In this section, we consider the family \(\mathcal{R}\) of Rfunctions introduced by Roldán López de Hierro and Shahzad in [14]. Precisely, a function \(\eta: [0,+\infty[\, \times[0,+\infty[\, \to\mathbb {R}\) is called Rfunction if the following conditions hold:
 (\(\eta_{1}\)):

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 \(n \in\mathbb{N}\).
Now, we use Rfunctions to define a new class of contractions. Let \((Z,d)\) be a metric space. Denote by Λ the family of lower semicontinuous functions \(\lambda:Z \to[0,+\infty[\). Let \(h : Z \to Z\) be a selfmapping and \(\lambda\in\Lambda\). In the sequel, we will use the following notation
$$D(u,v;\lambda): =d(u,v)+\lambda(u)+\lambda(v)\quad\mbox{for all } u,v \in Z. $$
Now, we define the new family of contractions.
Definition 3.1
Let \((Z,d)\) be a metric space and let \(h : Z \to Z\) be a mapping. The mapping h is a Rλcontraction if there exist an Rfunction \(\eta: [0,+\infty[\, \times[0,+\infty[\, \to\mathbb{R}\) and a function \(\lambda\in\Lambda\) such that
$$ \eta \bigl(D( hu, hv; \lambda),D( u, v;\lambda) \bigr)>0 $$
(3)
for all \(u,v \in Z\) with \(D( u, v;\lambda)>0\).
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
Let
\((Z,d)\)
be a complete metric space and let
\(h : Z \to Z\)
be a
Rλcontraction. Assume that, at least, one of the following conditions holds:

(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
We fix arbitrarily a point \(z_{0}\) of Z and we consider the Picard sequence \(\{z_{i}\}\) of h starting at \(z_{0}\), that is, the sequence defined by \(z_{i}= hz_{i1}\) for all \(i \in\mathbb{N}\). If for some \(j \in\mathbb{N} \cup\{0\}\) we have \(z_{j +1} = z_{j}\), then \(z_{j}\) is evidently a fixed point of h. Also, we claim that \(\lambda(z_{j})=0\). First, from \(z_{j}=z_{j+1}\), we deduce that \(z_{i}=z_{j}\) for all \(i \in \mathbb{N}\cup\{0\}\) with \(i \geq j\). Assume \(\lambda(z_{j})> 0\) and let \(t_{i} := D(z_{j+i},z_{j+i+1};\lambda)\), that is, a positive real number for all \(i \in\mathbb{N}\). Since h by hypothesis is an Rλcontraction, we get
$$\eta(t_{i+1},t_{i})= \eta \bigl(D(hz_{j+i},hz_{j+i+1}; \lambda ),D(z_{j+i},z_{j+i+1};\lambda) \bigr)>0 \quad\mbox{for all } i \in\mathbb{N}. $$
By property (\(\eta_{1}\)) of the function η it follows that \(\lambda (z_{j})=\lambda(z_{j+i}) \to0\) as \(i \to+ \infty\) and so \(\lambda (z_{j})=0\) and hence the conclusion follows if \(z_{j +1} = z_{j}\) for some \(j \in\mathbb{N}\cup\{0\}\). Therefore, we can suppose that \(z_{i1} \neq z_{i}\) for all \(i \in\mathbb{N}\).
We shall divide the proof in three parts. First, we show that
$$ \lim_{i\to+\infty} d(z_{i1},z_{i})=0 \quad \mbox{and}\quad\lim_{i\to+\infty} \lambda(z_{i})=0. $$
(4)
From \(z_{i1}\neq z_{i}\) for all \(i\in\mathbb{N}\), we deduce that
$$t_{i1} =D(z_{i1},z_{i};\lambda)>0 \quad \mbox{for all } i\in\mathbb{N}. $$
Thus the sequence \(\{t_{i}\}\subset\, ]0,+ \infty[\). Since h is an Rλcontraction, from (3) with \(u=z_{i}\) and \(v=z_{i+1}\), we get
$$\begin{aligned} \eta( t_{i+1},t_{i}) &= \eta \bigl(D( z_{i+1}, z_{i+2};\lambda), D( z_{i}, z_{i+1}; \lambda) \bigr) \\ &= \eta \bigl(D( hz_{i}, hz_{i+1}; \lambda), D( z_{i}, z_{i+1}; \lambda) \bigr)>0 \end{aligned}$$
for all \(i \in\mathbb{N} \cup\{0\}\). The property \((\eta_{1})\) of the function η allows one to state that \(t_{i} \to0 \) as \(i \to+ \infty\). Consequently, \(d(z_{i1}, z_{i}) \to0\) and \(\lambda(z_{i}) \to0\), that is, (4) holds.
The second part is to show that the sequence \(\{z_{i}\}\) is Cauchy. Let us assume that \(\{z_{i}\}\) is not a Cauchy sequence. Then there exist \(\sigma>0\) and two subsequences \(\{z_{j(k)}\}\) and \(\{z_{i(k)}\}\) of \(\{z_{i}\}\) with \(k\leq j(k) < i(k)\) and
$$d( z_{j(k)}, z_{i(k)1}) \leq\sigma< d( z_{j(k)}, z_{i(k)}) $$
for all \(k \in\mathbb{N}\). The above restrictions and \(\lim_{i \to+ \infty}d(z_{i1},z_{i})=0\) imply
$$\lim_{k \to+\infty} d( z_{j(k)}, z_{i(k)}) = \lim _{k \to+\infty} d( z_{j(k)1}, z_{i(k)1}) = \sigma. $$
Since \(\lambda(z_{i}) \to0\) as \(i \to+ \infty\), we get
$$\begin{aligned} \sigma=\lim_{k \to+\infty} D( z_{j(k)}, z_{i(k)}; \lambda)= \lim_{k \to+\infty} D( z_{j(k)1}, z_{i(k)1}; \lambda). \end{aligned}$$
The previous equality allows us to assume \(D( z_{j(k)1}, z_{i(k)1}; \lambda) >0\) for each \(k \in\mathbb{N}\). Now, we consider the sequences \(\{t_{k}\}, \{s_{k}\}\) given by
$$t_{k} :=D( z_{j(k)}, z_{i(k)}; \lambda)\quad \mbox{and} \quad s_{k} :=D( z_{j(k)1}, z_{i(k)1}; \lambda)\quad\mbox{for all } k \in\mathbb{N}. $$
From (3) with \(u=z_{j(k)1}\) and \(v=z_{i(k)1}\), we obtain
$$\begin{aligned} \qquad\eta(t_{k},s_{k}) &= \eta \bigl(D( z_{j(k)}, z_{i(k)}; \lambda) , D( z_{j(k)1}, z_{i(k)1}; \lambda) \bigr) \\ &= \eta \bigl(D(hz_{j(k)1}, hz_{i(k)1}; \lambda),D( z_{j(k)1}, z_{i(k)1}; \lambda) \bigr)>0 \end{aligned}$$
(5)
for all \(k \in\mathbb{N}\). Let \(L=\sigma\); from \(L= \sigma< d( z_{j(k)}, z_{i(k)})\leq D( z_{j(k)}, z_{i(k)}; \lambda)=t_{k} \) and (5), by property \(( \eta _{2})\) of the function η, we obtain \(\sigma= L = 0\), which is a contradiction. Hence \(\{z_{i}\}\) is a Cauchy sequence. As \((Z,d)\) is by hypothesis a complete metric space, there exists \(x \in Z\) such that \(z_{i} \to x\) as \(i \to+ \infty\). The hypothesis that λ is lower semicontinuous implies that
$$ 0 \leq\lambda(x) \leq\liminf_{i \to+ \infty} \lambda(z_{i})=0, $$
that is, \(\lambda(x)=0\). The third part is to prove that x is a fixed point of h. We consider the following three steps.
First step. h is a continuous mapping, that is, condition \((1)\) holds. From \(z_{i+1}=hz_{i} \to hx \), we get \(x=hx\).
Second step. Hypothesis \((2)\) holds. If there exists a subsequence \(\{ z_{i(k)}\}\) of \(\{z_{i}\}\) such that \(hz_{i(k)}=hx\) for all \(k \in \mathbb{N}\), then x is a fixed point of h. If this does not happen, then we can assume that \(z_{i} \neq x\) and \(hz_{i} \neq hx\) for all \(i \in \mathbb{N}\). Now, consider the sequences
$$t_{i} := D(hz_{i}, hx; \lambda)\quad\mbox{and}\quad s_{i}:= D(z_{i},x; \lambda) $$
for all \(i \in\mathbb{N}\). Such a choice ensures that \(\{t_{i}\}, \{ s_{i}\} \subset\, ]0,+\infty[\). Clearly, by (4) and \(\lambda (x)=0\), \(s_{i} \to0\) and since h is a Rϕcontraction, we have also
$$\begin{aligned} \eta(t_{i},s_{i})= \eta \bigl(D( hz_{i}, hx; \lambda) , D(z_{i},x; \lambda ) \bigr)>0\quad \mbox{for all } i \in\mathbb{N}. \end{aligned}$$
Then, by condition \((2)\), we get \(t_{i} \to0\). This allows one to state that
$$d(z_{i+1},hx)=d(hz_{i},hx) \to0 $$
and hence \(x=hx\).
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.
Finally, let us to verify that x is a unique fixed point of h. Proceeding by contradiction, we suppose that there exists \(z\neq x \) such that \(z=hz\). Let \(t_{i}:=D(z,x; \lambda) >0\) for all \(i \in\mathbb {N}\). Therefore
$$\begin{aligned} \eta(t_{i+1},t_{i})& =\eta \bigl(D(z,x; \lambda),D(z,x; \lambda) \bigr) \\ &= \eta \bigl(D(hz,hx; \lambda),D(z,x; \lambda) \bigr)>0, \end{aligned}$$
for all \(i \in\mathbb{N}\). Then by the property \((\eta_{1})\) of the function η, we obtain \(t_{i} \to0\), which contradicts the fact that \(d(z,x) \neq0\). Therefore \(z=x\) and so h has a unique fixed point. □
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
Let
\((Z, d)\)
be a complete metric space and
\(h : Z \rightarrow Z\)
be a mapping. Suppose that there exists a function
\(\lambda\in\Lambda\)
such that
$$D(hu, hv;\lambda) \leq\psi \bigl(D(u, v;\lambda) \bigr)D(u, v;\lambda) \quad \textit{for all } u, v \in Z \textit{ with } D(u, v;\lambda)>0 , $$
where
\(\psi: [0, +\infty[\, \to[0, 1[\)
is a function such that
\(\lim_{i \to+ \infty} \psi(t_{i}) = 1\)
implies
\(\lim_{i \to+ \infty} t_{i} = 0\), for all
\(\{t_{i}\} \subset[0, +\infty[\). Then
h
has a unique fixed point
\(x \in Z\)
such that
\(\lambda(x)=0\)
and, for any choice of the initial point
\(z_{0} \in Z\), the sequence
\(\{z_{i} \}\)
defined by
\(z_{i} = h z_{i1}\)
for each
\(i \in\mathbb{N}\)
converges to the point
x.
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
Let
\((Z, d)\)
be a complete metric space and
\(h : Z \to Z\)
be a mapping. Suppose that there exists a function
\(\lambda\in\Lambda\)
such that
$$D(h u, h v;\lambda) \leq\psi \bigl(D(u, v;\lambda) \bigr)D(u, v;\lambda) \quad \textit{for all } u, v \in Z \textit{ with } D(u, v;\lambda)>0, $$
where
\(\psi: [0, +\infty[\, \to[0, 1[\)
is a function such that
\(\limsup_{t \to r^{+}} \psi(t) < 1\), for all
\(r > 0\). Then
h
has a unique fixed point
\(x \in Z\)
such that
\(\lambda(x)=0\)
and, for any choice of the initial point
\(z_{0} \in Z\), the sequence
\(\{z_{i} \}\)
defined by
\(z_{i} = h z_{i1}\)
for each
\(i \in\mathbb{N}\)
converges to the point
x.
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
Let
\((Z, d)\)
be a complete metric space and
\(h : Z \to Z\)
be a mapping. Suppose that there exists a function
\(\lambda\in\Lambda\)
such that
$$\psi \bigl(D(h u, h v;\lambda) \bigr) \leq D(u, v;\lambda) \quad\textit{for all } u, v \in Z \textit{ with } D(u, v;\lambda)>0, $$
where
\(\psi: [0, +\infty[\, \to[0, 1[\)
is a right continuous function such that
\(\psi(t) >t\), for all
\(t > 0\). Then
h
has a unique fixed point
\(x \in Z\)
such that
\(\lambda(x)=0\)
and, for any choice of the initial point
\(z_{0} \in Z\), the sequence
\(\{z_{i} \}\)
defined by
\(z_{i} = h z_{i1}\)
for each
\(i \in\mathbb{N}\)
converges to the point
x.
From the previous corollary, we deduce the following result of integral type.
Corollary 3.4
Let
\((Z, d)\)
be a complete metric space and
\(h : Z \to Z\)
be a mapping. Suppose that there exists a function
\(\lambda\in\Lambda\)
such that
$$ \int_{0}^{D(hu, hv;\lambda)} \xi(\tau)\,d\tau \leq D(u, v;\lambda) \quad\textit{for all } u, v \in Z \textit{ with } D(u, v;\lambda)>0, $$
(6)
where
\(\xi:[0, +\infty[\, \to[0, +\infty[\)
is a function such that
\(\int _{0}^{t} \xi(\tau)\,d\tau\)
exists and
\(\int_{0}^{t} \xi(\tau)\,d\tau>t\), for every
\(t > 0\). Then
h
has a unique fixed point
\(x \in Z\)
such that
\(\lambda (x)=0\)
and, for any choice of the initial point
\(z_{0} \in Z\), the sequence
\(\{z_{i} \}\)
defined by
\(z_{i} = h z_{i1}\)
for each
\(i \in\mathbb{N}\)
converges to the point
x.
Example 3.1
Let \(Z =[0,\frac{15}{8}] \cup\{2\}\) endowed with the usual metric \(d(u, v)= uv\) for all \(u,v \in Z\). Obviously, \((Z, d)\) is a complete metric space. Consider the function \(h: Z \to Z\) defined by
$$h u = \left \{ \textstyle\begin{array}{@{}l@{\quad}l} \frac{u}{2} & \mbox{if }u \in[0,\frac{15}{8}],\\ \frac{3}{2} & \mbox{if }u =2. \end{array}\displaystyle \right . $$
Clearly, h satisfies condition (6) with respect to the function \(\xi:[0, +\infty[\, \to[0, +\infty[\) given by
$$\xi(t)= 1 +\frac{1}{(t+1)^{2}}\quad \mbox{for all } t \in[0,+\infty[ $$
and the lower semicontinuous function \(\lambda: Z \to[0,+ \infty[\) defined by \(\lambda(u)=u\) for all \(u \in Z\). Indeed, if \(u \leq v\) and \(u,v \in[0,\frac{15}{8}]\), then
$$\int_{0}^{D(hu, hv;\lambda)} \xi(\tau)\,d\tau= \frac{v+2}{v+1}v \leq2v = D(u, v;\lambda). $$
If \(u \in[0,\frac{15}{8}]\) and \(v=2\), or \(u=v=2\), then
$$\int_{0}^{D(hu, h2;\lambda)} \xi(\tau)\,d\tau= \frac{3+2}{3+1}3 \leq4 = D(u, 2;\lambda). $$
Since all the conditions of Corollary 3.4 are satisfied, the mapping T has a unique fixed point \(x=0\) in Z. Clearly, \(\lambda(x)=0\).
From \(d(h0,h2)= 3/2\) and \(d(0,2)=2\), we deduce that
$$\int_{0}^{d(h0, h2)} \xi(\tau)\,d\tau= \frac{21}{10} \geq2 = d(0, 2). $$
Thus h is not a Rcontraction with respect to the Rfunction \(\eta: [0,+\infty[\, \times[0,+\infty[\, \to\mathbb{R}\) defined by
$$\eta(t,s)= s  \int_{0}^{t} \xi(\tau)\,d\tau, \quad\mbox{for all } t,s \in [0,+\infty[. $$
It follows that Theorem 27 of [14] cannot be used to deduce that h has a fixed point with respect to this Rfunction. The previous consideration also shows that the role of the function λ is decisive in enlarging the class of selfmappings satisfying condition (6) and hence condition (3).