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Generalized contraction mappings in double controlled metric type space and related fixed point theorems
Journal of Inequalities and Applications volume 2023, Article number: 87 (2023)
Abstract
In this article, we introduce two new types of generalized contraction mappings in double controlled metric type spaces: Θdouble controlled contraction mapping and ĆirićReichRustypeΘdouble controlled contraction mapping. For each contraction mapping, we establish the existence and uniqueness of the fixed point theorems on the complete double controlled metric type space and provide examples. We present an application of our results and demonstrate how our results generalize several existing fixed point theorems in the literature.
1 Introduction
The fixed point theorem established the existence of a solution to an integral equation and was known as the Banach contraction principle [1]. It rapidly developed into a typical device for solving several existing problems in various areas of mathematics, including nonlinear analysis and its application. The fixed point theory advanced in two directions. One is to change the space under consideration, and the other is to change the contraction condition. Banach’s original thesis dealt with the standard metric space [1]. Many researchers generalized the structure of the metric space by changing the space under consideration; for example, bmetric space was introduced by Bakhtin [2] as an extension of standard metric spaces. Later, Branciari [3] presented the notion of generalized metric (Branciari metric) spaces, where the triangle inequality is replaced with \(d(a, b) \leq d(a,u) + d(u, v) + d(v, b)\); for all distinct pairwise points \(a, b,u, v \in W\), the map d is called a Branciari distance. Numerous fixed point results were established on such spaces [4, 5]. Subsequently, the notion of extended bmetric spaces appeared [6], and fixed point results on these spaces were established in many articles [7, 8]. Later, a more general concept of controlled metrictype spaces was introduced [9]. A further generalization resulted in the concepts of double controlled metric type spaces [10] and double controlled metriclike spaces [11, 12]. Several authors have studied various generalizations of the Banach contraction mapping principle. One of the prominent generalizations of the Banach contraction was presented by Kannan [13], who introduced a new contraction now known as the Kannan contraction. By utilizing the Kannan contraction, fixed point results have been established on bmetric spaces [14] and extended bmetric spaces [15].
Some researchers weakened the contraction property of the operator using many other contraction mappings (e.g., [16–25]); for example, Ćirić [18] and Reich [20] independently proved the existence and uniqueness theorem on complete metric space, when the operator has a certain contraction property. The contraction mapping was later known as a ĆirićReichRustype contraction. Jleli et al. ([26, 27]) introduced the concept of Θcontraction to establish a generalization of the Banach fixed point theorem in the situation of Branciari metric spaces. Recently, Abdeljawad et al. [28] proposed the notion of ΘBranciari contraction to establish a fixed point theorem in the extended Branciari bdistance space using the ĆirićReichRustype contraction mapping.
In this article, by focusing on two contraction mappings, ĆirićReichRustype and Θcontraction, we introduce them on double controlled metric type spaces. Hence, we denote the first contraction mapping by Θdouble controlled contraction mapping, while the second is denoted by the ĆirićReichRustype Θdouble controlled contraction mapping. Furthermore, we establish the existence and uniqueness of the fixed point theorems on complete doublecontrolled metric type spaces. We also present examples and demonstrate how our result generalizes many existing fixed point theorems. We end the article with an application of our results.
2 Preliminaries
Kamran et al. [6] defined the concept of extended bmetric spaces, and their work generalized many results [29–32].
Definition 2.1
([6])
Consider a mapping \(\omega : \mathcal{S}\times \mathcal{ S} \rightarrow [1,\infty )\), where \(\mathcal{S} \neq \emptyset \). The function \(d_{\omega}:\mathcal{S} \times \mathcal{S} \rightarrow \mathbbm{[}0, \infty )\), is called an extended bmetric if for all \(\hat{x},\hat{y}, \hat{z} \in \mathcal{S}\),

1.
\(d_{\omega}(\hat{x},\hat{z})=0 \) iff \(\hat{x}= \hat{z}\);

2.
\(d_{\omega}(\hat{x},\hat{z}) = d_{\omega}(\hat{z},\hat{x})\), symmetric;

3.
\(d_{\omega}(\hat{x},\hat{z}) \leq \omega (\hat{x},\hat{z}) [d_{\omega}( \hat{x},\hat{y}) + d_{\omega}(\hat{y},\hat{z})]\),
The pair \((\mathcal{S},d_{\omega})\) is called an extended bmetric space.
Branciari introduced the concept of generalized metric spaces [3], while the extended Branciari bdistance space was introduced in [28] and defined as follows.
Definition 2.2
([28])
Let S be a nonempty set, and let \(\omega : S\times S\rightarrow [1,\infty )\) be a mapping. The function \(d:S\times S\rightarrow \mathbbm{[}0,\infty )\), is called an extended Branciari bdistance if for all \(x, y \in S\) and all \(u, v \in S \) distinct from x, y, this holds:

1.
\(d(x,y) = 0\) iff \(x= y\);

2.
\(d(x,y) = d(y,x)\), symmetric;

3.
\(d(x,y) \leq \omega (x,y) [d(x,u) + d(u,v) + d(v,y)]\),
The pair \((S,d)\) is called an extended Branciari bdistance space.
An extension of the bmetric space into a controlled metric type space was introduced by Mlaiki et al. [9].
Definition 2.3
Let \(\beta : Z_{\beta }\times Z_{\beta }\rightarrow [1,\infty )\) be a mapping, where \(Z_{\beta }\neq \emptyset \). The function \(d_{\beta}:Z_{\beta }\times Z_{\beta }\rightarrow \mathbbm{[}0,\infty )\), is called a controlled metric if, for all \(\hat{x},\hat{y} ,\hat{z} \in Z_{\beta}\), the following conditions hold:
(d1) \(d_{\beta}(\hat{x},\hat{z})=0\) iff \(\hat{x}=\hat{z}\);
(d2) \(d_{\beta}(\hat{x},\hat{z})=d_{\beta}(\hat{z},\hat{x})\), symmetric;
(d3) \(d_{\beta}(\hat{x},\hat{z})\leq \beta (\hat{x},\hat{y}) d_{\beta}( \hat{x},\hat{y})+\beta (\hat{y},\hat{z}) d_{\beta}(\hat{y},\hat{z})\),
then, the pair \((Z_{\beta}, d_{\beta})\) is called a controlled metric type space.
Then a more general concept of a bmetric type space called a double controlled metric type space was introduced in [10] as follows.
Definition 2.4
([10])
Consider two noncomparable functions \(\beta ,\mu : Z\times Z\rightarrow [1,\infty )\), defined on a nonempty set Z. The mapping \(d_{\beta ,\mu}:Z\times Z\rightarrow \mathbbm{[}0,\infty )\) is called a double controlled metric type space by β and μ if, for all \(z_{1}, z_{2}, z_{3} \in Z\), the following conditions hold:
(Q1) \(d_{\beta ,\mu}(z_{1},z_{2})=0\) iff \(z_{1} = z_{2}\);
(Q2) \(d_{\beta ,\mu}(z_{1}, z_{2})=d_{\beta ,\mu}(z_{2}, z_{1})\), symmetric;
(Q3) \(d_{\beta ,\mu}(z_{1}, z_{2})\leq \beta (z_{1}, z_{3}) d_{\beta ,\mu}(z_{1}, z_{3})+\mu (z_{3}, z_{2}) d_{\beta ,\mu}(z_{3},z_{2})\).
The pair \((Z,d_{\beta ,\mu})\) is called a double controlled metric type space.
Remark 2.5
The class of double controlled metric type spaces is larger than that of controlled metric type spaces, which, in turn, is larger than the class of extended bmetric spaces. Moreover, the class of extended bmetric spaces is larger than that of bmetric spaces. All these classes are larger than the class of standard metric spaces. Every extended bmetric space is a controlled metric type space and double controlled metric type space, but the converse is invalid. Also, every controlled metric type space is, in fact, a double controlled metric type space, but the converse is invalid (Fig. 1).
The following example shows a controlled metric type space that is not an extended bmetric space [9].
Example 2.6
Let \(Z = \mathbb{N}\), define \(d_{\alpha}:Z \times Z \rightarrow \mathbbm{[}0,\infty )\), by
Then, \(( Z , d_{\alpha})\) is a controlled metric type space, with \(\alpha :Z \times Z \rightarrow \mathbbm{[}1,\infty )\), defined by
The next example illustrates a double controlled metric type space, which is not a controlled metric type space [10].
Example 2.7
([10])
Let \(Z = [0, +\infty )\), define the mapping \(d_{\beta , \mu}:Z\times Z \rightarrow \mathbbm{[}0,+\infty )\) by
Let \(\beta ,\mu : Z\times Z\rightarrow [1,+\infty )\) be two functions defined by
and
To see \(( Z , d_{\beta , \mu})\) is a double controlled metric type space. Observe that conditions \((Q1)\) and \((Q2)\) hold. To illustrate condition \((Q3)\) holds, note that if either \(z = u\) or \(z = v\), then \((Q3)\) holds. Thus, suppose \(u \neq v\), which means \(u \neq v \neq z\). We consider cases:
Case 1: if \(u \geq 1\) and \(v \in [0, 1)\), or \(v \geq 1\) and \(u \in [0, 1)\), then for any z, clearly condition \((Q3)\) is satisfied.
Case 2: if \(u, v > 1\), and \(z \geq 1\), then one can easily observe that
While, if \(z \in [0,1)\), then we obtain
Case 3: if \(u, v < 1\), and \(z \geq 1\), we obtain
While, if \(z \in [0,1)\), then easily condition \((Q3)\) holds. Therefore, \(( Z , d_{\beta , \mu})\) is a double controlled metric type space, which is not a controlled metric type space, by taking \(\beta = \mu \), note
Definition 2.8
Let \((Z,d_{\beta ,\mu})\) be a double controlled metric type space, where \(Z \neq \emptyset \), given any \(\varepsilon > 0\), the open ball \(B(y,\varepsilon )\) is defined as
Let \(\{y_{n}\}_{n\ge 0}\) be any sequence in Z. Then
(1) \(\{y_{n}\}\) converges to some w in Z if for every \(\varepsilon >0\), there exists \(N \in \mathbb{N}\), such that \(d_{\beta ,\mu} (y_{n},w)< \varepsilon \) for all \(n\geq N\), i.e. \(\lim_{n\rightarrow \infty} d_{\beta ,\mu}(y_{n}, w) = 0\).
(2) \(\{y_{n}\}\) is a Cauchy sequence if for every \(\varepsilon >0\), there exists \(N \in \mathbb{N}\) such that \(d_{\beta ,\mu}(y_{m},y_{n})<\varepsilon \), for all \(m,n\geq N\).
(3) The space \((Z,d_{\beta ,\mu})\) is called a complete double controlled metric type space if every Cauchy sequence in Z is convergent.
(4) A mapping \(T:Z \rightarrow Z\) is said to be continuous at \(y \in Z\) if for all \(\varepsilon >0\), there exists \(\delta >0\), such that \(T(B(y,\delta ))\subseteq B(Ty,\varepsilon )\). Thus if \(\{y_{n}\}_{n\ge 0}\) is any sequence which converges to u, i.e., \(\lim_{n\rightarrow \infty} d_{\beta ,\mu}(y_{n}, u) = 0\), then \(\lim_{n\rightarrow \infty} d_{\beta ,\mu}( Ty_{n},T u) = 0\).
Let \((X, d)\) be a metric space, and let \(T : X\rightarrow X\) be a mapping, then T is called a contraction [1] if there exists \(r \in [0, 1)\) such that this holds for all \(x, y \in X\),
Kannan [13] generalized the Banach contraction by introducing a new contraction on a metric space now known as the Kannan contraction. Thus, the mapping \(T: X\rightarrow X\) is called the Kannan contraction if there exists \(r \in [0, 1/2)\) such that
for all \(x, y \in X\). When \((X, d)\) is complete, then every contraction and every Kannan contraction have a unique fixed point; see [1, 13]. Later many authors utilized Kannantype contraction and established fixed point results on bmetric spaces [14], extended bmetric spaces [15], and more recently into double controlled dislocated quasimetric type spaces [24].
The concept of Θcontraction was proposed in [26] to extend some results on the fixed point theorem in the framework of the Branciari distance space. We recall the definition of the Θ set of functions.
Definition 2.9
([26])
Let Θ be the set of all functions \({\theta}:(0,\infty ) \rightarrow (1,\infty )\) obeying the following conditions:

(I)
θ is nondecreasing;

(II)
for each sequence \(\{ t_{m} \}\) of positive real numbers, this holds
$$ \lim_{m\rightarrow \infty} t_{m} = 0^{+} \quad \iff \quad \lim _{m\rightarrow \infty} \theta (t_{m}) = 1; $$ 
(III)
there exist k, with \(0 < k < 1\), and \(M \in (0, \infty ]\), such that this holds
$$ \lim_{t\rightarrow 0^{+}} \frac{\theta (t) 1}{t^{k}} = M. $$
It should be observed that Θ contains a large class of functions, for example, the functions \(\theta _{i}:(0,\infty ) \rightarrow (1,\infty )\), defined by \(\theta _{1} (t) = e^{\sqrt{t}}\), and \(\theta _{2} (t) = e^{\sqrt{ t e^{t}}}\) belongs to Θ.
3 Main results
Our work focuses on two types of contraction mappings: ĆirićReichRustype and Θcontraction. Following are two subsections for each type of contraction mappings on complete double controlled metric type spaces.
3.1 Θdouble controlled contraction mapping and fixed point theorems
With Θ as in Definition 2.9 and inspired by [26], we present the notion of Θdouble controlled contraction mapping and establish a fixed point theorem on complete double controlled metric type spaces.
Definition 3.1
Let \((Z,d_{\beta ,\mu})\) be a double controlled metric type space. Let \(T: Z \rightarrow Z\) be a self mapping. Then T is said to be Θdouble controlled contraction mapping if there exists a function \(\theta \in \Theta \) and an \(r \in (0,1)\) such that the following holds:
Lemma 3.2
Let \((Z,d_{\beta ,\mu})\) be a double controlled metric type space, and let \(T :Z\rightarrow Z\) be a Θdouble controlled contraction mapping. Then T is continuous.
Proof
As T is Θdouble controlled contraction mapping, hence this holds
for some \(\theta \in \Theta \) and \(r \in (0,1)\), and for any \(x, y \in Z\) with \(Tx \neq Ty\). Applying ln on both sides of equation (3.2), we obtain
As θ is nondecreasing, we obtain
Hence T is continuous. □
Our first main result of the fixed point theorem.
Theorem 3.3
Let \((Z,d_{\beta ,\mu})\) be a complete double controlled metric type space with \(\beta ,\mu : Z\times Z\rightarrow [1,\infty )\) two noncomparable functions, defined on a nonempty set Z, and let \(T : Z\rightarrow Z\) be a Θdouble controlled contraction self mapping satisfying the following:
(1)
for some \(p \in (0,1)\), and
(2) For any \(z \in Z\), both \(\lim_{n\rightarrow \infty}\beta (z,z_{n})\) and \(\lim_{n \rightarrow \infty}\mu (z_{n},z)\) exist and are finite, where the sequence \(\{ z_{n}\}\) is defined as \(z_{n} = T^{n} z_{0}\) for some \(z_{0} \in Z\).
Then T has a unique fixed point in Z.
Proof
Let \(z_{0}\) be any arbitrary point in Z, then we have a sequence \(\{z_{n}\}_{n\geq 0}\), with \(T^{n} z_{0} = z_{n}\), for all \(n \in \mathbb{N}\).
If for some \(m \in \mathbb{N}\), \(T^{m} z_{0}=T^{m+1}z_{0}\), then this implies that \(T^{m} z_{0}\) is a fixed point of the mapping T. Thus, without loss of generality, we may assume that \(z_{n}\neq z_{n+1}\), i.e., \(d_{\beta ,\mu}(T^{n} z_{0},T^{n+1} z_{0}) > 0\), for all \(n \in \mathbb{N}\).
Applying (3.1) recursively, we obtain
As \(\theta (t) > 1\), we have
Since \(0 < r < 1\), hence letting n tends to infinity in (3.6), we get
Employing property (II) in Definition 2.9, we obtain
In a similar method, one can show that
By (III) of Definition 2.9, there exists \(k \in (0,1)\) and \(M \in (0, \infty ]\) such that
Case 1: Let \(0 < M < \infty \), and define \(L = \frac{M}{2}\), from equation (3.9), there exists some \(n_{0} \in \mathbb{N}\), such that for all \(n \geq n_{0}\) we obtain
which implies that
Hence, for all \(n \geq n_{0}\), we have
By employing (3.5), we obtain
Letting \(n \rightarrow \infty \), in the above inequality, we get
Case 2: \(M = \infty \), in this case, let \(L > 0\) be any arbitrary number. Thus, by the definition of the limit, we can find some \(n_{1} \in \mathbb{N}\) such that
which gives
Again employing (3.5) in the above inequality and then letting \(n \rightarrow \infty \), we obtain
Thus, from equations (3.10) and (3.11), we deduce that for any \(M \in (0, \infty ]\) and \(0 < k < 1\), there exists some \(\hat{N} \in \mathbb{N}\), where \(\hat{N} = \max\{n_{0}, n_{1} \}\) such that
Next, we prove that T has a fixed point; we consider two cases.
Case 1: Suppose \(T^{n} z_{0} = z_{n} = z_{m} = T^{m} z_{0} \) for some \(m \neq n \in \mathbb{N}\). Assume \(m > n\), we obtain \(T^{mn} (z_{n}) = z_{n}\). Denote \(z_{n}\) by x, and \(q = m  n\), then we have \(T^{q} x = x\), which means x is a periodic point of T. Hence
Following the above argument, one can easily show that \(d_{\beta ,\mu}(x, Tx) = 0\), i.e., \(T x = x\), implying that x is a fixed point of T.
Case 2: Suppose that \(z_{n} = T^{n} z_{0} \neq T^{m} z_{0} = z_{m}\) for all \(m, n \in \mathbb{N}\) and assume \(m > n\).
Claim: \(\{z_{n} \}\) is a Cauchy sequence in Z.
For all \(m,n\in \hat{N}\) with \(m > n\), and by (Q3) of Definition 2.4, we have:
Hence
Note that the series
converges, by employing (3.12) and using conditions (1) and (2), we obtain
which is convergent as \(\frac{1}{k} > 1\). Let S and \(S_{n}\) be defined as
and
Therefore, (3.13) can be written as
letting \(n, m \rightarrow \infty \), and by incorporating equations (3.7) and (3.8), and the fact the above series is convergent, we obtain
Thus, \(\{ z_{n} \}\) is a Cauchy sequence in a complete double controlled metric type space \((Z,d_{\beta ,\mu})\); therefore, it converges to some \(\hat{x} \in Z\), i.e., \(\lim_{n \rightarrow \infty} d_{\beta ,\mu} (z_{n}, \hat{x}) = 0\).
Next, we show x̂ is a fixed point of T, i.e., \(T\hat{x} = \hat{x}\). By Lemma 3.2, T is continuous; thus, \(\lim_{n \rightarrow \infty} d_{\beta ,\mu} (T z_{n}, T\hat{x}) = 0\), hence we have \(T z_{n} = z_{n+1} \rightarrow T \hat{x}\). Using the triangular property of Definition 2.4, we have
Taking the limit as n tends to infinity in the above inequality and using condition (2), we obtain \(d_{\beta ,\mu}(\hat{x},T \hat{x})=0\); that is, \(T\hat{x} = \hat{x}\).
To prove the uniqueness of the fixed point, assume that T has two fixed points x̂, ŷ such that \(\hat{x}\neq \hat{y}\),
which is a contradiction, hence \(\hat{x} = \hat{y}\), so T has a unique fixed point. □
Next, we illustrate Theorem 3.3 by the following example.
Example 3.4
Let \(Z=\{0,1,2\}\). Consider the symmetric metric \(d_{\beta ,\mu}:Z \times Z \rightarrow \mathbbm{[}0,\infty )\), defined by
and
The two symmetric functions \(\beta ,\mu : Z\times Z\rightarrow [1,\infty )\), are given by
and
One can easily show that \((d_{\beta ,\mu},Z) \) is a complete double controlled metric type space.
Consider the self mapping \(T: Z \rightarrow Z\), defined by \(T(0) = 2\), and \(T(1) = T(2) = 1\), also let \(\theta : (0, \infty ) \rightarrow (1, \infty )\) be given by \(\theta (t) = e^{\sqrt{t}}\), then θ satisfies the properties in Definition 2.9, with \(k = 1/2\).
First, we show that T is Θdouble controlled contraction mapping as in Definition 3.1 with \(r = \frac{3}{5} \in (0,1)\).
For any \(z, w \in Z\), such that \(d_{\beta ,\mu}(T z,T w) \neq 0\), we investigate if \(\theta ( d_{\beta ,\mu}(T z,T w)) \leq [\theta (d_{\beta ,\mu}(z,w))]^{3/5}\). It is enough to explore these:
Next, we explore if conditions (1) and (2) of Theorem 3.3 are satisfied with \(p = 1/3 \in (0,1)\). To show (1) holds, i.e.,
take, for example, \(z_{0} = 0\), then \(z_{1} = T(0) =2\) and \(z_{2} = T(2) = 1\), \(z_{3} = T(1) = 1\), \(z_{j} = 1\) for \(j > 3\), hence
As \(z_{j} = 1\) for \(j > 3\), then \(\beta (z_{3},z_{4})\mu (z_{3},z)\) reduces to case (c). If we take \(0 \neq z_{0} \), then similarly we get \(\sup_{m\geq 1}\lim_{n\rightarrow \infty}\beta (z_{n+1},z_{n+2}) \mu (z_{n+1},z_{m})< 3\). Therefore, T is Θdouble controlled contraction mapping, satisfying conditions (1) and (2) of Theorem 3.3, so it has a unique fixed point \(z = 1\).
By taking \(\beta (z, w) = \mu (z, w)\) in Theorem 3.3 and modifying the two conditions (1) and (2) accordingly, we deduce the following corollary for the case of controlled metric type space.
Corollary 3.5
Let \((Z,d_{\beta})\) be a complete controlled metric type space, and let \(T :Z\rightarrow Z\) be a Θcontraction self mapping. Then T has a unique fixed point in Z.
In case \(\beta (z, w) = \mu (z, w) = 1\), we get the following corollary.
Corollary 3.6
Let \((Z,d)\) be a complete metric space, and let \(T :Z\rightarrow Z\) be a self mapping. Suppose there exists a function \(\theta \in \Theta \) and an \(r \in (0,1)\), such that the following holds:
Then T has a unique fixed point.
3.2 ĆirićReichRustype Θdouble controlled contraction mapping and fixed point theorem
In 1971, Ćirić [18] generalized Banach’s contraction principle theorem to a more general contraction. Then in 1974, Ćirić [19] generalized his own result [18] by introducing the quasicontraction and obtained the fixed point theorem under the below condition.
Definition 3.7
([19])
Let \((Z, d)\) be a metric space. If \(T: Z \rightarrow Z\) satisfies the quasicontraction condition
for all \(x, y \in Z\), and for some \(p \in [0, 1)\), then T has a unique fixed point in Z.
During the same time, Reich [20] stated the following result.
Definition 3.8
Let \((Z, \rho )\) be a complete metric space. If \(T: Z \rightarrow Z\) satisfies:
For all \(x,y \in Z\) and \(\lambda \in [0, 1/3) \), then T has a unique fixed point.
It should be noted that Ćirić and Reich proved the result independently. That is why in the literature it is referred to as a ĆirićReichRustype contraction. In [28], Abdeljawad et al. gave the following definition of the ĆirićReichRustype contraction mapping, and they used it to prove the existence and uniqueness of the fixed point when the extended Branciari bdistance space is complete.
Definition 3.9
([28])
Let \((Z, d)\) be an extended Branciari bdistance space, then a mapping \(T:Z \rightarrow Z\), is called a ĆirićReichRustype contraction mapping, if there exists a function \(\theta \in \Theta \), and \(p \in (0,1)\), such that
where
with \(\lim \sup_{n,m \rightarrow \infty} \omega (x_{n}, x_{m}) < 1/p\), here \(x_{n} = T^{n} x_{0}\) for some \(x_{0} \in Z\).
In the following, inspired by Definition 3.9, we generalize the ĆirićReichRustype contraction mapping in the setting of a double controlled metric type space. Therefore, we introduce the concept of ĆirićReichRustype Θdouble controlled contraction mapping (for short CRRΘdouble controlled contraction), with Θ as in Definition 2.9, and we establish the fixed point theorem.
Definition 3.10
Let \((Z,d_{\beta ,\mu})\) be a double controlled metric type space, and let \(T: Z \rightarrow Z\) be a self mapping. Then T is a CRRΘdouble controlled contraction mapping, if there exists a function \(\theta \in \Theta \) such that θ is continuous, and \(r \in (0,1)\), so that the following holds:
for all \(z, w \in Z\), such that \(d_{\beta ,\mu}(T z,T w) \neq 0 \), where
Next, our second main result of the fixed point theorem.
Theorem 3.11
Let \((Z,d_{\beta ,\mu})\) be a complete double controlled metric type space, and let \(\beta ,\mu : Z\times Z\rightarrow [1,\infty )\) be two noncomparable functions, defined on a nonempty set Z. Let \(T: Z\rightarrow Z\) be a CRRΘdouble controlled contraction self mapping satisfying the following:
(M1)
for some \(p \in (0,1)\), and
(M2) For any \(z \in Z\), both \(\lim_{n\rightarrow \infty}\beta (z,z_{n})\) and \(\lim_{n\rightarrow \infty}\mu (z_{n},z)\) exist and are finite, where the sequence \(\{ z_{n}\}\) is defined as \(z_{n} = T^{n} z_{0}\) for some \(z_{0} \in Z\).
Then T has a unique fixed point in Z.
Proof
Let \(z_{0}\) be any arbitrary point in Z, then we have a sequence \(\{z_{n}\}_{n\geq 0}\), with \(T^{n} z_{0} = z_{n}\), for all \(n \in \mathbb{N}\).
If for some \(q \in \mathbb{N}\), \(T^{q} z_{0}=T^{q+1}z_{0}\), then this implies that \(T^{q} z_{0}\) is a fixed point of the mapping T. Thus, without loss of generality, we may assume that \(d_{\beta ,\mu} (z_{n}, z_{n+1}) = d_{\beta ,\mu}(T^{n} z_{0},T^{n+1} z_{0}) > 0\), for all \(n \in \mathbb{N}\).
By Definition 3.10, we obtain
where
In case \(M(z_{n},z_{n1}) = d_{\beta ,\mu}(z_{n}, z_{n+1})\), then the inequality in (3.17) turns into
which is a contradiction, since \(0 < r < 1\). Therefore, we have
using (3.18), the inequality in (3.17) becomes
That is the same inequality as (3.5) in Theorem 3.3. Consequently, following the same steps of proof in Theorem 3.3 and utilizing conditions (M1) and (M2), we conclude that \(\{z_{n} \}\) is a Cauchy sequence in a complete double controlled metric type space \((Z,d_{\beta ,\mu})\). Therefore, it converges to some \(\hat{z} \in Z\), i.e.,
We claim that \(T(\hat{z}) = \hat{z}\). First, note that if there exists an integer N such that \(z_{N} = \hat{z}\), then ẑ is a fixed point since by (3.17), we have
Utilizing (3.18) and then repeating the steps as in (3.19), we obtain \(\theta (d_{\beta ,\mu}(T z_{N}, z_{N}) ) \leq [\theta (d_{\beta ,\mu}(z_{0},z_{1}))]^{r^{N}}\); applying the steps as in (3.6) and then taking the limit as N tends to infinity as in (3.7), we obtain \(d_{\beta ,\mu}(T z_{N}, z_{N}) = 0\), i.e., \(\hat{z} = z_{N} = T z_{N}\) is a fixed point. Thus, without loss of generality assume that \(z_{n} \neq \hat{z}\) for all n, to show ẑ is a fixed point, observe that continuity of T together with (3.20) implies
Utilizing (3.20), we obtain
By (Q3) of Definition 2.4, we deduce
Upon taking the limit as n tends to infinity and using (3.21), (3.22), and \((M2)\), we reach \(d_{\beta ,\mu}(\hat{z}, T \hat{z}) = 0\), hence \(T \hat{z} = \hat{z}\), proving ẑ is a fixed point.
Next, we show the uniqueness of the fixed point. Assume that T has two fixed points ẑ, ŵ such that \(\hat{z}\neq \hat{w}\), thus \(d_{\beta ,\mu}(\hat{z}, \hat{w} ) = d_{\beta ,\mu}(T \hat{z}, T \hat{w} ) > 0\). Utilizing (3.16), we obtain
which is a contradiction, hence \(\hat{z} = \hat{w}\), so T has a unique fixed point. □
Next, we present a supporting example for Theorem 3.11 and also refer to [28].
Example 3.12
Let \(Z = \{ S_{n} : n \in \mathbb{N} \}\), where
Define the mapping \(d_{\beta , \mu}:Z\times Z \rightarrow \mathbbm{[}0,+\infty )\) by \(d_{\beta , \mu} (x, y ) = \vert x  y \vert ^{2}\), and let \(\beta ,\mu : Z\times Z\rightarrow [1,+\infty )\) be defined by
and \(\mu (x, y) = \frac{1}{y} + 1\). Then \((Z, d_{\beta , \mu})\) is a complete double controlled metric type space. The mapping \(T:Z \rightarrow Z\) is given by
To show T is CRRΘdouble controlled contraction mapping with \(\theta (t) = e^{t}\), we need to show (3.16) holds, i.e.,
Note that if the above equation holds, it yields, \(e^{d_{\beta ,\mu}(T x,T y) } \leq [ e^{\theta (M(x, y))}]^{r}\). Applying ln on both sides, we get
where \(M(x, y) = \max \{d_{\beta ,\mu}(x, y), d_{\beta ,\mu}(x, Tx), d_{ \beta ,\mu}(y, Ty) \}\).
Hence to show T is CRRΘdouble controlled contraction mapping, it is enough to show that (3.23) holds.
Case 1: \(n = 1\) and \(m > 2\), then
Observe that
Since
Thus,
Case 2: For \(m > n > 1\), we have
While
Since
and
Hence
Thus, we have shown T is CRRΘdouble controlled contraction mapping. Next, we show that conditions (M1) and (M2) of Theorem 3.11 hold. We form a sequence \(\{ z_{n} \}\) by taking \(z_{0} = S_{n}\), with \(n > 2\). As \(T(S_{n}) = S_{n1}\), thus \(z_{1} =S_{n1}\), \(z_{2} = S_{n2}\), so \(z_{j} = S_{nj}\). Note that
and
Then,
take \(p = 1/4 \in (0, 1)\). Clearly for any \(x \in Z\), then \(\lim_{n\rightarrow \infty}\beta (x,S_{n})\) and \(\lim_{n\rightarrow \infty}\mu (S_{n}, x)\) exist and are finite. Therefore, T satisfies the properties of Theorem 3.11, hence T has a unique fixed point \(S_{1}\).
By taking \(\beta = \mu \) in Theorem 3.11 and modifying the conditions (M1) and (M2) accordingly, we obtain the following immediate result for the case of controlled metric type space.
Corollary 3.13
Let \((Z,d_{\beta})\) be a complete controlled metric type space, and let \(T :Z\rightarrow Z\) be a CRR Θdouble controlled contraction self mapping satisfying the following:
(C1)
for some \(p \in (0,1)\), and
(C2) For any \(z \in Z\), then \(\lim_{n\rightarrow \infty}\beta (z, z_{n})\) and \(\lim_{n\rightarrow \infty}\beta (z_{n},z)\) exist and are finite, where the sequence \(\{ z_{n}\}\) is defined as \(z_{n} = T^{n} z_{0}\) for some \(z_{0} \in Z\).
Then T has a unique fixed point in Z.
In case \(\beta (z, w) = \mu (z, w) = 1\) in Theorem 3.11, we get the following corollary.
Corollary 3.14
Let \((Z,d)\) be a complete metric space, let \(T :Z\rightarrow Z\) be a self mapping. Suppose there exists a continuous function \(\theta \in \Theta \) and an \(r \in (0,1)\), such that the following holds:
for all \(z, w \in Z\), such that \(d_{\beta ,\mu}(T z,T w) \neq 0 \), where
Then T has a unique fixed point.
Corollary 3.15
Let \((Z,d_{\beta ,\mu})\) be a complete double controlled metric type space, and let \(T :Z\rightarrow Z\) be a self mapping. Suppose there exists \(\gamma \in (0,1)\) such that the following holds:
\((S1)\)
(S2)
for some \(p \in (0,1)\), and
(S3) For any \(z \in Z\), then \(\lim_{n\rightarrow \infty}\beta (z,z_{n})\) and \(\lim_{n\rightarrow \infty}\mu (z_{n},z)\) exist and are finite, where the sequence \(\{ z_{n}\}\) is defined as \(z_{n} = T^{n} z_{0}\) for some \(z_{0} \in Z\).
Then T has a unique fixed point in Z.
Proof
Let \({\theta}:(0,\infty ) \rightarrow (1,\infty )\) be defined by \(\theta (t) = e^{\sqrt{t}}\), then \(\theta \in \Theta \) as in Definition 2.9, and it is continuous. Property (S1) implies
Let \(r = \sqrt{\gamma} \in (0,1)\) Therefore, the existence and uniqueness of the fixed point follow from Theorem 3.11. □
4 Application
In the closing, we would like to illustrate the importance of our Theorem 3.3 in finding a unique real solution for an mth degree polynomial. There are many other methods for root finding problems, such as numerical methods, but utilizing the fixed point results, it becomes quite easy, as it is presented below.
Theorem 4.1
For any natural number \(m \geq 3\), the equation
has a unique real solution in the interval \([1,1]\).
Proof
If \(\vert \xi \vert > 1\), then equation (4.1) does not have a solution, and therefore \(\vert \xi \vert \leq 1\). Let \(Z = [1, 1]\), then for any \(\xi , \nu \in Z\), let the metric be defined by \(d_{\beta ,\mu} (\xi , \nu ) = \vert \xi  \nu \vert \).
The two functions \(\beta , \mu : Z \times Z \rightarrow [1, \infty )\) are defined by
One can easily show that \((Z, d_{\beta ,\mu})\) is a complete double controlled metric type space. Let \(T: Z \rightarrow Z\) be a mapping defined by
and let \({\theta}:(0,\infty ) \rightarrow (1,\infty )\) be defined by \(\theta (t) = e^{\sqrt{t}}\), then \(\theta \in \Theta \).
We will show that T is Θdouble controlled contraction mapping.
As \(m \geq 3\), we can conclude that \(m^{4} \geq 81\). Therefore,
This yields
where \(r = {\sqrt{e^{\tau}}} \in (0,1)\), since \(3 < \tau \leq 5\). Hence
Next, we show conditions (1) and (2) of Theorem 3.3 hold by taking \(p = 1/4 \in (0,1)\). Hence for any \(z_{0} \in Z\), we define the sequence \(\{z_{n} \} \in Z\) by
Using the definitions of β, μ, and (4.3), we obtain
Also, both \(\lim_{n\rightarrow \infty}\beta (z,z_{n})\) and \(\lim_{n\rightarrow \infty}\mu (z_{n},z)\) exist and are finite. Thus, all the conditions of Theorem 3.3 are satisfied, and therefore T has a unique fixed point in Z, which is a unique real solution of equation (4.1). □
5 Conclusion
Our space was a double controlled metric type space; there, we introduced two new types of generalized contraction mappings. In the first one, inspired by the work [26], we introduced Θdouble controlled contraction mapping, while in the second one, inspired by the work [28], we introduced a ĆirićReichRustype Θdouble controlled contraction mapping. Under these mappings, we established the existence and uniqueness of the fixed point theorems on complete double controlled metric type spaces and presented some examples.
Karapinar introduced the notion of an interpolative Kannantype contraction in [33]. Recently, Aydi et al. initiated the concept of ωinterpolative ĆirićReichRustype contractions and established fixed point results [34, 35]. We propose some suggestions for future research directions, such as utilizing ωinterpolative ĆirićReichRustype contractions on complete double controlled metric type space and exploring fixed point results.
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Azmi, F.M. Generalized contraction mappings in double controlled metric type space and related fixed point theorems. J Inequal Appl 2023, 87 (2023). https://doi.org/10.1186/s1366002302999x
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DOI: https://doi.org/10.1186/s1366002302999x
Keywords
 Fixed point
 Double controlled metric type space
 ĆirićReichRustype contraction
 Generalized Contraction mapping
 Θ contraction
 Zeros of high degree polynomials