 Research
 Open Access
Some fixed point results via Rfunctions
 Antonella Nastasi^{1},
 Pasquale Vetro^{1} and
 Stojan Radenović^{2}Email author
https://doi.org/10.1186/s136630160572x
© Nastasi et al. 2016
 Received: 9 February 2016
 Accepted: 20 July 2016
 Published: 29 July 2016
Abstract
We establish existence and uniqueness of fixed points for a new class of mappings, by using Rfunctions and lower semicontinuous functions in the setting of metric spaces. As consequences of this results, we obtain several known fixed point results, in metric and partial metric spaces. An example is given to support the new theory. A homotopy result for operators on a set endowed with a metric is given as application.
Keywords
 Rfunction
 Rλcontraction
 fixed point
 metric space
 partial metric space
MSC
 47H10
 54H25
1 Introduction
Metric fixed point theory is a fundamental topic, which gives basic methods and notions for establish practical problems in mathematics and the other sciences. As an example, we consider the existence of solutions of mathematical problems reducible to equivalent fixed point problems. Thus, we recall that Banach contraction principle [1] is at the foundation of this theory. However, the potentiality of fixed point approaches attracted many scientists and hence there is a wide literature available for interested reader; see for instance [2–7]. We give some details on the notions and ideas used in this study.
First, the notion of partial metric space was introduced in 1994 by Matthews [8] as a part of the study of denotational semantics of data for networks. Clearly, this setting is a generalization of the classical concept of metric space. Also, some authors discussed the existence of several connections between partial metrics and topological aspects of domain theory; see for instance [9–12].
Second, the notion of \(\mathcal{Z}\)contraction was introduced in 2014 by Khojasteh et al. [13]. This concept is a new type of nonlinear contraction defined by using a specific function, called simulation function. Consequently, they proved the existence and uniqueness of fixed points for \(\mathcal{Z}\)contraction mappings (see [13], Theorem 2.8). The notion of Rcontraction was introduced in 2015 by Roldán López de Hierro and Shahzad [14]. Also this notion is a new type of nonlinear contraction defined by using a specific function called Rfunction. Naturally, they proved the existence and uniqueness of fixed points for Rcontraction mappings (see [14], Theorem 27). We point out that the advantage of these methods is in providing a unifying point of view for several fixed point problems; see recent results in [15–17].
Finally, Samet et al. [18], and Vetro and Vetro [19] discussed fixed point results, by using semicontinuous functions in metric spaces, that generalize and improve many existing fixed point theorems in the literature. As an application of presented results, the authors gave some theorems in the setting of partial metric spaces. In this paper, we use the ideas in [18, 19] and the notion of Rfunction to establishing the existence and uniqueness of fixed points that belong to the zero set of a certain function. As consequences of this study, we deduce several related fixed point results, in metric and partial metric spaces. Also, an example is given to support the new theory. As application, a homotopy result for operators on a set endowed with a metric is given.
2 Preliminaries
We will start with a brief recollection of basic notions and results in partial metric spaces that can be found in [8, 11, 20, 21].
 (p_{1}):

\(u = v \Leftrightarrow p(u, u) = p(u, v) = p(v, v)\);
 (p_{2}):

\(p(u, u) \leq p(u, v)\);
 (p_{3}):

\(p(u, v) = p(v, u)\);
 (p_{4}):

\(p(u, v) \leq p(u, w)+ p(w, v)p(w, w)\).
A partial metric space is a pair \((Z, p)\), where Z is a nonempty set and p is a partial metric on Z.
Every partial metric \(p : Z \times Z \to[0,+\infty[\) generates a \(T_{0}\) topology \(\tau_{p}\) on Z, which has as a base the family of open pballs \(\{U_{p}(u, \rho): u \in Z, \rho > 0\}\), where \(U_{p}(u, \rho) = \{v \in Z: p(u, v) < p(u, u)+\rho\}\) for all \(u \in Z\) and \(\rho> 0\).
 (i)
\(\{u_{j}\}\) converges to a point \(u \in Z\) if and only if \(p(u, u) = \lim_{j \to+\infty} p(u, u_{j})\);
 (ii)
\(\{u_{j}\}\) is called a Cauchy sequence if there exists \(\lim_{i,j \to+\infty}p(u_{i}, u_{j})\) (and it is finite);
 (iii)
\((Z, p)\) is said to be complete if every Cauchy sequence \(\{u_{j}\}\) in Z converges, with respect to \(\tau_{p}\), to a point \(u \in Z\) such that \(p(u, u) = \lim_{i,j \to+\infty}p(u_{i}, u_{j})\).
The following lemma shows that the function \(\lambda:Z \to[0,+\infty [\) defined by \(\lambda(u)=p(u,u)\) for all \(u \in Z\) is continuous in \((Z,d^{p})\).
Lemma 2.1
Let \((Z, p)\) be a partial metric space and let \(\lambda:Z \to [0,+\infty[\) be defined by \(\lambda(u)=p(u,u)\) for all \(u \in Z\). Then the function λ is continuous in the metric space \((Z,d^{p})\).
Proof
The following lemma correlates the Cauchy sequences of the spaces \((Z,p)\) and \((Z,d^{p})\).
Lemma 2.2
 (1)
\(\{u_{j}\}\) is a Cauchy sequence in \((Z, p)\) if and only if it is a Cauchy sequence in the metric space \((Z, d^{p})\);
 (2)
a partial metric space \((Z, p)\) is complete if and only if the metric space \((Z, d^{p})\) is complete.
3 New fixed point theorems in complete metric spaces
 (\(\eta_{1}\)):

for each sequence \(\{t_{n}\} \subset\,]0,+\infty[\) such that \(\eta(t_{n+1}, t_{n}) > 0\) for all \(n \in\mathbb{N}\), we have \(\lim_{n \to+ \infty}t_{n} = 0\);
 (\(\eta_{2}\)):

for every two sequences \(\{t_{n}\}, \{s_{n}\} \subset\, ]0,+\infty[\) such that \(\lim_{n \to+\infty}t_{n} = \lim_{n \to+\infty }s_{n} = L \geq0\), then \(L = 0\) whenever \(L < t_{n}\) and \(\eta(t_{n}, s_{n}) > 0\) for all \(n \in\mathbb{N}\).
Definition 3.1
In the following theorem, we establish a result of existence and uniqueness of a fixed point for Rλcontractions that belong to \(\{x \in Z: \lambda(x)=0\}\).
Theorem 3.1
 (1)
h is continuous;
 (2)
for every two sequences \(\{t_{i}\}, \{s_{i}\} \subset\, ]0,+\infty[\) such that \(\lim_{i \to+ \infty}s_{i} = 0\) and \(\eta(t_{i}, s_{i}) > 0\) for all \(i \in\mathbb{N}\), then \(\lim_{i \to+ \infty}t_{i} = 0\);
 (3)
\(\eta(t,s) \leq st\) for all \(t,s \in]0,+\infty[\).
Then h has a unique fixed point \(x \in Z\) such that \(\lambda(x)=0\) and, for any choice of the starting point \(z_{0} \in Z\), the sequence \(\{z_{n}\}\) defined by \(z_{n} = h z_{n1}\) for each \(n \in\mathbb{N}\) converges to the point x.
Proof
First step. h is a continuous mapping, that is, condition \((1)\) holds. From \(z_{i+1}=hz_{i} \to hx \), we get \(x=hx\).
Third step. Hypothesis (3) holds, that is, \(\eta(t,s) \leq st\) for all \(t,s \in\, ]0, +\infty[\). Since (3) ensures that condition (2) holds, we conclude that x is a fixed point of h.
Now, we present some particular results of fixed point in metric spaces, by choosing an appropriate Rfunction. The first corollary is a generalization of Geraghty’s fixed point theorem [22] and it is obtained by taking in Theorem 3.1 as Rfunction \(\eta(t,s) = \psi (s) s t\) for all \(t,s \in [0,+\infty[\), where ψ is endowed with a suitable property.
Corollary 3.1
Remark 3.1
From Corollary 3.1, we obtain Geraghty fixed point theorem [22], if the function \(\lambda\in\Lambda\) is defined by \(\lambda(u)=0\) for all \(u \in Z\). Clearly, the Geraghty result is a generalization of Banach’s contraction principle.
In the following corollary we give a result inspired by wellknown results in [4, 23, 24]. It is obtained by taking in Theorem 3.1 as Rfunction \(\eta(t,s) = \psi(s) s t\) for all \(t,s \in [0,+\infty[\), where ψ is endowed with a suitable property.
Corollary 3.2
If in Theorem 3.1 we consider as Rfunction \(\eta(t,s) = s \psi (t)\) for all \(t,s \in[0,+\infty[\), where ψ is a right continuous function, then we deduce the following corollary.
Corollary 3.3
From the previous corollary, we deduce the following result of integral type.
Corollary 3.4
Example 3.1
Since all the conditions of Corollary 3.4 are satisfied, the mapping T has a unique fixed point \(x=0\) in Z. Clearly, \(\lambda(x)=0\).
4 Fixed points in partial metric spaces
In this section, from our Theorem 3.1, we deduce easily various fixed point theorems on partial metric spaces including the Matthews fixed point theorem.
Theorem 4.1
 (j)
h is continuous with respect to metric \(d^{p}\);
 (jj)
for every two sequences \(\{t_{i}\}, \{s_{i}\} \subset\, ]0,+\infty[\) such that \(\lim_{i \to+ \infty}s_{i} = 0\) and \(\eta(t_{i}, s_{i}) > 0\) for all \(i \in\mathbb{N}\), then \(\lim_{i \to+ \infty}t_{i} = 0\);
 (jjj)
\(\eta(t,s) \leq st\) for all \(t,s \in\, ]0,+\infty[\).
Proof
From Theorem 4.1 if consider as Rfunction \(\eta(t,s) = k s t\) for all \(t,s \in[0,+\infty[\) with \(k \in[0,1[\), we obtain the Matthews fixed point theorem.
Corollary 4.1
From Theorem 4.1 if consider as Rfunction \(\eta(t,s) = \psi(s)\,s t\) for all \(t,s \in[0,+\infty[\), then we obtain a result of Geraghty type in partial metric spaces.
Corollary 4.2
Other known results of fixed point in the setting of partial metric spaces we can get considering suitable simulation functions.
5 An application to homotopy
Theorem 5.1
 (i)
\(u \neq Q(u, s)\) for each \(u \in V \setminus U\) and all \(s \in[0, 1]\);
 (ii)there exists \(\rho\in\Gamma\) such that, for each \(s \in[0, 1]\) and all \(u, v \in V\), we have$$ d \bigl(Q(u,s),Q(v,s) \bigr) \leq\rho \bigl(d(u,v) \bigr); $$(10)
 (iii)there exists a continuous function \(f: [0, 1] \to\mathbb {R}\) such thatfor all \(t, s \in[0, 1]\) and every \(u \in V\).$$ d \bigl(Q(u, t), Q(u, s) \bigr) \leq\biglf(t)f(s)\bigr $$
Proof
This ensures that \(Q(\cdot, s)\) has a fixed point in \(\overline{B}(u_{0}, \sigma)\) and hence in U, since all hypotheses of Theorem 3.1 are satisfied. So \(]s_{0} \varepsilon, s_{0} + \varepsilon[ \subset A \) and thus we see that A is an open subset of \([0, 1]\). □
6 Conclusions
Fixed point theory in various metric settings is largely studied as a useful tool for solving problems arising in mathematics and the other sciences. Here, we proved existence and uniqueness of fixed point by using the notion of an Rfunction in metric and partial metric spaces. This kind of result is helpful to cover existing theorems in the literature from a unifying point of view. An homotopy result for certain operators supports the new theory.
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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