1 Introduction and preliminaries
Throughout this paper, we denote \(\mathbb{R}^{+}_{0} = [0, +\infty)\) and \(\mathbb{N}_{0} = \mathbb{N} \cup\{0\}\), where \(\mathbb{N}\) is the set of all positive integers. First, we recall some basic concepts and notation.
The concept of b-metric was introduced by Czerwik [1] as a generalization of metric (see also Bakhtin [2, 3]) to extend the celebrated Banach contraction mapping principle. Following the initial paper of Czerwik [1], a number of researchers in nonlinear analysis investigated the topology of the paper and proved several fixed point theorems in the context of complete b-metric spaces (see [4‐8] and references therein).
Anzeige
Definition 1.1
[1]
Let X be a nonempty set, and \(s\geq1\) be a given real number. A mapping \(d : X \times X\to[0, +\infty)\) is said to be a b-metric if for all \(x, y, z \in X\), the following conditions are satisfied:
In this case, the pair \((X, d)\) is called a b-metric space (with constant s).
(b1)
\(d(x, y) =0\) if and only if \(x = y\);
(b2)
\(d(x, y) = d(y,x)\);
(b3)
\(d(x, z)\leq s[d(x, y) + d(y, z)]\).
Definition 1.2
[9]
Let X be a nonempty set, and \(s\geq1\) be a given real number. A mapping \(d : X \times X\to[0, +\infty)\) is said to be a quasi-b-metric if for all \(x, y, z \in X\), the following conditions are satisfied:
In this case, the pair \((X,d)\) is called a quasi-b-metric space (with constant s).
(bm1)
\(d(x, y) =0\) if and only if \(x = y\);
(bm2)
\(d(x, z)\leq s[d(x, y) + d(y, z)]\).
Anzeige
Definition 1.3
[10]
Let X be a nonempty set, and \(s\geq1\) be a given real number. A mapping \(d : X \times X\to[0, +\infty)\) is said to be a quasi-b-metric-like if for all \(x, y, z \in X\), the following conditions are satisfied:
In this case, the pair \((X, d)\) is called a quasi-b-metric-like space (with constant s).
(bM1)
\(d(x, y) =0\) implies \(x = y\);
(bM2)
\(d(x, z)\leq s[d(x, y) + d(y, z)]\).
Example 1.4
Let \(X=\{0,\frac{1}{2},\frac{1}{3}\} \cup [1, \infty)\), and let \(d : X \times X\to[0, +\infty)\) be defined as It is clear that \((X,d)\) is a quasi-b-metric-like space with constant \(s=9\).
$$d(x,y)=\left \{ \textstyle\begin{array}{l@{\quad}l} 6& \text{if } x=y=0, \\ 3 &\text{if } x=y=\frac{1}{3}, \\ 2 &\text{if } x=0, y=\frac{1}{2}, \\ \frac{1}{2} &\text{if } x=0, y=\frac{1}{3}, \\ \frac{3}{2} &\text{if } x=\frac{1}{3}, y=0, \\ |x-y| &\text{otherwise}. \end{array}\displaystyle \right . $$
Definition 1.5
(see e.g. [10])
Let \((X, d)\) be a quasi-b-metric-like space. Then:
(i)a
a sequence \(\{x_{n}\}\) in X is called a left-Cauchy sequence if and only if for every \(\varepsilon>0\), there exists a positive integer \(N=N(\varepsilon)\) such that \(d(x_{n}, x_{m})<\varepsilon\) for all \(n>m>N\);
(ii)b
a sequence \(\{x_{n}\}\) in X is called a right-Cauchy sequence if and only if for every \(\varepsilon>0\), there exists a positive integer \(N=N(\varepsilon)\) such that \(d(x_{n}, x_{m})<\varepsilon\) for all \(m>n>N\);
(iii)a
a quasi-partial metric space is said to be left-complete if every left-Cauchy sequence \(\{x_{n}\}\) in X converges with respect to d to a point \(u \in X\) such that
$$\lim_{n \to\infty}d( x_{n},u) = d(u,u) =\lim _{n,m\to\infty}d(x_{m}, x_{n})=0,\quad \text{where } m\geq n; $$
(iii)b
a quasi-partial metric space is said to be right-complete if every left-Cauchy sequence \(\{x_{n}\}\) in X converges with respect to d to a point \(u \in X\) such that
$$\lim_{n \to\infty}d(u, x_{n}) = d(u,u) =\lim _{n,m\to\infty}d(x_{n}, x_{m})=0,\quad \text{where } m\geq n. $$
Let \((X, d)\) and \((Y, \alpha)\) be quasi-b-metric-like spaces, and let \(f : X \to Y\) be a continuous mapping. Then
$$\lim_{n \to\infty} x_{n}=u \quad \Rightarrow\quad \lim _{n \to\infty} fx_{n}=fu. $$
In 2012, Samet et al. [11] introduced the concept of α-admissible mappings, and in 2013, Karapınar et al. [12] improved this notion as triangular α-admissible mappings.
Definition 1.6
Let \(\alpha: X \times X \rightarrow[0, +\infty)\) be a function. A self-mapping f is called an α-admissible mapping if for all \(x,y \in X\). If, further, f satisfies the condition for all \(x,y,z \in X\), then it is called triangular α-admissible mapping.
$$\alpha(x,y)\geq1 \quad \Rightarrow \quad \alpha(fx,fy)\geq1 $$
$$\alpha(x,z)\geq1 \quad \text{and}\quad \alpha(z,y)\geq1 \quad \Rightarrow \quad \alpha(x,y)\geq1 $$
Very recently, Popescu [13] improved these notions as follows.
Definition 1.7
[13]
Let \(\alpha:X\times X\rightarrow[0,\infty)\) be a function. If \(f: X\to X\) satisfies the condition for all \(x\in X\), then it is called a right-α-orbital admissible mapping. If f satisfies the condition for all \(x\in X\), then it is called a left-α-orbital admissible mapping. Furthermore, if f is both right-α-orbital admissible and left-α-orbital admissible, then f is called an α-orbital admissible mapping.
$$ (\mathrm{T}1)'\quad \alpha(x,fx)\geq1\quad \Rightarrow\quad \alpha \bigl(fx,f^{2}x\bigr) \geq1 $$
$$ (\mathrm{T}1)'' \quad \alpha(fx,x)\geq1 \quad \Rightarrow\quad \alpha\bigl(f^{2}x,fx\bigr) \geq1 $$
Triangular α-admissible mappings defined by Popescu [13] impose the following definitions.
Definition 1.8
[13]
Let \(f:X\rightarrow X\) be a self-mapping, and \(\alpha:X\times X\rightarrow[0,\infty)\) be a function. Then f is said to be triangular right-α-orbital admissible if f is right-α-orbital admissible and and is said to be triangular left-α-orbital admissible if f is α-orbital admissible and If T satisfies both (T2)′ and (T2)″, then it is called triangular α-orbital admissible.
$$ (\mathrm{T}2)'\quad \alpha(x,y)\geq1\quad \text{and}\quad \alpha(y,fy)\geq 1 \quad \Rightarrow\quad \alpha(x,fy) \geq1 $$
$$ (\mathrm{T}2)'' \quad \alpha(fx,x)\geq1 \quad \text{and} \quad \alpha(x,y)\geq 1\quad \Rightarrow\quad \alpha(fx,y) \geq1. $$
It is easy to conclude that each α-admissible mapping is an α-orbital admissible mapping and each triangular α-admissible mapping is a triangular α-orbital admissible mapping. However, the converses of the statements are false. In the following example, we see that a mapping that is triangular α-orbital admissible need not be triangular α-admissible.
Example 1.9
Let \(X = \{x_{i}: i=1,\ldots, n\}\) for some \(n\geq4\), and \(d : X \times X \to\mathbb{R}_{0}^{+}\) with \(d(x, y) = |x-y|\). We define a self-mapping \(f : X \to X \) such that \(fx_{i}=x_{i}\) for \(i=1,2\), \(fx_{i}=x_{j}\) for \(i,j \in\{ 3,4\}\), \(i\neq j\), \(fx_{i}=x_{i+1}\) for \(i \in\{5,\ldots, n-1\}\), and \(fx_{n}=fx_{5}\). Moreover, let \(\alpha: X \times X \to\mathbb{R}_{0}^{+}\) be such that
$$\alpha(x, y) = \left \{ \textstyle\begin{array}{l@{\quad}l} 1 & \text{if } (x, y) \in \{(x_{1},x_{3}), (x_{1},x_{4}), (x_{3},x_{3}), (x_{4},x_{4}), \\ &\hphantom{\text{if } (x, y) \in{}}{}(x_{3},x_{4}), (x_{4},x_{3}), (x_{3},x_{2}), (x_{4},x_{2}) \}, \\ 0 & \text{otherwise}. \end{array}\displaystyle \right . $$
Note that f is α-orbital admissible since \(\alpha(x_{3},fx_{3}) = \alpha(x_{3},x_{4}) = 1 \) and \(\alpha(x_{4},fx_{4}) = \alpha(x_{4},x_{3}) =1 \). On the other hand, we have \(\alpha(x_{1},x_{3}) = \alpha(x_{3},x_{2}) = 1 \), but \(\alpha(x_{1},x_{2})=0\). Hence, T is not triangular α-admissible.
Definition 1.10
[13]
Let \((X,d)\) be a quasi-b-metric-like space. Then X is said to be α-regular if for every sequence \(\{x_{n}\}\) in X such that \(\alpha(x_{n},x_{n+1})\geq1\) for all n and \(x_{n}\rightarrow x\in X\) as \(n\rightarrow\infty\), there exists a subsequence \(\{x_{n(k)}\}\) of \(\{x_{n}\}\) such that \(\alpha(x_{n(k)},x)\geq1\) for all k.
2 Main result
The notion of \((b)\)-comparison was introduced by Berinde [14] in order to extend the notion of \((c)\)-comparison.
Definition 2.1
[14]
Let \(s\geq1\) be a real number. A mapping \(\psi:\mathbb{R}_{0}^{+}\to\mathbb{R}_{0}^{+}\) is called a \((b)\)-comparison function if the following conditions are fulfilled:
(1)
ψ is monotone increasing;
(2)
there exist \(k_{0}\in\mathbb{N}\), \(a\in(0,1)\), and a convergent series of nonnegative terms \(\sum_{k=1}^{\infty}v_{k}\) such that \(s^{k+1}\psi ^{k+1}(t)\leq a s^{k}\psi^{k}(t)+v_{k}\) for all \(k\geq k_{0}\) and \(t\in[0,\infty)\).
The class of \((b)\)-comparison functions will be denoted by \(\Psi_{b}\). Notice that the notion of a \((b)\)-comparison function reduces to the concept of a \((c)\)-comparison function if \(s=1\).
The following lemma will be used in the proof of our main result.
Lemma 2.2
Let
\(s\geq1\)
be a real number. If
\(\psi :\mathbb{R}_{0}^{+}\to\mathbb{R}_{0}^{+}\)
is a
\((b)\)-comparison function, then:
(1)
the series
\(\sum_{k=0}^{\infty}s^{k}\psi^{k}(t)\)
converges for any
\(t\in\mathbb{R}_{0}^{+}\);
(2)
the function
\(p_{s}:[0,\infty)\to[0,\infty)\)
defined by
is increasing and continuous at 0.
$$p_{s}(t)= \sum_{k=0}^{\infty}s^{k}\psi^{k}(t)\quad \textit{for all } t\in[0,\infty) $$
Remark 2.3
It is easy to see that if \(\psi(t) \in\Psi _{b}\), then \(\psi(t)< t\) for all \(t>0\). In fact, if there is a \(t^{*}>0\) such that \(\psi(t^{*})\geq t^{*}\), then we have \(\psi^{2}(t^{*})\geq\psi(t^{*})\geq t^{*}\) (since ψ is increasing). Continuing in the same manner, we get \(\psi^{n}(t^{*})\geq t^{*}>0\), \(n\in \mathbb{N}\). This contradicts Lemma 2.2.
Definition 2.4
Let \((X,d)\) be a complete quasi-b-metric-like space with a constant \(s \geq1\). A self-mapping \(f: X \to X\) is called \((\alpha ,\psi)\)-contractive mapping if there exist two functions \(\psi\in \Psi_{b}\) and \(\alpha: X \times X \to[0,\infty)\) satisfying the following condition: for all \(x,y \in X\).
$$ \alpha(x,y)d(fx,fy) \leq\psi\bigl(d(x,y)\bigr) $$
(2.1)
Theorem 2.5
Let
\((X,d) \)
be a complete quasi-b-metric-like space, and let
\(f: X \to X\)
be an
\((\alpha,\psi)\)-contractive mapping. Suppose also that
Then
f
has a fixed point
u
in
X, and
\(d(u,u)=0\).
(i)
f
is
α-orbital admissible;
(ii)
there exists
\(x_{0}\in X\)
such that
\(\alpha(x_{0},fx_{0}) \geq1\)
and
\(\alpha(fx_{0},x_{0})\geq1\);
(iii)
f
is continuous.
Proof
By (ii) there exists \(x_{0}\in X\) such that \(\alpha(x_{0},fx_{0})\geq1\) and \(\alpha(fx_{0},x_{0})\geq1\). Define the iterative sequence \(\{x_{n}\}\) in X by \(x_{n+1}=fx_{n}\) for all \(n\in\mathbb{N}_{0}\). Note that if there exists \(n_{0} \in\mathbb{N}_{0}\) such that \(x_{n_{0}}= x_{n_{0}+1}\), then \(x_{n_{0}}\) becomes a fixed point, which completes the proof. Hence, throughout the proof, we suppose that \(x_{n}\neq x_{n+1}\) for all \(n\in\mathbb{N}_{0}\). Regarding the fact that f is α-orbital admissible, from (ii) we derive that Inductively, we get that Analogously, again by (ii) and the fact that f is α-orbital admissible we find that Consequently, we observe that From (2.1), by taking \(x=x_{n}\) and \(y=x_{n-1}\), we find that In view of Remark 2.3, we get that By analogy, again by (2.1) and by substituting \(x=x_{n-1}\) and \(y=x_{n}\), we have Consequently, From (2.4) and (2.5) we derive that By Lemma 2.2(1) and letting \(n \to\infty\) in (2.6), we have \(\lim_{n\rightarrow\infty}d(x_{n},x_{n+1})=\lim_{n\rightarrow \infty}d(x_{n+1}, x_{n})=0\).
$$\alpha(x_{0},x_{1})= \alpha(x_{0}, fx_{0}) \geq1 \quad \Rightarrow \quad \alpha(fx_{0},fx_{1})= \alpha(x_{1},x_{2}) \geq1. $$
$$ \alpha(x_{n},x_{n+1})\geq1\quad \text{for all } n\in\mathbb{N}_{0}. $$
(2.2)
$$\alpha(x_{1},x_{0})= \alpha(fx_{0}, x_{0}) \geq1 \quad \Rightarrow \quad \alpha(fx_{1},fx_{0})= \alpha(x_{2},x_{1}) \geq1. $$
$$ \alpha(x_{n+1},x_{n})\geq1 \quad \text{for all } n\in\mathbb{N}_{0}. $$
(2.3)
$$\begin{aligned} d(x_{n+1},x_{n})&= d(fx_{n},fx_{n-1}) \\ &\leq\alpha(x_{n},x_{n-1})d(fx_{n},fx_{n-1}) \\ & \leq\psi\bigl(d(x_{n},x_{n-1})\bigr). \end{aligned}$$
$$ d(x_{n+1},x_{n}) \leq\psi\bigl(d(x_{n},x_{n-1}) \bigr) < d(x_{n},x_{n-1}) \quad \text{for all } n\in\mathbb {N}. $$
(2.4)
$$\begin{aligned} d(x_{n},x_{n+1})&= d(fx_{n-1},fx_{n}) \\ &\leq\alpha(x_{n-1},x_{n })d(fx_{n-1},fx_{n }) \\ & \leq\psi\bigl(d(x_{n-1},x_{n})\bigr). \end{aligned}$$
$$ d(x_{n},x_{n+1})\leq\psi\bigl(d(x_{n-1},x_{n}) \bigr) < d(x_{n-1},x_{n}) \quad \text{for all } n\in\mathbb {N}. $$
(2.5)
$$ d(x_{n},x_{n+1})\leq\psi^{n} \bigl(d(x_{0},x_{1})\bigr)\quad \text{and}\quad d(x_{n+1},x_{n})\leq\psi^{n}\bigl(d(x_{1},x_{0}) \bigr) \quad \text{for all } n\in\mathbb {N}. $$
(2.6)
We further prove that the sequence \(\{x_{n}\}\) is right-Cauchy and left-Cauchy. For all \(n,p\in\mathbb{N}\), we have By letting \(n,p \to\infty\) we get that that is, the sequence \(\{x_{n}\}\) is right-Cauchy.
$$\begin{aligned} d(x_{n},x_{n+p})&\leq \sum_{i=1}^{p-1} s^{i}d(x_{n+i-1},x_{n+i})+s^{p-1}d(x_{n+p-1},x_{n+p}) \\ &< \sum_{i=1}^{p} s^{i} \psi^{n+i-1}\bigl(d(x_{0},x_{1})\bigr) \\ &= \frac{1}{s^{n-1} } \sum^{n+p-1}_{k=n} s^{k} \psi^{k}\bigl(d(x_{0},x_{1}) \bigr). \end{aligned}$$
$$ \lim_{n,p\to\infty} d(x_{n},x_{n+p})=0, $$
Analogously, that is, the sequence \(\{x_{n}\}\) is left-Cauchy. As a result, the sequence \(\{x_{n}\}\) is a Cauchy sequence. Since \((X,d)\) is complete, there exists a point \(u \in X\) such that Since f is continuous, we have □
$$ \lim_{n,p\to\infty} d(x_{n+p},x_{n})=0, $$
$$ \lim_{n \to\infty}d(u, x_{n})=\lim _{n \to\infty}d( x_{n},u) = d(u,u) =\lim_{n,m\to\infty}d(x_{n}, x_{m})= \lim_{n,m\to\infty}d(x_{m}, x_{n})=0. $$
(2.7)
$$u=\lim_{n \to\infty} x_{n+1}= \lim_{n \to\infty} fx_{n}=fu. $$
Example 2.6
Let \((X,d)\) be a quasi b-metric like space defined in Example 1.4, and let the mapping \(f:X\mapsto X\) be defined as Let \(\psi(t)=\frac{t}{2}\), \(t\geq0\), and let \(\alpha:X\times X\to [0,\infty)\) be defined as Then \(\psi\in\Psi_{b}\), and f is an \((\alpha,\psi)\)-contractive mapping. Since the conditions of Theorem 2.5 are satisfied, it follows that f has a fixed point in X.
$$fx=\left \{ \textstyle\begin{array}{l@{\quad}l} \frac{1}{3} &\text{if } x=0, \\ \frac{1}{2} &\text{if } x=\frac{1}{3}, \\ \frac{1}{2} &\text{if } x=\frac{1}{2}, \\ x+1 &\text{if } x\geq1. \end{array}\displaystyle \right . $$
$$\alpha(x,y)=\left \{ \textstyle\begin{array}{l@{\quad}l} 1 &\text{if } x,y\in\{0,\frac{1}{3},\frac{1}{2}\}, \\ 0 &\text{otherwise}. \end{array}\displaystyle \right . $$
It is possible to remove the heavy condition of continuity of the self-mapping f in Theorem 2.5. For this purpose, we need the following result, which is inspired from the results in [17].
Lemma 2.7
Let
\((X,d)\)
be a quasi-b-metric-like space with constant
s
and assume that
\(\{x_{n}\}\)
and
\(\{y_{n}\}\)
are sequences in
X
converging to
x
and
y, respectively. Then
In particular, if
\(d(x,y)=0 \), then
\(\lim_{n\rightarrow \infty} d(x_{n},y_{n})=0 \).
$$\begin{aligned} \frac{1}{s^{2}} d(x,y)-\frac{1}{s} d(x,x)-d(y,y) &\leq\liminf _{n\rightarrow\infty} d(x_{n},y_{n}) \leq \limsup _{n\rightarrow\infty} d(x_{n},y_{n}) \\ &\leq s d(x,x) + s^{2} d(y,y) + s^{2} d(x,y) . \end{aligned}$$
Moreover, for each
\(z\in X \), we have
If
\(d(x,x)=0\), then
$$ \frac{1}{s} d(x,z)- d(x,x) \leq\liminf_{n\rightarrow\infty} d(x_{n},z) \leq\limsup_{n\rightarrow\infty} d(x_{n},z) \leq s d(x,z) + s d(x,x). $$
(2.8)
$$\frac{1}{s} d(x,z) \leq\liminf_{n\rightarrow\infty} d(x_{n},z) \leq\limsup_{n\rightarrow\infty} d(x_{n},z) \leq s d(x,z). $$
Theorem 2.8
Let
\((X,d) \)
be a complete quasi-b-metric-like space, and let
\(f: X \to X\)
be an
\((\alpha,\psi)\)-contractive mapping. Suppose also that
Then
f
has a fixed point
u
in
X, and
\(d(u,u)=0\).
(i)
f
is
α-orbital admissible;
(ii)
there exists
\(x_{0}\in X\)
such that
\(\alpha(x_{0},fx_{0}) \geq1\)
and
\(\alpha(fx_{0},x_{0})\geq1\);
(iii)
X
is
α-regular.
Proof
By verbatim of the proof of Theorem 2.5 we find an iterative sequence \(\{x_{n}\}\) that converges to a point \(u \in X\) such that (2.7) holds.
Since \(d(u,u)=0\), by Lemma 2.7 we have By letting \(n \to\infty\) in these inequalities we derive that \(\frac {1}{s} d(u,fu) =0\) and hence \(fu=u\). □
$$\begin{aligned} \frac{1}{s} d(u,fu) &\leq\liminf_{n\rightarrow\infty} d(x_{n+1},fu) \\ &\leq\limsup_{n\rightarrow\infty} d(x_{n+1},fu) \\ &=\limsup_{n\rightarrow\infty} d(fx_{n},fu) \\ &\leq\limsup_{n\rightarrow\infty}\alpha (x_{n},u)d(fx_{n},fu) \\ & \leq\limsup_{n\rightarrow\infty}\psi\bigl(d(x_{n},u)\bigr). \end{aligned}$$
It is natural to consider the uniqueness of a fixed point of an \((\alpha,\psi)\)-contractive mapping. We notice that we need to add an additional condition to guarantee the uniqueness. Here, \(\operatorname{Fix}(f)\) denotes the set of all fixed points of f.
(U)
For all \(x,y\in\operatorname{Fix}(f)\), either \(\alpha(x,y)\geq1\) or \(\alpha (y,x)\geq1\).
Theorem 2.9
Proof
Suppose that \(x^{*}\) and \(y^{*}\) are two distinct fixed points of f, so that \(d(x^{*},y^{*})>0\).
If, for example, \(\alpha(x^{*},y^{*})\geq1\), then which is a contradiction. □
$$\begin{aligned} d\bigl(x^{*},y^{*}\bigr)&= d\bigl(fx^{*},fy^{*}\bigr) \\ &\leq\alpha\bigl(x^{*},y^{*}\bigr)d\bigl(fx^{*},fy^{*}\bigr) \\ &\leq\psi\bigl(d\bigl(x^{*},y^{*}\bigr)\bigr) \\ &< d\bigl(x^{*},y^{*}\bigr), \end{aligned}$$
Definition 2.10
Let \((X,d)\) be a complete quasi-b-metric-like space with a constant \(s \geq1\). A self-mapping \(f: X \to X\) is called a generalized \((\alpha,\psi)\)-contractive mapping of type \((A)\) if there exist two functions \(\psi\in\Psi_{b}\) and \(\alpha: X \times X \to[0,\infty)\) satisfying the following condition: for all \(x,y \in X\), where
$$ \alpha(x,y) d(fx,fy) \leq\psi\bigl(M(x,y)\bigr) $$
(2.9)
$$ M(x,y)=\max \bigl\{ d(x,y),d(x,fx),d(y,fy) \bigr\} . $$
(2.10)
Theorem 2.11
Let
\((X,d) \)
be a complete quasi-b-metric-like space, and let
\(f: X \to X\)
be a generalized
\((\alpha,\psi)\)-contractive mapping of type
\((A)\). Assume that
Then
f
has a fixed point
u
in
X, and
\(d(u,u)=0\).
(i)
f
is
α-orbital admissible;
(ii)
there exists
\(x_{0}\in X\)
such that
\(\alpha(x_{0},fx_{0}) \geq1\)
and
\(\alpha(fx_{0},x_{0})\geq1\);
(iii)
f
is continuous.
Proof
As in the proof of Theorem 2.5, we construct an iterative sequence \(x_{n+1}=fx_{n}\), \(n\in\mathbb{N}_{0}\), where the existence of \(x_{0}\in X\) is guaranteed by (ii). By the same reason as in the proof of Theorem 2.5, we may assume that \(x_{n}\neq x_{n+1}\) for all \(n\in\mathbb{N}_{0}\), and we can conclude that From (2.9) we have for all \(n \in\mathbb{N} \), where If \(M(x_{n-1},x_{n})=d(x_{n},x_{n+1})\), then since we assumed that \(x_{n}\neq x_{n+1}\), which is a contradiction. It allows us to conclude that \(M(x_{n-1},x_{n})=d(x_{n-1},x_{n})\), \(n\in\mathbb{N}\).
$$ \alpha(x_{n},x_{n+1})\geq1 \quad \text{and} \quad \alpha (x_{n+1},x_{n})\geq1\quad \text{for all } n\in \mathbb{N}_{0}. $$
(2.11)
$$\begin{aligned} \begin{aligned} d(x_{n},x_{n+1})&= d(fx_{n-1},fx_{n}) \\ &\leq\alpha(x_{n-1},x_{n})d(fx_{n-1},fx_{n}) \\ &\leq\psi\bigl(M(x_{n-1},x_{n})\bigr) \end{aligned} \end{aligned}$$
$$\begin{aligned} M(x_{n-1},x_{n})&=\max \bigl\{ d(x_{n-1},x_{n }),d(x_{n},fx_{n}),d(x_{n-1},fx_{n-1}) \bigr\} \\ &=\max \bigl\{ d(x_{n-1},x_{n}),d(x_{n},x_{n+1}) \bigr\} . \end{aligned}$$
$$d(x_{n},x_{n+1})\leq\psi\bigl(d(x_{n},x_{n+1}) \bigr)< d(x_{n},x_{n+1}), $$
Thus, and
$$d(x_{n},x_{n+1})\leq\psi\bigl(d(x_{n-1},x_{n}) \bigr)< d(x_{n-1},x_{n})\quad \text{for all } n\in\mathbb{N} $$
$$ d(x_{n},x_{n+1})\leq\psi^{n} \bigl(d(x_{0},x_{1})\bigr)\quad \text{for all } n\in \mathbb{N}. $$
(2.12)
Analogously, letting \(x=x_{n}\) and \(y=x_{n-1}\) in (2.9), we get for all \(n \in\mathbb{N} \), where For the estimation of \(d(x_{n+1},x_{n})\), we will consider three different cases.
$$\begin{aligned} d(x_{n+1},x_{n}) =& d(fx_{n},fx_{n-1}) \\ \leq& \alpha(x_{n},x_{n-1})d(fx_{n},fx_{n-1}) \\ \leq& \psi\bigl(M(x_{n},x_{n-1})\bigr) \end{aligned}$$
(2.13)
$$\begin{aligned} M(x_{n},x_{n-1}) =&\max \bigl\{ d(x_{n},x_{n-1}),d(x_{n},fx_{n}),d(x_{n-1},fx_{n-1}) \bigr\} \\ =&\max \bigl\{ d(x_{n},x_{n-1}),d(x_{n},x_{n+1}),d(x_{n-1},x_{n}) \bigr\} . \end{aligned}$$
Case 1. If \(M(x_{n},x_{n-1})=d(x_{n-1},x_{n})\), then, by (2.13),
$$ d(x_{n+1},x_{n})\leq\psi\bigl(d(x_{n-1},x_{n}) \bigr). $$
(2.14)
Case 2. If \(M(x_{n},x_{n-1})=d(x_{n},x_{n+1})\), then By Remark 2.3 we find that
$$ d(x_{n+1},x_{n})\leq\psi\bigl(d(x_{n},x_{n+1}) \bigr). $$
$$d(x_{n+1},x_{n})\leq\psi\bigl(d(x_{n},x_{n+1}) \bigr)< \psi^{n+1} \bigl(d(x_{0},x_{1}) \bigr). $$
Case 3. Otherwise, \(M(x_{n},x_{n-1})=d(x_{n},x_{n-1})\) and Observing (2.14) and (2.15), it follows that, for any \(n\in \mathbb{N}\), Obviously, in all considered cases, we deduce that Since ψ is an increasing function, let Consequently, we have that \(d(x_{n+1},x_{n})\leq\psi^{n}(v)\) and \(d(x_{n},x_{n+1})\leq\psi^{n}(v)\). By applying (bM2) for any \(n,p\in\mathbb{N}\) it follows that Therefore, \(\lim_{n,p\to\infty} d(x_{n},x_{n+p})=0\) and, likewise, \(\lim_{n,p\to\infty} d(x_{n+p},x_{n})=0\). Since, X is complete, there exists \(u\in X\) such that \(\lim_{n\rightarrow \infty}x_{n}=u\) and Furthermore, f is a continuous mapping, and hence \(u=\lim_{n\rightarrow\infty} x_{n}=\lim_{n\rightarrow\infty}fx_{n-1}=fu\). □
$$ d(x_{n+1},x_{n})\leq\psi\bigl(d(x_{n},x_{n-1}) \bigr). $$
(2.15)
$$ d(x_{n+1},x_{n})\leq\max \bigl\{ \psi^{n} \bigl(d(x_{0},x_{1})\bigr), \psi ^{n} \bigl(d(x_{1},x_{0})\bigr) \bigr\} . $$
(2.16)
$$\lim_{n\rightarrow\infty}d(x_{n+1},x_{n})=\lim _{n\rightarrow\infty }d(x_{n},x_{n+1}) =0. $$
$$v=\max \bigl\{ d(x_{0},x_{1}), d(x_{1},x_{0}) \bigr\} . $$
$$\begin{aligned} d(x_{n},x_{n+p}) \leq&\sum_{i=1}^{p-1} s^{i}d(x_{n+i-1},x_{n+i})+s^{p-1}d(x_{n+p-1},x_{n+p}) \\ \leq&\sum_{i=1}^{p} s^{i}d(x_{n+i-1},x_{n+i}) \\ \leq&\sum_{i=1}^{p} s^{i} \psi^{n+i-1}(v) \\ =&\frac{1}{s^{n-1}}\sum_{i=1}^{p} s^{n+i-1}\psi^{n+i-1}(v). \end{aligned}$$
$$ \lim_{n\rightarrow\infty}d(u,x_{n})=\lim _{n\rightarrow\infty }d(x_{n},u)=d(u,u)=0. $$
(2.17)
Theorem 2.12
Proof
Suppose that \(fx^{*}=x^{*}\) and \(fy^{*}=y^{*}\). Then so that \(d(x^{*},y^{*})=0 \Rightarrow x^{*}=y^{*}\). □
$$\begin{aligned} d\bigl(x^{*},y^{*}\bigr)&= d\bigl(fx^{*},fy^{*}\bigr) \\ &\leq\alpha\bigl(x^{*},y^{*}\bigr)\psi\bigl( d\bigl(fx^{*},fy^{*}\bigr)\bigr) \\ &\leq\psi\bigl(M\bigl(x^{*},y^{*}\bigr)\bigr) \\ &=\psi\bigl(d\bigl(x^{*},y^{*}\bigr)\bigr), \end{aligned}$$
In the following example, we show the existence of a function satisfying conditions of Theorem 2.11 but not satisfying conditions of Theorem 2.5.
Example 2.13
Let \((X,d)\) be a quasi-b-metric-like space described in Example 1.4, and \(f:X\mapsto X\) the mapping Let \(\psi(t)=\frac{t}{2}\), \(t\geq0\), and let \(\alpha:X\times X\to [0,\infty)\) be defined as Then (2.1) does not hold, for example, for \(x=0\) and \(y=\frac {1}{3}\), but (2.9) holds, f has a unique fixed point \(u=\frac {1}{2}\), and \(d(u,u)=0\).
$$fx=\left \{ \textstyle\begin{array}{l@{\quad}l} \frac{1}{2} &\text{if } x=0, \\ 0, &\text{if } x=\frac{1}{3}, \\ \frac{1}{2} &\text{if } x=\frac{1}{2}, \\ x+1 &\text{if } x\geq1. \end{array}\displaystyle \right . $$
$$\alpha(x,y)=\left \{ \textstyle\begin{array}{l@{\quad}l} 1 &\text{if } (x,y)\in \{(0,\frac{1}{3}),(0,\frac {1}{2}),(\frac{1}{2},0),(\frac{1}{2},\frac{1}{3}),(\frac {1}{2},\frac{1}{2}) \}, \\ 0 &\text{otherwise}. \end{array}\displaystyle \right . $$
Definition 2.14
Let \((X,d)\) be a complete quasi-b-metric-like space with a constant \(s \geq1\). A self-mapping \(f: X \to X\) is called a generalized \((\alpha,\psi)\)-contractive mapping of type \((B)\) if there exist two functions \(\psi\in\Psi_{b}\) and \(\alpha: X \times X \to[0,\infty)\) satisfying the following condition: for all \(x,y \in X\), where
$$ \alpha(x,y) d(fx,fy) \leq\psi\bigl(N(x,y)\bigr) $$
(2.18)
$$ N(x,y)=\max \biggl\{ d(x,y), \frac{d(x,fx)+d(y,fy)}{2} \biggr\} . $$
(2.19)
The following theorem can be deduced from the inequality \(N(x,y) \leq M(x,y)\) for all x, y, together with the monotonicity of ψ.
Theorem 2.15
Let
\((X,d) \)
be a complete quasi-b-metric-like space, and let
\(f: X \to X\)
be a generalized
\((\alpha,\psi)\)-contractive mapping of type
\((B)\). Assume that
Then
f
has a fixed point
u
in
X, and
\(d(u,u)=0\).
(i)
f
is
α-orbital admissible;
(ii)
there exists
\(x_{0}\in X\)
such that
\(\alpha(x_{0},fx_{0}) \geq1\)
and
\(\alpha(fx_{0},x_{0})\geq1\);
(iii)
f
is continuous.
Definition 2.16
Let \((X,d)\) be a complete quasi-b-metric-like space with a constant \(s \geq1\). A self-mapping \(f: X \to X\) is called a generalized \((\alpha,\psi)\)-contractive mapping of type \((C)\) if there exist two functions \(\psi\in\Psi_{b}\) and \(\alpha: X \times X \to[0,\infty)\) satisfying the following condition: for all \(x,y \in X\), where
$$ s\alpha(x,y) d(fx,fy) \leq\psi\bigl(M(x,y)\bigr) $$
(2.20)
$$ M(x,y)=\max \bigl\{ d(x,y),d(x,fx),d(y,fy) \bigr\} . $$
(2.21)
The following theorem is easily observed from Theorem 2.11 since inequality (2.9) can be easily derived from inequality (2.20).
Theorem 2.17
Let
\((X,d) \)
be a complete quasi-b-metric-like space, and let
\(f: X \to X\)
be a generalized
\((\alpha,\psi)\)-contractive mapping of type
\((C)\). Assume that
Then
f
has a fixed point
u
in
X, and
\(d(u,u)=0\).
(i)
f
is
α-orbital admissible;
(ii)
there exists
\(x_{0}\in X\)
such that
\(\alpha(x_{0},fx_{0}) \geq1\)
and
\(\alpha(fx_{0},x_{0})\geq1\);
(iii)
f
is continuous.
In the next theorems, we establish a fixed point result for a generalized \((\alpha,\psi)\)-contractive mapping of type \((C)\) without any continuity assumption on the mapping f.
Theorem 2.18
Let
\((X,d) \)
be a complete quasi-b-metric-like space, and let
\(f: X \to X\)
be a generalized
\((\alpha,\psi)\)-contractive mapping of type
\((C)\). Suppose that
Then
f
has a fixed point
u
in
X, and
\(d(u,u)=0\).
(i)
f
is
α-orbital admissible;
(ii)
there exists
\(x_{0}\in X\)
such that
\(\alpha(x_{0},fx_{0}) \geq1\)
and
\(\alpha(fx_{0},x_{0})\geq1\);
(iii)
X
is
α-regular.
Proof
As in the proof of Theorem 2.5, we consider an iterative sequence \(\{x_{n}\}\), and we obtain the existence of \(u\in X\) such that (2.17) holds. By Lemma 2.7 we get where
$$\begin{aligned} d(u,fu) \leq& s \liminf_{n\to\infty} d(x_{n+1},fu) \\ \leq& s\limsup_{n\to\infty} d(x_{n+1},fu) \\ \leq& s\limsup_{n\to\infty} \alpha(x_{n},u) d(fx_{n},fu) \\ \leq& \limsup_{n\to\infty} \psi\bigl(M(x_{n},u)\bigr), \end{aligned}$$
$$M(x_{n},u)=\max \bigl\{ d(x_{n-1},u), d(x_{n-1},x_{n}),d(u,fu) \bigr\} . $$
According to (2.17) and the fact that \(\lim_{n\rightarrow\infty }d(x_{n-1},x_{n})=0\), it remains to discuss only the case \(M(x_{n},u)=d(u,fu)\) because otherwise it follows \(d(u,fu)=0 \Rightarrow u=fu\).
Notice that, under this assumption, \(d(u,fu)\leq\psi(d(u,fu))\) also implies \(d(u,fu)=0\) since \(\psi(t)< t\) for any \(t>0\). Hence, u is a fixed point of the mapping f. □
Theorem 2.19
Example 2.20
Let \((X,d)\) be a quasi b-metric like space defined in Example 1.4, and let the mapping \(f:X\mapsto X\) be defined as Let \(\psi(t)=\frac{t}{2}\), \(t\geq0\), and \(\alpha:X\times X\to [0,\infty)\) be defined as Then \(\psi\in\Psi_{b}\), and f is a generalized \((\alpha,\psi )\)-contractive mapping of type \((C)\). Since the conditions of Theorem 2.18 are satisfied, it follows that f has a fixed point in X.
$$fx=\left \{ \textstyle\begin{array}{l@{\quad}l} 1 &\text{if } x=0, \\ \frac{1}{2} &\text{if } x=\frac{1}{3}, \\ \frac{1}{2} &\text{if } x=\frac{1}{2}, \\ x+1 &\text{if } x\geq1. \end{array}\displaystyle \right . $$
$$\alpha(x,y)=\left \{ \textstyle\begin{array}{l@{\quad}l} 1 &\text{if } (x,y)\in \{(\frac{1}{3},\frac {1}{2}),(\frac{1}{2},\frac{1}{2}) \}, \\ \frac{1}{9} &\text{if } (x,y)\in \{0,\frac{1}{3},\frac {1}{2} \}\times \{0,\frac{1}{3},\frac{1}{2} \} \setminus \{(\frac{1}{3},\frac{1}{2}),(\frac{1}{2},\frac {1}{2}) \}, \\ 0 &\text{otherwise}. \end{array}\displaystyle \right . $$
Definition 2.21
Let \((X,d)\) be a complete quasi-b-metric-like space with a constant \(s \geq1\). A self-mapping \(f: X \to X\) is called a generalized \((\alpha,\psi)\)-contractive mapping of type \((D)\) if there exist two functions \(\psi\in\Psi_{b}\) and \(\alpha: X \times X \to[0,\infty)\) satisfying the following condition: for all \(x,y \in X\), where
$$ \alpha(x,y) d(fx,fy) \leq\psi\bigl(L(x,y)\bigr) $$
(2.22)
$$ L(x,y)=\max \biggl\{ d(x,y), \frac{d(x,fx)+d(y,fy)}{2s} \biggr\} . $$
(2.23)
Theorem 2.22
Let
\((X,d)\)
be a complete quasi-b-metric-like space, and let
\(f: X \to X\)
be a generalized
\((\alpha,\psi)\)-contractive mapping of type
\((D)\). Suppose that
Then
f
has a fixed point in
X, that is, there exists
\(u \in X\)
such that
\(fu=u\)
and
\(d(u,u)=0\).
(i)
f
is triangular
α-orbital admissible;
(ii)
there exists
\(x_{0}\in X\)
such that
\(\alpha(x_{0},fx_{0}) \geq1\)
and
\(\alpha(fx_{0},x_{0})\geq1\);
(iii)
X
is
α-regular.
Proof
As in the proof of Theorem 2.11, we consider an iterative sequence \(\{x_{n}\}\) and obtain the existence of \(u\in X\) such that (2.17) holds. By Lemma 2.7 we get where If \(N(x_{n},u)=d(x_{n-1},u)\), then we conclude the result due to (2.17). Taking \(\lim_{n\rightarrow\infty}d(x_{n-1}, x_{n})=0\) into account, we deduce that \(\lim_{n\rightarrow\infty} N(x_{n},u)=\frac{ d(u,fu)}{2s}\). Notice that, under this assumption, \(\frac{ 1}{s}d(u,fu)\leq\psi (\frac{ d(u,fu)}{2s})\) also implies \(d(u,fu)=0\) since \(\psi(t)< t\) for any \(t>0\). Hence, u is a fixed point of the mapping f. □
$$\begin{aligned} \frac{ 1}{s} d(u,fu) \leq& \liminf_{n\to\infty} d(x_{n+1},fu) \\ \leq& \limsup_{n\to\infty} d(x_{n+1},fu) \\ \leq& \limsup_{n\to\infty} \alpha(x_{n},u) d(fx_{n},fu) \\ \leq& \limsup_{n\to\infty} \psi\bigl(N(x_{n},u)\bigr), \end{aligned}$$
$$N(x_{n},u)=\max \biggl\{ d(x_{n-1},u), \frac {d(x_{n-1},x_{n})+d(u,fu)}{2s} \biggr\} . $$
3 Consequences
In this section, we will list some consequences of our main results.
3.1 For standard quasi-b-metric-like
Corollary 3.1
Let
\((X,d)\)
be a complete quasi-b-metric-like space, and let
\(f: X \to X\)
be a mapping such that
for all
\(x,y \in X\), where
\(\psi\in\Psi_{b}\). If
f
is continuous, then
f
has a fixed point
u
in
X, and
\(d(u,u)=0\).
$$ d(fx,fy) \leq\psi\bigl(\max \bigl\{ d(x,y),d(x,fx),d(y,fy) \bigr\} \bigr) $$
(3.1)
Proof
The proof of Corollary 3.1 follows from Theorem 2.12 by taking \(\alpha(x,y)=1\) for all \(x,y \in X\), so (ii) is satisfied for any \(x_{0}\in X\), f is obviously an α-orbital admissible, and (U) holds. Inequality (3.1) allows us to conclude that f is a generalized \((\alpha,\psi)\)-contractive mapping of type \((A)\). □
Corollary 3.2
Let
\((X,d)\)
be a complete quasi-b-metric-like space, and let
\(f: X \to X\)
be a continuous mapping such that
for all
\(x,y \in X\), where
\(\psi\in\Psi_{b}\). Then
f
has a fixed point
u
in
X, and
\(d(u,u)=0\).
$$ d(fx,fy) \leq\psi \biggl(\max \biggl\{ d(x,y),\frac {d(x,fx)+d(y,fy)}{2} \biggr\} \biggr) $$
(3.2)
Proof
Notice that the continuity condition of f in Corollary 3.1 can be removed by adding an extra term s.
Corollary 3.3
Let
\((X,d) \)
be a complete quasi-b-metric-like space, and let
\(f: X \to X\)
be a mapping such that
for all
\(x,y \in X\), where
\(\psi\in\Psi_{b}\). Then
f
has a fixed point
u
in
X
such that
\(d(u,u)=0\).
$$ d(fx,fy) \leq s \psi\bigl(\max \bigl\{ d(x,y),d(x,fx),d(y,fy) \bigr\} \bigr) $$
(3.3)
Proof
The proof of Corollary 3.3 follows from Theorem 2.18 by taking \(\alpha(x,y)=1\) for all \(x,y \in X\). Then f is an α-orbital admissible mapping, and both inequalities in (ii) hold for any \(x_{0}\in X\). Notice that since \(\alpha(x,y)=1\), any constructive sequence turns to be regular, and thus X is α-regular. □
Corollary 3.4
Let
\((X,d) \)
be a complete quasi-b-metric-like space, and let
\(f: X \to X\)
be a mapping such that
for all
\(x,y \in X\), where
\(\psi\in\Psi_{b}\). Then
f
has a fixed point
u
in
X
such that
\(d(u,u)=0\).
$$ d(fx,fy) \leq\psi\bigl(d(x,y)\bigr) $$
(3.4)
3.2 For standard quasi-b-metric-like spaces with a partial order
In this section, we deduce various fixed point results on a quasi-b-metric-like space endowed with a partial order. We, first, recollect some basic notions and notation.
Definition 3.5
Let \((X,\preceq)\) be a partially ordered set, and \(f: X\to X\) be a given mapping. We say that f is nondecreasing with respect to ⪯ if for all \(x,y\in X\),
$$x\preceq y\quad \Rightarrow \quad fx\preceq fy. $$
Definition 3.6
Let \((X,\preceq)\) be a partially ordered set. A sequence \(\{x_{n}\} \subseteq X\) is said to be nondecreasing (respectively, nonincreasing) with respect to ⪯ if \(x_{n}\preceq x_{n+1}\), \(n\in\mathbb{N}\) (respectively, \(x_{n+1}\preceq x_{n}\), \(n\in\mathbb{N}\)).
Definition 3.7
Let \((X,\preceq)\) be a partially ordered set, and d be a b-metric-like on X. We say that \((X,\preceq,d)\) is regular if for every nondecreasing (respectively, nonincreasing) sequence \(\{x_{n}\} \subseteq X\) such that \(x_{n}\to x\in X\) as \(n\to\infty\), there exists a subsequence \(\{ x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \(x_{n_{k}}\preceq x\) (respectively, \(x\preceq x_{n_{k}}\)) for all k.
We have the following result.
Corollary 3.8
Let
\((X,\preceq)\)
be a partially ordered set (which does not contain an infinite totally unordered subset), and
d
be a
b-metric-like on
X
with constant
\(s\geq1\)
such that
\((X,d)\)
is complete. Let
\(f: X\to X\)
be a nondecreasing mapping with respect to ⪯. Suppose that there exists
\(\psi\in\Psi_{b}\)
such that
for all
\(x,y \in X\)
with
\(x\preceq y\)
or
\(y\preceq x\), where
\(M(x,y)\)
is defined as in (2.10). Suppose also that the following conditions hold:
Then
f
has a fixed point
\(u \in X\)
with
\(d(u,u)=0\). Moreover, if for all
\(x,y\in X\), there exists
\(z\in X\)
such that
\(x\preceq z\)
and
\(y\preceq z\), then
f
has a unique fixed point.
$$ d(fx,fy) \leq\psi\bigl(M(x,y)\bigr) $$
(3.5)
(i)
there exists
\(x_{0}\in X\)
such that
\(x_{0}\preceq fx_{0}\)
and
\(fx_{0}\preceq x_{0}\);
(ii)
f
is continuous or
(ii)′
\((X,\preceq,d)\)
is regular, and
d
is continuous.
Proof
Define the mapping \(\alpha: X\times X\to[0,\infty)\) by Clearly, f satisfies (2.20), that is, for all \(x,y\in X\). From condition (i) we have \(x_{0}\preceq fx_{0}\) and \(fx_{0}\preceq x_{0}\). Moreover, for all \(x,y\in X\), from the monotone property of f we have Hence, the self-mapping f is α-admissible. Similarly, we can prove that f is triangular α-admissible and so triangular α-orbital admissible. Now, if f is continuous, then the existence of a fixed point follows from Theorem 2.17.
$$ \alpha(x,y)=\left \{ \textstyle\begin{array}{l@{\quad}l} 1 &\mbox{if } x\preceq y \mbox{ or } x\succeq y, \\ 0 &\mbox{otherwise}. \end{array}\displaystyle \right . $$
$$\alpha(x,y)s d(fx, fy)\leq\psi\bigl(M(x,y)\bigr) $$
$$\begin{aligned}& \alpha(x,y)\geq1\quad \Rightarrow \quad x\succeq y\quad \mbox{or} \\& x \preceq y \quad \Rightarrow \quad fx\succeq fy \quad \mbox{or} \\& fx\preceq fy \quad \Rightarrow \quad \alpha(fx,fy)\geq1. \end{aligned}$$
Suppose that \((X,\preceq,d)\) is regular. Let \(\{x_{n}\}\) be a sequence in X such that \(\alpha(x_{n},x_{n+1})\geq1\) for all n and \(x_{n} \rightarrow x\in X\) as \(n\rightarrow\infty\). By the regularity hypothesis, since X does not contain an infinite totally unordered subset, there exists a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \(x_{n_{k}}\preceq x\) or \(x\preceq x_{n_{k}}\) for all k.
This implies from the definition of α that \(\alpha (x_{n_{k}},x)\geq1\) for all k. In this case, the existence of a fixed point follows again from Theorem 2.18. □
Corollary 3.9
Let
\((X,\preceq)\)
be a partially ordered set (which does not contain an infinite totally unordered subset), and
d
be a
b-metric-like on
X
with constant
\(s\geq1\)
such that
\((X,d)\)
is complete. Let
\(f: X\to X\)
be a nondecreasing mapping with respect to ⪯. Suppose that there exists
\(\psi\in\Psi_{b}\)
such that
for all
\(x,y \in X\)
with
\(x\succeq y\)
or
\(y\succeq x\). Suppose also that the following conditions hold:
Then
T
has a fixed point
\(u \in X\)
with
\(d(u,u)=0\). Moreover, if for all
\(x,y\in X\), there exists
\(z\in X\)
such that
\(x\preceq z\)
and
\(y\preceq z\), we have the uniqueness of a fixed point.
$$ d(fx,fy) \leq\psi\bigl(d(x,y)\bigr) $$
(3.6)
(i)
there exists
\(x_{0}\in X\)
such that
\(x_{0}\preceq fx_{0}\)
and
\(fx_{0}\preceq x_{0}\);
(ii)
f
is continuous.
Acknowledgements
The authors are grateful to the reviewers for their careful reviews and useful comments. The first and third authors were supported by Grant No. 174025 of the Ministry of Science, Technology and Development, Republic of Serbia.
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.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.