A Fixed Point Theorem for Contraction Type Maps in Partially Ordered Metric Spaces and Application to Ordinary Differential Equations

(1)

Volume 2012, Article ID 981517,8pages 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. Kim

²

1Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran

2Department of Mathematics, Kangnam University, Gyeonggi, Yongin 446-702, Republic of Korea

Correspondence should be addressed to G. H. Kim,[email protected] Received 21 November 2011; Accepted 20 December 2011

Academic Editor: Mingshu Peng

Copyrightq2012 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 see1–7. 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 in8, where some applications to matrix equations were studied. This fixed point theorem was refined and extended in7,9 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

ut ft, ut ift∈I 0, T,

u0 uT, 1.1

whereT >0 andf:I×R → Ris continuous function.

(2)

Definition 1.1. A lower solution for1.1is a functionu∈C¹I,Rsuch that ut≤ft, ut fort∈I 0, T,

u0≤uT. 1.2

Very recently, Harjani and Sadarangani4proved the following existence theorem.

Theorem 1.2. Consider problem1.1withf : I×R → Ris continuous and suppose that there existsλ >0 such that forx, y∈Rwithy≥x

0≤f t, y

λy−

ft, x λx

≤λφ y−x

, 1.3

whereφ : 0,∞ → 0,∞can be written byφx x − ψxwithψ :0,∞ → 0,∞con- tinuous increasing positive in0,∞,ψ0 0 and lim_{t→ ∞}ψt ∞. Then the existence of a lower solution of 1.1provides the existence of a unique solution of1.1.

InSection 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

LetSdenote the class of those functionsα:0,∞ → 0,1which satisfies the condition lim sup

s→t

αs<1, ∀t∈0,∞. 2.1

We prove the main theorem of the paper as follows.

Theorem 2.1. LetX,be a partially order metric space that there exists a metricdinXsuch that X, dis a complete metric space. Letf : X → Xbe an increasing mapping such that there exists x₀∈Xwithx₀ fx0. Suppose that there existsα∈Ssuch that

d

fx, f y

≤α d

x, y d

x, y

, 2.2

for all comparablex, y∈X. If

f iscontinuous 2.3

or

if an increasing sequence{xn} → x in X, then x_n x, ∀n∈N. 2.4

(3)

Besides, if

for each x, y∈X, there exists z∈X which is comparable to x and y, 2.5

thenfhave a unique fixed point.

Proof. We first show thatf has a fixed point. Sincex0 fx0andf is increasing function, we obtain by induction that

x0fx0f²x0f³x0· · ·fⁿx0fⁿ¹x0· · ·. 2.6

Putx_nfⁿx0,n1,2, . . .. Sincex_nx_n1for eachn∈Nthen by2.2 dx_n1, x_n2 d

fⁿ¹x0, fⁿ²x0

≤αdxn, x_n1dxn, x_n1

≤dxn, x_n1.

2.7

And so the sequence {dxn1, xn} is nonincreasing and bounded below. Thus there exists τ ≥ 0 such that limn→ ∞dx_n1, xn τ. Since lim sup_s_→ταs < 1 andατ < 1 then there existr ∈0,1and >0 such thatαs< rfor alls∈τ, τ. We can takeν∈Nsuch that τ ≤dxn1, xn≤τfor alln∈Nwithn≥ν. Then since

dx_n1, x_n2≤αdxn, x_n1dxn, x_n1≤rdxn, x_n1, 2.8

for alln∈Nwithn≥νwe have ∞

n1

dxn, x_n1≤^ν

n1

dxn, x_n1 ∞ n1

rⁿdxν, x_ν1<∞, 2.9

and hence {xn} is a Cauchy sequence. SinceX is complete,{xn} converges to some point z∈X. To prove thatzis a fixed point off, iffis a continuous, then

z lim

n→ ∞x_n lim

n→ ∞fⁿx0 lim

n→ ∞fⁿ¹x0 f lim

n→ ∞fⁿx0

fz; 2.10

hencezfz. If case2.4holds then d

fz, z

≤d

fz, fxn d

fxn, z

≤αdxn, zdxn, z dx_n1, z

≤dxn, z dxn1, z.

2.11

(4)

Sincedxn, z → 0 then we getfz z. To prove the uniqueness of the fixed point, lety be another fixed point off. From2.5there existsx ∈ X which is comparable toyandz.

Monotonicity implies thatfⁿxis comparable tofⁿy yandfⁿz zforn 0,1, . . ..

Moreover,

d

z, fⁿx d

fⁿz, fⁿx

≤α d

fⁿ⁻¹z, fⁿ⁻¹x d

fⁿ⁻¹z, fⁿ⁻¹x

≤d

fⁿ⁻¹z, fⁿ⁻¹x d

z, fⁿ⁻¹x .

2.12

Consequently, the sequence ζ_n^z : dz, fⁿx is nonnegative and decreasing and so lim_n_{→ ∞}dz, fⁿx ζ_z ∈ R. Similarly we can show that the sequenceζ^y_n : dy, fⁿxis nonnegative and decreasing and so lim_n_{→ ∞}dy, fⁿx ζ_y ∈R. Now similarly the above method we can chooser1, r2in0,1andτ1∈Nsuch that

d

z, fⁿx

≤α d

z, fⁿ⁻¹x d

z, fⁿ⁻¹x

≤r₁d

z, fⁿ⁻¹x , d

y, fⁿx

≤α d

y, fⁿ⁻¹x d

y, fⁿ⁻¹x

≤r₂d

y, fⁿ⁻¹x ,

2.13

for alln∈Nwithn > τ₁. Finally d

z, y

≤d

z, fⁿx d

fⁿx, y

≤r₁^n−τ¹d

z, f^τ¹x₀

r₂^n−τ¹d

y, f^τ¹x₀

, 2.14

for alln∈Nwithn > τ1. Therefor if in2.14takingn → ∞yieldsdz, y 0.

3. Application to Ordinary Differential Equation

In this section we present an example where Theorem 2.1can be applied. This example is inspired in2,4,7.

Definition 3.1. LetBdenote the class of those functionsφ :0,∞ → 0,∞which satisfies the following condition:

iφis increasing,

iifor eachx >0,φx< x, iiiβx φx/x∈S.

For example,φx ax, where 0≤a <1,φx x/x1, andφx ln1xare inB.

Theorem 3.2. Consider problem1.1withf : I×R → Ris continuous and suppose that there existsλ >0 such that forx, y∈Rwith y≥x

0≤f t, y

λy−

ft, x λx

≤λφ y−x

, 3.1

(5)

whereφ∈B. Then the existence of a lower solution of 1.1provides the existence of a unique solution of 1.1.

Proof. Problem1.1is equivalent to the integral equation

ut _T

0

Gt, s

fs, us λus

ds, 3.2

where

Gt, s

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

e^λTs−t

e^λT−1 , 0≤s < t≤T, e^λs−t

e^λT−1, 0≤t < s≤T.

3.3

DefineF :CI,R → CI,Rby Fut

_T

0

Gt, s

fs, us λus

ds. 3.4

Note that ifu∈CI,Ris a fixed point ofF thenu∈C¹I,Ris a solution of1.1. Now, we check that hypotheses inTheorem 2.1are satisfied. Indeed,XCI,Ris a partially ordered set if we define the following order relation inX:

x, y∈CI,R, x≤y iﬀxt≤yt, ∀t∈I. 3.5

The mappingFis increasing since, by hypotheses, foru≥v

ft, u λu≥ft, v λv 3.6

which implies fort∈I, using thatGt, s>0 fort, s∈I×I, that

Fut _T

0

Gt, s

fs, us λus ds

≥ _T

0

Gt, s

fs, vs λvs

ds Fvt.

3.7

Beside, foru≥v

dFu, Fv sup

t∈I |Fut−Fvt|

≤sup

t∈I

_T

0

Gt, sfs, us λus−

fs, vs λvsds

≤sup

t∈I

_T

0

Gt, s·λφus−vsds.

3.8

(6)

As the functionφxis increasing andu≥vthenφus−vs≤φdu, vwe obtain dFu, Fv≤sup

t∈I

_T

0

Gt, s·λφus−vsds

≤λφdu, vsup

t∈I

_T

0

Gt, sds

λφdu, vsup

t∈I

1 e^λT−1

1

λe^λTs−t^t₀1

λe^λs−t^T_t

λφdu, v· 1

λ

e^λT−1

φdu, v.

3.9

Then foru≥v

dFu, Fv≤αdu, vdu, v. 3.10

Finally, letβtbe a lower solution of1.1, and we will show thatβ≤Fβ.

Indeed,

βt λβt≤f t, βt

λβt, fort∈I. 3.11 Multiplying bye^λtwe get

βte^λt

≤ f

t, βt

λβt

e^λt, fort∈I, 3.12

and this gives us

βte^λt ≤β0 _t

0

f s, βs

λβs

e^λsds, fort∈I 3.13

which implies that

β0e^λT≤βTe^λT≤β0 _T

0

f s, βs

λβs

e^λsds, 3.14

and so

β0≤ _T

0

e^λs e^λT−1

f s, βs

λβs

ds. 3.15

From this equality and3.13we obtain

βte^λt≤ _t

0

e^λTs e^λT−1

f s, βs

λβs ds

_T

t

e^λs e^λT−1

f s, βs

λβs

ds, 3.16

(7)

and, consequently,

βt≤ _t

0

e^λTs−t e^λT−1

f s, βs

λβs ds

_T

t

e^λs−t e^λT−1

f s, βs

λβs

ds. 3.17

Hence

βt≤ _T

0

Gt, s f

s, βs

λβs ds

F β

t, fort∈I. 3.18

Finally,Theorem 2.1gives thatFhas a unique fixed point.

Example 3.3. Letφ0:0,∞ → 0,∞be a defined

φ₀t

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

0, 0≤t≤3, 3t−9, 3< t≤4, 3

4t, 4< t.

3.19

Letf :I×R → Rbe continuous and suppose that there existsλ >0 such that forx, y ∈R withy≥x

0≤f t, y

λy−

ft, x λx

≤λφ₀ y−x

. 3.20

Then beTheorem 2.1, the existence of a lower solution for1.1provides the existence of a unique solution of1.1.

The example discussed above cannot be the result of Harjani and Sadarangani noted Theorem 1.2, becauseψx x−φ₀xis 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 TechnologyGrant no.: 2010-0010243.

References

1 R. P. Agarwal, M. A. El-Gebeily, and D. O’Regan, “Generalized contractions in partially ordered metric spaces,” Applicable Analysis, vol. 87, no. 1, pp. 109–116, 2008.

2 A. Amini-Harandi and H. Emami, “A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary diﬀerential equations,” Nonlinear Analysis, vol. 72, no. 5, pp. 2238–2242, 2010.

3 T. Gnana Bhaskar and V. Lakshmikantham, “Fixed point theorems in partially ordered metric spaces and applications,” Nonlinear Analysis, vol. 65, no. 7, pp. 1379–1393, 2006.

4 J. Harjani and K. Sadarangani, “Fixed point theorems for weakly contractive mappings in partially ordered sets,” Nonlinear Analysis, vol. 71, no. 7-8, pp. 3403–3410, 2009.

(8)

5 V. Lakshmikantham and L. B. ´Ciri´c, “Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces,” Nonlinear Analysis, vol. 70, no. 12, pp. 4341–4349, 2009.

6 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.

7 J. J. Nieto and R. Rodr´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.

8 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.

9 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.

(9)

Submit your manuscripts at http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

^{Journal of}

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations

International Journal of

Volume 2014

Applied Mathematics^{Journal of}

Mathematical PhysicsAdvances in

Complex Analysis

^{Journal of}

Optimization

^{Journal of}

Combinatorics

Journal of

Function Spaces

Abstract and Applied Analysis

International Journal of Mathematics and Mathematical Sciences

The Scientific World Journal

Discrete Dynamics in Nature and Society

Discrete Mathematics

^{Journal of}

A Fixed Point Theorem for Contraction Type Maps in Partially Ordered Metric Spaces and Application to Ordinary Differential Equations

Research Article