Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2012, Article ID 981517, 8 pages
doi:10.1155/2012/981517
Research Article
A Fixed Point Theorem for Contraction Type Maps
in Partially Ordered Metric Spaces and Application
to Ordinary Differential Equations
M. Eshaghi Gordji,1 H. Baghani,1 and G. H. Kim2
1
2
Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran
Department of Mathematics, Kangnam University, Gyeonggi, Yongin 446-702, Republic of Korea
Correspondence should be addressed to G. H. Kim, ghkim@kangnam.ac.kr
Received 21 November 2011; Accepted 20 December 2011
Academic Editor: Mingshu Peng
Copyright q 2012 M. Eshaghi Gordji 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 present a fixed point theorem for generalized contraction in partially ordered complete metric
spaces. As an application, we give an existence and uniqueness for the solution of a periodic
boundary value problem.
1. Introduction
The contraction mapping theorem and the abstract monotone iterative technique are well
known and are applicable to a variety of situations. Recently, there has been a trend to weaken
the requirement on the contraction by considering metric spaces endowed with partial order
see 1–7. It is of interest to determine if it is still possible to establish the existence of a
unique fixed point assuming that the operator considered is monotone in such a setting. Such
a fixed point theorem is useful, for example, in establishing the existence of a unique solution
to periodic boundary value problems, besides many others.
That approach was initiated by Ran and Reurings in 8, where some applications to
matrix equations were studied. This fixed point theorem was refined and extended in 7, 9
and applied to the periodic boundary value problem in the monotone case. In this paper, we
consider a special case of the following periodic boundary value problem
u t ft, ut
if t ∈ I 0, T ,
u0 uT ,
where T > 0 and f : I × R → R is continuous function.
1.1
2
Discrete Dynamics in Nature and Society
Definition 1.1. A lower solution for 1.1 is a function u ∈ C1 I, R such that
u t ≤ ft, ut for t ∈ I 0, T ,
1.2
u0 ≤ uT .
Very recently, Harjani and Sadarangani 4 proved the following existence theorem.
Theorem 1.2. Consider problem 1.1 with f : I × R → R is continuous and suppose that there
exists λ > 0 such that for x, y ∈ R with y ≥ x
0 ≤ f t, y λy − ft, x λx ≤ λφ y − x ,
1.3
where φ : 0, ∞ → 0, ∞ can be written by φx x − ψx with ψ : 0, ∞ → 0, ∞ continuous increasing positive in 0, ∞, ψ0 0 and limt → ∞ ψt ∞. Then the existence of a lower
solution of 1.1 provides the existence of a unique solution of 1.1.
In Section 2, we prove a new fixed point theorem in partially ordered complete metric
spaces. In Section 3, existence of a unique solution for problem 1.1 is obtained under
suitable conditions.
2. Fixed Point Theorem
Let S denote the class of those functions α : 0, ∞ → 0, 1 which satisfies the condition
lim sup αs < 1,
s → t
∀t ∈ 0, ∞.
2.1
We prove the main theorem of the paper as follows.
Theorem 2.1. Let X, be a partially order metric space that there exists a metric d in X such that
X, d is a complete metric space. Let f : X → X be an increasing mapping such that there exists
x0 ∈ X with x0 fx0 . Suppose that there exists α ∈ S such that
d fx, f y ≤ α d x, y d x, y ,
2.2
f is continuous
2.3
for all comparable x, y ∈ X. If
or
if an increasing sequence {xn } → x in X, then xn x,
∀n ∈ N.
2.4
Discrete Dynamics in Nature and Society
3
Besides, if
for each x, y ∈ X, there exists z ∈ X which is comparable to x and y,
2.5
then f have a unique fixed point.
Proof. We first show that f has a fixed point. Since x0 fx0 and f is increasing function,
we obtain by induction that
x0 fx0 f 2 x0 f 3 x0 · · · f n x0 f n1 x0 · · · .
2.6
Put xn f n x0 , n 1, 2, . . .. Since xn xn1 for each n ∈ N then by 2.2
dxn1 , xn2 d f n1 x0 , f n2 x0 2.7
≤ αdxn , xn1 dxn , xn1 ≤ dxn , xn1 .
And so the sequence {dxn1 , xn } is nonincreasing and bounded below. Thus there exists
τ ≥ 0 such that limn → ∞ dxn1 , xn τ. Since lim sups → τ αs < 1 and ατ < 1 then there
exist r ∈ 0, 1 and > 0 such that αs < r for all s ∈ τ, τ . We can take ν ∈ N such that
τ ≤ dxn1 , xn ≤ τ for all n ∈ N with n ≥ ν. Then since
dxn1 , xn2 ≤ αdxn , xn1 dxn , xn1 ≤ rdxn , xn1 ,
2.8
for all n ∈ N with n ≥ ν we have
∞
dxn , xn1 ≤
n1
ν
∞
dxn , xn1 r n dxν , xν1 < ∞,
n1
2.9
n1
and hence {xn } is a Cauchy sequence. Since X is complete, {xn } converges to some point
z ∈ X. To prove that z is a fixed point of f, if f is a continuous, then
z lim xn lim f n x0 lim f n1 x0 f
n→∞
n→∞
n→∞
lim f n x0 n→∞
fz;
2.10
hence z fz. If case 2.4 holds then
d fz, z ≤ d fz, fxn d fxn , z
≤ αdxn , zdxn , z dxn1 , z
≤ dxn , z dxn1 , z.
2.11
4
Discrete Dynamics in Nature and Society
Since dxn , z → 0 then we get fz z. To prove the uniqueness of the fixed point, let y
be another fixed point of f. From 2.5 there exists x ∈ X which is comparable to y and z.
Monotonicity implies that f n x is comparable to f n y y and f n z z for n 0, 1, . . ..
Moreover,
d z, f n x d f n z, f n x
≤ α d f n−1 z, f n−1 x d f n−1 z, f n−1 x
≤ d f n−1 z, f n−1 x
2.12
d z, f n−1 x .
Consequently, the sequence ζnz : dz, f n x is nonnegative and decreasing and so
y
limn → ∞ dz, f n x ζz ∈ R. Similarly we can show that the sequence ζn : dy, f n x is
nonnegative and decreasing and so limn → ∞ dy, f n x ζy ∈ R. Now similarly the above
method we can choose r1 , r2 in 0, 1 and τ1 ∈ N such that
d z, f n x ≤ α d z, f n−1 x d z, f n−1 x ≤ r1 d z, f n−1 x ,
d y, f n x ≤ α d y, f n−1 x d y, f n−1 x ≤ r2 d y, f n−1 x ,
2.13
for all n ∈ N with n > τ1 . Finally
d z, y ≤ d z, f n x d f n x, y ≤ r1n−τ1 d z, f τ1 x0 r2n−τ1 d y, f τ1 x0 ,
2.14
for all n ∈ N with n > τ1 . Therefor if in 2.14 taking n → ∞ yields dz, y 0.
3. Application to Ordinary Differential Equation
In this section we present an example where Theorem 2.1 can be applied. This example is
inspired in 2, 4, 7.
Definition 3.1. Let B denote the class of those functions φ : 0, ∞ → 0, ∞ which satisfies
the following condition:
i φ is increasing,
ii for each x > 0, φx < x,
iii βx φx/x ∈ S.
For example, φx ax, where 0 ≤ a < 1, φx x/x 1, and φx ln1 x are in B.
Theorem 3.2. Consider problem 1.1 with f : I × R → R is continuous and suppose that there
exists λ > 0 such that for x, y ∈ R with y ≥ x
0 ≤ f t, y λy − ft, x λx ≤ λφ y − x ,
3.1
Discrete Dynamics in Nature and Society
5
where φ ∈ B. Then the existence of a lower solution of 1.1 provides the existence of a unique solution
of 1.1.
Proof. Problem 1.1 is equivalent to the integral equation
ut T
Gt, s fs, us λus ds,
3.2
0
where
⎧ λT s−t
e
⎪
⎪
⎪
⎨ eλT − 1 ,
Gt, s ⎪
⎪ eλs−t
⎪
⎩
,
eλT − 1
0 ≤ s < t ≤ T,
3.3
0 ≤ t < s ≤ T.
Define F : CI, R → CI, R by
Fut T
Gt, s fs, us λus ds.
3.4
0
Note that if u ∈ CI, R is a fixed point of F then u ∈ C1 I, R is a solution of 1.1. Now, we
check that hypotheses in Theorem 2.1 are satisfied. Indeed, X CI, R is a partially ordered
set if we define the following order relation in X:
x, y ∈ CI, R,
x≤y
iff xt ≤ yt,
∀t ∈ I.
3.5
The mapping F is increasing since, by hypotheses, for u ≥ v
ft, u λu ≥ ft, v λv
3.6
which implies for t ∈ I, using that Gt, s > 0 for t, s ∈ I × I, that
Fut T
Gt, s fs, us λus ds
0
≥
T
Gt, s fs, vs λvs ds Fvt.
3.7
0
Beside, for u ≥ v
dFu, Fv sup|Fut − Fvt|
t∈I
≤ sup
t∈I
≤ sup
t∈I
T
Gt, sfs, us λus − fs, vs λvs ds
0
T
0
Gt, s · λφus − vsds.
3.8
6
Discrete Dynamics in Nature and Society
As the function φx is increasing and u ≥ v then φus − vs ≤ φdu, v we obtain
dFu, Fv ≤ sup
T
Gt, s · λφus − vsds
0
t∈I
≤ λφdu, vsup
t∈I
T
Gt, sds
0
1
1 λT s−t t 1 λs−t T
e
λφdu, vsup λT
0 e
t
λ
−1 λ
t∈I e
3.9
1
λφdu, v · λT
eλT − 1 φdu, v.
λ e −1
Then for u ≥ v
dFu, Fv ≤ αdu, vdu, v.
3.10
Finally, let βt be a lower solution of 1.1, and we will show that β ≤ Fβ.
Indeed,
β t λβt ≤ f t, βt λβt,
for t ∈ I.
3.11
Multiplying by eλt we get
βteλt
≤
f t, βt λβt eλt ,
for t ∈ I,
3.12
and this gives us
βteλt ≤ β0 t
f s, βs λβs eλs ds,
for t ∈ I
3.13
f s, βs λβs eλs ds,
3.14
0
which implies that
β0e
λT
≤ βT e
λT
≤ β0 T
0
and so
β0 ≤
T
0
eλs f s, βs λβs ds.
λT
e −1
3.15
From this equality and 3.13 we obtain
βte ≤
λt
t
0
eλT s f s, βs λβs ds eλT − 1
T
t
eλs f s, βs λβs ds,
eλT − 1
3.16
Discrete Dynamics in Nature and Society
7
and, consequently,
βt ≤
t
0
eλT s−t f s, βs λβs ds eλT − 1
T
t
eλs−t f s, βs λβs ds.
eλT − 1
3.17
Hence
βt ≤
T
Gt, s f s, βs λβs ds F β t,
for t ∈ I.
3.18
0
Finally, Theorem 2.1 gives that F has a unique fixed point.
Example 3.3. Let φ0 : 0, ∞ → 0, ∞ be a defined
⎧
⎪
0,
⎪
⎪
⎪
⎨
φ0 t 3t − 9,
⎪
⎪
⎪
⎪
⎩ 3 t,
4
0 ≤ t ≤ 3,
3 < t ≤ 4,
3.19
4 < t.
Let f : I × R → R be continuous and suppose that there exists λ > 0 such that for x, y ∈ R
with y ≥ x
0 ≤ f t, y λy − ft, x λx ≤ λφ0 y − x .
3.20
Then be Theorem 2.1, the existence of a lower solution for 1.1 provides the existence of a
unique solution of 1.1.
The example discussed above cannot be the result of Harjani and Sadarangani noted
Theorem 1.2, because ψx x − φ0 x is not increasing.
Acknowledgment
The third author of this work was partially supported by Basic Science Research Program
through the National Research Foundation of Korea NRF funded by the Ministry of
Education, Science and Technology Grant no.: 2010-0010243.
References
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8
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