Su Fixed Point Theory and Applications 2014, 2014:227
http://www.fixedpointtheoryandapplications.com/content/2014/1/227
RESEARCH
Open Access
Contraction mapping principle with
generalized altering distance function in
ordered metric spaces and applications to
ordinary differential equations
Yongfu Su*
*
Correspondence:
[email protected]
Department of Mathematics, Tianjin
Polytechnic University, Tianjin,
300387, P.R. China
Abstract
The aim of this paper is to present the definition of a generalized altering distance
function and to extend the results of Yan et al. (Fixed Point Theory Appl. 2012:152,
2012) and some others, and to prove a new fixed point theorem of generalized
contraction mappings in a complete metric space endowed with a partial order by
using generalized altering distance functions. The results of this paper can be used to
investigate a large class of nonlinear problems. As an application, we discuss the
existence of a solution for a periodic boundary value problem.
Keywords: contraction mapping principle; partially ordered metric spaces; fixed
point; generalized altering distance function; ordinary differential equations
1 Introduction
The Banach contraction mapping principle is a classical and powerful tool in nonlinear
analysis. Weak contractions are generalizations of the Banach contraction mapping, which
have been studied by several authors. In [–], the authors prove some types of weak contractions in complete metric spaces, respectively. In particular, the existence of a fixed
point for weak contractions and generalized contractions was extended to partially ordered metric spaces in [, –]. Among them, some involve altering distance functions.
Such functions were introduced by Khan et al. in [], where they present some fixed point
theorems with the help of such functions. First, we recall the definition of an altering distance function.
Definition . An altering distance function is a function ψ : [, ∞) → [, ∞) which satisfies:
(a) ψ is continuous and non-decreasing.
(b) ψ =  if and only if t = .
Recently, Harjani and Sadarangani proved some fixed point theorems for weak contractions and generalized contractions in partially ordered metric spaces by using the altering
distance function in [, ], respectively. Their results improve the theorems of [, ].
© 2014 Su; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
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Theorem . ([]) Let (X, ≤) be a partially ordered set and suppose that there exists a
metric d ∈ X such that (X, d) is a complete metric space. Let f : X → X be a continuous and
non-decreasing mapping such that
d f (x), f (y) ≤ d(x, y) – ψ d(x, y) ,
for x ≥ y,
where ψ : [, ∞) → [, ∞) is continuous and non-decreasing function such that ψ is positive in (, ∞), ψ() =  and limt→∞ ψ(t) = ∞. If there exists x ∈ X with x ≤ f (x ), then f
has a fixed point.
Theorem . ([]) Let (X, ≤) be a partially ordered set and suppose that there exists a
metric d ∈ X such that (X, d) is a complete metric space. Let f : X → X be a continuous and
non-decreasing mapping such that
ψd f (x), f (y) ≤ ψ d(x, y) – φ d(x, y) ,
for x ≥ y,
where ψ and φ are altering distance functions. If there exists x ∈ X with x ≤ f (x ), then
f has a fixed point.
Subsequently, Amini-Harandi and Emami proved another fixed point theorem for contraction type maps in partially ordered metric spaces in []. The following class of functions is used in [].
Let denote the class of those functions β : [, ∞) → [, ) which satisfy the condition:
β(tn ) →  ⇒ tn → .
Theorem . ([]) Let (X, ≤) be a partially ordered set and suppose that there exists a
metric d such that (X, d) is a complete metric space. Let f : X → X be an increasing mapping
such that there exists an element x ∈ X with x ≤ f (x ). Suppose that there exists β ∈ such that
d f (x), f (y) ≤ β d(x, y) d(x, y) for each x, y ∈ X with x ≥ y.
Assume that either f is continuous or M is such that if an increasing sequence xn → x ∈ X,
then xn ≤ x, ∀n. Besides, if for each x, y ∈ X there exists z ∈ m which is comparable to x
and y, then f has a unique fixed point.
In , Yan et al. proved the following result.
Theorem . ([]) Let X be a partially ordered set and suppose that there exists a metric
d in x such that (X, d) is a complete metric space. Let T : X → X be a continuous and
non-decreasing mapping such that
ψ d(Tx, Ty) ≤ φ d(x, y) ,
∀x ≥ y,
where ψ is an altering distance function and φ : [, ∞) → [, ∞) is a continuous function
with the condition ψ(t) > φ(t) for all t > . If there exists x ∈ X such that x ≤ Tx , then T
has a fixed point.
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The aim of this paper is to present the definition of generalized altering distance function
and to extend the results of Yan et al. [] and some others, and to prove a new fixed point
theorem of generalized contraction mappings in a complete metric space endowed with a
partial order by using generalized altering distance functions. The results of this paper can
be used to investigate a large class of nonlinear problems. As an application, we discuss
the existence of a solution for a periodic boundary value problem.
2 Main results
We first give the definition of generalized altering distance function as follows.
Definition . A generalized altering distance function is a function ψ : [, ∞) → [, ∞)
which satisfies:
(a) ψ is non-decreasing;
(b) ψ =  if and only if t = .
We first recall the following notion of a monotone non-decreasing function in a partially
ordered set.
Definition . If (X, ≤) is a partially ordered set and T : X → X, we say that T is monotone non-decreasing if x, y ∈ X, x ≤ y ⇒ T(x) ≤ T(y).
This definition coincides with the notion of a non-decreasing function in the case where
X = R and ≤ represents the usual total order in R.
In what follows, we prove the following theorem, which is the generalized type of Theorems .-..
Theorem . Let X be a partially ordered set and suppose that there exists a metric d
in x such that (X, d) is a complete metric space. Let T : X → X be a continuous and nondecreasing mapping such that
ψ d(Tx, Ty) ≤ φ d(x, y) ,
∀x ≥ y,
where ψ is a generalized altering distance function and φ : [, ∞) → [, ∞) is a right upper
semi-continuous function with the condition: ψ(t) > φ(t) for all t > . If there exists x ∈ X
such that x ≤ Tx , then T has a fixed point.
Proof Since T is a non-decreasing function, we obtain by induction that
x ≤ Tx ≤ T  x ≤ T  x ≤ · · · ≤ T n x ≤ T n+ x ≤ · · · .
()
Put xn+ = Txn . Then, for each integer n ≥ , from () and, as the elements xn+ and xn are
comparable, we get
ψ d(xn+ , xn ) = ψ d(Txn , Txn– ) ≤ φ d(xn , xn– ) .
()
Using the condition of Theorem . we have
d(xn+ , xn ) < d(xn , xn– ).
()
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Hence the sequence {d(xn+ , xn )} is decreasing and, consequently, there exists r ≥  such
that
d(xn+ , xn ) → r+ ,
as n → ∞. Consider the properties of ψ and φ, letting n → ∞ in () we get
ψ(r) ≤ lim ψ d(xn+ , xn ) ≤ lim φ d(xn , xn– ) ≤ φ(r).
n→∞
n→∞
By using the condition: ψ(t) > φ(t) for all t > , we have r = , and hence
d(xn+ , xn ) → ,
()
as n → ∞. In what follows, we will show that {xn } is a Cauchy sequence. Suppose that
{xn } is not a Cauchy sequence. Then there exists ε >  for which we can find subsequences
{xnk } with nk > mk > k such that
d(xnk , xmk ) ≥ ε
()
for all k ≥ . Further, corresponding to mk we can choose nk in such a way that it is the
smallest integer with nk > mk and satisfying (). Then
d(xnk– , xmk– ) < ε.
()
From () and (), we have
ε ≤ d(xnk , xmk ) ≤ d(xnk ), xnk– + d(xnk– , xmk ) < d(xnk , xnk– ) + ε.
Letting k → ∞ and using (), we get
lim d(xnk , xmk ) = ε.
k→∞
()
By using the triangular inequality we have
d(xnk , xmk ) ≤ d(xnk , xnk– ) + d(xnk– , xmk– ) + d(xmk– , xmk ),
d(xnk– , xmk– ) ≤ d(xnk– , xnk ) + d(xnk , xmk ) + d(xmk , xmk– ).
Letting k → ∞ in the above two inequalities and using () and (), we have
lim d(xnk– , xmk– ) = ε.
k→∞
As nk > mk and xnk– and xmk– are comparable, using () we have
ψ d(xnk , xmk ) ≤ φ d(xnk– , xmk– ) .
()
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Consider the properties of ψ and φ, letting k → ∞ and taking into account () and (), we
have
ψ(ε) ≤ φ(ε).
From the condition ψ(t) > φ(t) for all t > , we get ε = , which is a contradiction. This
shows that {xn } is a Cauchy sequence and, since X is a complete metric space, there exists
z ∈ X such that xn → z as n → ∞. Moreover, the continuity of T implies that
z = lim xn+ = lim Txn = Tz,
n→∞
n→∞
and this proves that z is a fixed point. This completes the proof.
In what follows, we prove that Theorem . is still valid for T being not necessarily
continuous, assuming the following hypothesis in X:
If (xn ) is a non-decreasing sequence in X such that xn → x
then xn ≤ x for all n ∈ N.
()
Theorem . Let (X, ≤) be a partially ordered set and suppose that there exists a metric d
in X such that (X, d) is a complete metric space. Assume that X satisfies (). Let T : X → X
be a non-decreasing mapping such that
ψ d(Tx, Ty) ≤ φ d(x, y) ,
∀x ≥ y,
where ψ is a generalized altering distance functions and φ: [, ∞) → [, ∞) is a right upper
semi-continuous function with the condition ψ(t) > φ(t) for all t > . If there exists x ∈ X
such that x ≤ Tx , then T has a fixed point.
Proof Following the proof of Theorem . we only have to check that T(z) = z. As (xn ) is
a non-decreasing sequence in X and limn→∞ xn = z the condition () gives us that xn ≤ z
for every n ∈ N and consequently,
ψ d xn+ , T(z) = ψ d T(xn ), T(z) ≤ φ d(xn , z) .
Letting n → ∞ and taking into account that ψ is an altering distance function, we have
ψ d z, T(z) ≤ φ().
Using condition of theorem we have φ() = , this implies (d(z, T(z))) = . Thus,
d(z, T(z)) =  or equivalently, T(z) = z.
Now, we present an example where it can be appreciated that the hypotheses in Theorems . and Theorems . do not guarantee uniqueness of the fixed point. An example
appears in [].
Let X = {(, ), (, )} ⊂ R and consider the usual order (x, y) ≤ (z, t) ⇔ x ≤ z, y ≤ t.
Thus, (x, y) is a partially ordered set whose different elements are not comparable. Besides
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(X, d ) is a complete metric space considering d the Euclidean distance. The identity map
T(x, y) = (x, y) is trivially continuous and non-decreasing and condition () of Theorem .
is satisfied, since the elements in X are only comparable to themselves. Moreover, (, ) ≤
T(, ) = (, ) and T has two fixed points in X.
In what follows, we give a sufficient condition for the uniqueness of the point in Theorems . and .. This condition is:
for x, y ∈ X there exists a lower bound or an upper bound.
()
In [] it is proved that condition () is equivalent to:
for x, y ∈ X there exists z ∈ X which is comparable to x and y.
()
Theorem . Adding condition () to the hypotheses of Theorem . (resp. Theorem .)
we obtain the uniqueness of the fixed point of T.
Proof Suppose that there exist z, y ∈ X which are fixed points. We distinguish two cases.
Case . If y is comparable to z then T n (y) = y is comparable to T n (z) = z for n = , , , . . .
and
ψ d(z, y) = ψ d T n (z), T n (y)
≤ φ d T n– (z), T n– (y)
≤ φ d(z, y) .
As we have the condition ψ(t) > φ(t) for t >  we obtain d(z, y) =  and this implies z = y.
Case . If y is not comparable to z then there exists x ∈ X comparable to y and z. Monotonicity of T implies that T n (x) is comparable to T n (y) and to T n (z) = z, for n = , , , . . .
Moreover,
ψ d z, T n (x) = ψ d T n (z), T n (x)
≤ φ d T n– (z), T n– (x)
= φ d z, T n– (x) .
()
Hence, ψ is a generalized altering distance function and we have the condition ψ(t) >
φ(t) for t > , this gives us that {d(z, f n (x))} is a non-negative decreasing sequence and,
consequently, there exists γ such that
lim d z, T n (x) = γ .
n→∞
Letting n → ∞ in () and, taking into account the properties of ψ and φ, we obtain
ψ(γ ) ≤ φ(γ ).
This and the condition ψ(t) > φ(t) for t >  imply γ = . Analogously, it can be proved that
lim d y, T n (x) = .
n→∞
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Finally, as
lim d z, T n (x) = lim d y, T n (x) = 
n→∞
n→∞
the uniqueness of the limit gives us y = z. This finishes the proof.
Remark . Under the assumption of Theorem ., it can be proved that for every x ∈ X,
limn→∞ T n (x) = z, where z is the fixed point (i.e. the operator f is Picard).
Remark . Theorem . is a particular case of Theorem . for ψ being the identity
function, and φ(t) = t – ψ(t). Theorem . is a particular case of our Theorem . for φ(t)
being replaced by ψ(t) – φ(t). Theorem . is a particular case of Theorem . for ψ being
the identity function, and φ(t) = β(t)t. Theorem . is also a particular case of Theorem .
for ψ and φ being continuous.
Example . The following are some generalized altering distance functions:
⎧
⎨,
t = ,
⎩[t] + , t > ,
⎧
⎨,
t = ,
ψ (t) =
⎩λ([t] + ), t > ,
ψ (t) =
where α >  is a constant.
ψ (t) =
⎧
⎨t,
 ≤ t < ,
⎩αt  , t ≥ ,
where α ≥  is a constant.
We choose ψ(t) = ψ (t) and
⎧
⎨t  ,  ≤ t < ,
φ(t) =
⎩βt, t ≥ ,
where  < β < α is a constant. By using Theorem ., we can get the following result.
Theorem . Let X be a partially ordered set and suppose that there exists a metric d
in x such that (X, d) is a complete metric space. Let T : X → X be a continuous and nondecreasing mapping such that
 ≤ d(Tx, Ty) < 
d(Tx, Ty) ≥ 
⇒

d(Tx, Ty) ≤ d(x, y) ,

α d(Tx, Ty) ≤ βd(x, y)
⇒
for any x, y ∈ X. If there exists x ∈ X such that x ≤ Tx , then T has a fixed point.
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3 Application to ordinary differential equations
In this section we present two examples where our Theorems . and . can be applied.
The first example is inspired by []. We study the existence of a solution for the following
first-order periodic problem:
⎧
⎨u (t) = f (t, u(t)), t ∈ [, T],
⎩u() = u(T),
()
where T >  and f : I × R → R is a continuous function. Previously, we considered the
space C(I) (I = [, T]) of continuous functions defined on I. Obviously, this space with the
metric given by
d(x, y) = sup x(t) – y(t) : t ∈ I ,
for x, y ∈ C(I),
is a complete metric space. C(I) can also be equipped with a partial order given by
x, y ∈ C(I),
x≤y
⇔
x(t) ≤ y(t)m,
for t ∈ I.
Clearly, (C(I), ≤) satisfies condition (), since for x, y ∈ C(I) the functions max{x, y} and
min{x, y} are least upper and greatest lower bounds of x and y, respectively. Moreover, in
[] it is proved that (C(I), ≤) with the above mentioned metric satisfies condition ().
Now we give the following definition.
Definition . A lower solution for () is a function α ∈ C () (I) such that
⎧
⎨α (t) ≤ f (t, α(t)), for t ∈ I,
⎩α() ≤ α(T).
Theorem . Consider problem () with f : I × R → R continuous and suppose that there
exist λ, α >  with
α≤
λ(eλT – )
T(eλT + )

such that for x, y ∈ R with x ≥ y
 ≤ f (t, x) + λx – f (t, y) + λy ≤ α g(x – y),
where g(t) : [, +∞) → [, +∞) is a light upper semi-continuous function with g() = ,
g(t) < t  , ∀t > . Then the existence of a lower solution for () provides the existence of an
unique solution of ().
Proof Problem () can be written as
⎧
⎨u (t) + λu(t) = f (t, u(t)) + λu(t), for t ∈ I = [, T],
⎩u() = u(T).
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This problem is equivalent to the integral equation
T
u(t) =
G(t, s) f s, u(s) + λu(s) ds,

where G(t, s) is the Green function given by
⎧
⎨ eλ(T+s–t) ,  ≤ s < t ≤ T,
eλT –
G(t, s) = λ(s–t)
⎩e
,
 ≤ t < s ≤ T.
eλT –
Define F : C(I) → C(I) by
T
(Fu)(t) =
G(t, s) f s, u(s) + λu(s) ds.

Note that if u ∈ C(I) is a fixed point of F then u ∈ C  (I) is a solution of (). In what
follows, we check that the hypotheses in Theorems . and . are satisfied. The mapping
F is non-decreasing, since we have u ≥ v, and using our assumption. We can obtain
f (t, u) + λu ≥ f (t, v) + λv
which implies, since G(t, s) > , that for t ∈ I
T
G(t, s) f s, u(s) + λu(s) ds
T
G(t, s) f s, v(s) + λv(s) ds = (Fv)(t).
(Fu)(t) =

≥

Besides, for u ≥ v, we have
d(Fu, Fv) = sup(Fu)(t) – (Fv)(t)
t∈I
= sup (Fu)(t) – (Fv)(t)
t∈I
T
= sup
t∈I

T
≤ sup
t∈I
G(t, s) f s, u(s) + λu(s) – f s, v(s) – λv(s) ds
G(t, s)α g u(s) – v(s) ds.
()

Using the Cauchy-Schwarz inequality in the last integral we get
T
G(t, s)α g u(s) – v(s) ds

≤
T

 
α g u(s) – v(s) ds
G(t, s) ds

T


.
()
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The first integral gives us
T

t
G(t, s) ds =
T
G(t, s) ds +

G(t, s) ds
t
t
=

eλ(T+s–t)
ds +
(eλT – )
=

e(λT–)
λ(eλT – )
=
eλT + 
.
λ(eλT – )
t
T
eλ(s–t)
ds
(eλT – )
()
The second integral in () gives the following estimate:
T
α  g u(s) – v(s) ds ≤ α  g u – v · T

= α  g d(u, v) · T.
()
Taking into account ()-() we have
λT


e +
· α  g d(u, v) · T 
d(Fu, Fv) ≤ sup
λT – )
λ(e
t∈I
λT

√ 
e +
=
· α · T · g d(u, v) 
λT
λ(e – )
and from the last inequality we obtain
d(Fu, Fv) ≤
eλT + 
· α  · T · g d(u, v)
λT
λ(e – )
or, equivalently.
λ eλT –  d(Fu, Fv) ≤ eλT +  · α  · T · g d(u, v) .
By our assumption, as
α≤
λ(eλT – )
T(eλT + )

,
the last inequality gives us
λ eλT –  d(Fu, Fv) ≤ λ eλT –  · g d(u, v) ,
and, hence,
d(Fu, Fv) ≤ g d(u, v) .
()
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Put ψ(t) = t  and φ(t) = g(t). Obviously, ψ is a generalized altering distance function, ψ(t)
and φ(t) satisfy the condition of ψ(t) > φ(t) for t > . From (), we obtain for u ≥ v
ψ d(Fu, Fv) ≤ φ d(u, v) .
Finally, let α(t) be a lower solution for (); we claim that α ≤ F(α). In fact
α (t) + λα(t) ≤ f t, α(t) + λα(t),
for t ∈ I.
We multiply by eλt ,
α(t)eλt ≤ f t, α(t) + λα(t) eλt ,
for t ∈ I,
and this gives us
t
f s, α(s) + λα(s) eλs ds,
α(t)e ≤ α() +
λt
for t ∈ I.
()

As α() ≤ α(T), the last inequality gives us
α()eλt ≤ α(T)eλT ≤ α() +
T
f s, α(s) + λα(s) eλs ds,

and so
T
α() ≤

eλs f s, α(s) + λα(s) ds.
λT
e –
This and () give us
t
α(t)e ≤
λt

eλ(T+s) f s, α(s) + λα(s) ds +
eλT – 
T
t
eλs f s, α(s) + λα(s) ds
eλT – 
and, consequently,
T
G(t, s) f s, α(s) + λα(s) ds

=

t
eλ(s–t) f s, α(s) + λα(s) ds
eλT – 
eλ(T+s–t)
ds +
eλT – 
α(t) ≤
t

= (Fα)(t),
for t ∈ I.
Finally, Theorems . and . show that F has an unique fixed point.
Example . In Theorem ., we can choose the function g(t) as follows:
() g (t) = ln(t  + );
()
⎧

⎪
⎪
⎨t ,  ≤ t < ,
g (t) =  , t = ,
⎪
⎪
⎩
t,  < t < +∞.
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()
g (t) =
⎧
⎨t  ,
⎩t –  ,

 ≤ t ≤  ,


< t < +∞.
The functions g (t), g (t) are continuous and non-decreasing. The function g (t) is right
upper semi-continuous. If we choose g(t) = g (t) in Theorem ., we obtain the result of
[].
Example . Consider the following first-order periodic problem:
⎧
⎨u (t) =
sin t
et
– βx, t ∈ [, T],
()
⎩u() = u(T).
Let
f (t, x) =
sin t
– βx,
et
x ∈ [, ∞), t ∈ [, ],
then f (t, x) is continuous. Further, for x ≥ y, we have
x–y
.
f (t, x) + λx – f (t, y) + λy = (λ – β)

We chose β ∈ [, λ] such that
(λ – β) ≤
λ(eλT – )
T(eλT + )

.
Taking g(t) = ( t ) for all t ∈ [, +∞), we have
f (t, x) + λx – f (t, y) + λy = (λ – β) g(x – y).
By using Theorem ., we know that the first-order periodic problem () has a unique
solution.
A second example where our results can be applied is the following two-point boundary
value problem of the second order differential equation:
⎧
⎨– d x = f (t, x),
x ∈ [, ∞), t ∈ [, ],
dt 
⎩x() = x() = .
()
It is well known that x ∈ C  [, ] is a solution of () that is equivalent to x ∈ C[, ] being
a solution of the integral equation

x(t) =

G(t, s)f s, x(s) ds,
for t ∈ [, ],
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where G(t, s) is the Green function given by
⎧
⎨t( – s),  ≤ t ≤ s ≤ ,
G(t, s) =
⎩s( – t),  ≤ s ≤ t ≤ .
()
Theorem . Consider problem () with f : I × R → [, ∞) continuous and nondecreasing with respect to the second variable and suppose that there exists  ≤ α ≤ 
such that for x, y ∈ R with x ≥ y
f (t, x) – f (t, y) ≤ α g(x – y),
()
where g(t) : [, +∞) → [, +∞) is a light upper semi-continuous function with g() = ,
g(t) < t  , ∀t > . Then our problem () has a unique non-negative solution.
Proof Consider the cone
P = x ∈ C[, ] : x(t) ≥  .
Obviously, (P, d) with d(x, y) = sup{|x(t) – y(t)| : t ∈ [, ]} is a complete metric space. Consider the operator given by

(Tx)(t) =
G(t, s)f s, x(s) ds,
for x ∈ P,

where G(t, s) is the Green function appearing in ().
As f is non-decreasing with respect to the second variable, for x, y ∈ P with y ≥ x and
t ∈ [, ], we have
(Ty)(t) =


G(t, s)f s, y(s) ds ≥

G(t, s)f s, x(s) ds ≥ (Tx)(t),

and this proves that T is a non-decreasing operator.
Besides, for y ≥ x and taking into account (), we obtain
d(Ty, Tx) = sup (Tx)(t) – (Ty)(t)
t∈[,]
= sup (Tx)(t) – (Ty)(t)
t∈[,]

= sup
G(t, s) f s, x(s) – f s, y(s) ds
t∈[,] 
≤ sup

G(t, s)α g x(s) – y(s)

G(t, s)α g x(s) – y(s) ds
t∈[,] 
≤ sup
t∈[,] 

= α ln y – x +  sup
G(t, s) ds.
t∈[,] 
()
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It is easy to verify that

G(t, s) ds =

–t  t
+


and that


G(t, s) ds = .

sup
t∈[,] 
These facts, the inequality (), and the hypothesis  < α ≤  give us
α
g(x – y)

≤ g x – y = g d(x, y) .
d(Tx, Ty) ≤
Hence
d(Ty, Tx) ≤ g d(x, y) .
Put ψ(t) = t  , φ(t) = g(t), obviously ψ is an altering distance function, ψ and φ satisfy the
condition of ψ(t) > φ(t), for t > . From the last inequality, we have
ψ d(Tx, Ty) ≤ φ d(x, y) .
Finally, as f and G are non-negative functions
T =

G(t, s)f (s, ) ds ≥ 

and Theorems . and . tell us that F has a unique non-negative solution.
Remark . In Theorem ., we can choose g(t) as g (), g (t), and g (t) as well as in
Theorem ..
Example . Consider the following two-point boundary value problem of the second
order differential equation:
⎧
⎨– d x = sin t + x , x ∈ [, ∞), t ∈ [, ],
+cos tπ
et
dt 
⎩x() = x() = .
()
Let
f (t, x) =
sin t
x
+
,
t
e
 + cos tπ
x ∈ [, ∞), t ∈ [, ],
then f (t, x) is continuous and non-decreasing with respect to the second variable. Further,
for x ≥ y, we have
y
x
–
≤
f (t, x) – f (t, y) =
 + cos tπ  + cos tπ
x–y


.
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Taking g(t) = t for all t ∈ [, +∞). By using Theorem ., we know that the two-point
boundary value problem () has a unique non-negative solution.
Competing interests
The author declares that they have no competing interests.
Received: 29 May 2014 Accepted: 8 October 2014 Published: 3 November 2014
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doi:10.1186/1687-1812-2014-227
Cite this article as: Su: Contraction mapping principle with generalized altering distance function in ordered metric
spaces and applications to ordinary differential equations. Fixed Point Theory and Applications 2014 2014:227.
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