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

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.

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 [8–23]).

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

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

• (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;

• (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

• (a) x is unique,

• (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

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

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

  $$\lim_{n \to \infty} \sigma_{b}(x_{n},x_{n}) = \lim_{n \to \infty} \sigma_{b}(x_{n + 1},x_{n + 1}) = 0; $$

• (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

• (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.

• (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.

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

• (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:

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

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

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

• (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:

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

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

• (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

• (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.

• (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.

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

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

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

• (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:

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

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

• (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:

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

• (ii) there exist functions $\psi,\varphi \in \Psi$ such that

  $$\psi \bigl( \alpha ( x,y )\sigma_{b}(fx,fy) \bigr) \le \psi \bigl( N(x,y) \bigr) - \varphi \bigl( N(x,y) \bigr); $$

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

• (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:

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

• (ii) there exist functions $\psi,\varphi \in \Psi$ such that

  $$\psi \bigl( \alpha ( x,y )\sigma_{b}(fx,fy) \bigr) \le \beta \bigl( N(x,y) \bigr) \bigl( N(x,y) \bigr); $$

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

• (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:

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

• (ii) there exist functions $\varphi \in \Psi$ such that

  $$\alpha ( x,y )\sigma_{b}(fx,fy) \le \varphi \bigl( N(x,y) \bigr); $$

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

• (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:

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

• (ii) there exists a function $\varphi \in \Psi$ such that

  $$\alpha ( x,y )\sigma_{b}(fx,fy) \le N(x,y) - \varphi \bigl( N(x,y) \bigr); $$

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

• (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

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

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

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

• (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.