Remarks on contractive type mappings
 Jarosław Górnicki^{1}Email author
DOI: 10.1186/s1366301706014
© The Author(s) 2017
Received: 9 December 2016
Accepted: 4 May 2017
Published: 18 May 2017
Abstract
We present an analog of Banach’s fixed point theorem for \(CJM\) contractions in preordered metric spaces. These results substantially extend theorems of Ran and Reurings’ (Proc. Am. Math. Soc. 132(5): 14351443, 2003) and Nieto and RodríguezLópez (Order 22: 223239, 2005).
Keywords
contractive map fixed point preordered space Gmetric spaceMSC
47H10 54H251 Introduction
In 2003, Ran and Reurings [1] established a fixed point theorem that extends the Banach contraction principle (BCP) to the setting of partially ordered metric spaces. In the original version, Ran and Reurings used a continuous function. In 2006, Nieto and RodríguezLópez [2] established a similar result replacing the continuity of the nonlinear operator by monotonicity. The key feature in this theorems is that the contractivity condition on the nonlinear map is only assumed to hold on elements that are comparable in the partial order.
As an application the authors obtained a theorem on the existence of a unique solution for periodic boundary problems relative to ordinary differential equations. Similar applications for a mixed monotone mapping were given by Gnana Bhaskar and Lakshmikantham [3]. Further improvements of the above results were found by Petruşel and Rus [4] and Jachymski [5].
In this paper we extend fixed point theorems established by Ran and Reurings and Nieto and RodríguezLópez to \(CJM\) contractions on preordered metric spaces, where a preordered binary relation is weaker than a partial order.
2 Definitions
We have to introduce many types of contractions.
Definition 2.1
 1.T is said to be a (usual) contraction (C, for short) [6], if there exists \(\lambda\in[0,1)\) such that$$d(Tx,Ty)\leqslant\lambda d(x,y) \quad \mbox{for any }\quad x,y\in X. $$
 2.T is said to be a Browder contraction (\(Bro\), for short) [7], if there exists a function φ from \((0,\infty)\) into itself satisfying the following:
 (a)
φ is nondecreasing and right continuous,
 (b)
\(\varphi(t)< t\) for any \(t\in(0,\infty)\),
 (c)
\(d(Tx,Ty)\leqslant\varphi(d(x,y))\) for any \(x,y\in X\).
 (a)
 3.T is said to be a BoydWong contraction (\(BoWo\), for short) [8], if there exists a function φ from \((0,\infty)\) into itself satisfying the following:
 (a)
φ is upper semicontinuous from the right, i.e., \(\lambda_{i}\downarrow\lambda\geqslant0\Rightarrow\limsup_{i\rightarrow\infty}\varphi(\lambda_{i})\leqslant\varphi (\lambda)\),
 (b)
\(\varphi(t)< t\) for any \(t\in(0,\infty)\),
 (c)
\(d(Tx,Ty)\leqslant\varphi(d(x,y))\) for any \(x,y\in X\).
 (a)
 4.T is said to be an Ri contraction (Ri, for short) [9], if there exists a function φ from \([0,\infty)\) into itself satisfying the following:
 (a)
\(\limsup_{s\rightarrow t+}\varphi(s)< t\) for any \(t\in(0,\infty)\),
 (b)
\(\varphi(t)< t\) for any \(t\in(0,\infty)\),
 (c)
\(d(Tx,Ty)\leqslant\varphi(d(x,y))\) for any \(x,y\in X\).
 (a)
 5.T is said to be a MeirKeeler contraction (\(MeKe\), for short) [10], if for any \(\varepsilon>0\), there exists \(\delta>0\) such that, for all \(x,y\in X\),$$\varepsilon\leqslant d(x,y)< \varepsilon+\delta \quad\mbox{implies }\quad d(Tx,Ty)< \varepsilon. $$
 6.T is said to be a Matkowski contraction (\(Mat\), for short) [11], if there exists a function φ from \((0,\infty)\) into itself satisfying the following:
 (a)
φ is nondecreasing,
 (b)
\(\lim_{n\rightarrow\infty}\varphi^{n}(t)=0\) for every \(t\in(0,\infty)\),
 (c)
\(d(Tx,Ty)\leqslant\varphi(d(x,y))\) for any \(x,y\in X\).
 (a)
 7.T is said to be a Wardowski contraction (\(War\), for short) [12], if there exists a function \(F:(0,\infty )\rightarrow\mathbb{R}\) satisfying the following:
 (a)
F is strictly increasing,
 (b)for any sequence \(\{\alpha_{n}\}\) of positive numbers,$$\lim_{n\rightarrow\infty}\alpha_{n}=0 \quad\mbox{iff }\quad \lim _{n\rightarrow\infty} F(\alpha_{n})=\infty, $$
 (c)
there exists \(k\in(0,1)\) such that \(\lim_{\alpha \rightarrow0+}\alpha^{k}F(\alpha)=0\),
 (d)for some \(t>0\), if \(Tx\neq Ty\), then$$t+F \bigl(d(Tx,Ty) \bigr)\leqslant F \bigl(d(x,y) \bigr). $$
 (a)
 8.T is said to be a ĆirićJachymskiMatkowski contraction (\(CJM\), for short) [13–15], if the following hold:
 (a)for every \(\varepsilon>0\), there exists \(\delta>0\) such that, for all \(x,y\in X\),$$d(x,y)< \varepsilon+ \delta \quad \mbox{implies } \quad d(Tx,Ty)\leqslant \varepsilon, $$
 (b)
\(x\neq y\) implies \(d(Tx,Ty)< d(x,y)\).
 (a)
Remark 2.2
It is not difficult to prove that conditions (a) and (d) of Definition 2.1(7) and the assumption that F is continuous or upper semicontinuous from the right is sufficient for the existence and uniqueness of fixed point of T if T maps a complete metric space \((X,d)\) into itself (compare [12] Theorem 2.1).
3 Fixed point theorems
In this section we extended fixed point theorems established by Ran and Reurings and Nieto and RodríguezLópez to \(CJM\) contractions on preordered metric spaces, where a preordered binary relation is weaker than a partial order.
Definition 3.1
 (a)
reflexive if \(x \preccurlyeq x\) for all \(x\in X\),
 (b)
transitive if \(x\preccurlyeq z\) for all \(x,y,z\in X\) such that \(x\preccurlyeq y\) and \(y\preccurlyeq z\).
Example 3.2
Definition 3.3
An preordered metric space is a triple \((X,d,\preccurlyeq)\) where \((X,d)\) is a metric space and ≼ is a preorder on X.
One of the most important hypotheses that we shall use in this section is the monotonicity of the involved mappings.
Definition 3.4
Let ≼ be a binary relation on X and \(T:X\rightarrow X\) be a mapping. We say that T is ≼nondecreasing if \(Tx\preccurlyeq Ty\) for all \(x,y\in X\) such that \(x\preccurlyeq y\).
Definition 3.5
Let \((X,d)\) be a metric space, let \(A\subset X\) be a nonempty subset and let ≼ be a binary relation on X. Then the triple \((A,d,\preccurlyeq)\) is said to be nondecreasing regular if for all sequences \(\{x_{n}\}\subset A\) such that \(\{x_{n}\}\rightarrow x\in A\) and \(x_{n}\preccurlyeq x_{n+1}\) for all \(n\in\mathbb{N}\), we have \(x_{n}\preccurlyeq x\) for all \(n\in\mathbb {N}\).
The following result is the extension of Ran and Reurings’ result to \(CJM\) contraction on preordered metric spaces.
Theorem 3.6
 (i)
\((X,d)\) is complete,
 (ii)
T is ≼nondecreasing,
 (iii)
T is continuous,
 (iv)
there exists \(x_{0}\in X\) such that \(x_{0}\preccurlyeq Tx_{0}\),
 (v)for all \(x,y\in X\) with \(x\succcurlyeq y\),
 (a)for every \(\varepsilon>0\), there exists \(\delta >0\) such that,$$d(x,y)< \varepsilon+ \delta \quad \textit{implies } \quad d(Tx,Ty)\leqslant \varepsilon, $$
 (b)
\(x\neq y\) implies \(d(Tx,Ty)< d(x,y)\).
 (a)
Proof
To prove uniqueness, we assume that \(v\in X\) is another fixed point of T such that \(u\neq v\). By hypothesis, there exists \(w\in X\) such that \(u\preccurlyeq w\) and \(v\preccurlyeq w\).
Let \(\{w_{n}=Tw_{n1}\}\) be the Picard sequence of T based on \(w_{0}=w\). As T is ≼nondecreasing, \(v=Tv\preccurlyeq Tw=w_{1}\) and \(u=Tu\preccurlyeq Tw=w_{1}\). By induction, \(v\preccurlyeq w_{n}\) and \(u\preccurlyeq w_{n}\) for all \(n\geqslant0\).
Case 1. If \(v=w_{n_{0}}\) for some \(n_{0}\geqslant0\), then \(v=Tv=Tw_{n_{0}}=w_{n_{0}+1}\) and by induction, \(w_{n}=v\) for all \(n\geqslant n_{0}\), so \(\{w_{n}\}\rightarrow v\).
Case 2. If \(v\prec w_{n}\) for all \(n\geqslant0\), then by \((b)\), \(d(v,w_{n+1})=d(Tv,Tw_{n})< d(v,w_{n})\). Mimicking the previous part of the proof, we get \(\lim_{n\rightarrow\infty}d(v,w_{n})=0\), so \(\{w_{n}\}\rightarrow v\).
Thus \(\{w_{n}\}\rightarrow v\) and \(\{w_{n}\}\rightarrow u\). The uniqueness of the limit concludes that \(u=v\), so T has a unique fixed point. □
Remark 3.7
 (v)for all \(x,y\in X\) with \(x\succcurlyeq y\) ,
 (a)for every \(\varepsilon>0\) , there exists \(\delta >0\) such that$$\max \biggl\{ d(x,y),d(x,Tx),d(y,Ty),\frac{1}{2} \bigl[d(x,Ty)+d(y,Tx) \bigr] \biggr\} < \varepsilon + \delta\quad \mbox{implies }\quad d(Tx,Ty)\leqslant \varepsilon, $$
 (b)\(x\neq y\) implies$$d(Tx,Ty)< \max \biggl\{ d(x,y),d(x,Tx),d(y,Ty),\frac{1}{2} \bigl[d(x,Ty)+d(y,Tx) \bigr] \biggr\} . $$
 (a)
From Theorem 3.6 we get the following corollary.
Corollary 3.8
[1], Theorem 2.1
 (i)
\((X,d)\) is complete,
 (ii)
T is ≼nondecreasing,
 (iii)
T is continuous,
 (iv)
there exists \(x_{0}\in X\) such that \(x_{0}\preccurlyeq Tx_{0}\),
 (v)there exists \(\lambda\in[0,1)\) such that, for all \(x,y\in X\) with \(x\succcurlyeq y\),$$\begin{aligned} d(Tx,Ty)\leqslant\lambda d(x,y). \end{aligned}$$(5)
Observe that the BCP is stronger than Corollary 3.8, which only requires the inequality for comparable points, that is, for all \(x,y\in X\) such that \(x\succcurlyeq y\).
Example 3.9
After the appearance of the Ran and Reurings’ result, Nieto and RodríguezLópez exchanged the continuity of the mapping T with the condition nondecreasing regularity (Definition 3.5). The following result is an extension of Nieto and RodríguezLópez’ theorem to a \(CJM\) contraction on preordered metric spaces.
Theorem 3.10
 (i)
\((X,d)\) is complete,
 (ii)
T is ≼nondecreasing,
 (iii)
\((X,d,\preccurlyeq)\) is nondecreasing regular,
 (iv)
there exists \(x_{0}\in X\) such that \(x_{0}\preccurlyeq Tx_{0}\),
 (v)for all \(x,y\in X\) with \(x\succcurlyeq y\),
 (a)for every \(\varepsilon>0\), there exists \(\delta >0\) such that$$d(x,y)< \varepsilon+ \delta \quad\textit{implies }\quad d(Tx,Ty)\leqslant \varepsilon, $$
 (b)
\(x\neq y\) implies \(d(Tx,Ty)< d(x,y)\).
 (a)
Proof
Corollary 3.11
[2], Theorem 2.2
 (i)
\((X,d)\) is complete,
 (ii)
T is ≼nondecreasing,
 (iii)
\((X,d,\preccurlyeq)\) is nondecreasing regular,
 (iv)
there exists \(x_{0}\in X\) such that \(x_{0}\preccurlyeq Tx_{0}\),
 (v)there exists \(\lambda\in[0,1)\) such that, for all \(x,y\in X\) with \(x\succcurlyeq y\),$$d(Tx,Ty)\leqslant\lambda d(x,y). $$
4 Gmetric spaces
We show the natural extension of RanReurings’ and NietoRodríguezLópez’ results to the setting of Gmetric spaces.
In 2006, Mustafa and Sims [17] introduced a new class of generalized metric spaces which are called Gmetric spaces as a generalization of metric spaces. Subsequently, many fixed point results on such spaces appeared; see [18]. Here, we present the necessary definitions and results, which will be useful for the rest of the paper. However, for more details, we refer to [18].
Definition 4.1
[17]
 (\(G_{1}\)):

\(G(x,y,z)=0\) if \(x=y=z\),
 (\(G_{2}\)):

\(G(x,x,y)>0\) for all \(x,y\in X\) with \(x\neq y\),
 (\(G_{3}\)):

\(G(x,x,y)\leqslant G(x,y,z)\) for all \(x,y,z\in X\) with \(z\neq y\),
 (\(G_{4}\)):

\(G(x,y,z)=G(x,z,y)=G(y,z,x)=\cdots\) (symmetry in all three variables),
 (\(G_{5}\)):

\(G(x,y,z)\leqslant G(x,a,a)+G(a,y,z)\) for all \(x,y,z,a\in X\)
Definition 4.2
[17]
Let \((X,G)\) be a Gmetric space, and let \(\{x_{n}\}\) be a sequence of points of X, therefore, we say that \(\{x_{n}\}\) is Gconvergent to \(x\in X\) if \(\lim_{n,m\rightarrow\infty}G(x,x_{n},x_{m})=0\), that is, for any \(\varepsilon>0\), there exists \(N\in\mathbb{N}\) such that \(G(x,x_{n},x_{m})<\varepsilon\) for all \(n,m\geqslant N\). We call x the limit of the sequence and write \(x_{n}\rightarrow x\) or \(\lim_{n\rightarrow\infty}x_{n}=x\).
Proposition 4.3
[17]
 (1)
\(\{x_{n}\}\) is Gconvergent to x,
 (2)
\(G(x_{n},x_{n},x)\rightarrow0\) as \(n\rightarrow\infty\),
 (3)
\(G(x_{n},x,x)\rightarrow0\) as \(n\rightarrow\infty\),
 (4)
\(G(x_{n},x_{m},x)\rightarrow0\) as \(n,m\rightarrow\infty\).
Definition 4.4
[17]
Let \((X,G)\) be a Gmetric space. A sequence \(\{x_{n}\}\) in X is called a GCauchy sequence if for any \(\varepsilon>0\) there is \(N\in\mathbb {N}\) such that \(G(x_{n},x_{m},x_{k})<\varepsilon\) for all \(n,m,k\geqslant N\), that is, \(G(x_{n},x_{m},x_{k})\rightarrow0\) as \(n,m,k\rightarrow\infty \).
Definition 4.5
[17]
A Gmetric space \((X,G)\) is called Gcomplete if every GCauchy sequence is Gconvergent in \((X,G)\).
Definition 4.6
[19]
Let \((X,G)\) and \((X',G')\) be Gmetric spaces and \(f:(X,G)\rightarrow(X',G')\) be a function, then f is said to be Gcontinuous at a point \(a\in X\) if and only if, given \(\varepsilon>0\), there exists \(\delta >0\) such that \(x,y\in X\) and \(G(a,x,y)<\delta\) implies \(G'(f(a),f(x),f(y))<\varepsilon\). A function f is Gcontinuous at X if and only if it is Gcontinuous at all \(a\in X\).
Definition 4.7
[17]
A Gmetric space \((X,G)\) is said to be symmetric if \(G(x,y,y)=G(y,x,x)\) for all \(x,y\in X\).
In the symmetric case, many fixed point theorems on Gmetric spaces are particular cases of existing fixed point theorems in metric spaces. Also in the nonsymmetry case (such spaces have a quasimetric structure), many fixed point theorems follows directly from existing fixed point theorems on metric spaces. A key role in this case is played by the following theorem (see [20, 21]).
Theorem 4.8
 (1)
\((X,\delta_{G})\) is a metric space,
 (2)
\(\{x_{n}\}\subset X\) is Gconvergent to \(x\in X\) if and only if \(\{x_{n}\}\) is convergent to x in \((X,\delta_{G})\),
 (3)
\(\{x_{n}\}\subset X\) is GCauchy if and only if \(\{ x_{n}\}\) is Cauchy in \((X,\delta_{G})\),
 (4)
\((X,G)\) is Gcomplete if and only if \((X,\delta _{G})\) is complete.
Definition 4.9
An preordered Gmetric space is a triple \((X,G,\preccurlyeq)\) where \((X,G)\) is a Gmetric space and ≼ is a preordered on X.
The following result can be considered as the natural extension of Ran and Reurings’ result to \(CJM\) contractions on preordered Gmetric spaces.
Theorem 4.10
 (i)
\((X,G)\) is Gcomplete,
 (ii)
T is ≼nondecreasing,
 (iii)
T is Gcontinuous,
 (iv)
there exists \(x_{0}\in X\) such that \(x_{0}\preccurlyeq Tx_{0}\),
 (v)for all \(x,y,z\in X\) with \(x\succcurlyeq y\succcurlyeq z\),
 (a)for every \(\varepsilon>0\), there exists \(\delta >0\) such that$$G(x,y,z)< \varepsilon+ \delta \quad \textit{implies }\quad G(Tx,Ty,Tz)\leqslant \varepsilon, $$
 (b)
\(G(x,y,z)>0\) implies \(G(Tx,Ty,Tz)< G(x,y,z)\).
 (a)
Proof
Corollary 4.11
[18], Theorem 5.2.1
 (i)
\((X,G)\) is Gcomplete,
 (ii)
T is ≼nondecreasing,
 (iii)
T is Gcontinuous,
 (iv)
there exists \(x_{0}\in X\) such that \(x_{0}\preccurlyeq Tx_{0}\),
 (v)there exists \(\lambda\in[0,1)\) such that$$\begin{aligned} G(Tx,Ty,Ty)\leqslant\lambda G(x,y,y) \quad \textit{for all }\quad x,y \in X \textit{ with } x\succcurlyeq y. \end{aligned}$$(6)
Definition 4.12
Let \((X,G)\) be a Gmetric space, let \(A\subset X\) be a nonempty subset and let ≼ be a binary relation on X. Then the triple \((A,G,\preccurlyeq)\) is said to be nondecreasing regular if for all sequences \(\{x_{n}\}\subset A\) such that \(\{x_{n}\}\rightarrow x\in A\) and \(x_{n}\preccurlyeq x_{n+1}\) for all \(n\in\mathbb{N}\), we have \(x_{n}\preccurlyeq x\) for all \(n\in \mathbb{N}\).
The following result can be considered as the natural extension of Nieto and RodríguezLópez’ result to \(CJM\) contractions on preordered Gmetric spaces.
Theorem 4.13
 (i)
\((X,G)\) is Gcomplete,
 (ii)
T is ≼nondecreasing,
 (iii)
\((X,G,\preccurlyeq)\) is nondecreasing regular,
 (iv)
there exists \(x_{0}\in X\) such that \(x_{0}\preccurlyeq Tx_{0}\),
 (v)for all \(x,y,z\in X\) with \(x\succcurlyeq y\succcurlyeq z\),
 (a)for every \(\varepsilon>0\), there exists \(\delta >0\) such that$$G(x,y,z)< \varepsilon+ \delta \quad\textit{implies }\quad G(Tx,Ty,Tz)\leqslant \varepsilon, $$
 (b)
\(G(x,y,z)>0\) implies \(G(Tx,Ty,Tz)< G(x,y,z)\).
 (a)
Corollary 4.14
[18], Theorem 5.2.2
 (i)
\((X,G)\) is Gcomplete,
 (ii)
T is ≼nondecreasing,
 (iii)
\((X,G,\preccurlyeq)\) is nondecreasing regular,
 (iv)
there exists \(x_{0}\in X\) such that \(x_{0}\preccurlyeq Tx_{0}\),
 (v)there exists \(\lambda\in[0,1)\) such that, for all \(x,y\in X\) with \(x\succcurlyeq y\),$$G(Tx,Ty,Ty)\leqslant\lambda G(x,y,y). $$
Declarations
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Authors’ Affiliations
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