# Weak contractions on chains in a generalized metric space with a partial order

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DOI: 10.1007/s13370-013-0146-6

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- Choudhury, B.S. & Maity, P. Afr. Mat. (2014) 25: 745. doi:10.1007/s13370-013-0146-6

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## Abstract

Weak contraction mapping principle is a generalization of the Banach contraction mapping principle. Weakly contractive mappings are intermediate to contraction mappings and nonexpansive mappings. They have been studied in several contexts. Metric fixed point theory in partially ordered spaces have rapidly developed in recent times. In this paper we extend the concept of weak contraction to subset of a partially ordered generalized metric space which are chains by themselves. It is noted that this weak contraction is different from weak contraction on the whole space. We prove here that under certain assumptions the weakly contractive mapping on certain chains will have a fixed point. Two illustrative examples are given.

### Keywords

\(G\)-metric spacePartially ordered setWeak contractionFixed pointOrbitMonotone property### Mathematics Subject Classification (2000)

54H25## 1 Introduction

\(G\)-metric spaces were introduced by Mustafa and Sims [1, 2]. This is a generalization of metric spaces in which every triplet of elements is assigned to a non-negative real number. Analysis of the structure of this space was done in some detail in [2]. Fixed point theory in this space was initiated in [3]. Particularly, Banach contraction mapping principle was established in this work. After that several fixed point results were proved in this space. Some of these works are noted in ([4–14]). Several other studies relevant to metric spaces are being extended to \(G\)-metric spaces as, for instances, a best approximation result in these spaces has been established in [15] and the concept of \(\omega \)-distance, which is relevant to minimization problems in metric spaces [16], has been extended to G-metric spaces [10].

In this paper we establish the weak contraction principle in \(G\)-metric spaces. A weak contraction is a generalization of Banach contraction. Banach contraction principle has been generalized by various authors. Over the years, and presently also, it remains an active area of research. Some of the very recent examples from this line of research are noted in [17–20]. In particular, weak contraction principle was introduced in Hilbert spaces by [21] and was extended to metric spaces by [22].

The weak contraction mappings are weaker than the contraction mappings but stronger than the nonexpansive mappings. The weak contraction principle states that every weak contraction on a complete metric space necessarily has a unique fixed point. There are several works on weak contractions and weakly contractive type mappings, some of these are noted in [23–29].

Fixed point theory in partially ordered metric spaces has rapidly developed in recent times. Some instances of these works are noted in [30–35]. One of the reasons of the widespread interest in these problems is that they utilize both analytic and order theoretic aspects of fixed point theory. G-metric spaces with a partial order has been treated in [10].

In this work we have defined the weak contraction on certain subsets of a G-metric spaces with a partial order which are chains by themselves. We have shown that whenever a mapping is a weak contraction on every chain containing a specific orbit, it will have a fixed point. We have given two examples to illustrate our ideas. With the help of one of these examples it is shown that that the weak contraction mentioned above is different from that defined on the whole space.

## 2 Mathematical preliminaries

**Definition 2.1**

*metric space*[2]) Let \(X\) be a nonempty set and let \(G:X \times X \times X \longrightarrow R^{+}\) be a function satisfying the following properties:

- (G1)
\(G(x, y, z)=0\) if \(x=y=z\);

- (G2)
\(0<G(x,x,y)\); for all \(x,y\in X\) with \(x\ne y\);

- (G3)
\(G(x,x,y)\le G(x,y,z),\) for all \(x,y,z\in X\) with \(z\ne y\);

- (G4)
\(G(x,y,z)=G(x,z,y)=G(y,z,x)=......\),(symmetry in all three variables);

- (G5)
\(G(x,y,z)\le G(x,a,a)+G(a,y,z)\), for all \(x,y,z,a\in X\) (rectangle inequality).

**Definition 2.2**

[2] Let \((X,\,G)\) be a \(G\)-metric space and let \(\{x_{n}\}\) be a sequence of points of \(X\), a point \(x\in X\) is said to be the limit of the sequence \(\{x_{n}\}\) if \(\underset{n,m\rightarrow \infty }{\lim } G(x,x_{n},x_{m})=0\) and one says that the sequence \(\{x_{n}\}\) is \(G\)-convergent to x.

Thus, if \(x_{n}\rightarrow x\) in a \(G\)-metric space \((X,\,G)\), then for any \(\epsilon >0\), there exists a positive integer \(N\) such that \(G(x,x_{n},x_{m})<\epsilon \), for all \(n,m\ge N\).

It has been shown in [2] that the \(G\)-metric induces a Housdorff topology and the convergence described in the above definition is relative to this topology. The topology being Housdorff, a sequence can converge at most to one point.

**Definition 2.3**

[2] Let \((X,\,G)\) be a \(G\)-metric space, a sequence \( \{x_{n}\}\) is called \(G\)-cauchy if for every \(\epsilon >0\), there is a positive integer \(N\) such that \(G(x_{n},x_{m},x_{l})<\epsilon \), for all \(n,m\ge N\), that is, if \(G(x_{n},x_{m},x_{l})\rightarrow 0\) as \(n,m\rightarrow \infty \).

We next state the following lemmas.

**Lemma 2.4**

- (1)
\(\{x_{n}\}\) is \(G\)-convergent to x.

- (2)
\(G(x_{n},x_{n},x)\rightarrow 0\), as \(n\rightarrow \infty .\)

- (3)
\(G(x_{n},x,x)\rightarrow 0,\) as \(n\rightarrow \infty .\)

- (4)
\(G(x_{m},x_{n},x)\rightarrow 0,\) as \(m,n\rightarrow \infty .\)

**Lemma 2.5**

- (1)
The sequence \(\{x_{n}\}\) is \(G\)-cauchy.

- (2)
For every \(\epsilon >0\), there exists a positive integer N such that \(G(x_{n},x_{m},x_{m})<\epsilon \), for all \(n,m\ge N.\)

**Lemma 2.6**

[2] If \((X, G)\) is a \(G\)-metric space then \(G(x,y,y)\le 2G(y,x,x)\) for all \(x,y\in X\).

**Lemma 2.7**

If \((X,G)\) is a \(G\)-metric space then \(\{x_{n}\}\) is a \(G\)-cauchy sequence if and only if for every \(\epsilon >0\), there exists a positive integer N such that \(G(x_{n},x_{m},x_{m})<\epsilon \), for all \(m>n\ge N.\)

**Lemma 2.8**

[1] For every \(G\)-metric space \((X, G)\) we can define a metric space \((X, d_{G})\) by \(d_{G}(x,y) = G(x,y,y)+ G(y,x,x)\) ,for all \(x, y\in X \).

**Definition 2.9**

[2] A \(G\)-metric space \((X,\,G)\) is called symmetric \(G\)-metric space if \(G(x,y,y)=G(y,x,x)\) for all \(x,y\in X\).

**Definition 2.10**

[2] A \(G\)-metric space \((X,\,G)\) is said to be \(G\)-complete ( or complete \(G\)-metric space) if every \(G\)-cauchy sequence in \((X,\,G)\) is convergent in X.

The following is the Banach contraction principle in the \(G\)-metric space proved by [3].

**Theorem 2.11**

[3] Let \((X, G)\) be a complete \(G\)-metric space and let \(T:X\rightarrow X\) be a mapping satisfying the following condition:

\(G(T(x),T(y),T(z))\le aG(x,y,z)\) for all \(x,y,z\in X\) where \(0\le a<1\), then \(T\) has a unique fixed point.

In fact it is a special case of the more general result proved by the above mentioned authors (Theorem 2.1 of [3]).

**Definition 2.12**

**Definition 2.13**

Let \((X,\preceq )\) be a partially ordered set and \(T:X \rightarrow X\) be a mapping. The mapping \(T\) is said to have monotone property if for any \(x_{1},x_{2}\in X,\)\(x_{1}\preceq x_{2}\) implies \(Tx_{1}\preceq Tx_{2}\).

**Definition 2.14**

A subset \(S\) of a partially ordered set \((X, \preceq )\) is a chain if \( x\preceq y\) or \(y\preceq x\) whenever \(x, y\) in \(S\).

Next we define weak contraction on a chain in a partially ordered \(G\)-metric space.

**Definition 2.15**

*Weak contraction on a chain*) A map \(T:C \rightarrow C\), where \(C\) is a subset of a partially ordered \(G\)-metric space \((X, G)\) such that \(C\) is a chain, is called weakly contractive on the chain \(C\) if for all \(x,y,z\in C\) with \(x\preceq y\preceq z\) and \(z\ne y\),

## 3 Main results

**Theorem 3.1**

Let \((X,\preceq )\) be a partially ordered set and \(G\) be a \(G\)-metric on \(X\) which has the property that whenever a monotone increasing sequence \(\{x_{n}\}\) converges to a point \(p\), it will follow that \(x_{n}\preceq p\). Let \(T:X\rightarrow X\) be a mapping with the monotone property on \(X\). Let \(x_{0}\in X\) be such that \(x_{0}\preceq Tx_{0}\), then the orbit \(O(x_{0})\) is a chain. If \(T\) is weakly contractive on every chain \(C \supseteq O(x_{0})\), then \(T\) has a fixed point.

*Proof*

Let \(x_{0}\in X\) and \(\{x_{n}\}\) be the sequence defined by \(x_{n+1}=T{x_{n}},n\ge 1\).

Let \(\rho _{n} = G(x_{n},x_{n},x_{n+1}).\)

Suppose \(\rho ^{*}>0\). Since \(\Psi \) is nondecreasing, \(\Psi (\rho _{n})\ge \Psi (\rho ^{*})>0.\) Hence, from (3.3) we have, \(\rho _{n+1}\le \rho _{n}-\Psi (\rho ^{*})\), for all \(n\ge 0\).

Adding over \(n = m, m+1,....N\), we have \(\rho _{N+m}\le \rho _{m}-N\Psi (\rho ^{*})\).

If \(\rho ^{*} \ne 0\), then \(\Psi (\rho ^{*}) \ne 0\), which implies that \(N\Psi (\rho ^{*})\) increases infinitely \(N\) tends to infinity. Also \(\{\rho _{n}\}\) is bounded. Then for large \(N\), the above inequality gives a contradiction. Therefore \(\rho ^{*} = 0\).

For given \(\epsilon >0\) and a positive integer \(N\), let \(B(N,\epsilon )=\{x:G(x_{N},x_{N},x)<\epsilon \) and \(x_{N}\preceq x \}\).

We next show that for sufficiently large N, \(T\) is a mapping from \(B(N,\epsilon )\) to \(B(N,\epsilon )\).

If \(x= x_{N}\), then \(G(x_{N},x_{N},Tx)= G(x_{N},x_{N},x_{N+1})<min\{\frac{\epsilon }{2} ,\Psi (\frac{\epsilon }{2})\}< \epsilon .\)

Hence \(Tx\in B(N,\epsilon )\). So we assume that \(x\ne x_{N}\).

Then we have the following two cases.

**Case-I**\(G(x_{N},x_{N},x)\le \frac{\epsilon }{2}\).

**Case-II**\(\frac{\epsilon }{2}<G(x_{N},x_{N},x)<\epsilon \).

Therefore, \(Tx_{N}= x_{N+1}\preceq Tx\) for all \(x\in B(N,\epsilon )\).

Then from (3.1), \(x_{N}\preceq x_{N+1}\preceq Tx\).

The above argument thus implies that, \(Tx\in B(N,\epsilon )\), that is, \(T\) is a selfmap of \(B(N,\epsilon )\) for sufficiently large \(N\).

Let \(\underset{n\rightarrow \infty }{\lim }x_{n}=p\).

Next we show that \(p\) is a fixed point of \(T\).

By our construction, \(\{x_{n}\}\) is a monotone increasing sequence and hence, by our assumption, \(x_{n}\preceq p\) for all \(n\ge 0\). If \(p= x_{m}\) for some \(m\), then it follows that \(p= x_{n}\) for all \(n\ge m\). In particular, we have \(p= x_{m+1}= Tx_{m}= Tp\) , that is, \(p\) is a fixed point of \(T\).

So we assume that \(p\ne x_{n}\) for all \(n\).

This completes the proof of the theorem.

*Example 3.1*

Let a partial order \(^{\prime }\preceq ^{\prime }\) on \(X\) be defined as follows:

\(y \preceq z\) whenever \(y\ge z\), for all \(y,z \in X\) and \((y - z)\) is divisible by 2.

It then easily follows that the inequality (2.1) satisfied. Thus all the conditions of the Theorem 3.1 are satisfied. Hence, by Theorem 3.1, a fixed point of \(T\) exists.

Here \(z = 0\) is a fixed point of \(T\).

*Remark 3.1*

It is noted that in the above example the inequality (2.1) is satisfied on the chain \(0\preceq 2 \preceq 4\preceq ......\), but is not satisfied on the chain \(0 \preceq 1 \preceq 3 \preceq .....\).. This can be seen by assuming \(x=3=y\), \(z=1\), which leads to a violation of (2.1). Hence \(T\) is not weakly contractive over the whole space.

*Example 3.2*

Let \(X = \{0,2,4\}\) and \(G:X\times X\times X\rightarrow R^{+}\) be defined as follows:

\(G(0,2,2)=3.5\), \(G(0,0,2)=4.5\), \(G(0,4,4)=7.5\), \(G(0,0,4)=7.5\), \(G(0,2,4)=7.5\) and \(G(x,y,z)= \mid x-y \mid + \mid y-z \mid + \mid z-x \mid \), for the rest.

Then \(G\) is a non symmetric \(G\)-metric space.

Let a partial order \(^{\prime }\preceq ^{\prime }\) on \(X\) be defined as follows: \(4 \preceq 2 \preceq 0\).

*Remark 3.2*

The condition \(x\preceq y\preceq z\) with \(y\ne z\) is essential in the definition of the weak contraction on a chain in a partially ordered \(G\)-metric space in order to make the definition meaningful. Otherwise, the inequality is reducible to a weak contraction inequality in \((X,d_{G})\) where \(d_{G}\) is the metric induced by the \(G\)-metric as described in Lemma 2.8 and the result of the theorem can be obtained by an application of a result in metric space. In Example 3.2, the inequality (2.1) is not valid with \(z=y= 2\), \(x=4\). It is worth mentioning that several fixed point problems in \(G\)-metric spaces are reducible to metric fixed point problems. Our case is outside this category.

## 4 An application to a boundary value problem

The above problem, with some conditions imposed on \(f\), was considered by [33] and later by [32].

Particular choices of \(\phi \) satisfying the required conditions are \(\phi (t) = kt, \, 0<k<1\) and \(\phi (t) = log (1 + t)\).

**Theorem 4.1**

*Proof*

Since \(K(t,s)\ge 0\), (4.10) implies that \(Sx(t)\ge Sy(t)\) for all \(t\in [0,1]\) whenever \(x(t)\ge y(t)\) for all \(t\in [0,1]\), that is, \(Sy\preceq Sx\) whenever \(y\preceq x\). This proves that \(S\) is monotone.

Next we show that the lower solution \(x_0\) satisfies \(x_0\preceq Sx_0\). We follow the argument of [33] in the following.

By (4.9) and the properties of \(\Phi \), we have that \(\psi \) is continuous, monotonically increasing and \(\psi (t) = 0\) if and only if \(t =0\).

Thus all the conditions of the Theorem 3.1 are satisfied by \(S\). Hence \(S\) must have a fixed point, that is, there exists \(x(t)\) such that \(x(t)= Sx(t)= \int ^{T}_{0}G(t,s)[f(s,x(s))+ \lambda x(s)]\).

This proves that the system of differential Eqs. (4.1)–(4.2) has a solution.

## Acknowledgments

The work is supported by the Council of Scientific and Industrial Research, Government of India, under Research Project No - 25(0168)/09/EMR-II. The support is gratefully acknowledged. The authors acknowledge the suggestions of the learned referees.