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

Authors: M. Eshaghi Gordji; H. Baghani; G. H. Kim
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

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.


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## Body

## 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
(1.1)u′(t)=f(t,u(t))ift∈I=[0,T],u(0)=u(T),
where T>0 and f:I×ℝ→ℝ is continuous function.Definition 1.1.
A lower solution for (1.1) is a function u∈C1(I,ℝ) such that
(1.2)u′(t)≤f(t,u(t))fort∈I=[0,T],u(0)≤u(T).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(1.3)0≤f(t,y)+λy-[f(t,x)+λx]≤λϕ(y-x),
where  ϕ:[0,∞)→[0,∞) can be written by  ϕ(x)=x-ψ(x) with  ψ:[0,∞)→[0,∞) continuous increasing positive in  (0,∞),   ψ(0)=0 and limt→∞ψ(t)=∞. Then the existence of a lower solution of (1.1) provides the existence of a unique solution of (1.1).In Section2, 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

LetS denote the class of those functions α:[0,∞)→[0,1) which satisfies the condition
(2.1)limsup⁡s→t+α(s)<1,∀t∈[0,∞).
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≼f(x0). Suppose that there exists  α∈S such that
(2.2)d(f(x),f(y))≤α(d(x,y))d(x,y),
for all comparable  x,y∈X. If
(2.3)fiscontinuous
or
(2.4)ifanincreasingsequence{xn}→xinX,thenxn≼x,∀n∈N.
Besides, if
(2.5)foreachx,y∈X,thereexistsz∈Xwhichiscomparabletoxandy,
then  f have a unique fixed point.Proof.
We first show thatf has a fixed point. Since x0≼f(x0) and f is increasing function, we obtain by induction that
(2.6)x0≼f(x0)≼f2(x0)≼f3(x0)≼⋯≼fn(x0)≼fn+1(x0)⋯.
Put xn=fn(x0),  n=1,2,…. Since xn≼xn+1 for each n∈ℕ then by (2.2)
(2.7)d(xn+1,xn+2)=d(fn+1(x0),fn+2(x0))≤α(d(xn,xn+1))d(xn,xn+1)≤d(xn,xn+1).
And so the sequence {d(xn+1,xn)} is nonincreasing and bounded below. Thus there exists τ≥0 such that limn→∞d(xn+1,xn)=τ. Since limsups→τ+α(s)<1 and α(τ)<1 then there exist r∈[0,1) and ϵ>0 such that α(s)<r for all s∈[τ,τ+ϵ]. We can take ν∈ℕ such that τ≤d(xn+1,xn)≤τ+ϵ for all n∈ℕ with n≥ν. Then since
(2.8)d(xn+1,xn+2)≤α(d(xn,xn+1))d(xn,xn+1)≤rd(xn,xn+1),
for all n∈ℕ with n≥ν we have
(2.9)∑n=1∞d(xn,xn+1)≤∑n=1νd(xn,xn+1)+∑n=1∞rnd(xν,xν+1)<∞,
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
(2.10)z=lim⁡n→∞xn=lim⁡n→∞fn(x0)=lim⁡n→∞fn+1(x0)=f(lim⁡n→∞fn(x0))=f(z);
hence z=f(z). If case (2.4) holds then
(2.11)d(f(z),z)≤d(f(z),f(xn))+d(f(xn),z)≤α(d(xn,z))d(xn,z)+d(xn+1,z)≤d(xn,z)+d(xn+1,z).
Since d(xn,z)→0 then we get f(z)=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 fn(x) is comparable to fn(y)=y and fn(z)=z for n=0,1,…. Moreover,
(2.12)d(z,fn(x))=d(fn(z),fn(x))≤α(d(fn-1(z),fn-1(x)))d(fn-1(z),fn-1(x))≤d(fn-1(z),fn-1(x))=d(z,fn-1(x)).
Consequently, the sequence ζnz:=d(z,fn(x)) is nonnegative and decreasing and so limn→∞d(z,fn(x))=ζz∈ℝ. Similarly we can show that the sequence ζny:=d(y,fn(x)) is nonnegative and decreasing and so limn→∞d(y,fn(x))=ζy∈ℝ. Now similarly the above method we can choose r1,r2 in [0,1) and τ1∈ℕ such that
(2.13)d(z,fn(x))≤α(d(z,fn-1(x)))d(z,fn-1(x))≤r1d(z,fn-1(x)),d(y,fn(x))≤α(d(y,fn-1(x)))d(y,fn-1(x))≤r2d(y,fn-1(x)),
for all n∈ℕ with n>τ1. Finally
(2.14)d(z,y)≤d(z,fn(x))+d(fn(x),y)≤r1n-τ1d(z,fτ1x0)+r2n-τ1d(y,fτ1x0),
for all n∈ℕ with n>τ1. Therefor if in (2.14) taking n→∞ yields d(z,y)=0.


## 3.  Application to Ordinary Differential Equation

In this section we present an example where Theorem2.1 can be applied. This example is inspired in [2, 4, 7].Definition 3.1.
Let𝔅 denote the class of those functions ϕ:[0,∞)→[0,∞) which satisfies the following condition:(i)
ϕ is increasing,(ii)
for eachx>0,  ϕ(x)<x,(iii)
β(x)=ϕ(x)/x∈S.For example,ϕ(x)=ax, where 0≤a<1, ϕ(x)=x/(x+1), and ϕ(x)=ln(1+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(3.1)0≤f(t,y)+λy-[f(t,x)+λx]≤λϕ(y-x),
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
(3.2)u(t)=∫0TG(t,s)[f(s,u(s))+λu(s)]ds,
where
(3.3)G(t,s)={eλ(T+s-t)eλT-1,0≤s<t≤T,eλ(s-t)eλT-1,0≤t<s≤T.
Define F:C(I,ℝ)→C(I,ℝ) by
(3.4)(Fu)(t)=∫0TG(t,s)[f(s,u(s))+λu(s)]ds.
Note that if u∈C(I,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=C(I,ℝ) is a partially ordered set if we define the following order relation in X:
(3.5)x,y∈C(I,R),x≤yiffx(t)≤y(t),∀t∈I.
The mapping F is increasing since, by hypotheses, foru≥v(3.6)f(t,u)+λu≥f(t,v)+λv
which implies for t∈I, using that G(t,s)>0 for (t,s)∈I×I, that
(3.7)(Fu)(t)=∫0TG(t,s)[f(s,u(s))+λu(s)]ds≥∫0TG(t,s)[f(s,v(s))+λv(s)]ds=(Fv)(t).
Beside, for u≥v(3.8)d(Fu,Fv)=sup⁡t∈I|(Fu)(t)-(Fv)(t)|≤sup⁡t∈I∫0TG(t,s)|f(s,u(s))+λu(s)-(f(s,v(s))+λv(s))|ds≤sup⁡t∈I∫0TG(t,s)⋅λϕ(u(s)-v(s))ds.
As the function ϕ(x) is increasing and u≥v then ϕ(u(s)-v(s))≤ϕ(d(u,v)) we obtain
(3.9)d(Fu,Fv)≤sup⁡t∈I∫0TG(t,s)⋅λϕ(u(s)-v(s))ds≤λϕ(d(u,v))sup⁡t∈I∫0TG(t,s)ds=λϕ(d(u,v))sup⁡t∈I1eλT-1(1λeλ(T+s-t)]0t+1λeλ(s-t)]tT)=λϕ(d(u,v))⋅1λ(eλT-1)(eλT-1)=ϕ(d(u,v)).
Then for u≥v(3.10)d(Fu,Fv)≤α(d(u,v))d(u,v).
Finally, let β(t) be a lower solution of (1.1), and we will show that β≤F(β).
Indeed,(3.11)β′(t)+λβ(t)≤f(t,β(t))+λβ(t),fort∈I.
Multiplying by eλt we get
(3.12)(β(t)eλt)′≤[f(t,β(t))+λβ(t)]eλt,fort∈I,
and this gives us
(3.13)β(t)eλt≤β(0)+∫0t[f(s,β(s))+λβ(s)]eλsds,fort∈I
which implies that
(3.14)β(0)eλT≤β(T)eλT≤β(0)+∫0T[f(s,β(s))+λβ(s)]eλsds,
and so
(3.15)β(0)≤∫0TeλseλT-1[f(s,β(s))+λβ(s)]ds.
From this equality and (3.13) we obtain
(3.16)β(t)eλt≤∫0teλ(T+s)eλT-1[f(s,β(s))+λβ(s)]ds+∫tTeλseλT-1[f(s,β(s))+λβ(s)]ds,
and, consequently,
(3.17)β(t)≤∫0teλ(T+s-t)eλT-1[f(s,β(s))+λβ(s)]ds+∫tTeλ(s-t)eλT-1[f(s,β(s))+λβ(s)]ds.
Hence
(3.18)β(t)≤∫0TG(t,s)[f(s,β(s))+λβ(s)]ds=(F(β))(t),fort∈I.
Finally, Theorem 2.1 gives that F has a unique fixed point.Example 3.3.
Letϕ0:[0,∞)→[0,∞) be a defined
(3.19)ϕ0(t)={0,0≤t≤3,3t-9,3<t≤4,34t,4<t.
Let f:I×ℝ→ℝ be continuous and suppose that there exists λ>0 such that for x,y∈ℝ withy≥x(3.20)0≤f(t,y)+λy-[f(t,x)+λx]≤λϕ0(y-x).
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, becauseψ(x)=x-ϕ0(x) is not increasing.



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*Source: 981517-2012-03-11.xml*