Open Access

Some generalizations for \((\alpha - \psi,\phi )\)-contractions in b-metric-like spaces and an application

Fixed Point Theory and Applications20172017:26

https://doi.org/10.1186/s13663-017-0620-1

Received: 15 August 2017

Accepted: 2 November 2017

Published: 6 December 2017

Abstract

In this paper, we introduce a new class of \(\alpha_{qs^{p}}\)-admissible mappings and provide some fixed point theorems involving this class of mappings satisfying some new conditions of contractivity in the setting of b-metric-like spaces. Our results extend, unify, and generalize classical and recent fixed point results for contractive mappings.

Keywords

\(\alpha_{qs^{p}}\)-admissible mappings \((\alpha_{qs^{p}} - \psi,\phi )\) contractive mapping b-metric-like spacefixed point

MSC

47H1054H25

1 Introduction

In the past years extensions of a metric fixed point theory to generalized structures have received much attention. Also in these structures the concepts of fixed point theorems and contractions have appeared with a remarkable influence on applications in the theory of differential and integral equations, and giving appropriate mathematical models for solving a variety of applied problems in the mathematical sciences and engineering. Some generalizations are b-metric spaces introduced by Bakhtin [1] (and later extensively used by Czerwik [2]), partial metric spaces by Matthews [3], b-partial metric spaces by Shukla [4], metric-like spaces by Harandi [5], and b-metric-like spaces by Alghmandi et al. [6]. Later, Hussain [7] discussed the topological structure of b-metric-like spaces.

Also these generalizations have been associated with new and generalized classes of contractive mappings. In this direction, Samet et al. [8] introduced the concept of α-admissible, α-contractive, and \(\alpha - \psi\)-contractive mappings, further extended to the \((\alpha,\beta ) \)-contractive mappings. Many papers dealing with these notions have been considered to prove fixed point results (for example, see [823]).

In this paper, working in this direction, we introduce the concept of an \(\alpha_{qs^{p}} \)-admissible mapping and provide some fixed point results involving \(\alpha_{qs^{p}} - \lambda\) contractions and generalized \((\alpha_{qs^{p}} - \psi,\phi )\) contractive mappings in the larger framework of b-spaces, precisely, in the setting of b-metric-like spaces. The presented theorems improve, extend, generalize, and unify a number of existing results in the literature.

2 Preliminaries

Definition 2.1

([2])

Let X be a nonempty set. A mapping \(d:X \times X \to [ 0,\infty )\) is called a b-metric if the following conditions hold for all \(x,y,z \in X\) and for some \(s \ge 1\):
$$\begin{gathered} d ( x,y ) = 0\quad \mbox{if and only if}\quad x = y; \\ d ( x,y ) = d ( y,x ); \\ d ( x,y ) \le s \bigl[ d ( x,z ) + d ( z,y ) \bigr]. \end{gathered} $$
The pair \(( X,d )\) is called a b-metric space with parameter s.

Definition 2.2

([3])

Let X be a nonempty set. A mapping \(p:X \times X \to [ 0,\infty )\) is called a partial metric if the following conditions hold for all \(x,y,z \in X\) and \(s \ge 1\): \(x = y \Leftrightarrow p ( x,x ) = p ( x,y ) = p ( y,y )\);
$$\begin{gathered} p ( x,x ) \le p ( x,y ); \\ p ( x,y ) = p ( y,x ); \\ p ( x,y ) \le p ( x,z ) + p ( z,y ) - p ( z,z ). \end{gathered} $$
The pair \(( X,p )\) is called a partial metric space.

Definition 2.3

([4])

Let X be a nonempty set. A mapping \(p_{b}:X \times X \to [ 0,\infty )\) is called a partial b-metric if, for any real number \(s \ge 1\) and for all \(x,y,z \in X\):
$$\begin{gathered} x = y\quad \Leftrightarrow\quad p_{b} ( x,x ) = p_{b} ( x,y ) = p_{b} ( y,y ); \\ p_{b} ( x,x ) \le p_{b} ( x,y ); \\ p_{b} ( x,y ) = p_{b} ( y,x ); \\ p_{b} ( x,y ) \le s \bigl[ p_{b} ( x,z ) + p_{b} ( z,y ) \bigr] - p_{b} ( z,z ). \end{gathered} $$
The pair \(( X,p_{b} )\) is called a partial b-metric space.

Definition 2.4

([5])

Let X be a nonempty set. A mapping \(\sigma:X \times X \to [ 0,\infty )\) is called metric-like if the following conditions hold for all \(x,y,z \in X\):
$$\begin{gathered} \sigma ( x,y ) = 0 \quad \mbox{implies}\quad x = y; \\ \sigma ( x,y ) = \sigma ( y,x ); \\ \sigma ( x,y ) \le \sigma ( x,z ) + \sigma ( z,y ). \end{gathered} $$
The pair \(( X,\sigma )\) is called a metric-like space.

Definition 2.5

([6])

Let X be a nonempty set. A mapping \(\sigma_{b}:X \times X \to [ 0,\infty )\) is called b-metric-like if the following conditions hold for all \(x,y,z \in X\) and for some \(s \ge 1\):
$$\begin{gathered} \sigma_{b} ( x,y ) = 0 \quad \mbox{implies}\quad x = y; \\ \sigma_{b} ( x,y ) = \sigma_{b} ( y,x ); \\ \sigma_{b} ( x,y ) \le s \bigl[ \sigma_{b} ( x,z ) + \sigma_{b} ( z,y ) \bigr]. \end{gathered} $$
The pair \(( X,\sigma_{b} )\) is called a b-metric-like space.

In a b-metric-like space \(( X,\sigma_{b} )\), if \(x,y \in X\) and \(\sigma_{b} ( x,y ) = 0\), then \(x = y\), but the converse need not be true, and \(\sigma_{b} ( x,x )\) may be positive for \(x \in X\).

Remark 2.6

The class of b-metric-like spaces is larger than either metric-like spaces or b-metric-spaces, since a b-metric-like space is a metric-like space when \(s = 1\) and since every b-metric space is a b-metric-like space with the same parameter s. However, the converse implications do not hold.

Example 2.7

([6])

Let \(X = R^{ +} \cup \{ 0\}\). Define the function \(\sigma_{b}:X^{2} \to [ 0,\infty )\) by \(\sigma_{b}(x,y) = (x + y)^{2}\) for all \(x,y \in X\). Then \(( X,\sigma_{b} )\) is a b-metric-like space with parameter \(s = 2\).

Example 2.8

([24])

Let \(X = R^{ +} \cup \{ 0\}\). Define the function \(\sigma_{b}:X^{2} \to [ 0,\infty )\) by \(\sigma_{b}(x,y) = (\max \{ x,y \})^{2}\) for all \(x,y \in X\). Then \(( X,\sigma_{b} )\) is a b-metric-like space with parameter \(s = 2\). Clearly, \(( X,\sigma_{b} )\) is not a b-metric or metric-like space.

Definition 2.9

([6])

Let \(( X,\sigma_{b} )\) be a b-metric-like space with parameter s, let \(\{ x_{n} \}\) be any sequence in X, and let \(x \in X\). Then
  1. (a)

    The sequence \(\{ x_{n} \}\) is said to converge to x if \(\lim_{n \to \infty} \sigma_{b} ( x_{n},x ) = \sigma_{b} ( x,x )\);

     
  2. (b)

    The sequence \(\{ x_{n} \}\) is said to be a Cauchy sequence in \(( X,\sigma_{b} )\) if \(\lim_{n,m \to \infty} \sigma_{b} ( x_{n},x_{m} )\) exists and is finite;

     
  3. (c)

    \(( X,\sigma_{b} )\) is said to be a complete b-metric-like space if, for every Cauchy sequence \(\{ x_{n} \}\) in X, there exists \(x \in X\) such that \(\lim_{n,m \to \infty} \sigma_{b} ( x_{n},x_{m} ) = \lim_{n \to \infty} \sigma_{b} ( x_{n},x ) = \sigma_{b} ( x,x )\).

     

The limit of a sequence in a b-metric-like space need not be unique.

Proposition 2.10

([6])

Let \(( X,\sigma_{b} )\) be a b-metric-like space with parameter s, and let \(\{ x_{n} \}\) be any sequence in X with \(x \in X\) such that \(\lim_{n \to \infty} \sigma_{b} ( x_{n},x ) = 0\).Then
  1. (a)

    x is unique,

     
  2. (b)

    \(\sigma_{b} ( x,y ) / s \le \lim_{n \to \infty} \sigma_{b} ( x_{n},y ) \le s\sigma_{b} ( x,y )\) for all \(y \in X\).

     

In 2012, Samet et al. [8] introduced the class of α-admissible mappings.

Definition 2.11

Let X be a nonempty set, \(f:X \to X\), and \(\alpha:X \times X \to R^{ +} \). We say that f is an α-admissible mapping if \(\alpha ( x,y ) \ge 1\) implies that \(\alpha ( fx,fy ) \ge 1\) for all \(x,y \in X\).

Since, in general, a b-metric-like space is not continuous, we quote the following lemmas about the convergence of sequences.

Lemma 2.12

([7])

Let \(( X,\sigma_{b} )\) be a b-metric-like space with parameter \(s \ge 1\), and suppose that \(\{ x_{n} \}\) and \(\{ y_{n} \}\) are \(\sigma_{b}\)-convergent to x and y, respectively. Then we have
$$\begin{aligned} \frac{1}{s^{2}}\sigma_{b} ( x,y ) - \frac{1}{s}\sigma_{b} ( x,x ) - \sigma_{b} ( y,y ) & \le \mathop{\lim \inf}_{n \to \infty} \sigma_{b} ( x_{n},y_{n} ) \\ &\le \mathop{\lim \sup}_{n \to \infty} \sigma_{b} ( x_{n},y_{n} ) \le s\sigma_{b} ( x,x ) + s^{2}\sigma_{b} ( y,y ) + s^{2} \sigma_{b} ( x,y ). \end{aligned} $$
In particular, if \(\sigma_{b} ( x,y ) = 0\), then we have \(\lim_{n \to \infty} \sigma_{b} ( x_{n},y_{n} ) = 0\).
Moreover, for each \(z \in X\), we have
$$\begin{aligned} \frac{1}{s}\sigma_{b} ( x,z ) - \sigma_{b} ( x,x ) &\le \mathop{\lim \inf}_{n \to \infty} \sigma_{b} ( x_{n},z ) \\ &\le \mathop{\lim \sup}_{n \to \infty} \sigma_{b} ( x_{n},z ) \le s\sigma_{b} ( x,z ) + s\sigma_{b} ( x,x ). \end{aligned} $$
In particular, if \(\sigma_{b} ( x,x ) = 0\), then
$$\begin{aligned} \frac{1}{s}\sigma_{b} ( x,z ) &\le\mathop{\lim \inf} _{n \to \infty} \sigma_{b} ( x_{n},z ) \\ &\le \mathop{\lim \sup}_{n \to \infty} \sigma_{b} ( x_{n},z ) \le s\sigma_{b} ( x,z ). \end{aligned} $$

The following result is useful.

Lemma 2.13

Let \(( X,\sigma_{b} )\) be a b-metric-like space with parameter \(s \ge 1\). Then
  1. (a)

    If \(\sigma_{b}(x,y) = 0\), then \(\sigma_{b}(x,x) = \sigma_{b}(y,y) = 0\);

     
  2. (b)
    If \((x_{n})\) is a sequence such that \(\lim_{n \to \infty} \sigma_{b}(x_{n},x_{n + 1}) = 0\), then we have
    $$\lim_{n \to \infty} \sigma_{b}(x_{n},x_{n}) = \lim_{n \to \infty} \sigma_{b}(x_{n + 1},x_{n + 1}) = 0; $$
     
  3. (c)

    If \(x \ne y\), then \(\sigma_{b}(x,y) > 0\).

     

Proof

The proof is obvious. □

Lemma 2.14

Let \(( X,\sigma_{b} )\) be a complete b-metric-like space with parameter \(s \ge 1\), and let \(\{ x_{n} \}\) be a sequence such that
$$ \lim_{n \to \infty} \sigma_{b} ( x_{n},x_{n + 1} ) = 0. $$
(2.1)
If \(\{ x_{n} \}\) is not Cauchy, then there exist \(\varepsilon > 0\) and two subsequences \(\{ x_{m_{k}} \}\) and \(\{ x_{n_{k}} \}\) of \(\{ x_{n} \}\) with \(n _{k} > m _{k} >k\) (positive integers) such that \(\sigma_{b}(x_{m_{k}},x_{n_{k}}) \ge \varepsilon\), \(\sigma_{b}(x_{m_{k}},x_{n_{k} - 1}) < \varepsilon\), \(\varepsilon / s^{2} \le \lim \sup_{k \to \infty} \sigma_{b}(x_{m_{k} - 1},x_{n_{k} - 1}) \le \varepsilon s\), \(\varepsilon / s \le \lim \sup_{k \to \infty} \sigma_{b}(x_{n_{k} - 1},x_{m_{k}}) \le \varepsilon s^{2}\), and \(\varepsilon / s \le \lim \sup_{k \to \infty} \sigma_{b}(x_{m_{k} - 1},x_{n_{k}}) \le \varepsilon s^{2}\).

Proof

If \(\{ x_{n} \}\) is not a \(\sigma_{b}\)-Cauchy sequence, then there exists \(\varepsilon > 0\) for which we can find two subsequences \(\{ x_{m_{k}} \}\) and \(\{ x_{n_{k}} \}\) of \(\{ x_{n} \}\) such that \(n_{k}\) is the smallest index for which
$$ n_{k} > m_{k} > k,\quad \sigma_{b}(x_{m_{k}},x_{n_{k}}) \ge \varepsilon. $$
(2.2)
This means that
$$ \sigma_{b}(x_{m_{k}},x_{n_{k} - 1}) < \varepsilon. $$
(2.3)
From (2.2) and property (c) of Definition 2.4 we have
$$ \begin{aligned}[b] \varepsilon &\le \sigma_{b}(x_{m_{k}},x_{n_{k}}) \le s\sigma_{b}(x_{m_{k}},x_{m_{k} - 1}) + s \sigma_{b}(x_{m_{k} - 1},x_{n_{k}}) \\ &\le s\sigma_{b}(x_{m_{k}},x_{m_{k} - 1}) + s^{2} \sigma_{b}(x_{m_{k} - 1},x_{n_{k} - 1}) + s^{2} \sigma_{b}(x_{n_{k} - 1},x_{n_{k}}). \end{aligned} $$
(2.4)
Taking the upper limit as \(k \to \infty\) in (2.4) and using (2.1), (2.2), and (2.3), we get
$$ \frac{\varepsilon}{s^{2}} \le \mathop{\lim \sup}_{k \to \infty} \sigma_{b}(x_{m_{k} - 1},x_{n_{k} - 1}). $$
(2.5)
By the triangle inequality we have
$$\sigma_{b}(x_{m_{k} - 1},x_{n_{k} - 1}) \le s \sigma_{b}(x_{m_{k} - 1},x_{m_{k}}) + s\sigma_{b}(x_{m_{k}},x_{n_{k} - 1}), $$
so, taking the upper limit as \(k \to \infty\) and using (2.1), we get
$$ \mathop{\lim \sup}_{k \to \infty} \sigma_{b}(x_{m_{k} - 1},x_{n_{k} - 1}) \le \varepsilon s. $$
(2.6)
By (2.5) and (2.6) we have
$$ \frac{\varepsilon}{s^{2}} \le \mathop{\lim \sup}_{k \to \infty} \sigma_{b}(x_{m_{k} - 1},x_{n_{k} - 1}) \le \varepsilon s. $$
(2.7)
Also, we have
$$\varepsilon \le \sigma_{b}(x_{m_{k}},x_{n_{k}}) \le s \sigma_{b}(x_{m_{k}},x_{m_{k} - 1}) + s\sigma_{b}(x_{m_{k} - 1},x_{n_{k}}), $$
and, taking the upper limit as \(k \to \infty\), we get
$$ \frac{\varepsilon}{s} \le \mathop{\lim \sup}_{k \to \infty} \sigma_{b}(x_{m_{k} - 1},x_{n_{k}}). $$
(2.8)
Again
$$\varepsilon \le \sigma_{b}(x_{m_{k}},x_{n_{k}}) \le s \sigma_{b}(x_{m_{k}},x_{n_{k} - 1}) + s\sigma_{b}(x_{n_{k} - 1},x_{n_{k}}). $$
Taking the upper limit as \(k \to \infty\) and using (2.1), we get
$$ \frac{\varepsilon}{s} \le \mathop{\lim \sup}_{k \to \infty} \sigma_{b}(x_{n_{k} - 1},x_{m_{k}}). $$
(2.9)
Since \(\sigma_{b} ( x_{n_{k} - 1},x_{m_{k}} ) \le s\sigma_{b} ( x_{n_{k} - 1},x_{m_{k} - 1} ) + s\sigma_{b} ( x_{m_{k} - 1},x_{m_{k}} )\), from (2.1) and (2.7) we have
$$ \mathop{\lim \sup}_{k \to \infty} \sigma_{b}(x_{n_{k} - 1},x_{m_{k}}) \le s\mathop{\lim \sup}_{k \to \infty} \sigma_{b}(x_{n_{k} - 1},x_{m_{k} - 1}) \le \varepsilon s^{2}. $$
(2.10)
Consequently,
$$ \frac{\varepsilon}{s} \le \mathop{\lim \sup}_{k \to \infty} \sigma_{b}(x_{n_{k} - 1},x_{m_{k}}) \le \varepsilon s^{2}. $$
(2.11)
Also,
$$\sigma_{b} ( x_{m_{k} - 1},x_{n_{k}} ) \le s \sigma_{b} ( x_{m_{k} - 1},x_{n_{k} - 1} ) + s \sigma_{b} ( x_{n_{k} - 1},x_{n_{k}} ). $$
Then from (2.7), (2.8), and (2.1) we have
$$\mathop{\lim \sup}_{k \to \infty} \sigma_{b}(x_{m_{k} - 1},x_{n_{k}}) \le s\mathop{\lim \sup}_{k \to \infty} \sigma_{b}(x_{m_{k} - 1},x_{n_{k} - 1}) \le \varepsilon s^{2}. $$
Consequently,
$$ \frac{\varepsilon}{s} \le \mathop{\lim \sup}_{k \to \infty} \sigma_{b}(x_{m_{k} - 1},x_{n_{k}}) \le \varepsilon s^{2}. $$
(2.12)
This completes proof. □

3 Main results

We begin this section with the following definition.

Definition 3.1

Let \(( X,\sigma_{b} )\) be a b-metric-like space with parameter \(s \ge 1\), let \(\alpha:X \times X \to [ 0,\infty )\) be a function, and let \(q \ge 1\) and \(p \ge 2\) be arbitrary constants. A mapping \(f:X \to X\) is \(\alpha_{qs^{p}} \)-admissible if \(\alpha ( x,y ) \ge qs^{p}\) implies \(\alpha ( fx,fy ) \ge qs^{p}\) for all \(x,y \in X\).

Remark 3.2

  1. (i)

    Taking \(q = 1\) in this definition, we obtain an \(\alpha_{s^{p}} \)-admissible mapping defined in a b-metric-like space or in a b-metric space.

     
  2. (ii)

    Note that, for \(s = 1\), the definition reduces to an \(\alpha_{q} \)-admissible mapping defined in a metric space or in a metric-like space.

     
  3. (iii)

    For \(s = 1\) and \(q = 1\), the definition reduces to the definition of an α-admissible mapping in a metric space [8].

     
  4. (iv)

    The class of \(\alpha_{qs^{p}} \)-admissible mappings is strictly larger, and, more generally, because the constant \(p \ge 2\), it is not restricted to some certain values.

     

We further consider the following properties.

Let \(( X,\sigma_{b} )\) be a complete b-metric-like space with parameter \(s \ge 1\), and let \(\alpha:X \times X \to [ 0,\infty )\) be a function. Then:
(\(H_{qs^{p}}\)): 

If \(\{ x_{n} \}\) is a sequence in X such that \(x_{n} \to x \in X\) as \(n \to \infty\) and \(\alpha ( x_{n},x_{n + 1} ) \ge qs^{p}\), then there exists a subsequence \(\{ x_{n_{k}} \}\) of \(\{ x_{n} \}\) such that \(\alpha ( x_{n_{k}},x ) \ge qs^{p}\) for all \(k \in N\).

(\(U_{qs^{p}}\)): 

For all \(x,y \in \operatorname{Fix} ( f )\), we have \(\alpha ( x,y ) \ge qs^{p}\), where \(\operatorname{Fix} ( f )\) denotes the set of fixed points of f.

Example 3.3

Let \(X = ( 0, + \infty )\). Define \(f:X \to X\) and \(\alpha:X \times X \to [ 0, + \infty )\) by \(fx = \ln x\) for all \(x \in X\), and let
$$\alpha ( x,y ) = \textstyle\begin{cases} 2s^{2},& x \ne y, \\ 0,& x = y \end{cases}\displaystyle \quad \mbox{for any } s \ge 1. $$
Then, f is \(\alpha_{qs^{p}} \)-admissible.

Example 3.4

Let \(X = ( 0, + \infty )\). Define \(f:X \to X\) and \(\alpha:X \times X \to [ 0,\infty )\) by \(fx = 3x\) for all \(x \in X\) and
$$\alpha ( x,y ) = \textstyle\begin{cases} 2,& x \ne y, \\ 0, &x = y \end{cases}\displaystyle \quad \mbox{for all }x,y \in X. $$
Then f is \(\alpha_{qs^{p}} \)-admissible.

Based on the definition of quasi-contraction from Ćirić, we introduce the following definition in the setting of a b-metric-like space.

Definition 3.5

Let \(( X,\sigma_{b} )\) be a complete b-metric-like space with parameter \(s \ge 1\), and let \(f:X \to X\) be a given mapping. We say that f is a generalized \(\alpha_{qs^{p}} - \lambda\)-quasi-contraction if f is an \(\alpha_{qs^{p}} \)-admissible mapping such that
$$ \alpha ( x,y )\sigma_{b} ( fx,fy ) \le \lambda \max \left \{ \begin{matrix} \sigma_{b} ( x,y ),\sigma_{b} ( x,fx ),\sigma_{b} ( y,fy ),\sigma_{b} ( x,fy ), \\ \sigma_{b} ( y,fx ),\sigma_{b} ( x,x ),\sigma_{b} ( y,y ) \end{matrix} \right \} $$
(3.1)
for all \(x,y \in X\) and \(\lambda \in [ 0,1 / 2 )\).

Remark 3.6

If we take \(\alpha ( x,y ) = s^{2}\) (\(p = 2\) and \(q = 1\)), then the definition reduces to the definition of an \(s - \lambda\) quasi-contraction, and if we take \(s = 1\), then the definition reduces to the λ-quasi-contraction in the setting of metric spaces.

Theorem 3.7

Let \(( X,\sigma_{b} )\) be a complete b-metric-like space with parameter \(s \ge 1\), let \(f:X \to X\) be a self-mapping, and let \(\alpha:X \times X \to R^{ +} \) be a given function. Suppose that the following conditions are satisfied:
  1. (i)

    f is an \(\alpha_{qs^{p}} \)-admissible mapping;

     
  2. (ii)

    f is an \(\alpha_{qs^{p}} - \lambda\) contractive mapping;

     
  3. (iii)

    there exists \(x_{0} \in X\) such that \(\alpha ( x_{0},fx_{0} ) \ge qs^{p}\);

     
  4. (iv)

    either f is continuous, or property \(H_{qs^{p}}\) is satisfied.

     

Then f has a fixed point. Moreover, f has a unique fixed point if property \(U_{qs^{p}}\) is satisfied.

Proof

By hypothesis (iii) there exists \(x_{0} \in X\) such that \(\alpha ( x_{0},fx_{0} ) \ge qs^{p}\). We define the sequence \(\{ x_{n} \}\) in X by \(x_{n} = fx_{n - 1}\) for all \(n \in N\). If \(x_{n} = x_{n + 1}\) for some \(n \in N\), then \(u = x_{n}\) is a fixed point for f. Consequently, we suppose that \(x_{n} \ne x_{n + 1}\) (\(\sigma_{b} ( x_{n},x_{n + 1} ) > 0\)) for all \(n \in N\).

Since f is an \(\alpha_{qs^{p}} \)-admissible mapping, we have
$$\begin{gathered} \alpha ( x_{0},x_{1} ) = \alpha ( x_{0},fx_{0} ) \ge qs^{p},\qquad \alpha ( fx_{0},fx_{1} ) = \alpha ( x_{1},x_{2} ) \ge qs,\quad \mbox{and} \\ \alpha ( fx_{1},fx_{2} ) = \alpha ( x_{2},x_{3} ) \ge qs^{p}. \end{gathered} $$
Hence, by induction we get
$$\alpha ( x_{n},x_{n + 1} ) \ge qs^{p}\quad \mbox{for all } n \in {N}. $$
By condition (3.1) we have:
$$ \begin{aligned}[b] &qs^{p}\sigma_{b} ( x_{n},x_{n + 1} ) \\ &\quad = qs^{p}\sigma_{b} ( fx_{n - 1},fx_{n} ) \le \alpha ( x_{n - 1},x_{n} )\sigma_{b} ( fx_{n - 1},fx_{n} ) \\ &\quad \le \lambda \max \left \{ \begin{matrix} \sigma_{b} ( x_{n - 1},x_{n} ),\sigma_{b} ( x_{n - 1},fx_{n - 1} ),\sigma_{b} ( x_{n},fx_{n} ),\sigma_{b} ( x_{n - 1},fx_{n} ), \\ \sigma_{b} ( x_{n},fx_{n - 1} ),\sigma_{b} ( x_{n - 1},x_{n - 1} ),\sigma_{b} ( x_{n},x_{n} ) \end{matrix} \right \} \\ &\quad = \lambda \max \left \{ \begin{matrix} \sigma_{b} ( x_{n - 1},x_{n} ),\sigma_{b} ( x_{n - 1},x_{n} ),\sigma_{b} ( x_{n},x_{n + 1} ),\sigma_{b} ( x_{n - 1},x_{n + 1} ), \\ \sigma_{b} ( x_{n},x_{n} ),\sigma_{b} ( x_{n - 1},x_{n - 1} ),\sigma_{b} ( x_{n},x_{n} ) \end{matrix} \right \} \\ &\quad \le \lambda \max \left \{ \begin{matrix} \sigma_{b} ( x_{n - 1},x_{n} ),\sigma_{b} ( x_{n - 1},x_{n} ),\sigma_{b} ( x_{n},x_{n + 1} ),s [ \sigma_{b} ( x_{n - 1},x_{n} ) + \sigma_{b} ( x_{n},x_{n + 1} ) ], \\ 2s\sigma_{b} ( x_{n},x_{n - 1} ),2s\sigma_{b} ( x_{n - 1},x_{n} ),2s\sigma_{b} ( x_{n},x_{n - 1} ) \end{matrix} \right \} \\ &\quad = \lambda \max \left \{ \begin{matrix} \sigma_{b} ( x_{n - 1},x_{n} ),\sigma_{b} ( x_{n - 1},x_{n} ),\sigma_{b} ( x_{n},x_{n + 1} ),s [ \sigma_{b} ( x_{n - 1},x_{n} ) + \sigma_{b} ( x_{n},x_{n + 1} ) ], \\ 2s\sigma_{b} ( x_{n},x_{n - 1} ) \end{matrix} \right \}. \end{aligned} $$
(3.2)
If \(\sigma_{b} ( x_{n - 1},x_{n} ) < \sigma_{b} ( x_{n},x_{n + 1} )\) for some \(n \in {N}\), then from inequality (3.2) we have \(\sigma_{b} ( x_{n}, x_{n + 1} ) \le 2\lambda / qs^{p - 1}\sigma_{b} ( x_{n},x_{n + 1} )\), a contradiction since \(2\lambda / qs^{p - 1} < 1\).
Hence, for all \(n \in N\), \(\sigma_{b} ( x_{n},x_{n + 1} ) \le \sigma_{b} ( x_{n - 1},x_{n} )\), and also by inequality (3.2) we get
$$ \sigma_{b} ( x_{n},x_{n + 1} ) \le \frac{2\lambda}{qs^{p - 1}}\sigma_{b} ( x_{n - 1},x_{n} ). $$
(3.3)
Similarly, by the contractive condition of theorem we have:
$$ \sigma_{b} ( x_{n - 1},x_{n} ) \le \frac{2\lambda}{qs^{p - 1}}\sigma_{b} ( x_{n - 2},x_{n - 1} ). $$
(3.4)
Generally, from (3.3) and (3.4) we have, for all n,
$$ \sigma_{b} ( x_{n},x_{n + 1} ) \le c \sigma_{b} ( x_{n - 1},x_{n} ) \le\cdots \le c^{n}\sigma_{b} ( x_{0},x_{1} ), $$
(3.5)
where \(0 \le c = 2\lambda / qs^{p - 1} < 1\). Taking limit as \(n \to \infty\) in (3.5), we have
$$ \sigma_{b} ( x_{n},x_{n + 1} ) \to 0. $$
(3.6)
Now we prove that \(\{ x_{n} \}\) is a Cauchy sequence. To do this, let \(m,n > 0\) be such that \(m > n\).
Using Definition 2.4(c), we have
$$\begin{aligned} \sigma_{b} ( x_{n},x_{m} ) &\le s \bigl[ \sigma_{b} ( x_{n},x_{n + 1} ) + \sigma_{b} ( x_{n + 1},x_{m} ) \bigr] \\ &\le s\sigma_{b} ( x_{n},x_{n + 1} ) + s^{2}\sigma_{b} ( x_{n + 1},x_{n + 2} ) + s^{3}\sigma_{b} ( x_{n + 2},x_{n + 3} ) + \cdots \\ &\le sc^{n}\sigma_{b} ( x_{0},x_{1} ) + s^{2}c^{n + 1}\sigma_{b} ( x_{0},x_{1} ) + s^{3}c^{n + 2}\sigma_{b} ( x_{0},x_{1} ) +\cdots \\ &= sc^{n}\sigma_{b} ( x_{0},x_{1} ) \bigl[ 1 + sc + ( sc )^{2} + ( sc )^{3} +\cdots \bigr] \\ &\le \frac{sc^{n}}{1 - sc}\sigma_{b} ( x_{0},x_{1} ). \end{aligned} $$
Taking the limit as \(n,m \to \infty\), we have \(\sigma_{b} ( x_{n},x_{m} ) \to 0\), since \(0 \le cs = 2\lambda s / qs^{p - 1} = 2\lambda / qs^{p - 2} < 1\). Therefore \(\{ x_{n} \}\) is a Cauchy sequence in the complete b-metric-like space \(( X,\sigma_{b} )\). Thus there is some \(u \in X\) such that \(\{ x_{n} \}\) converges to u.
If f is a continuous mapping, then we get:
$$f(u) = f\Bigl(\lim_{n \to \infty} x_{n}\Bigr) = \lim _{n \to \infty} f(x_{n}) = \lim_{n \to \infty} (x_{n + 1}) = u. $$
Thus u is a fixed point of f.

On the other hand, if f is not a continuous function and property \(H_{qs^{p}}\) holds, then there exists a subsequence \(\{ x_{n_{k}} \}\) of \(\{ x_{n} \}\) such that \(\alpha ( x_{n_{k}},u ) \ge qs^{p}\) for all \(k \in {N}\).

Since \(\alpha ( x_{n_{k}},u ) \ge qs^{p}\), applying condition (3.1) with \(x = x_{n_{k}}\) and \(y = u\), we obtain
$$ \begin{aligned}[b] qs^{p}\sigma_{b} ( x_{n_{k} + 1},fu ) &= qs^{p}\sigma_{b} ( fx_{n_{k}},fu ) \le \alpha ( x_{n_{k}},u )\sigma_{b} ( fx_{n_{k}},fu ) \\ &\le \lambda \max \left \{ \begin{matrix} \sigma_{b} ( x_{n_{k}},u ),\sigma_{b} ( x_{n_{k}},fx_{n_{k}} ),\sigma_{b} ( u,fu ),\sigma_{b} ( x_{n_{k}},fu ), \\ \sigma_{b} ( u,fx_{n_{k}} ),\sigma_{b} ( x_{n_{k}},x_{n_{k}} ),\sigma_{b} ( u,u ) \end{matrix} \right \} \\ &= \lambda \max \left \{ \begin{matrix} \sigma_{b} ( x_{n_{k}},u ),\sigma_{b} ( x_{n_{k}},x_{n_{k} + 1} ),\sigma_{b} ( u,fu ), \\ \sigma_{b} ( x_{n_{k}},fu ),\sigma_{b} ( u,x_{n_{k} + 1} ),\sigma_{b} ( x_{n_{k}},x_{n_{k}} ),\sigma_{b} ( u,u ) \end{matrix} \right \}. \end{aligned} $$
(3.7)
Taking the upper limit as \(k \to \infty\) in (3.7) and using (3.6), and Lemmas 2.12 and 2.13, we have
$$ qs^{p - 1}\sigma_{b} ( u,fu ) = qs^{p} \frac{1}{s}\sigma_{b} ( u,fu ) \le 2\lambda s \sigma_{b} ( u,fu ). $$
(3.8)
From (3.8) we get \(\sigma_{b} ( u,fu ) = 0\), which implies that \(fu = u\). Hence u is a fixed point of f.
Further, suppose that u and v are two fixed points of f, where \(fu = u\) and \(fv = v\) for some \(u \ne v\). Since property \(U_{qs^{p}}\) is satisfied, we have \(\alpha ( u,v ) \ge qs^{p}\). Hence, from (3.1) we have
$$ \begin{aligned}[b] qs^{p}\sigma_{b} ( u,v ) &= qs^{p}\sigma_{b} ( fu,fv ) \le \alpha ( u,v ) \sigma_{b} ( fu,fv ) \\ &\le \lambda \max \left \{ \begin{matrix} \sigma_{b} ( u,v ),\sigma_{b} ( u,fu ),\sigma_{b} ( v,fv ),\sigma_{b} ( u,fv ), \\ \sigma_{b} ( v,fu ),\sigma_{b} ( u,u ),\sigma_{b} ( v,v ) \end{matrix} \right \} \\ &= \lambda \max \left \{ \begin{matrix} \sigma_{b} ( u,v ),\sigma_{b} ( u,u ),\sigma_{b} ( v,v ),\sigma_{b} ( u,v ), \\ \sigma_{b} ( v,u ),\sigma_{b} ( u,u ),\sigma_{b} ( v,v ) \end{matrix} \right \} \\ &\le 2\lambda s\sigma_{b} ( u,v ). \end{aligned} $$
(3.9)
So \(\sigma_{b} ( u,v ) = 0\), and since \(0 \le c = 2\lambda / qs^{p - 1} < 1\), we get \(\sigma_{b} ( u,v ) = 0\). Hence the fixed point is unique. □

The following theorem is a version of the Hardy-Rogers result.

Theorem 3.8

Let \(( X,\sigma_{b} )\) be a complete b-metric-like space with parameter \(s \ge 1\), and let \(f:X \to X\) be a given self-mapping. Suppose that there exists a function \(\alpha:X \times X \to [ 0,\infty )\) such that
$$\alpha ( x,y )\sigma_{b} ( fx,fy ) \le \alpha_{1} \sigma_{b} ( x,y ) + \alpha_{2}\sigma_{b} ( x,fx ) + \alpha_{3}\sigma_{b} ( y,fy ) + \alpha_{4} \sigma_{b} ( x,fy ) + \alpha_{5}\sigma_{b} ( y,fx ), $$
for all \(x,y \in X\) and the constants \(a_{i} \ge 0\), \(i = 1,\ldots,5\), where \(a_{1} + a_{2} + a_{3} + a_{4} + a_{5} < 1 / 2\). Assume also that:
  1. (i)

    f is an \(\alpha_{qs^{p}} \)-admissible mapping;

     
  2. (ii)

    there exists \(x_{0} \in X\) such that \(\alpha ( x_{0},fx_{0} ) \ge qs^{p}\);

     
  3. (iii)

    either f is continuous, or property \(H_{qs^{p}}\) is satisfied.

     

Then f has a fixed point. Moreover, f has a unique fixed point if property \(U_{qs^{p}}\) is satisfied.

Proof

This theorem can be considered as a corollary of Theorem 3.7, since, for all \(x,y \in X\), we have
$$\begin{gathered} \alpha_{1}\sigma_{b} ( x,y ) + \alpha_{2}\sigma_{b} ( x,fx ) + \alpha_{3} \sigma_{b} ( y,fy ) + \alpha_{4}\sigma_{b} ( x,fy ) + \alpha_{5}\sigma_{b} ( y,fx ) \\ \quad \le ( \alpha_{1} + \alpha_{2} + \alpha_{3} + \alpha_{4} + \alpha_{5} )\max \bigl\{ \sigma_{b} ( x,y ),\sigma_{b} ( x,fx ),\sigma_{b} ( y,fy ),\sigma_{b} ( x,fy ),\sigma_{b} ( y,fx ) \bigr\} \\ \quad = k\max \bigl\{ \sigma_{b} ( x,y ),\sigma_{b} ( x,fx ),\sigma_{b} ( y,fy ),\sigma_{b} ( x,fy ), \sigma_{b} ( y,fx ) \bigr\} , \end{gathered} $$
where \(0 < k = a_{1} + a_{2} + a_{3} + a_{4} + a_{5} < 1 / 2\). □

Corollary 3.9

Let \(( X,\sigma_{b} )\) be complete b-metric-like space with parameter \(s \ge 1\). If \(f:X \to X\) is a self-mapping and there exist constants \(a_{i} \ge 0\), \(i = 1,\ldots,5\), with \(a_{1} + a_{2} + a_{3} + a_{4} + a_{5} < 1 / 2\) such that
$$qs^{p}\sigma_{b} ( fx,fy ) \le \alpha_{1} \sigma_{b} ( x,y ) + \alpha_{2}\sigma_{b} ( x,fx ) + \alpha_{3}\sigma_{b} ( y,fy ) + \alpha_{4} \sigma_{b} ( x,fy ) + \alpha_{5}\sigma_{b} ( y,fx ), $$
for all \(x,y \in X\) and a constant \(p \ge 2\), then f has a unique fixed point in X.

Proof

In Theorem 3.8, take the function \(\alpha ( x,y ) = qs^{p}\). □

Remark 3.10

Theorem 3.7 generalizes Theorem 18 in [7]. For \(\alpha ( x,y ) = s^{2}\) and for all \(x,y \in X\), Theorems 3.7 and 3.8 reduce to Theorems 3.2 and 3.13 of [19]. In Theorem 3.7 and Corollary 3.9, by choosing the constants \(a_{i}\) in certain manner, we obtain, as particular cases, certain classes of \(\alpha_{qs^{p}} \)-types of Kannan, Chatterjea, Reich, and Zamfirescu contractions.

The notion of \(\alpha - \psi\) contractive mappings is defined in a complete metric space in [8]. Thereafter, many authors provided various fixed point theorems for such a class of mappings. In the following definition, we extend and generalize the notions of \(\alpha - \psi\) and \((\psi - \phi )\)-contractive mappings in the context of larger spaces, such as b-metric-like spaces. The aim of this section is to extend and generalize the main classical result and other existing results in the literature on b-metric and metric-like spaces.

Let \(( X,\sigma_{b} )\) be a b-metric-like space with parameter \(s \ge 1\). For a self-mapping \(f:X \to X\), we define \(N ( x,y )\) by
$$ N(x,y) = \max \biggl\{ \sigma_{b}(x,y),\sigma_{b}(x,fx), \sigma_{b}(y,fy),\frac{\sigma_{b}(x,fy) + \sigma_{b}(y,fx)}{4s}\biggr\} $$
(3.10)
for all \(x,y \in X\).
The families Ψ, Φ with altering distance functions are defined as follows:
$$\begin{gathered} \psi: [ 0,\infty ) \to [ 0,\infty )\quad \mbox{an increasing and continuous function}; \\ \phi: [ 0,\infty ) \to [ 0,\infty )\quad \mbox{is continuous, and }\phi ( t ) < \psi ( t )\mbox{ for all }t > 0. \end{gathered} $$
Let \(\mathbb{S}\) be the set of all mappings \(\beta: [ 0,\infty ) \to [ 0,1 )\) satisfying the condition
$$\beta ( t_{n} ) \to 1\quad \mbox{as }n \to \infty\mbox{ implies that }t_{n} \to 0\mbox{ as }n \to \infty. $$

Definition 3.11

Let \(( X,\sigma_{b} )\) be a b-metric-like space with parameter \(s \ge 1\), and let \(f:X \to X\) be a self-mapping. Also, let \(\alpha:X \times X \to [ 0,\infty )\) and \(q \ge 1\), \(p \ge 2\). We say that f is an \(( \alpha_{qs^{p}} - \psi,\phi )\) generalized contractive mapping if there exist \(\psi \in \Psi\), \(\phi \in \Phi\) such that
$$ \psi \bigl( \alpha ( x,y )\sigma_{b}(fx,fy) \bigr) \le \phi \bigl( N(x,y) \bigr) $$
(3.11)
for all \(x,y \in X\) with \(\alpha ( x,y ) \ge qs^{p}\), where \(N ( x,y )\) is defined by (3.10).

Remark 3.12

  1. (i)

    Taking \(q = 1\) in the definition, we obtain \(\alpha_{s} - (\psi,\phi )\) admissible mappings defined in a b-metric-like space or in a b-metric space.

     
  2. (ii)

    Note that, for \(\alpha ( x,y ) = q\), the definition reduces to an \(\alpha_{q} \)-admissible mapping defined in a metric space or in a metric-like space.

     
  3. (iii)

    For \(s = 1\) and \(q = 1\), the definition reduces to the definition of an α-admissible mapping in a metric space.

     
  4. (iv)

    The definition reduces to a \((\psi,\phi ) \)-contractive mapping if we take \(\alpha ( x,y ) = 1\).

     
  5. (v)

    The definition reduces to an \(\alpha_{qs^{p}} - \phi\) contractive mapping if we take \(\psi ( t ) = t\).

     
  6. (vi)

    The definition reduces to an \(\alpha_{qs^{p}} - \lambda\) contractive mapping if we take \(\psi ( t ) = t\) and \(\phi ( t ) = \lambda t\) for \(\lambda \in ( 0,1 )\).

     

We now present the following theorem.

Theorem 3.13

Let \(( X,\sigma_{b} )\) be a complete b-metric-like space with parameter \(s \ge 1\), and let \(f:X \to X\) be an \(( \alpha_{qs^{p}} - \psi,\phi )\) generalized contractive mapping. Suppose that the following conditions are satisfied:
  1. (i)

    f is an \(\alpha_{qs^{p}} \)-admissible mapping;

     
  2. (ii)

    there exists \(x_{0} \in X\) such that \(\alpha ( x_{0},fx_{0} ) \ge qs^{p}\);

     
  3. (iii)

    either f is continuous, or property \(H_{qs^{p}}\) is satisfied.

     

Then f has a fixed point \(x \in X\). Moreover, f has a unique fixed point if property \(U_{qs^{p}}\) is satisfied.

Proof

By assumption (ii) there exists a point \(x_{0} \in X\) such that \(\alpha ( x_{0},fx_{0} ) \ge qs^{p}\). We construct a sequence \(\{ x_{n} \}\) in X by \(x_{n} = f^{n}x_{0} = f ( x_{n - 1} )\) for all \(n \in {N}\). If we suppose that \(\sigma_{b} ( x_{n},x_{n + 1} ) = 0\) for some n, then \(x_{n + 1} = x_{n}\), and the proof is completed, since \(u = x_{n} = x_{n + 1} = f ( x_{n} ) = fu\). Consequently, throughout the proof, we assume that
$$ \sigma_{b} ( x_{n},x_{n + 1} ) > 0\quad \mbox{for all } n \in {N}. $$
(3.12)
Since f is an \(\alpha_{qs^{p}} \)-admissible mapping, we observe that
$$\begin{gathered} \alpha ( x_{0},x_{1} ) = \alpha ( x_{0},fx_{0} ) \ge qs^{p},\qquad \alpha ( fx_{0},fx_{1} ) = \alpha ( x_{1},x_{2} ) \ge qs\quad \mbox{and}\\ \alpha ( fx_{1},fx_{2} ) = \alpha ( x_{2},x_{3} ) \ge qs^{p}. \end{gathered} $$
In general, by induction we derive that
$$ \alpha ( x_{n},x_{n + 1} ) \ge qs^{p}\quad \mbox{for all } n \in {N}. $$
(3.13)
By (3.13) and condition (3.11) we have:
$$ \begin{aligned}[b] \psi \bigl( \sigma_{b} ( x_{n},x_{n + 1} ) \bigr) &\le \psi \bigl( qs^{p}\sigma_{b} ( x_{n},x_{n + 1} ) \bigr) = \psi \bigl( qs^{p} \sigma_{b} ( fx_{n - 1},fx_{n} ) \bigr) \\ &\le \psi \bigl( \alpha ( x_{n - 1},x_{n} ) \sigma_{b} ( fx_{n - 1},fx_{n} ) \bigr) \\ &\le \phi \bigl( N ( x_{n - 1},x_{n} ) \bigr) < \psi \bigl( N ( x_{n - 1},x_{n} ) \bigr), \end{aligned} $$
(3.14)
where
$$ \begin{aligned}[b] N ( x_{n - 1},x_{n} ) &= \max \left \{ \begin{matrix} \sigma_{b} ( x_{n - 1},x_{n} ),\sigma_{b} ( x_{n - 1},fx_{n - 1} ),\sigma_{b} ( x_{n},fx_{n} ), \\ \frac{\sigma_{b} ( x_{n - 1},fx_{n} ) + \sigma_{b} ( x_{n},fx_{n - 1} )}{4s} \end{matrix} \right \} \\ &= \max \left \{ \begin{matrix} \sigma_{b} ( x_{n - 1},x_{n} ),\sigma_{b} ( x_{n - 1},x_{n} ),\sigma_{b} ( x_{n},x_{n + 1} ), \\ \frac{\sigma_{b} ( x_{n - 1},x_{n + 1} ) + \sigma_{b} ( x_{n},x_{n} )}{4s} \end{matrix} \right \} \\ &\le \max \left \{ \begin{matrix} \sigma_{b} ( x_{n - 1},x_{n} ),\sigma_{b} ( x_{n - 1},x_{n} ),\sigma_{b} ( x_{n},x_{n + 1} ), \\ \frac{s [ \sigma_{b} ( x_{n - 1},x_{n} ) + \sigma_{b} ( x_{n},x_{n + 1} ) ] + 2s\sigma_{b} ( x_{n - 1},x_{n} )}{4s} \end{matrix} \right \}. \end{aligned} $$
(3.15)
If we assume that, for some \(n \in {N}\),
$$\sigma_{b} ( x_{n - 1},x_{n} ) < \sigma_{b} ( x_{n},x_{n + 1} ), $$
then from inequality (3.15) we get
$$ N ( x_{n - 1},x_{n} ) \le \sigma_{b} ( x_{n},x_{n + 1} ). $$
(3.16)
Again, by (3.13) and condition (3.11) we have:
$$ \begin{aligned}[b] \psi \bigl( \sigma_{b} ( x_{n},x_{n + 1} ) \bigr) &\le \psi \bigl( qs^{p}\sigma_{b} ( x_{n},x_{n + 1} ) \bigr) = \psi \bigl( qs^{p} \sigma_{b} ( fx_{n - 1},fx_{n} ) \bigr) \\ &\le \psi \bigl( \alpha ( x_{n - 1},x_{n} ) \sigma_{b} ( fx_{n - 1},fx_{n} ) \bigr) \\ &\le \phi \bigl( N ( x_{n - 1},x_{n} ) \bigr) < \psi \bigl( N ( x_{n - 1},x_{n} ) \bigr). \end{aligned} $$
(3.17)
By the property ψ inequality (3.17) implies that
$$ \sigma_{b} ( x_{n},x_{n + 1} ) \le N ( x_{n - 1},x_{n} ). $$
(3.18)
From (3.16) and (3.18) we have
$$ N ( x_{n - 1},x_{n} ) = \sigma_{b} ( x_{n},x_{n + 1} ). $$
(3.19)
From (3.17), using (3.19), we obtain
$$ \begin{aligned}[b] \psi \bigl( \sigma_{b} ( x_{n},x_{n + 1} ) \bigr) &\le \psi \bigl( qs^{p}\sigma_{b} ( x_{n},x_{n + 1} ) \bigr) = \psi \bigl( qs^{p} \sigma_{b} ( fx_{n - 1},fx_{n} ) \bigr) \\ &\le \psi \bigl( \alpha ( x_{n - 1},x_{n} ) \sigma_{b} ( fx_{n - 1},fx_{n} ) \bigr) \\ &\le \phi \bigl( N ( x_{n - 1},x_{n} ) \bigr) = \phi \bigl( \sigma_{b} ( x_{n},x_{n + 1} ) \bigr) \\ &< \psi \bigl( \sigma_{b} ( x_{n},x_{n + 1} ) \bigr), \end{aligned} $$
(3.20)
which gives a contradiction, since we have assumed that \(\sigma_{b} ( x_{n},x_{n + 1} ) > 0\) and \(\phi ( t ) < \psi ( t )\) for all \(t > 0\). Hence, for all \(n \in {N}\), \(\sigma_{b} ( x_{n},x_{n + 1} ) \le \sigma_{b} ( x_{n - 1},x_{n} )\), and the sequence \(\{ \sigma_{b} ( x_{n},x_{n + 1} ) \}\) is decreasing and bounded below. Hence there exists \(l \ge 0\) such that \(\sigma_{b} ( x_{n},x_{n + 1} ) \to l\). Also,
$$\lim_{n \to \infty} \sigma_{b} ( x_{n},x_{n + 1} ) = \lim_{n \to \infty} N ( x_{n - 1},x_{n} ) = l. $$
We shall prove that \(l = 0\).
Consider
$$ \begin{aligned}[b] \psi \bigl( \sigma_{b} ( x_{n},x_{n + 1} ) \bigr) &\le \psi \bigl( qs^{p}\sigma_{b} ( x_{n},x_{n + 1} ) \bigr) = \psi \bigl( qs^{p} \sigma_{b} ( fx_{n - 1},fx_{n} ) \bigr) \\ &\le \psi \bigl( \alpha ( x_{n - 1},x_{n} ) \sigma_{b} ( fx_{n - 1},fx_{n} ) \bigr) \\ &\le \phi \bigl( N ( x_{n - 1},x_{n} ) \bigr) = \phi \bigl( \sigma_{b} ( x_{n},x_{n + 1} ) \bigr). \end{aligned} $$
(3.21)
If we assume that \(l > 0\), taking the limit in (3.21), we have
$$\psi ( l ) \le \phi ( l ), $$
which is a contradiction since \(\psi ( t ) > \phi ( t )\) for \(t > 0\). Hence \(l = 0\), and
$$ \lim_{n \to \infty} \sigma_{b} ( x_{n},x_{n + 1} ) = \lim_{n \to \infty} N ( x_{n - 1},x_{n} ) = 0. $$
(3.22)
Next, we shall show that \(\{ x_{n} \}\) is a Cauchy sequence in X. Suppose, on the contrary, that \(\{ x_{n} \}\) is not a Cauchy sequence. Then by Lemma 2.14 there exist \(\varepsilon > 0\) and two subsequences \(\{ x_{m_{k}} \}\) and \(\{ x_{n_{k}} \}\) of \(\{ x_{n} \}\), with \(n_{k} > m_{k} > k\), such that
$$ \begin{gathered} \sigma_{b}(x_{m_{k}},x_{n_{k}}) \ge \varepsilon,\qquad \sigma_{b}(x_{m_{k}},x_{n_{k} - 1}) < \varepsilon, \\ \frac{\varepsilon}{ s^{2}} \le \mathop{\lim \sup}_{k \to \infty} \sigma_{b}(x_{m_{k} - 1},x_{n_{k} - 1}) \le \varepsilon s,\\ \frac{\varepsilon}{s} \le \mathop{\lim \sup}_{k \to \infty} \sigma_{b}(x_{n_{k} - 1},x_{m_{k}}) \le \varepsilon s^{2},\quad \mbox{and} \\ \frac{\varepsilon}{s} \le \mathop{\lim \sup}_{k \to \infty} \sigma_{b}(x_{m_{k} - 1},x_{n_{k}}) \le \varepsilon s^{2}. \end{gathered} $$
(3.23)
From the definition of \(N ( x,y )\) we have
$$ \begin{aligned}[b] N ( x_{m_{k} - 1},x_{n_{k} - 1} ) &= \max \left \{ \begin{matrix} \sigma_{b}(x_{m_{k} - 1},x_{n_{k} - 1}),\sigma_{b}(x_{m_{k} - 1},fx_{m_{k} - 1}),\sigma_{b}(x_{n_{k} - 1},fx_{n_{k} - 1}), \\ \frac{\sigma_{b}(x_{m_{k} - 1},fx_{n_{k} - 1}) + \sigma_{b}(x_{n_{k} - 1},fx_{m_{k} - 1})}{4s} \end{matrix} \right \} \\ &= \max \left \{ \begin{matrix} \sigma_{b}(x_{m_{k} - 1},x_{n_{k} - 1}),\sigma_{b}(x_{m_{k} - 1},x_{m_{k}}),\sigma_{b}(x_{n_{k} - 1},x_{n_{k}}), \\ \frac{\sigma_{b}(x_{m_{k} - 1},x_{n_{k}}) + \sigma_{b}(x_{n_{k} - 1},x_{m_{k}})}{4s} \end{matrix} \right \}. \end{aligned} $$
(3.24)
Taking the upper limit as \(k \to \infty\) in (3.24) and using (3.22), (3.23), we get
$$ \begin{aligned}[b] &\mathop{\lim \sup}_{k \to \infty} N ( x_{m_{k} - 1},x_{n_{k} - 1} ) \\ &\quad = \mathop{\lim \sup}_{k \to \infty} \max \left \{ \begin{matrix} \sigma_{b}(x_{m_{k} - 1},x_{n_{k} - 1}),\sigma_{b}(x_{m_{k} - 1},x_{m_{k}}),\sigma_{b}(x_{n_{k} - 1},x_{n_{k}}), \\ \frac{\sigma_{b}(x_{m_{k} - 1},x_{n_{k}}) + \sigma_{b}(x_{n_{k} - 1},x_{m_{k}})}{4s} \end{matrix} \right \} \\ &\quad \le \max \biggl\{ \varepsilon s,0,0,\frac{\varepsilon s}{2} \biggr\} \le \varepsilon s. \end{aligned} $$
(3.25)
Using the \(\alpha_{qs^{p}}\)-weak contractive condition, we have
$$ \begin{aligned}[b] \psi \bigl( qs^{p}\sigma_{b}(x_{m_{k}},x_{n_{k}}) \bigr) &\le \psi \bigl( qs^{p}\sigma_{b}(fx_{m_{k} - 1},fx_{n_{k} - 1}) \bigr) \\ &\le \psi \bigl( \alpha ( x_{m_{k} - 1},x_{n_{k} - 1} ) \sigma_{b}(fx_{m_{k} - 1},fx_{n_{k} - 1}) \bigr) \\ &\le \phi \bigl( N ( x_{m_{k} - 1},x_{n_{k} - 1} ) \bigr). \end{aligned} $$
(3.26)
Taking the upper limit in (3.26), using (3.23) and (3.25), we obtain
$$\begin{aligned} \psi ( \varepsilon s ) &\le \psi \bigl( q\varepsilon s^{p - 1} \bigr) = \psi \biggl( qs^{p}\frac{\varepsilon}{s} \biggr) \le \psi \Bigl( \mathop{\lim \sup}_{k \to \infty} \sigma_{b}(x_{m_{k}},x_{n_{k}}) \Bigr) \\ &\le \phi \Bigl( \mathop{\lim \sup}_{k \to \infty} \bigl( N ( x_{m_{k} - 1},x_{n_{k} - 1} ) \bigr) \Bigr) \le \phi ( \varepsilon s ) \\ &< \psi ( \varepsilon s ), \end{aligned} $$
which is a contradiction, since \(\varepsilon > 0\). Therefore \(\{ x_{n} \}\) is a Cauchy sequence in the complete b-metric-like space \(( X,\sigma_{b} )\). Thus, there is some \(u \in X\) such that \(\{ x_{n} \}\) converges to u. If f is a continuous mapping, we get:
$$f(u) = f\Bigl(\lim_{n \to \infty} x_{n}\Bigr) = \lim _{n \to \infty} f(x_{n}) = \lim_{n \to \infty} (x_{n + 1}) = u, $$
and u is a fixed point of f.
If the self-map f is not continuous, then from (3.13) and condition \(H_{qs^{p}}\), there exists a subsequence \(\{ x_{n_{k}} \}\) of \(\{ x_{n} \}\) such that \(\alpha ( x_{n_{k}},u ) \ge qs^{p}\) for all \(k \in {N}\). Since \(\alpha ( x_{n_{k}},u ) \ge qs^{p}\), applying contractive condition (3.11), with \(x = x_{n_{k}}\) and \(y = u\), we obtain
$$ \begin{aligned}[b] \psi \bigl( qs^{p}\sigma_{b} ( x_{n_{k} + 1},fu ) \bigr) &= \psi \bigl( qs^{p}\sigma_{b} ( fx_{n_{k}},fu ) \bigr) \\ &\le \psi \bigl( \alpha ( x_{n_{k}},u )\sigma_{b} ( fx_{n_{k}},fu ) \bigr) \\ &\le \phi \bigl( N ( x_{n_{k}},u ) \bigr), \end{aligned} $$
(3.27)
where
$$ \begin{aligned}[b] N ( x_{n_{k}},u ) &= \max \left \{ \begin{matrix} \sigma_{b} ( x_{n_{k}},u ),\sigma_{b} ( x_{n_{k}},fx_{n_{k}} ),\sigma_{b} ( u,fu ), \\ \frac{\sigma_{b} ( x_{n_{k}},fu ) + \sigma_{b} ( u,fx_{n_{k}} )}{4s} \end{matrix} \right \} \\ &= \max \left \{ \begin{matrix} \sigma_{b} ( x_{n_{k}},u ),\sigma_{b} ( x_{n_{k}},x_{n_{k} + 1} ),\sigma_{b} ( u,fu ), \\ \frac{\sigma_{b} ( x_{n_{k}},fu ) + \sigma_{b} ( u,x_{n_{k} + 1} )}{4s} \end{matrix} \right \}. \end{aligned} $$
(3.28)
Taking the upper limit in (3.28) and using Lemma 2.13 and result (3.22), we obtain
$$ \mathop{\lim \sup}_{n \to \infty} N ( x_{n_{k}},u ) \le \max \biggl\{ 0,0,\sigma_{b} ( u,fu ),\frac{s\sigma_{b} ( u,fu )}{4s} \biggr\} = \sigma_{b} ( u,fu ). $$
(3.29)
Taking the upper limit as \(k \to \infty\) in (3.27) and using (3.29) and Lemma 2.13, we obtain
$$ \begin{aligned}[b] \psi \bigl( qs^{p - 1}\sigma_{b} ( u,fu ) \bigr) &= \psi \biggl( qs^{p}\frac{1}{s}\sigma_{b} ( u,fu ) \biggr) \le \psi \Bigl( qs^{p}\mathop{\lim \sup}_{k \to \infty} \sigma_{b} ( x_{n_{k}},fu ) \Bigr) \\ &\le \phi \Bigl( \mathop{\lim \sup}_{k \to \infty} N ( x_{n_{k}},u ) \Bigr) < \psi \Bigl( \mathop{\lim \sup}_{k \to \infty} N ( x_{n_{k}},u ) \Bigr) \\ &\le \psi \bigl( \sigma_{b} ( u,fu ) \bigr). \end{aligned} $$
(3.30)
From (3.30) we get \(\sigma_{b} ( u,fu ) = 0\), which implies that \(fu = u\). Hence u is a fixed point of f.
Suppose that u and v are two fixed points of f, where \(fu = u\) and \(fv = v\) are such that \(u \ne v\). Then, by hypothesis \(U_{qs^{p}}\), \(\alpha ( u,v ) \ge qs^{p}\), and applying (3.11), we have
$$ \begin{aligned}[b] \psi \bigl( qs^{p}\sigma_{b} ( u,u ) \bigr) &= \psi \bigl( qs^{p}\sigma_{b} ( fu,fu ) \bigr) \le \psi \bigl( \alpha ( u,u )\sigma_{b} ( fu,fu ) \bigr) \\ &\le \phi \bigl( N ( u,u ) \bigr) \le \phi \bigl( \sigma_{b} ( u,u ) \bigr), \end{aligned} $$
(3.31)
where
$$N ( u,u ) = \max \biggl\{ \sigma_{b} ( u,u ),\sigma_{b} ( u,u ),\sigma_{b} ( u,u ),\frac{\sigma_{b} ( u,u ) + \sigma_{b} ( u,u )}{4s} \biggr\} = \sigma_{b} ( u,u ). $$
From inequality (3.31) it follows that \(\sigma_{b} ( u,u ) = 0\) (also \(\sigma_{b} ( v,v ) = 0\)).
Again we have
$$ \begin{aligned}[b] \psi \bigl( qs^{p}\sigma_{b} ( u,v ) \bigr) &= \psi \bigl( qs^{p}\sigma_{b} ( fu,fv ) \bigr) \le \psi \bigl( \alpha ( u,v )\sigma_{b} ( fu,fv ) \bigr) \\ &\le \phi \bigl( N ( u,v ) \bigr) \le \phi \bigl( \sigma_{b} ( u,v ) \bigr), \end{aligned} $$
(3.32)
where \(N ( u,v ) = \sigma_{b} ( u,v )\).

Inequality (3.32) implies that \(\sigma_{b} ( u,v ) = 0\). Therefore \(u = v\), and the fixed point is unique. □

Remark 3.14

Our theorem extends Theorems 2.1, 2.2, and 2.7 of Aydi et al. [9].

By taking \(\phi ( t ) = \psi ( t ) - \varphi ( t )\), where \(\varphi \in \Psi\), in Theorem 3.13 we obtain the following result.

Corollary 3.15

Let \(( X,\sigma_{b} )\) be a complete b-metric-like space with parameter \(s \ge 1\), let \(f:X \to X\) be a self-mapping, and let \(\alpha:X \times X \to [ 0,\infty )\). Suppose that the following conditions are satisfied:
  1. (i)

    f is an \(\alpha_{qs^{p}} \)-admissible mapping;

     
  2. (ii)
    there exist functions \(\psi,\varphi \in \Psi\) such that
    $$\psi \bigl( \alpha ( x,y )\sigma_{b}(fx,fy) \bigr) \le \psi \bigl( N(x,y) \bigr) - \varphi \bigl( N(x,y) \bigr); $$
     
  3. (iii)

    there exists \(x_{0} \in X\) such that \(\alpha ( x_{0},fx_{0} ) \ge qs^{p}\);

     
  4. (iv)

    either f is continuous, or property \(H_{qs^{p}}\) is satisfied.

     

Then f has a fixed point \(x \in X\). Moreover, f has a unique fixed point if property \(U_{qs^{p}}\) is satisfied.

Remark 3.16

This corollary extends Theorems 3 and 4 of Roshan et al. [25].

By taking \(\psi ( t ) = t\) and \(\phi ( t ) = \beta ( t )t\) where \(\beta \in \mathbb{S}\) is as in Theorem 3.13, we obtain the following result.

Corollary 3.17

Let \(( X,\sigma_{b} )\) be a complete b-metric-like space with parameter \(s \ge 1\), let \(f:X \to X\) be a self-mapping, and let \(\alpha:X \times X \to [ 0,\infty )\). Suppose that the following conditions are satisfied:
  1. (i)

    f is an \(\alpha_{qs^{p}} \)-admissible mapping;

     
  2. (ii)
    there exist functions \(\psi,\varphi \in \Psi\) such that
    $$\psi \bigl( \alpha ( x,y )\sigma_{b}(fx,fy) \bigr) \le \beta \bigl( N(x,y) \bigr) \bigl( N(x,y) \bigr); $$
     
  3. (iii)

    there exists \(x_{0} \in X\) such that \(\alpha ( x_{0},fx_{0} ) \ge qs^{p}\);

     
  4. (iv)

    either f is continuous, or property \(H_{qs^{p}}\) is satisfied.

     

Then f has a fixed point \(x \in X\). Moreover, f has a unique fixed point if property \(U_{qs^{p}}\) is satisfied.

If we take \(\psi ( t ) = t\) in Theorem 3.13, then we get the following result.

Corollary 3.18

Let \(( X,\sigma_{b} )\) be a complete b-metric-like space with parameter \(s \ge 1\), let \(f:X \to X\) be a self-mapping, and let \(\alpha:X \times X \to [ 0,\infty )\). Suppose that the following conditions are satisfied:
  1. (i)

    f is an \(\alpha_{qs^{p}} \)-admissible mapping;

     
  2. (ii)
    there exist functions \(\varphi \in \Psi\) such that
    $$\alpha ( x,y )\sigma_{b}(fx,fy) \le \varphi \bigl( N(x,y) \bigr); $$
     
  3. (iii)

    there exists \(x_{0} \in X\) such that \(\alpha ( x_{0},fx_{0} ) \ge qs^{p}\);

     
  4. (iv)

    either f is continuous, or property \(H_{qs^{p}}\) is satisfied.

     

Then f has a fixed point \(x \in X\). If property \(U_{qs^{p}}\) is satisfied, then f has a unique fixed point.

Remark 3.19

Corollary 3.18 generalizes and extends Theorem 2.7 of Samet et al. [8].

Corollary 3.20

Let \(( X,\sigma_{b} )\) be a complete b-metric-like space with parameter \(s \ge 1\), let \(f:X \to X\) be a self-mapping, and let \(\alpha:X \times X \to [ 0,\infty )\). Suppose that the following conditions are satisfied:
  1. (i)

    f is an \(\alpha_{qs^{p}} \)-admissible mapping;

     
  2. (ii)
    there exists a function \(\varphi \in \Psi\) such that
    $$\alpha ( x,y )\sigma_{b}(fx,fy) \le N(x,y) - \varphi \bigl( N(x,y) \bigr); $$
     
  3. (iii)

    there exists \(x_{0} \in X\) such that \(\alpha ( x_{0},fx_{0} ) \ge qs^{p}\);

     
  4. (iv)

    either f is continuous, or property \(H_{qs^{p}}\) is satisfied.

     

Then f has a fixed point \(x \in X\). Moreover, f has a unique fixed point if property \(U_{qs^{p}}\) is satisfied.

Proof

It follows from Corollary 3.15 by taking \(\psi ( t ) = t\). □

Remark 3.21

Corollary 3.20 generalizes Theorem 2.7 of Harandi [5].

Corollary 3.22

Let \(( X,\sigma_{b} )\) be a complete b-metric like space with parameter \(s \ge 1\), and let \(f,g\) be two selfmaps of X with \(\psi \in \Psi\), \(\varphi \in \Phi\) satisfying the condition
$$\psi \bigl( \alpha_{qs^{p}}\sigma_{b} ( fx,fy ) \bigr) \le \lambda \psi \bigl( M ( x,y ) \bigr) $$
for all \(x,y \in X\), where \(M ( x,y )\) is defined in (3.15), and \(q > 1\). Then f and g have a unique common fixed point in X.

Proof

In Theorem 3.13, take \(\varphi (t) = \lambda \psi ( t )\) where \(0 < \lambda < 1\). □

Corollary 3.23

Let \(( X,\sigma_{b} )\) be a complete b-metric-like space with parameter \(s \ge 1\), and let \(f:X \to X\) be a self-mapping such that, for all \(x,y \in X\) and any arbitrary coefficient \(p \ge 1\),
$$qs^{p}\sigma_{b} ( fx,fy ) \le k\max \biggl\{ \sigma_{b} ( x,y ),\sigma_{b} ( x,fx ),\sigma_{b} ( y,fy ),\frac{\sigma_{b} ( x,fy ) + \sigma_{b} ( y,fx )}{4s} \biggr\} , $$
where \(k \in (0,1)\). Then f has a unique fixed point.

Proof

It follows from Corollary 3.15 by taking \(\alpha ( x,y ) = qs^{p}\), \(\psi ( t ) = t\), and \(\varphi (t) = ( 1 - k )t\) for all \(t \in [0,\infty )\) and \(k \in (0,1)\). □

Remark 3.24

It is clear that we can derive several corresponding results by replacing the b-metric-like space with some other spaces such as a b-metric space, a metric space, a metric-like space, and a partial metric space. Conditions (3.1) and (3.12) are more general than the analogues in the previous literature, and theorems related to those conditions have a more general character because of the parameter s and arbitrary coefficients q, p.

3.1 Application

In this section, we will use Corollary 3.23 to show that there is a solution to the following integral equation:
$$ x ( t ) = \int_{0}^{T} G \bigl( t,r,x ( r ) \bigr)\,dr. $$
(3.33)
Let \(X = C ( [ 0,T ] )\) be the set of real continuous functions defined on \([ 0,T ]\) for \(T > 0\).
We endow X with
$$\sigma_{b} ( x,y ) = \max_{t \in [ 0,1 ]} \bigl( \bigl\vert x ( t ) \bigr\vert + \bigl\vert y ( t ) \bigr\vert \bigr)^{m}\quad \mbox{for all } x,y \in X. $$
It is evident that \(( X,\sigma_{b} )\) is a complete b-metric-like space with parameter \(s = 2^{m - 1}\) with \(m > 1\).

Consider the mapping \(f:X \to X\) defined by \(fx ( t ) = \int_{0}^{T} G ( t,r,x ( r ) )\,dr\).

Theorem 3.25

Consider equation (3.33) and suppose that
  1. (a)

    \(G: [ 0,T ] \times [ 0,T ] \times R \to R^{ +} = [ 0,\infty )\) (that is, \(G ( t,r,x ( r ) ) \ge 0\)) is continuous;

     
  2. (b)

    there exists a continuous \(\gamma: [ 0,T ] \times [ 0,T ] \to R\);

     
  3. (c)

    \(\sup_{t \in [ 0,T ]}\int_{0}^{T} \gamma ( t,r ) \,dr \le 1\);

     
  4. (d)
    there exists a constant \(L \in ( 0,1 )\) such that, for all \(( t,r ) \in [ 0,T ]^{2}\) and \(x,y \in R\),
    $$\bigl\vert G \bigl( t,r,x ( r ) \bigr) + G \bigl( t,r,y ( r ) \bigr) \bigr\vert \le \biggl( \frac{L}{s^{3}} \biggr)^{\frac{1}{m}}\gamma ( t,r ) \bigl( \bigl\vert x ( r ) \bigr\vert + \bigl\vert y ( r ) \bigr\vert \bigr). $$
     
Then the integral equation (3.33) has a unique solution in \(x \in X\).

Proof

For \(x,y \in X\), from conditions (c) and (d), for all t, we have
$$\begin{aligned} qs\sigma_{b} \bigl( fx ( t ),fy ( t ) \bigr) &= qs \bigl( \bigl\vert fx ( t ) \bigr\vert + \bigl\vert fy ( t ) \bigr\vert \bigr)^{m} \\ &= qs \biggl( \biggl\vert \int_{0}^{T} G \bigl( t,r,x ( r ) \bigr)\,dr \biggr\vert + \biggl\vert \int_{0}^{T} G \bigl( t,r,y ( r ) \bigr)\,dr \biggr\vert \biggr)^{m} \\ &\le qs \biggl( \int_{0}^{T} \bigl\vert G \bigl( t,r,x ( r ) \bigr) \bigr\vert \,dr + \int_{0}^{T} \bigl\vert G \bigl( t,r,y ( r ) \bigr) \bigr\vert \,dr \biggr)^{m} \\ &\le qs \biggl( \int_{0}^{T} \biggl( \frac{L}{s^{3}} \biggr)^{\frac{1}{m}}\gamma ( t,r ) \bigl( \bigl( \bigl( \bigl\vert x ( r ) + y ( r ) \bigr\vert \bigr)^{m} \bigr)^{\frac{1}{m}} \bigr)\,dr \biggr)^{m} \\ &\le qs \biggl( \int_{0}^{T} \biggl( \frac{L}{s^{3}} \biggr)^{\frac{1}{m}}\gamma ( t,r )\sigma_{b}^{\frac{1}{m}} \bigl( x ( r ),y ( r ) \bigr)\,dr \biggr)^{m} \\ &\le qs \cdot \frac{L}{s^{3}}\sigma_{b} \bigl( x ( r ),y ( r ) \bigr) \biggl( \int_{0}^{T} \gamma ( t,r )\,dr \biggr)^{m} \\ &= \frac{qL}{s^{2}}\sigma_{b} \bigl( x ( r ),y ( r ) \bigr) \biggl( \int_{0}^{T} \gamma ( t,r )\,dr \biggr)^{m} \\ &\le \frac{qL}{s^{2}}\sigma_{b} \bigl( x ( r ),y ( r ) \bigr), \end{aligned} $$
which implies that
$$\begin{aligned} s\sigma_{b} \bigl( fx ( t ),fy ( t ) \bigr) &\le \frac{L}{s^{2}}\sigma_{b} \bigl( x ( r ),y ( r ) \bigr) \\ &\le k\max \biggl\{ \sigma_{b} ( x,y ),\sigma_{b} ( x,Tx ),\sigma_{b} ( y,Ty )\frac{\sigma_{b} ( x,Ty ) + \sigma_{b} ( y,Tx )}{4s} \biggr\} , \end{aligned} $$
where \(k = L / s^{2} \in ( 0,1 )\).

Therefore, all of the conditions of Corollary 3.23 are satisfied, and, as a result, the mapping f has a unique fixed point in X, which is a solution of the integral equation (3.33). □

4 Conclusions

In this paper, the class of \(\alpha_{qs^{p}} \)-admissible mappings is introduced in a larger structure such as a b-metric-like space. Some fixed point results dealing with \((\alpha - \psi,\phi )\) contractions are obtained, and they cover and unify a huge number of published results in the related literature.

Declarations

Acknowledgements

The authors gratefully acknowledge all the remarks and comments made by the anonymous referees.

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

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

(1)
Department of Mathematics and Computer Sciences, Faculty of Natural Sciences, University of Gjirokastra
(2)
Department of Mathematics, Indiana University
(3)
Faculty of Mechanical Engineering, University of Belgrade
(4)
State University of Novi Pazar

References

  1. Bakhtin, IA: The contraction mapping principle in quasimetric spaces. Funct. Anal., Ulyanovsk Gos. Ped. Inst. 30, 26-37 (1989) Google Scholar
  2. Czerwik, S: Contraction mappings in b-metric spaces. Acta Math. Inform. Univ. Ostrav. 1, 5-11 (1993) MathSciNetMATHGoogle Scholar
  3. Matthews, SG: Partial metric topology. Ann. N.Y. Acad. Sci. 728, 183-197 (1994) MathSciNetView ArticleMATHGoogle Scholar
  4. Shukla, S: Partial b-metric spaces and fixed point theorem. Mediterr. J. Math. 11(2) 703-711 (2013) View ArticleMATHGoogle Scholar
  5. Amini-Harandi, A: Metric-like spaces, partial metric spaces and fixed points. Fixed Point Theory Appl. 2012, 204 (2012) MathSciNetView ArticleMATHGoogle Scholar
  6. Alghmandi, MA, Hussain, N, Salimi, P: Fixed point and coupled fixed point theorems on b-metric-like spaces. J. Inequal. Appl. 2013, 402 (2013) MathSciNetView ArticleMATHGoogle Scholar
  7. Hussain, N, Roshan, JR, Parvaneh, V, Kadelburg, Z: Fixed points of contractive mappings in b-metric-like spaces. Sci. World J., 2014, 471827 (2014) Google Scholar
  8. Samet, B, Vetro, C, Vetro, P: Fixed point theorems for \(\alpha - \psi\)-contractive type mappings. Nonlinear Anal. 75, 2154-2165 (2012) MathSciNetView ArticleMATHGoogle Scholar
  9. Aydi, H, Karapinar, E: Fixed point results for generalized \(\alpha - \psi \)-contractions in metric-like spaces and applications. Electron. J. Differ. Equ. 2015, 133 (2015) MathSciNetView ArticleMATHGoogle Scholar
  10. Aydi, H, Jellali, M, Karapinar, E: Common fixed points for generalized α-implicit contractions in partial metric spaces, consequences and application. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 109(2), 367-384 (2015) MathSciNetView ArticleMATHGoogle Scholar
  11. Aydi, H, Felhi, A, Sahmim, S: On common fixed points for \(\alpha - \psi\)-contractions and generalized cyclic contractions in b-metric-like spaces and consequences. J. Nonlinear Sci. Appl. 9, 2492-2510 (2016) MathSciNetMATHGoogle Scholar
  12. Amiri, P, Rezapur, S, Shahzad, N: Fixed points of generalized \(\alpha-\psi\)-contractions. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 108, 519-526 (2014) View ArticleGoogle Scholar
  13. Doric, D: Common fixed point for generalized \((\psi, \varphi)\)-weak contractions. Appl. Math. Lett. 22, 1896-1900 (2009) MathSciNetView ArticleMATHGoogle Scholar
  14. Hussain, N, Vetro, C, Vetro, F: Fixed point results for α-implicit contractions with application to integral equations. Nonlinear Anal., Model. Control 21(3), 362-378 (2016) MathSciNetGoogle Scholar
  15. Karapinar, E: \(\alpha - \psi\)-Geraghty contraction type mappings and some related fixed point results. Filomat 28(1), 37-48 (2014) MathSciNetView ArticleMATHGoogle Scholar
  16. Karapinar, E, Samet, B: Generalized \(\alpha-\psi\) contractive type mappings and related fixed point theorems with applications. Abstr. Appl. Anal. 2012, Article ID 793486 (2012) MathSciNetMATHGoogle Scholar
  17. La Rosa, V, Vetro, P: Common fixed points for \(\alpha - \psi - \varphi\)-contractions in generalized metric spaces. Nonlinear Anal., Model. Control 19(1), 43-54 (2014) MathSciNetMATHGoogle Scholar
  18. Salimi, P, Hussain, N, Shukla, S: Fathollahi, S, Radenovic, S Fixed point results for cyclic \(\alpha - \psi - \varphi\)-contractions with application to integral equations. J. Comput. Appl. Math. 290, 445-458 (2015) MathSciNetView ArticleMATHGoogle Scholar
  19. Aydi, A, Felhi, A, Afshari, H: New Geraghty type contractions on metric-like spaces. J. Nonlinear Sci. Appl. 10(2), 780-788 (2017) MathSciNetView ArticleGoogle Scholar
  20. Aydi, H, Felhi, A, Sahmim, S: Ciric-Berinde fixed point theorems for multi-valued mappings on α-complete metric-like spaces. Filomat 31(12), 3727-3740 (2017) Google Scholar
  21. Aydi, H, Felhi, A, Sahmim, S: Common fixed points via implicit contractions on b-metric-like spaces. J. Nonlinear Sci. Appl. 10(4), 1524-1537 (2017) MathSciNetView ArticleMATHGoogle Scholar
  22. Ali, MU, Kamran, T, Karapınar, E: Further discussion on modified multivalued \(\alpha_{*} - \psi \)-contractive type mapping. Filomat 29(8), 1893-1900 (2015) MathSciNetView ArticleMATHGoogle Scholar
  23. Ali, MU, Kamran, T, Kiran, Q: Fixed point theorem for \(( \alpha - \psi,\varphi )\) contractive mappings with two metrics. J. Adv. Math. Stud. 7(2), 8-11 (2014) MathSciNetMATHGoogle Scholar
  24. Chen, C, Dong, J, Zhu, C: Some fixed point theorems in b-metric-like spaces. Fixed Point Theory Appl. 2015, 122 (2015) MathSciNetView ArticleMATHGoogle Scholar
  25. Roshan, JR, Parvaneh, V, Sedghi, S, Shbkolaei, N, Shatanawi, W: Common fixed points of almost generalized contractive mappings in ordered b-metric spaces. Fixed Point Theory Appl. 2013, 159 (2013) MathSciNetView ArticleMATHGoogle Scholar

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