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# A new contraction mapping principle in partially ordered metric spaces and applications to ordinary differential equations

- Fangfang Yan
^{1}, - Yongfu Su
^{1}Email author and - Qiansheng Feng
^{1}

**2012**:152

https://doi.org/10.1186/1687-1812-2012-152

© Yan et al.; licensee Springer 2012

**Received:**28 April 2012**Accepted:**30 August 2012**Published:**18 September 2012

## Abstract

The aim of this paper is to extend the results of Harjani and Sadarangani and some other authors and to prove a new fixed point theorem of a contraction mapping in a complete metric space endowed with a partial order by using altering distance functions. Our theorem 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
- altering distance function
- differential equation

## 1 Introduction

The Banach contraction principle is a classical and powerful tool in nonlinear analysis. Weak contractions are generalizations of Banach’s contraction mapping studied by several authors. In [1–8], the authors prove some types of weak contractions in complete metric spaces respectively. In particular, the existence of a fixed point for weak contraction and generalized contractions was extended to partially ordered metric spaces in [2, 9–18]. Among them, the altering distance function is basic concept. Such functions were introduced by Khan *et al*. in [1], where they present some fixed point theorems with the help of such functions. Firstly, we recall the definition of an altering distance function.

**Definition 1.1**An altering distance function is a function $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ which satisfies

- (a)
*ψ*is continuous and nondecreasing. - (b)
$\psi =0$ if and only if $t=0$.

Recently, Harjani and Sadarangani proved some fixed point theorems for weak contraction and generalized contractions in partially ordered metric spaces by using the altering distance function in [11, 19] respectively. Their results improve the theorems of [2, 3].

**Theorem 1.1** [11]

*Let*$(X,\le )$

*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*$f:X\to X$

*be a continuous and nondecreasing mapping such that*

*where* $\psi :[0,\mathrm{\infty})\u27f6[0,\mathrm{\infty})$ *is continuous and nondecreasing function such that* *ψ* *is positive in* $(0,\mathrm{\infty})$, $\psi (0)=0$ *and* ${lim}_{t\to \mathrm{\infty}}\psi (t)=\mathrm{\infty}$. *If there exists* ${x}_{0}\in X$ *with* ${x}_{0}\le f({x}_{0})$, *then* *f* *has a fixed point*.

**Theorem 1.2** [19]

*Let*$(X,\le )$

*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*$f:X\to X$

*be a continuous and nondecreasing mapping such that*

*where* *ψ* *and* *ϕ* *are altering distance functions*. *If there exists* ${x}_{0}\in X$ *with* ${x}_{0}\le f({x}_{0})$, *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 [10]. The following class of functions is used in [10].

Let ℜ denote the class of functions $\beta :[0,\mathrm{\infty})\u27f6[0,1)$ which satisfies the condition $\beta ({t}_{n})\u27f61\Rightarrow {t}_{n}\u27f60$.

**Theorem 1.3** [10]

*Let*$(X,\le )$

*be a partially ordered set*,

*and suppose that there exists a metric*

*d*

*such that*$(X,d)$

*is a complete metric space*.

*Let*$f:X\to X$

*be an increasing mapping such that there exists an element*${x}_{0}\in X$

*with*${x}_{0}\le f({x}_{0})$.

*Suppose that there exists*$\beta \in \mathrm{\Re}$

*such that*

*Assume that either* *f* *is continuous or* *M* *is such that if an increasing sequence* ${x}_{n}\to x\in X$, *then* ${x}_{n}\le x$, ∀*n*. *Besides*, *if for each* $x,y\in X$, *there exists* $z\in m$ *which is comparable to* *x* *and* *y*, *then* *f* *has a unique fixed point*.

The purpose of this paper is to extend the results of [10, 11, 19] and to obtain a new contraction mapping principle in partially ordered metric spaces. The result is more generalized than the results of [10, 11, 19] and other works. The main theorems can be used to investigate a large class of nonlinear problems. In this paper, we also present some applications to first- and second-order ordinary differential equations.

## 2 Main results

We first recall the following notion of a monotone nondecreasing function in a partially ordered set.

**Definition 2.1** If $(X,\le )$ is a partially ordered set and $T:X\to X$, we say that *T* is monotone nondecreasing if $x,y\in X$, $x\le y\Rightarrow T(x)\le T(y)$.

This definition coincides with the notion of a nondecreasing function in the case where $X=R$ and ≤ represents the usual total order in R.

We shall need the following lemma in our proving.

**Lemma 2.1** *If* *ψ* *is an altering distance function and* $\varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ *is a continuous function with the condition* $\psi (t)>\varphi (t)$ *for all* $t>0$, *then* $\varphi (0)=0$.

*Proof*Since $\varphi (t)<\psi (t)$ and

*ϕ*,

*ψ*are continuous, we have

This finishes the proof. □

In what follows, we prove the following theorem which is the generalized type of Theorem 1.1-1.3.

**Theorem 2.1**

*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*$T:X\to X$

*be a continuous and nondecreasing mapping such that*

*where* *ψ* *is an altering distance function and* $\varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ *is a continuous function with the condition* $\psi (t)>\varphi (t)$ *for all* $t>0$. *If there exists* ${x}_{0}\in X$ *such that* ${x}_{0}\le T{x}_{0}$, *then* *T* *has a fixed point*.

*Proof*Since

*T*is a nondecreasing function, we obtain, by induction, that

*X*is a complete metric space, there exists $z\in X$ such that ${x}_{n}\to z$ as $n\to \mathrm{\infty}$. Moreover, the continuity of

*T*implies that

and this proves that *z* is a fixed point. This completes the proof. □

*T*not necessarily being continuous, assuming the following hypothesis in

*X*:

**Theorem 2.2**

*Let*$(X,\le )$

*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*(9).

*Let*$T:X\to X$

*be a nondecreasing mapping such that*

*where* *ψ* *is an altering distance function and* $\varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ *is a continuous function with the conditions* $\psi (t)>\varphi (t)$ *for all* $t>0$. *If there exists* ${x}_{0}\in X$ *such that* ${x}_{0}\le T{x}_{0}$, *then* *T* *has a fixed point*.

*Proof*Following the proof of Theorem 2.1, we only have to check that $T(z)=z$. As $({x}_{n})$ is a nondecreasing sequence in

*X*and ${lim}_{n\to \mathrm{\infty}}{x}_{n}=z$, the condition (9) gives us that ${x}_{n}\le z$ for every $n\in N$, and consequently,

*ψ*is an altering distance function, we have

Using Lemma 2.1, we have $\varphi (0)=0$, which implies $\mathrm{\Psi}(d(z,T(z)))=0$. Thus $d(z,T(z))=0$ or equivalently, $T(z)=z$. □

Now, we present an example where it can be appreciated that the hypotheses in Theorems 2.1 and Theorems 2.2 do not guarantee the uniqueness of the fixed point. The example appears in [17].

Let $X=\{(1,0),(0,1)\}\subset {R}^{2}$ and consider the usual order $(x,y)\le (z,t)\iff x\le z$, $y\le t$. Thus, $(x,y)$ is a partially ordered set whose different elements are not comparable. Besides, $(X,{d}_{2})$ is a complete metric space and ${d}_{2}$ is the Euclidean distance. The identity map $T(x,y)=(x,y)$ is trivially continuous and nondecreasing, and the condition (9) of Theorem 2.2 is satisfied since the elements in *X* are only comparable to themselves. Moreover, $(1,0)\le T(1,0)=(1,0)$ and *T* has two fixed points in *X*.

**Theorem 2.3** *Adding the condition* (11) *to the hypotheses of Theorem* 2.1 (*resp*. *Theorem * 2.2), *we obtain the uniqueness of the fixed point of* *T*.

*Proof* Suppose that there exist $z,y\in X$ which are fixed points. We distinguish the following two cases:

*y*is comparable to

*z*, then ${T}^{n}(y)=y$ is comparable to ${T}^{n}(z)=z$ for $n=0,1,2,\dots $ and

By the condition $\psi (t)>\varphi (t)$ for $t>0$, we obtain $d(z,y)=0$ and this implies $z=y$.

*y*is not comparable to

*z*, then there exists $x\in X$ 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=0,1,2,\dots $ . Moreover,

*ψ*is an altering distance function and the condition of $\psi (t)>\varphi (t)$ for $t>0$. This gives us that $\{d(z,{f}^{n}(x))\}$ is a nonnegative decreasing sequence, and consequently, there exists

*γ*such that

*ψ*and Φ are continuous functions, we obtain

This and the condition of Theorem 2.1 implies $\varphi (\gamma )=0$, and consequently, $\gamma =0$.

the uniqueness of the limit gives us $y=z$. This finishes the proof. □

**Remark 2.1** Under the assumption of Theorem 2.3, it can be proved that for every $x\in X$, ${lim}_{n\to \mathrm{\infty}}{T}^{n}(x)=z$, where *z* is the fixed point (*i.e.*, the operator *f* is Picard).

**Remark 2.2** Theorem 1.1 is a particular case of Theorem 2.1 for *ψ*, the identity function, and $\varphi (x)=x-\psi (x)$.

Theorem 1.2 is a particular case of Theorem 2.1 for $\varphi (x)=\psi (x)\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}{\varphi}_{1.2}(x)$, ${\varphi}_{1.2}$ is an altering function in Theorem 1.2. Theorem 1.3 is a particular case of Theorem 2.1 for *ψ*, the identity function, and $\varphi (x)=\psi (x)x$.

## 3 Application to ordinary differential equations

*I*. Obviously, this space with the metric given by

Clearly, $(C(I),\le )$ satisfies the condition (10) since for $x,y\in C(I)$, the functions $max\{x,y\}$ and $min\{x,y\}$ are the least upper and the greatest lower bounds of *x* and *y*, respectively. Moreover, in [17] it is proved that $(C(I),\le )$ with the above mentioned metric satisfies the condition (9).

Now, we give the following definition.

**Definition 3.1**A lower solution for (13) is a function $\alpha \in {C}^{(1)}(I)$ such that

**Theorem 3.1**

*Consider the problem*(13)

*with*$f:I\times R\u27f6R$

*continuous*,

*and suppose that there exist*$\lambda ,\alpha >0$

*with*

*such that for*$x,y\in R$

*with*$x\ge y$,

*Then the existence of a lower solution for* (13) *provides the existence of a unique solution of *(13).

*Proof*The problem (13) can be written as

*F*, then $u\in {C}^{1}(I)$ is a solution of (13). In what follows, we check that the hypotheses in Theorems 2.2 and 2.3 are satisfied. The mapping

*F*is nondecreasing for $u\ge v$; using our assumption, we can obtain

*ψ*is an altering distance function, $\psi (x)$ and $\varphi (x)$ satisfy the condition of $\psi (x)>\varphi (x)$ for $x>0$. From (18), we obtain for $u\ge v$,

Finally, Theorems 2.2 and 2.3 give that F has an unique fixed point. □

**Theorem 3.2**

*Consider the problem*(20)

*with*$f:I\times R\to [0,\mathrm{\infty})$

*continuous and nondecreasing with respect to the second variable*,

*and suppose that there exists*$0\le \alpha \le 8$

*such that for*$x,y\in R$

*with*$y\ge x$,

*Then our problem* (20) *has a unique nonnegative solution*.

*Proof*Consider the cone

where $G(t,s)$ is the Green function appearing in (21).

*f*is nondecreasing with respect to the second variable, then for $x,y\in P$ with $y\ge x$ and $t\in [0,1]$, we have

and this proves that *T* is a nondecreasing operator.

*ψ*is an altering distance function,

*ψ*and

*ϕ*satisfy the condition $\psi (x)>\varphi (x)$, for $x>0$. From the last inequality, we have

*f*and

*G*are nonnegative functions,

Theorems 2.2 and 2.3 tell us that *F* has a unique nonnegative solution. □

## Declarations

### Acknowledgements

This project is supported by the National Natural Science Foundation of China under the grant (11071279).

## Authors’ Affiliations

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