# Contraction mapping principle with generalized altering distance function in ordered metric spaces and applications to ordinary differential equations

- Yongfu Su
^{1}Email author

**2014**:227

https://doi.org/10.1186/1687-1812-2014-227

© Su; licensee Springer. 2014

**Received: **29 May 2014

**Accepted: **8 October 2014

**Published: **3 November 2014

## 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

## 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 [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 contractions and generalized contractions was extended to partially ordered metric spaces in [2, 9–22]. Among them, some involve altering distance functions. Such functions were introduced by Khan *et al.* in [1], where they present some fixed point theorems with the help of such functions. First, 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 non-decreasing. - (b)
$\psi =0$ if and only if $t=0$.

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 [11, 23], respectively. Their results improve the theorems of [2, 3].

**Theorem 1.2** ([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 non*-

*decreasing mapping such that*

*where* $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ *is continuous and non*-*decreasing 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 1.3** ([23])

*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 non*-

*decreasing 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 those functions $\beta :[0,\mathrm{\infty})\to [0,1)$ which satisfy the condition: $\beta ({t}_{n})\to 1\Rightarrow {t}_{n}\to 0$.

**Theorem 1.4** ([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*.

In 2012, Yan *et al.* proved the following result.

**Theorem 1.5** ([24])

*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 non*-

*decreasing 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*.

The aim of this paper is to present the definition of generalized altering distance function and to extend the results of Yan *et al.* [24] 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 2.1**A generalized altering distance function is a function $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ which satisfies:

- (a)
*ψ*is non-decreasing; - (b)
$\psi =0$ if and only if $t=0$.

We first recall the following notion of a monotone non-decreasing function in a partially ordered set.

**Definition 2.2** If $(X,\le )$ is a partially ordered set and $T:X\to X$, we say that *T* is monotone non-decreasing if $x,y\in X$, $x\le y\Rightarrow T(x)\le T(y)$.

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 1.2-1.5.

**Theorem 2.3**

*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 non*-

*decreasing mapping such that*

*where* *ψ* *is a generalized altering distance function and* $\varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ *is a right upper semi*-*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 non-decreasing function, we obtain by induction that

*ψ*and

*ϕ*, letting $n\to \mathrm{\infty}$ in (2) we get

*ψ*and

*ϕ*, letting $k\to \mathrm{\infty}$ and taking into account (7) and (8), we have

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

*X*:

**Theorem 2.4**

*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 non*-

*decreasing mapping such that*

*where* *ψ* *is a generalized altering distance functions and* *ϕ*: $[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ *is a right upper semi*-*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*Following the proof of Theorem 2.3 we only have to check that $T(z)=z$. As $({x}_{n})$ is a non-decreasing 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 condition of theorem we have $\varphi (0)=0$, this 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 2.3 and Theorems 2.4 do not guarantee uniqueness of the fixed point. An example appears in [12].

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 considering ${d}_{2}$ the Euclidean distance. The identity map $T(x,y)=(x,y)$ is trivially continuous and non-decreasing and condition (1) of Theorem 2.4 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 2.5** *Adding condition* (11) *to the hypotheses of Theorem * 2.3 (*resp*. *Theorem * 2.4) *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 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

As we have 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 a generalized altering distance function and we have the condition $\psi (t)>\varphi (t)$ for $t>0$, this gives us that $\{d(z,{f}^{n}(x))\}$ is a non-negative decreasing sequence and, consequently, there exists

*γ*such that

*ψ*and

*ϕ*, we obtain

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

**Remark 2.6** 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 2.7** Theorem 1.2 is a particular case of Theorem 2.3 for *ψ* being the identity function, and $\varphi (t)=t-\psi (t)$. Theorem 1.3 is a particular case of our Theorem 2.3 for $\varphi (t)$ being replaced by $\psi (t)-\varphi (t)$. Theorem 1.4 is a particular case of Theorem 2.3 for *ψ* being the identity function, and $\varphi (t)=\beta (t)t$. Theorem 1.5 is also a particular case of Theorem 2.3 for *ψ* and *ϕ* being continuous.

**Example 2.8**The following are some generalized altering distance functions:

where $\alpha \ge 1$ is a constant.

where $0<\beta <\alpha $ is a constant. By using Theorem 2.3, we can get the following result.

**Theorem 2.9**

*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 non*-

*decreasing mapping such that*

*for any* $x,y\in X$. *If there exists* ${x}_{0}\in X$ *such that* ${x}_{0}\le T{x}_{0}$, *then* *T* *has a fixed point*.

## 3 Application to ordinary differential equations

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

Clearly, $(C(I),\le )$ satisfies condition (10), since for $x,y\in C(I)$ the functions $max\{x,y\}$ and $min\{x,y\}$ are least upper and greatest lower bounds of *x* and *y*, respectively. Moreover, in [17] it is proved that $(C(I),\le )$ with the above mentioned metric satisfies 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 3.2**

*Consider problem*(13)

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

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

*with*

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

*with*$x\ge y$

*where* $g(t):[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ *is a light upper semi*-*continuous function with* $g(0)=0$, $g(t)<{t}^{2}$, $\mathrm{\forall}t>0$. *Then the existence of a lower solution for* (13) *provides the existence of an unique solution of* (13).

*Proof*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 2.3 and 2.4 are satisfied. The mapping

*F*is non-decreasing, since we have $u\ge v$, and using our assumption. We can obtain

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

Finally, Theorems 2.3 and 2.4 show that *F* has an unique fixed point. □

**Example 3.3**In Theorem 3.2, we can choose the function $g(t)$ as follows:

- (1)
${g}_{1}(t)=ln({t}^{2}+1)$;

- (2)${g}_{2}(t)=\{\begin{array}{cc}{t}^{3},\hfill & 0\le t<1,\hfill \\ \frac{1}{2},\hfill & t=1,\hfill \\ t,\hfill & 1<t<+\mathrm{\infty}.\hfill \end{array}$
- (3)${g}_{3}(t)=\{\begin{array}{cc}{t}^{3},\hfill & 0\le t\le \frac{1}{2},\hfill \\ t-\frac{3}{8},\hfill & \frac{1}{2}<t<+\mathrm{\infty}.\hfill \end{array}$

The functions ${g}_{1}(t)$, ${g}_{2}(t)$ are continuous and non-decreasing. The function ${g}_{3}(t)$ is right upper semi-continuous. If we choose $g(t)={g}_{1}(t)$ in Theorem 3.2, we obtain the result of [5].

**Example 3.4**Consider the following first-order periodic problem:

By using Theorem 3.2, we know that the first-order periodic problem (20) has a unique solution.

**Theorem 3.5**

*Consider problem*(21)

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

*continuous and non*-

*decreasing with respect to the second variable and suppose that there exists*$0\le \alpha \le 8$

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

*with*$x\ge y$

*where* $g(t):[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ *is a light upper semi*-*continuous function with* $g(0)=0$, $g(t)<{t}^{2}$, $\mathrm{\forall}t>0$. *Then our problem* (21) *has a unique non*-*negative solution*.

*Proof*Consider the cone

where $G(t,s)$ is the Green function appearing in (22).

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

and this proves that *T* is a non-decreasing operator.

*ψ*is an altering distance function,

*ψ*and

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

*f*and

*G*are non-negative functions

and Theorems 2.3 and 2.4 tell us that *F* has a unique non-negative solution. □

**Remark 3.6** In Theorem 3.5, we can choose $g(t)$ as ${g}_{1}(1)$, ${g}_{2}(t)$, and ${g}_{3}(t)$ as well as in Theorem 3.2.

**Example 3.7**Consider the following two-point boundary value problem of the second order differential equation:

Taking $g(t)=\frac{t}{2}$ for all $t\in [0,+\mathrm{\infty})$. By using Theorem 3.2, we know that the two-point boundary value problem (25) has a unique non-negative solution.

## Declarations

## Authors’ Affiliations

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