 Research
 Open access
 Published:
Fixed point theorems for generalized \((\alpha , \phi )\)Meir–Keeler type hybrid contractive mappings via simulation function in bmetric spaces
Fixed Point Theory and Algorithms for Sciences and Engineering volume 2024, Article number: 4 (2024)
Abstract
In this paper, we introduce the notion of generalized \((\alpha , \phi )\)Meir–Keeler hybrid contractive mappings of type I and II via simulation function and establish fixed point theorems for such mappings in the setting of complete bmetric spaces. Our results extend and generalize many related fixed point results in the existing literature. Finally, we provide an example in support of our main finding.
1 Introduction
Fixed point theory is one of the most important topics in development of nonlinear and mathematical analysis in general. Also, fixed point theory has been effectively used in many other branches of science such as chemistry, physics, biology, economics, computer science, all engineering fields, and so on. In 1922, Banach [1] introduced a wellknown fixed point result, now called Banach contraction principle, which is one of the pivotal results in nonlinear analysis. Due to its importance and fruitful applications, several authors have obtained many interesting extensions and generalizations of the Banach contraction principle in several direction (see, e.g., [2, 3]). These generalizations are achieved either by using contractive conditions or by imposing some additional conditions on the ambient spaces. For example, one of the important and peculiar generalizations is due to Meir and Keeler [4]. The class of Meir–Keeler contractions consists of the class of Banach contractions and many other classes of nonlinear contractions (see, for example, [5]). Meir and Keeler’s theorem was the originator of further exploration in metric fixed point theory. Later on, Meir–Keeler contraction mapping has been generalized by several authors in several ways. For more works in this line of research, we refer to [6–8], as well as [9–14]. On the other hand, the notion of a bmetric space was introduced by Bakhtin [15] and Czerwik [16] as a generalization of metric spaces. Since then, several papers have been published on the fixed point theory in such spaces which are interesting extensions and generalizations of the Banach contraction principle. For further works in the setting of bmetric spaces and their generalization, we refer the readers to [17–42]. In 2020, Karapinar et al. [43] studied fixed point results for the Meir–Keeler contraction via simulation function in the setting of metric spaces. Inspired and motivated by the work of Karapinar et al. [43], the main objectives of this research is to introduce the notion of generalized \((\alpha , \phi )\)Meir–Keeler hybrid contractive mappings of type I and II via simulation function and establish fixed point theorems for the introduced mappings in the setting of bmetric spaces. The present results extend and generalize the results of Karapinar et al. [43] and many other related results in the existing literature.
2 Preliminaries
In what follows we recall basic definitions and results on the topics which we use in the sequel.
Notations 1
Throughout this paper, we denote \(\mathbb{R}^{+}\), \(\mathbb{R}\) and \(\mathbb{N}\) respectively by

\(\mathbb{R}^{+} = [0,\infty ) \) – the set of all nonnegative real numbers;

\(\mathbb{R}\) – the set of all real numbers;

\(\mathbb{N}\) – the set of all natural numbers.
Khojasteh et al. [44] introduced the notion of a simulation function as follows.
Definition 1
([44])
A weak simulation function is a mapping \(\zeta : \mathbb{R}^{+} \times \mathbb{R}^{+} \rightarrow \mathbb{R}\) satisfying the following conditions:
 \((\zeta _{1})\):

\(\zeta (0, 0) = 0\);
 \((\zeta _{2})\):

\(\zeta (t, s) < st\) for all \(t, s > 0\).
Note
Throughout this paper we denote by \(Z_{w}\) the family of all simulation functions \(\zeta : \mathbb{R}^{+} \times \mathbb{R}^{+} \rightarrow \mathbb{R}\). Due to the axiom \((\zeta _{2})\), we have \(\zeta (t, t) < 0\) for all \(t > 0\).
Recently, Suzuki [45] introduced the following class of mappings and proved the following interesting fixed point result to extend the coverage of Meir–Keeler theorem in the setting of metric spaces. Let \((X, d)\) be a metric space and \(T : X \rightarrow X\) be a selfmapping. Define a mapping \(M : X \times X \rightarrow \mathbb{R}^{+}\) as follows:
And let \(p : X \times X \rightarrow \mathbb{R}^{+}\) be a mapping satisfies the following conditions:
 \((P^{1}_{p}: M)\):

\(x\neq y\) and \(d(x, Tx) \leq d(x, y)\) imply \(p(x, y) \leq M(x, y)\);
 \((P^{2}_{p}: c)\):

\(x_{n} \neq y\), \(\lim_{n\rightarrow \infty}d(x_{n}, y) = 0\), and \(\lim_{n\rightarrow \infty}d(x_{n}, Tx_{n}) = 0\) imply
$$ \limsup_{n\rightarrow \infty}d(x_{n}, y)\leq cd(y, Ty),\quad \text{where } c \in [0, 1). $$
Theorem 1
([45])
Let T be a selfmapping on a complete metric space \((X, d)\). Let \(p : X \times X \rightarrow \mathbb{R}^{+} \) be mapping that satisfies the conditions \((P^{1}_{p} : M)\) and \((P^{2}_{p} : c)\) defined above. Suppose also that the following are satisfied:

(i)
For any \(\epsilon > 0\), there exists \(\delta (\epsilon ) > 0\) such that \(x \neq y\) and \(p(x, y) < \epsilon +\delta (\epsilon )\) imply \(d(Tx, Ty)\leq \epsilon \);

(ii)
\(x \neq y\) and \(p(x, y) > 0\) imply \(d(Tx, Ty) < p(x, y)\).
Then T has a unique fixed point z. Moreover, the sequence \(\{T^{n}x\}\) converges to z for all \(x \in X\).
Bakhtin [15] and Czerwik [16] defined a bmetric space as follows.
Definition 2
Let X be a nonempty set and \(s \geq 1 \) be a given real number. A function \(d:X \times X \rightarrow \mathbb{R}^{+} \) is said to be a bmetric if and only if for all \(x, y, z \in X \), the following conditions are satisfied:

(a)
\(d(x,y) = 0 \) if and only if \(x=y \);

(b)
\(d(x,y)=d(y,x) \);

(c)
\(d(x,z) \leq s[d(x,y)+d(y,z)] \).
The pair \((X,d) \) is called a bmetric space.
It should be noted that the class of bmetric spaces is effectively larger than that of metric spaces. A metric space is a bmetric with \(s = 1 \). But, in general, the converse is not true.
Example 1
([32])
Let \(X =\mathbb{R} \) and \(d:X \times X\rightarrow \mathbb{R}^{+} \) be given by \(d(x,y)=x  y^{2} \) for \(x ,y \in X \), then d is a bmetric on X with \(s = 2 \) but it is not a metric on X since for \(x=2\), \(y=4 \), and \(z=6 \), we have
Hence, the triangle inequality for a metric does not hold.
Definition 3
([46])
Let X be a bmetric space and \(\{x_{n}\} \) a sequence in X. We say that

1.
\(\{x_{n}\}\) is bconvergent to \(x \in X \) if \(d(x_{n},x )\rightarrow 0 \) as \(n \rightarrow \infty \).

2.
\(\{x_{n} \} \) is a bCauchy sequence if \(d(x_{n} ,x_{m} ) \rightarrow 0 \) as \(n,m \rightarrow \infty \).

3.
\((X,d )\) is bcomplete if every bCauchy sequence in X is bconvergent.
Definition 4
([47])
Let \((X,d)\) be a bmetric space with the coefficient \(s \geq 1 \) and let \(T\colon X\rightarrow X \) be a given mapping. We say that T is bcontinuous at \(x_{0} \in X \) if and only if for every sequence \({x_{n}} \in X\) such that \({x_{n}}\rightarrow x_{0} \) as \(n\rightarrow \infty \), we have \(Tx_{n} \rightarrow Tx_{0} \) as \(n\rightarrow \infty \). If T is bcontinuous at each point \(x \in X \), then we say that T is bcontinuous on X.
In general, a bmetric is not necessarily continuous.
Example 2
([48])
Let \(X= \mathbb{N} \cup \{\infty \} \).
Define a mapping \(d : X \times X \rightarrow \mathbb{R}^{+}\) as follows:
Observe that \(d(m,p)\leq \frac{3}{2}[d(m,n)+d(n,p)]\) for all \(m,n\), \(p\in X \).
Then \((X,d)\) is a bmetric space with \(s = \frac{3}{2} \).
If we choose \(x_{n} = 2n\) for each \(n\in \mathbb{N}\), then
that is, \(x_{n} \rightarrow \infty\) as \(n\rightarrow \infty \).
But \(\lim_{n\rightarrow \infty} d(x_{n},1) = 2\neq 5= d(\infty ,1)\). Hence, d is not continuous.
The following are definitions of αorbital admissible and triangular αorbital admissible mappings.
Definition 5
([49])
Let X be a nonempty set and \(\alpha : X \times X \rightarrow \mathbb{R}^{+}\) a function. A mapping \(T: X \rightarrow X\) is said to be αorbital admissible if, for all \(x\in X\), \(\alpha (x,Tx) \geq 1\) implies \(\alpha (Tx,T^{2} x) \geq 1\).
Definition 6
([49])
Let X be a nonempty set, \(T : X \rightarrow X\), and \(\alpha : X \times X \rightarrow \mathbb{R}^{+}\). We say that T is triangular αorbital admissible if:

(i)
T is αorbital admissible;

(ii)
for all \(x, y \in X\), \(\alpha (x, y) \geq 1\) and \(\alpha (y, T y) \geq 1\) imply that \(\alpha (x, T y) \geq 1\).
In 2020, Karapinar et al. [43] introduced the class of hybrid contraction mappings of type I and II and studied fixed point results for such mappings.
Definition 7
([43])
Let T be a selfmapping on a metric space \((X, d)\) and \(\zeta \in Z_{w}\). Suppose that \(p : X \times X \rightarrow \mathbb{R}^{+}\) is a function that satisfies only \((P^{1} p : M)\). Then T is called a hybrid contraction of type I if the following conditions are fulfilled:

(a)
For any \(\epsilon > 0\), there exists \(\delta (\epsilon ) > 0\) such that \(x \neq y\) and \(p(x, y) < \epsilon + \delta (\epsilon )\) imply \(d(Tx, Ty) \leq \epsilon \);

(b)
\(x \neq y\) and \(p(x, y) > 0\) imply \(\zeta (\alpha (x, y)d(Tx, Ty), p(x, y)) \geq 0\).
Let a mapping \(N : X \times X \rightarrow \mathbb{R}^{+}\) be defined as follows:
where T is a selfmapping defined on a metric space \((X, d)\). We notice that, for any \(x, y \in X\) with \(x = y\), we have \(0 = d(Tx, Ty) \leq N(x, y)\). Moreover, if \(x \neq y\), then \(N(x, y) > 0\).
Definition 8
([43])
Let T be a selfmapping on a metric space \((X, d)\) and \(\zeta \in Z_{w}\). Suppose that \(p : X \times X \rightarrow \mathbb{R}^{+}\) is a function that satisfies \((P^{1} p : N)\) and \((P^{2} p : c)\), for all \(c \in [0, 1)\). Then T is called a hybrid contraction of type II if the following conditions are satisfied:

(a)
For any \(\epsilon > 0\) there exists \(\delta (\epsilon ) > 0\) such that \(x \neq y\) and \(p(x, y) < \epsilon + \delta (\epsilon )\) imply \(d(Tx, Ty) \leq \epsilon \);

(b)
\(x \neq y\) and \(p(x, y) > 0\) imply \(\zeta (\alpha (x, y)d(Tx, Ty), p(x, y)) \geq 0\).
Theorem 2
([43])
Let \((X, d)\) be a complete metric space and \(T : X \rightarrow X\) be a hybrid contraction of type I. Assume that the following conditions are satisfied:

(i)
T is triangular αorbital admissible;

(ii)
there exists \(x_{0} \in X\) such that \(\alpha (x_{0}, Tx_{0}) \geq 1\);

(iii)
T is continuous.
Then T has a fixed point u. Moreover, \(\{T^{n}x\}\) converges to u for all \(x \in X\).
Theorem 3
([43])
Let \((X, d)\) be a complete metric space and \(T : X \times X\) be a hybrid contraction of type II. Assume that the following conditions are fulfilled:

(i)
T is triangular αorbital admissible;

(ii)
there exists \(x_{0} \in X\) such that \(\alpha (x_{0}, Tx_{0}) \geq 1\);

(iii)
either T is continuous;

(iv)
or \(T^{2}\) is continuous and \(\alpha (u,Tu)\geq 1\);

(v)
or \((X, d)\) is regular.
Then T has a fixed point u. Moreover, \(\{T^{n}x\}\) converges to u for all \(x \in X\).
3 Results
In this section, first we introduce generalized \((\alpha , \phi )\)Meir–Keeler hybrid contractive mapping of type I in the setting of bmetric spaces and prove fixed point results for such mappings.
Note
In this section, we denote the class of mappings Ψ by
Let \((X, d)\) be a bmetric space with \(s\geq 1\) and \(T : X \rightarrow X\) be a selfmapping. We define a mapping \(M_{s} : X \times X \rightarrow \mathbb{R}^{+}\) by
Let also \(p : X \times X \rightarrow \mathbb{R}^{+}\) be a mapping. The following conditions are used in this section:
 \((P^{1}_{p} : M_{s})\):

\(x\neq y\) and \(d(x, Tx) \leq d(x, y)\) imply \(p(x, y) \leq M_{s}(x, y)\);
 \((P^{2}_{p}: sc)\):

\(x_{n} \neq y\), \(\lim_{n\rightarrow \infty}d(x_{n}, y) = 0\) and \(\lim_{n\rightarrow \infty}d(x_{n}, Tx_{n}) = 0\) imply \(\limsup_{n\rightarrow \infty}(sd(x_{n}, y))\leq cd(y, Ty)\), where \(c \in [0, 1)\).
Definition 9
Let \((X,d)\) be a bmetric space with \(s\geq 1\), \(T:X\rightarrow X\), \(\alpha :X\times X\rightarrow \mathbb{R}^{+}\), \(p : X \times X \rightarrow \mathbb{R}^{+}\) satisfy \((P^{1}_{p} : M_{s})\), and \(\phi \in \varPsi \). Then the mapping T is said to be a generalized \((\alpha , \phi )\)Meir–Keeler hybrid contractive mapping of type I if it satisfies, for all \(x,y\in X\), the following conditions:

(i)
For any \(\epsilon > 0\), there exists \(\delta (\epsilon ) > 0\) such that \(x \neq y\) and \(p(x, y) < \epsilon +\delta (\epsilon )\) imply \(d(Tx, Ty)\leq \frac {\epsilon}{s}\);

(ii)
\(x \neq y\) and \(p(x, y) > 0\) imply \(\zeta (\alpha (x,y)\phi (d(Tx, Ty)), \phi (p(x, y)))\geq 0 \).
Remark 1
If T is a generalized \((\alpha , \phi )\)Meir–Keeler hybrid contractive mapping of type I, then
Indeed, we have \(d(x,y)>0\) since \(x \neq y\). If \(p(x,y)=0\), from (ii), we have \(\phi (d(Tx, Ty))<\epsilon \) for any \(\epsilon >0\). But \(\epsilon >0\) is arbitrary, thus we obtain \(Tx=Ty\). In this case, \(\alpha (x,y)\phi (d(Tx, Ty))=0\leq \phi (p(x, y))\). Otherwise, \(p(x, y)>0\), and if \(Tx\neq Ty\), then \(d(Tx, Ty)>0\). If \(\alpha (x,y)=0\), then (1) is satisfied. On the other hand, from \((\zeta _{2})\) and Definition 9(ii), we get
so (1) holds.
Now, we give our first main result as follows:
Theorem 4
Let \((X,d)\) be a complete bmetric space with \(s\geq 1\), \(T:X\rightarrow X\), \(\alpha :X\times X\rightarrow \mathbb{R}^{+}\) be mappings, and \(\phi \in \varPsi \). Suppose the following conditions hold:

(i)
T is generalized \((\alpha , \phi )\)Meir–Keeler hybrid contractive mapping of type I;

(ii)
T is a triangular αorbital admissible mapping;

(iii)
There exists \(x_{0}\in X\) such that \(\alpha (x_{0},Tx_{0})\geq 1\);

(iv)
T is bcontinuous.
Then T has a fixed point z. Moreover, \(\{T^{n}x\}\) converges to z for all \(x\in X\).
Proof
By (iii) above, there exists \(x_{0}\in X \) such that \(\alpha (x_{0},Tx_{0})\geq 1\). We construct an iterative sequence \(\{x_{n}\}\) in X by \(x_{n}=Tx_{n1}\) for \(n\in \mathbb{N}\). Suppose first that \(x_{n_{0}}=x_{n_{0}+1}\) for some \(n_{0}\in \mathbb{N}\). Since \(Tx_{n_{0}} =x_{n_{0}+1} =x_{n_{0}}\), the point \(x_{n_{0}}\) is a fixed point of T and this completes the proof. So from now on, we suppose that \(x_{n}\neq x_{n+1}\) for all \(n\in \mathbb{N}\cup \{0\} \). Since T is triangular αorbital admissible, \(\alpha (x_{0},Tx_{0} )=\alpha (x_{0},x_{1} )\geq 1\Rightarrow \alpha (Tx_{0},Tx_{1} )=\alpha (x_{1},x_{2} )\geq 1\Rightarrow \alpha (Tx_{1},Tx_{2} )=\alpha (x_{2},x_{3} )\geq 1\). Continuing in this manner, we get
Again, by using the assumption that T is triangular αorbital admissible, for all \(n\in \mathbb{N}\cup \{0\} \), (2) yields that \(\alpha (x_{n},x_{n+1})\geq 1\) and \(\alpha (x_{n+1},x_{n+2})\geq 1 \Rightarrow \alpha (x_{n},x_{n+1}) \geq 1\). Recursively, we conclude that \(\alpha (x_{n},x_{n+j})\geq 1\) for all \(n,j\in \mathbb{N}\). In what follows we prove that the sequence \(\{d(x_{n},x_{n+1})\}\) is monotone decreasing. Taking \(x=x_{n}\) and \(y=x_{n+1}\) in \((P^{1}_{p} : M_{s})\), we get
which implies
where
and, taking the btriangle inequality into account, we observe that
which gives
By Definition 9(ii), we get that
which is equivalent to
If \(M_{s}(x_{n}, x_{n+1})=d(x_{n+1}, x_{n+2})\), then (3) yields a contradiction. Thus, we have
Moreover, from (3), we get
which implies, using the monotonicity of ϕ,
that is, \(\{d(x_{n},x_{n+1})\}\) is a monotone decreasing sequence of nonnegative real numbers. Thus, there is some \(l\geq 0\) such that \(\lim_{n\rightarrow \infty} d(x_{n}, x_{n+1})=l\). We need to show \(l=0\). Suppose, on the contrary, that \(l>0\) and set \(0<\epsilon =l\). We also note that
On the other hand, from (3) and (4), we have
for n sufficiently large. So, applying Definition 9(i), we have
Combining (5) together with (6), we obtain
which is a contradiction. We conclude that \(\epsilon =0\), that is,
Now, we show that \(\{x_{n}\}\) is a bCauchy sequence. Let \(\epsilon _{1}>0\) be fixed. From (7), we can choose \(k\in \mathbb{N}\) large enough such that
for some \(\delta _{1}>0\). Without loss of generality, we assume that \(\delta _{1}=\delta _{1}(\epsilon _{1})<\epsilon _{1}\). By induction, we prove that
We already have (9) from (8), for \(m=1\). Suppose that (9) is satisfied for some \(m=j\). Now, we show that (9) holds for \(m=j+1\). On account of (8) and (9), we first observe that
Thus, we have
From the above inequality, we have
and, by Definition 9(i), we get
Now, using the btriangle inequality, as well as (8) and (10), we have
So, (9) holds for \(m=j+1\). Therefore,
In other words, for \(m>n\), we have \(\lim_{n,m \rightarrow \infty} d(x_{n},x_{m})=0\) and hence the sequence \(\{x_{n}\}\) is a bCauchy sequence. Since, \((X,d)\) is a complete bmetric space, there exists \(u\in X\) such that \(x_{n}\rightarrow u\) as \(n\rightarrow \infty \). By bcontinuity of T, we have
that is, u is a fixed point of T. □
Now, replacing continuity of T by continuity of \(T^{2}\) in Theorem 4, we prove the following fixed point result.
Theorem 5
Let \((X,d)\) be a complete bmetric space with \(s\geq 1\) and let \(T:X\rightarrow X\) be a generalized \((\alpha , \phi )\)Meir–Keeler hybrid contractive mapping of type I satisfying the following conditions:

(i)
T is a triangular αorbital admissible mapping;

(ii)
There exists \(x_{0}\in X\) such that \(\alpha (x_{0},Tx_{0})\geq 1\);

(iii)
\(T^{2}\) is continuous.
Then \(\{T^{n}x\}\) is converges to z for all \(x\in X\). Moreover, for \(\alpha (z,Tz)\geq 1\), z is a fixed point of T, and T is discontinuous at z if and only if \(\lim_{x\rightarrow z}M_{s}(x,z)\neq 0\).
Proof
Following the proof of Theorem 4, we see that the sequence \(\{x_{n}\}\) in X defined by \(x_{n}=Tx_{n1}\) for all \(n\in \mathbb{N}\) is convergent to \(z\in X\) and \(\alpha (x_{n}, x_{n+1})\geq 1\) for all \(n\in \mathbb{N}\cup \{0\}\). Regarding the fact that any subsequence of \(\{x_{n}\}\) converges to z, we get \(\lim_{n\rightarrow \infty}x_{n+1 }=\lim_{n \rightarrow \infty}Tx_{n}=z\) and \(\lim_{n\rightarrow \infty}x_{n+2 }=\lim_{n \rightarrow \infty}T^{2}x_{n}=z\). On the other hand, due to the continuity of \(T^{2}\),
We claim that \(Tz=z\). Suppose, on the contrary, that \(Tz\neq z\) and \(p(z,Tz)>0\). Then we have
Thus, using (1) together with the hypothesis \(\alpha (z,Tz)\geq 1\), we obtain
and also
which is a contradiction. So, \(z=Tz\), that is, z is a fixed point of T. □
Definition 10
A bmetric space \((X,d)\) is called regular if for any sequence \(\{x_{n}\}\) in X with \(\lim_{n\rightarrow \infty}d(x_{n},z)=0\) and \(\alpha (x_{n}, x_{n+1})\geq 1\) for all \(n\in \mathbb{N}\cup \{0\}\), one has \(\alpha (x_{n}, z)\geq 1\) for all \(n\in \mathbb{N}\cup \{0\}\).
In the following, we prove the following fixed point theorem, without continuity assumption of T and \(T^{2}\).
Theorem 6
Let \((X,d)\) be a complete bmetric space with \(s\geq 1\) and \(T:X\rightarrow X\) be a generalized \((\alpha , \phi )\)Meir–Keeler hybrid contractive mapping of type I. Suppose that \((P_{p}^{2}: sc)\) and the following conditions hold:

(i)
T is a triangular αorbital admissible mapping;

(ii)
There exists \(x_{0}\in X\) such that \(\alpha (x_{0},Tx_{0})\geq 1\);

(iii)
\((X,d)\) is regular.
Then \(\{T^{n}x\}\) is converges to z for all \(x\in X\). Moreover, z is a fixed point of T.
Proof
Following the proof of Theorem 4, we see that the sequence \(\{x_{n}\}\) in X defined by \(x_{n}=Tx_{n1}\) for all \(n\in \mathbb{N}\) is convergent to \(z\in X\) and \(\alpha (x_{n}, x_{n+1})\geq 1\) for all \(n\in \mathbb{N}\cup \{0\}\). We notice also that all adjacent terms in \(\{x_{n}\}\) are distinct. Moreover, we note \(T^{n}x\neq z\) for all \(n\in \mathbb{N} \cup \{0\}\). Regarding the limits \(\lim_{n\rightarrow \infty}d(x_{n},z)=0\) and \(\lim_{n\rightarrow \infty}d(x_{n},x_{n+1})=0\), we drive from \((P^{2}_{p}:sc)\) that
So, by assumption (iii), we get \(\alpha (x_{n},z)\geq 1\). Now, we prove that z is a fixed point of T. Suppose, on the contrary, that \(Tz\neq z\). Taking \(x=x_{n}\) and \(y=z\) in Definition 9(ii), we obtain that
which is equivalent to
Since ϕ is monotone, (12) yields
Applying the btriangle inequality and using (13), we have
Taking the limit as \(n\rightarrow \infty \) in (14) and using \((P^{2}_{p} :sc)\), we obtain that
which is a contradiction. Therefore, z is a fixed point of T. □
For the uniqueness of fixed point, we need the following additional condition.
Condition (U)
For all \(x,y \in \mathrm{Fix}(T)\), we have \(\alpha (x,y)\geq 1\), where \(\mathrm{Fix}(T)\) denotes the set of all fixed points of T.
Theorem 7
Adding Condition (U)to the hypotheses of Theorem 4 (resp. Theorems 5and 6), we prove the uniqueness of fixed point of T.
Proof
We argue by contradiction, that is, suppose there exist \(z,w \in X\) such that \(z=Tz\) and \(w=Tw\) with \(z\neq w\). By Condition (U), we have \(\alpha (z,w)\geq 1\). We notice first that the case \(p(z,w)=0\) is impossible since we have \(Tz=Tw\) and
which is a contradiction. Thus, we get that \(p(z,w)>0\). Since
by \((P^{1}_{p}:M_{s})\), we have
where
Using Definition 9(ii), we get
which imply
which is a contradiction. Hence, \(d(z,w)=0\), that is, the fixed point of T is unique. □
In the following, we introduce generalized \((\alpha , \phi )\)Meir–Keeler hybrid contractive mapping of type II and study fixed point results for such mappings.
Definition 11
Let \((X,d)\) be a bmetric space with \(s\geq 1\), \(T:X\rightarrow X\), \(\alpha :X\times X\rightarrow \mathbb{R}^{+}\), \(\zeta \in Z_{w}\), \(\phi \in \varPsi \), and suppose \(p :X\times X\rightarrow \mathbb{R}^{+}\) is a function that satisfies \((P^{1}_{p}:N_{s})\) and \((P^{2}_{p}: sc)\). The mapping T is said to be a generalized \((\alpha , \phi )\)Meir–Keeler hybrid contractive mapping of type II if it satisfies for all \(x,y\in X\) the following conditions:

(a)
For any \(\epsilon > 0\), there exists \(\delta (\epsilon ) > 0\) such that \(x \neq y\) and \(p(x, y) < \epsilon +\delta (\epsilon )\) imply \(d(Tx, Ty)\leq \frac{\epsilon}{s}\);

(b)
\(x \neq y\) and \(p(x, y) > 0 \) imply
$$ \zeta \bigl(\alpha (x,y)\phi \bigl(d(Tx, Ty)\bigr), \phi \bigl(p(x, y)\bigr) \bigr)\geq 0. $$(15)
We define a mapping \(N_{s}:X\times X \rightarrow \mathbb{R^{+}}\) by
We note that, for any \(x,y\in X\) with \(x=y\), we have \(0=d(Tx,Ty)\leq N_{s}(x,y)\). Moreover, if \(x\neq y\), then \(N_{s}(x,y)>0\).
Now, we state and prove the following fixed point theorem.
Theorem 8
Let \((X,d)\) be a complete bmetric space with \(s\geq 1\) and \(T:X\rightarrow X\) be a generalized \((\alpha , \phi )\)Meir–Keeler hybrid contractive mapping of type II satisfying the following conditions:

(i)
T is a triangular αorbital admissible mapping;

(ii)
there exists \(x_{0}\in X\) such that \(\alpha (x_{0},Tx_{0})\geq 1\);

(iii)
either T is continuous;

(iv)
or \(T^{2}\) is continuous and \(\alpha (z,Tz)\geq 1\);

(v)
or \((X,d)\) is regular.
Then T has a fixed point z. Moreover, \(\{T^{n}x\}\) is convergent to z for all \(x\in X\).
Proof
As in the proof of Theorem 4, we construct a recursive sequence \(\{x_{n}\}\) as follows:
One can conclude that \(\alpha (x_{n},x_{n+1})\geq 1\) for all \(n\in \mathbb{N} \cup \{0\}\), due to conditions (i) and (ii). Throughout the proof, we assume \(x_{n}\neq x_{n+1}\) for all \(n\in \mathbb{N} \cup \{0\}\). Indeed, as it was discussed in the proof of Theorem 4, the other case is trivial and is excluded. Now, by letting \(x=x_{n}\) and \(y=x_{n+1}\) in \((P^{1}_{p} : N_{s})\), we have
which implies
where
By Definition 11(b), we have
Consequently, the above inequality yields
where
Thus, from (16), (17) and the monotonicity of ϕ, for all \(n\in \mathbb{N} \cup \{0\}\), we have
that is, \(\{d(x_{n},x_{n+1})\}\) is nonincreasing sequence of nonnegative real numbers. Consequently, there exists a real number \(r\geq 0\) such that \(d(x_{n},x_{n+1})\rightarrow r\) as \(n\rightarrow \infty \). Suppose that \(r=\epsilon >0\). First, we note that \(r=\epsilon < d(x_{n},x_{n+1})\) for all \(n\in \mathbb{N} \cup \{0\}\). On the other hand, from (16), there exists \(\delta >0\) such that
for n sufficiently large. Keeping the observations above, Definition 11(a) yields that
Thus, we have
which is a contradiction. So, we derive that \(\epsilon =0\), that is, \(\lim_{n\rightarrow \infty}d(x_{n},x_{n+1})=0\). In what follows, we show that the sequence \(\{x_{n}\}\) is bCauchy. For this aim, let \(m\in \mathbb{N}\) be large enough to satisfy
Now, we show by induction that
Without loss of generality, we assume that \(\delta _{1}=\delta _{1}(\epsilon )<\epsilon \). We have already proved the claim for \(k=1\). Now, we consider the following two cases:
Case (i). If \(d(x_{m+k},x_{m+k+1})\leq d(x_{m},x_{m+k})\), then we get
and
Hence, we have
and so it follows from Definition 11(a) that
Thus, by the btriangle inequality, we have
Case (ii). If \(d(x_{m+k},x_{m+k+1})> d(x_{m},x_{m+k})\), then we get
Thus, by induction, (18) holds for every \(k\in \mathbb{N}\). Since \(\epsilon _{1}>0\) is arbitrary, we get
which implies that \(\{x_{n}\}\) is a bCauchy sequence in a complete bmetric space \((X,d)\). Hence, \(\{x_{n}\}\) bconverges to some \(z\in X\).
Next, we show that z is a fixed point of T. If T is continuous, then we have
that is, z is a fixed point of T.
If \(T^{2}\) is continuous, since \(x_{n} \rightarrow z\), we get that any subsequence of \(\{x_{n}\}\) converges to the same limit point z, so
On the other hand, due to the continuity of \(T^{2}\),
We claim that \(Tz=z\). To the contrary, if \(Tz\neq z\), then we have \(p(z,Tz)>0\) and
Therefore, together with the supplementary hypothesis \(\alpha (z,Tz)\geq 1\), we have
From the above inequality, we obtain
which is a contradiction. Hence, z is a fixed point of T.
If X is regular, we deduce that \(d(z,Tz)=0\), using the same arguments as in the proof of Theorem 6. That is, z is a fixed point of T. □
The uniqueness of fixed point of T can be deduced as in Theorem 7.
Now, we give an example to illustrate Theorem 8.
Example 3
Let \(X = [0, 4]\) and \(d : X \times X \rightarrow \mathbb{R}^{+}\) be defined by \(d(x, y) = x y^{2}\) for all \(x,y \in X\). Then \((X,d)\) is a complete bmetric space with \(s=2\) which is not a metric space. Let \(T : X \rightarrow X\) be defined by
Also, we define \(\alpha : X \times X \rightarrow \mathbb{R}^{+}\), \(q:X\times X \rightarrow \mathbb{R}^{+}\) and \(\phi :\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\) as follows:
\(q(x,y)=\max \{d(x,y),\frac {d(x,Tx)d(y,Ty)}{1+d(x,y)}, \frac {d(x,Tx)d(y,Ty)}{1+d(Tx,Ty)} \}\) and \(\phi (t)=\frac {t^{2}}{2}\). First, we note that q satisfies condition \((P^{1}_{q}: N_{s})\) and \(q(x, y) > 0\) for all \(x \neq y\). Since, for \(x = 0\) we have \(T0 = 1\) and \(\alpha (0, T0) = \alpha (0, 1) = 2 > 1\), assumption (ii) of Theorem 8 is satisfied. Also, it is easy to see that T is triangular αorbital admissible. Let \(\zeta \in Z_{w}\) be is given by \(\zeta (t,s) = \frac{2}{3}s t\). Now, we consider the following cases:

Case 1.
For \(x,y \in [0, 2)\), \(x \neq y\), we have \(d(Tx, Ty) = 0\), so
$$ \zeta \bigl( \alpha (x, y)\phi \bigl(d(Tx, Ty)\bigr), \phi \bigl(q(x, y)\bigr) \bigr) = \frac{2\phi (q(x, y))}{3}= \frac{ (q(x, y))^{2}}{3}>0. $$ 
Case 2.
For \(x, y \in [2, 4]\), \(x \neq y\), we have
$$ d(Tx, Ty) = \frac{xy}{2}, \qquad q(x, y) = \max \biggl\{ xy, \frac{\frac{x}{2}.\frac{y}{2}}{1+xy}, \frac{\frac{x}{2}.\frac{y}{2}}{1+\frac{xy}{2}} \biggr\} , $$so
$$\begin{aligned} \zeta \bigl(\alpha (x, y)\phi \bigl(d(Tx, Ty)\bigr), \phi \bigl(q(x, y)\bigr) \bigr) =& \frac{2\phi (q(x, y))}{3}\phi \biggl(\frac{xy}{2} \biggr) \\ =&\frac{(q(x, y))^{2}}{3}\frac{(xy)^{2}}{8}\geq 0. \end{aligned}$$ 
Case 3.
For \(x \in [0, 2)\) and \(y \in [2, 4]\), we have \(\alpha (x, y) = 0\) and
$$ \zeta \bigl(\alpha (x, y)\phi \bigl(d(Tx, Ty)\bigr), \phi \bigl(q(x, y)\bigr) \bigr) = \frac{2\phi (q(x, y))}{3}=\frac{(q(x, y))^{2}}{3} > 0. $$
Thus, due to the cases considered above, T satisfies all the conditions of Theorem 8 and has a unique fixed point \(x = 1\).
Now, we give some corollaries to our main findings.
Corollary 1
Let \((X,d)\) be a complete bmetric space with \(s\geq 1\) and let \(T:X\rightarrow X\) be a \((\alpha ,\phi )\)Meir–Keeler hybrid contractive mapping of type I with \(p(x,y)=d(x,y)\). Assume that the following conditions are satisfied:

(i)
T is triangular αorbital admissible;

(ii)
there exists \(x_{0} \in X\) such that \(\alpha (x_{0}, Tx_{0}) \geq 1\);

(iii)
either T is continuous or \(T^{2}\) is continuous and \(\alpha (u, Tu) \geq 1\) or \((X, d)\) is regular.
Then T has a fixed point u. Moreover, \(\{T^{n}x\}\) converges to u for all \(x \in X\).
Remark 2
Under the conditions of Corollary 1, since \(x \neq y\) implies \(d(x, y) > 0\), it is obvious that (b) from Definition 9 is equivalent to the following:
 (b′):

\(d(x,y)>0\) implies \(\zeta (\alpha (x,y)\phi (d(Tx,Ty)),\phi (d(x,y)))\geq 0\).
Proof
It is clear that d satisfies the conditions \((P^{1}_{d}: M_{s}) \), respectively \((P^{2}_{d} : 0)\), and so all the assumptions of Theorems 4, 5, and 6 are also satisfied. Thus, T has a fixed point. □
Corollary 2
Let \((X, d)\) be a complete bmetric space with \(s\geq 1\), and let \(T : X \rightarrow X\) be a \((\alpha ,\phi )\)Meir–Keeler hybrid contractive mapping of type I. Let \(\rho : X \times X \rightarrow \mathbb{R}^{+}\) be defined by
where \(a_{1}, a_{2}, a_{3} \in [0, \frac{1}{s})\), \(a_{1} + a_{2} \leq \frac{1}{2s}\) and \(a_{3} \leq \frac{1}{2s}\). Assume also that:

(i)
T is triangular αorbital admissible;

(ii)
there exists \(x_{0} \in X\) such that \(\alpha (x_{0}, Tx_{0}) \geq 1\);

(iii)
either T is continuous or \(T^{2}\) is continuous and \(\alpha (u, Tu) \geq 1\) or \((X, d)\) is regular.
Then T has a fixed point u. Moreover, \(\{T^{n}x\}\) converges to u for all \(x \in X\).
Proof
Let \(x, y \in X\) be such that \(x\neq y\) and \(d(x, Tx) \leq d(x, y)\). Then,
which shows that \((P^{1}_{\rho}: M_{s})\) holds. On the other hand, if \(x_{n} \neq y\), then
hold, and then we have
Thus, \((P^{2}_{\rho} : a_{3})\) holds. Hence, T has a fixed point. □
4 Conclusion
In 2020, Karapinar et al. [43] introduced a generalized Meir–Keeler contraction via a simulation function and studied fixed point results for the mappings introduced in the setting of metric spaces. In this work, we introduced generalized \((\alpha ,\phi )\)Meir–Keeler hybrid contractive mappings of type I and II in the setting of bmetric spaces and proved the existence and uniqueness of fixed points for such mappings. Our results extend and generalize the results of Karapinar et al. [43] and many other related fixed point results in the existing literature. Finally, we have also supported the main result of this work by an illustrative example.
Data availability
Not applicable.
References
Banach, S.: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 3(1), 133–181 (1922)
Taş, N.: Interpolative contractions and discontinuity at fixed point. Appl. Gen. Topol. 24(1), 145–156 (2023)
Edraoui, M., Semami, S.: Fixed points results for various types of interpolative cyclic contraction. Appl. Gen. Topol. 24(2), 247–252 (2023)
Meir, A., Keeler, E.: A theorem on contraction mappings. J. Math. Anal. Appl. 28, 326–329 (1969)
Kirk, W.A.: Contraction mappings and extensions. In: Handbook of Metric Fixed Point Theory, pp. 1–34 (2001)
Bisht, R.K.: An overview of the emergence of weaker continuity notions, various classes of contractive mappings and related fixed point theorems. J. Fixed Point Theory Appl. 25(1), 11 (2023)
Zhou, Z., Tan, B., Li, S.: Inertial algorithms with adaptive step sizes for split variational inclusion problems and their applications to signal recovery problem. Math. Methods Appl. Sci. (2023). https://doi.org/10.1002/mma.9436
Lu, N., He, F., Van Dung, N.: On a question concerning Meir–Keeler contractions in complete bmetric spaces. J. Math. Anal. Appl. 527, 127470 (2023)
Aydi, H., Banković, R., Mitrović, I., Nazam, M.: Nemytzki–Edelstein–Meir–Keeler type results in bmetric spaces. Discrete Dyn. Nat. Soc. 2018, 4745764 (2018)
Gholamian, N., Khanehgir, M.: Fixed points of generalized Meir–Keeler contraction mappings in bmetriclike spaces. Fixed Point Theory Appl. 2016, 34 (2016)
Gülyaz, S., Karapinar, E., Erhan, I.M.: Generalized αMeir–Keeler contraction mappings on Branciari bmetric spaces. Filomat 17(31), 445–5456 (2017)
Karapinar, E., Czerwik, S., Aydi, H.: \((\alpha , \varphi )\)Meir–Keeler contraction mappings in generalized bmetric spaces. J. Funct. Spaces 2018(4), 326–4620 (2018)
Mitrovi, Z.D., Radenovi, S.: On Meir–Keeler contraction in Branciari bmetric spaces. Trans. A. Razmadze Math. Inst. 173(1), 83–90 (2019)
Mlaiki, N., Souayah, N., Abodayeh, K., Abdeljawad, T.: Meir–Keeler contraction mappings in \(M_{b}\)metric spaces. J. Comput. Anal. Appl. 4(27) (2019). https://doi.org/10.1155/2014/756298
Bakhtin, I.: The contraction mapping principle in quasimetric spaces. Func. An., Gos. Ped. Inst. Unianowsk 30, 26–37 (1989)
Czerwik, S.: Contraction mappings in bmetric spaces. Acta Math. Inform. Univ. Ostrav. 1(1), 5–11 (1993)
Aghajani, A., Abbas, M., Roshan, J.R.: Common fixed point of generalized weak contractive mappings in partially ordered bmetric spaces. Math. Slovaca 64, 941–960 (2014)
Deshmukh, A., Gopal, D.: Topology of nontriangular metric spaces and related fixed point results. Filomat 35(11), 3557–3570 (2021)
Ding, H.S., Imdad, M., Radenović, S., Vujaković, J.: On some fixed point results in bmetric. Math. Sci. 22(2), 151–164 (2016)
Dosenović, T., Pavlović, M., Radenović, S.: Contractive conditions in bmetric spaces. Vojnotehnicki Glasnik/Mil. Tech. Cour. 65, 851–865 (2017)
Faraji, H., Nourouzi, K.: A generalization of Kannan and Chatterjea fixed point theorems on complete bmetric spaces. Sahand Commun. Math. Anal. 6, 77–86 (2017)
Faraji, H., Nourouzi, K.: Fixed and common fixed points for \((y, j)\)weakly contractive mappings in bmetric spaces. Sahand Commun. Math. Anal. 7, 49–62 (2017)
Faraji, H., Nourouzi, K., O’Regan, D.: A fixed point theorem in uniform spaces generated by a family of bpseudo metrics. Fixed Point Theory 20, 177–183 (2019)
Hussain, N., Mitrović, Z.D., Radenović, S.: A common fixed point theorem of Fisher in bmetric spaces. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113, 949–956 (2019)
Ilchev, A., Zlatanov, B.: On fixed points for Reich maps in bmetric spaces. In: Annual of Konstantin Preslavski University of Shumen, Faculty of Mathematics and Computer Science VI: Shumen, Bulgaria, pp. 77–88 (2016)
Jovanović, M., Kadelburg, Z., Radenović, S.: Common fixed point results in metrictype spaces. Fixed Point Theory Appl. 2010, 978121 (2010)
Khamsi, M.A., Hussain, N.: KKM mappings in metric type spaces. Nonlinear Anal. 73, 3123–3129 (2010)
Kirk, W., Shahzad, N.: Fixed Point Theory in Distance Spaces. Springer, Berlin (2014)
Koleva, R., Zlatanov, B.: On fixed points for Chatterjea’s maps in bmetric spaces. Turk. J. Anal. Number Theory 4, 31–34 (2016)
Miculescu, R., Mihail, A.: A generalization of Matkowski’s fixed point theorem and Istratescu’s fixed point theorem concerning convex contractions. J. Fixed Point Theory Appl. 19(2), 1525–1533 (2017)
Mitrović, Z.D., Hussain, N.: On weak quasicontractions in bmetric spaces. Publ. Math. (Debr.) 94(3–4), 289–298 (2019)
Roshan, J.R., Parvaneh, V., Altun, I.: Some coincidence point results in ordered bmetric spaces and applications in a system of integral equations. Appl. Math. Comput. 226, 725–737 (2014)
Sing, S.L., Czerwik, S., Krol, K., Singh, A.: Coincidences and fixed points of hybrid contractions. Tamsui Oxf. J. Math. Sci. 24, 401–416 (2008)
Suzuki, T.: Basic inequality on a bmetric space and its applications. J. Inequal. Appl. 2017, 256 (2017)
Zoto, K., Rhoades, B., Radenović, S.: Common fixed point theorems for a class of \((s, q)\)contractive mappings in bmetriclike spaces and applications to integral equations. Math. Slovaca 69, 233–247 (2019)
Ma, Z., Nazam, M., Khan, S.U., Li, X.: Fixed point theorems for generalizedcontractions with applications. J. Funct. Spaces 2018, 8368546 (2018)
Nazam, M., Arshad, M., Postolache, M.: Coincidence and common fixed point theorems for four mappings satisfying \((\alpha (s), F)\)contraction. Nonlinear Anal., Model. Control 23(5), 664–690 (2018)
Nazam, M., Hussain, N., Hussain, A., Arshad, M.: Fixed point theorems for weakly betaadmissible pair of Fcontractions with application. Nonlinear Anal., Model. Control 24(6), 898–918 (2019)
Nazam, M., Acar, Ö.: Fixed points of \((\alpha , \psi )\)contractions in Hausdorff partial metric spaces. Math. Methods Appl. Sci. 42(16), 5159–5173 (2019)
Kir, M., Kiziltunc, H.: On some well known fixed point theorems in bmetric spaces. Turk. J. Anal. Number Theory 1(1), 13–16 (2013)
Huang, H., Vujaković, J., Radenović, S.: A note on common fixed point theorems for isotone increasing mappings in ordered bmetric spaces. J. Nonlinear Sci. Appl. 8(5), 808–815 (2015)
Chandok, S.C., Jovanović, M.S., Radenović, S.N.: Ordered bmetric spaces and Geraghty type contractive mappings. Vojnotehnicki Glasnik/Mil. Tech. Cour. 65(2), 331–345 (2017)
Karapinar, E., Fulga, A., Kumam, P.: Revisiting the Meir–Keeler contraction via simulation function. Filomat 34(5), 1645–1657 (2020)
Khojasteh, F., Shukla, S., Radenovic, S.: A new approach to the study of fixed point theorems for simulation functions. Filomat 29(6), 1189–1194 (2015)
Suzuki, T.: Fixed point theorems for contractions of rational type in complete metric spaces. J. Nonlinear Sci. Appl. 11, 98–107 (2018)
Boriceanu, M., Bota, M., Petrusel, A.: Multivalued fractals in bmetric spaces. Cent. Eur. J. Math. 8(2), 367–377 (2010)
Boriceanu, M.: Strict fixed point theorems for multivalued operators in bmetric spaces. Int. J. Mod. Math. 4, 285–301 (2009)
Hussain, N., Doric, D., Kadelburg, Z., Radenović, S.: Suzukitype fixed point results in metric type spaces. Fixed Point Theory Appl. 2012, 126 (2012)
Popescu, O.: Some new fixed point theorems for αGeraghty contraction type maps in metric spaces. Fixed Point Theory Appl. 2014(1), 190 (2014)
Acknowledgements
The authors would like to thank the College of Natural Sciences, Jimma University for funding this research work.
Funding
Not applicable.
Author information
Authors and Affiliations
Contributions
MAM contributed to the conceptualization, formal analysis, methodology, writing, editing, and approving the manuscript. KKT involved in formal analysis, methodology and writing the original draft. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Abduletif Mamud, M., Koyas Tola, K. Fixed point theorems for generalized \((\alpha , \phi )\)Meir–Keeler type hybrid contractive mappings via simulation function in bmetric spaces. Fixed Point Theory Algorithms Sci Eng 2024, 4 (2024). https://doi.org/10.1186/s13663023007587
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13663023007587