, y
t
0
e
τs
ds ≤
1
τ
ϕ
d
n
x, y
e
τt
.
4.7
Hence, for τ ≥ 1weobtain
t, s, x
s
ds g
t
,t∈ . 4.9
Theorem 4.2. Consider 4.9. Suppose that
i K :
× ×
n
→
n
and g : →
n
are continuous;
ii Kt, s, · :
n
→
n
is increasing for each t, s ∈ ;
iii there exists a comparison function ϕ :
→
,withϕλt ≤ λϕt for each t ∈
and
t
−t
K
t, s, x
0
s
ds g
t
, for any t ∈
. 4.11
Then the integral equation 4.9 has a unique solution x
∗
in C ,
n
.
Proof. We consider the gauge space E :C
,
n
, D :d
n
n∈
where
d
n
−t
K
t, s, x
s
ds g
t
. 4.13
As before, consider on E the partial order defined by
x ≤ y iff x
t
≤ y
t
for any t ∈
. 4.14
Then E, D, ≤ is an ordered and complete gauge space. Moreover, for any increasing
sequence x
n
n∈
in E converging to a certain x
∗
s
− y
s
e
−τ|s|
e
τ|s|
ds
≤
t
−t
e
τ|s|
ϕ
x
s
− y
≤ ϕ
d
n
x, y
|t|
−
|
t
|
e
τ|s|
ds ≤
2
τ
ϕ
d
n
x, y
e
τ|t|
,t∈
Remark 4.3. It is worth mentioning that it could be of interest to extend the above technique
for other metrical fixed-point theorems, see 15, 16, and so forth.
References
1 J. Dugundji, Topology, Allyn and Bacon, Boston, Mass, USA, 1966.
2 A. C. M. Ran and M. C. B. Reurings, “A fixed point theorem in partially ordered sets and some
applications to matrix equations,” Proceedings of the American Mathematical Society, vol. 132, no. 5, pp.
1435–1443, 2004.
3 J. J. Nieto and R. Rodr
´
ıguez-L
´
opez, “Existence and uniqueness of fixed point in partially ordered
sets and applications to ordinary differential equations,” Acta Mathematica Sinica, vol. 23, no. 12, pp.
2205–2212, 2007.
4 A. Petrus¸el and I. A. Rus, “Fixed point theorems in ordered L-spaces,” Proceedings of the American
Mathematical Society, vol. 134, no. 2, pp. 411–418, 2006.
5 J. J. Nieto, R. L. Pouso, and R. Rodr
´
ıguez-L
´
opez, “Fixed point theorems in ordered abstract spaces,”
Proceedings of the American Mathematical Society, vol. 135, no. 8, pp. 2505–2517, 2007.
6 D. O’Regan and A. Petrus¸el, “Fixed point theorems for generalized contractions in ordered metric
spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 2, pp. 1241–1252, 2008.
7 J. J. Nieto and R. R odr
´
ıguez-L
´
opez, “Contractive mapping theorems in partially ordered sets and
applications to ordinary differential equations,” Order, vol. 22, no. 3, pp. 223–239, 2005.
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