Abstract
We study the existence and uniqueness of fixed points for self-operators defined in a b-metric space and belonging to the class of \((\alpha,\psi)\)-type contraction mappings. The obtained results generalize and unify several existing fixed point theorems in the literature.
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1 Introduction and preliminaries
Very recently, we studied in [1] the existence and uniqueness of fixed points for self-operators defined in a metric space and belonging to the class of \((\alpha,\psi)\)-type contraction mappings (see [2–5] for some works in this direction). We proved that the class of α-ψ-type contractions includes large classes of contraction-type operators, whose fixed points can be obtained by means of the Picard iteration. The aim of this paper is to extend the obtained results in [1] to self-operators defined in a b-metric space.
We start by recalling the following definition.
Definition 1.1
([6])
Let X be a nonempty set. A mapping \(d: X\times X \to[0,\infty)\) is called b-metric if there exists a real number \(b\geq1\) such that for every \(x,y,z\in X\), we have
-
(i)
\(d(x,y)=0\) if and only if \(x=y\);
-
(ii)
\(d(x,y)=d(y,x)\);
-
(iii)
\(d(x,z)\leq b[d(x,y)+d(y,z)]\).
In this case, the pair \((X,d)\) is called a b-metric space.
There exist many examples in the literature (see [6–8]) showing that the class of b-metrics is effectively larger than that of metric spaces.
The notions of convergence, compactness, closedness and completeness in b-metric spaces are given in the same way as in metric spaces. For works on fixed point theory in b-metric spaces, we refer to [9–12] and the references therein.
Definition 1.2
([13])
Let \(\psi:[0,\infty)\to[0,\infty)\) be a given function. We say that ψ is a comparison function if it is increasing and \(\psi^{n}(t)\to 0\), \(n\to\infty\), for any \(t\geq0\), where \(\psi^{n}\) is the nth iterate of ψ.
In [13, 14], several results regarding comparison functions can be found. Among these we recall the following.
Lemma 1.3
If \(\psi:[0,\infty)\to[0,\infty)\) is a comparison function, then
-
(i)
each iterate \(\psi^{k}\) of ψ, \(k\geq1\), is also a comparison function;
-
(ii)
ψ is continuous at zero;
-
(iii)
\(\psi(t)< t\) for any \(t >0\);
-
(iv)
\(\psi(0)=0\).
The following concept was introduced in [15].
Definition 1.4
Let \(b\geq1\) be a real number. A mapping \(\psi:[0,\infty)\to[0,\infty )\) is called a b-comparison function if
-
(i)
ψ is monotone increasing;
-
(ii)
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
$$b^{k+1}\psi^{k+1}(t)\leq a b^{k} \psi^{k}(t)+v_{k} $$for \(k\geq k_{0}\) and any \(t\geq0\).
The following lemma has been proved.
Lemma 1.5
Let \(\psi:[0,\infty)\to[0,\infty)\) be a b-comparison function. Then
-
(i)
the series \(\sum_{k=0}^{\infty}b^{k}\psi ^{k}(t)\) converges for any \(t\geq0\);
-
(ii)
the function \(s_{b}:[0,\infty)\to[0,\infty)\) defined by
$$s_{b}(t)=\sum_{k=0}^{\infty}b^{k}\psi^{k}(t), \quad t\geq0 $$is increasing and continuous at 0.
Lemma 1.6
([17])
Any b-comparison function is a comparison function.
Throughout this paper, for \(b\geq1\), we denote by \(\Psi_{b}\) the set of b-comparison functions.
Definition 1.7
Let \((X,d)\) be a b-metric space with constant \(b\geq1\), and let \(T: X\to X\) be a given mapping. We say that T is an α-ψ contraction if there exist a b-comparison function \(\psi\in\Psi_{b}\) and a function \(\alpha:X\times X\to\mathbb{R}\) such that
2 Main results
Let \(T: X\to X\) be a given mapping. We denote by \(\operatorname{Fix}(T)\) the set of its fixed points; that is,
For \(b\geq1\) and \(\psi\in\Psi_{b}\), let \(\Sigma_{\psi}^{b}\) be the set defined by
We have the following result.
Proposition 2.1
Let \((X,d)\) be a b-metric space with constant \(b\geq1\), and let \(T: X\to X\) be a given mapping. Suppose that there exist \(\alpha: X\times X \to\mathbb{R}\) and \(\psi\in\Psi_{b}\) such that T is an α-ψ contraction. Suppose that there exists \(\sigma\in\Sigma_{\psi}^{b}\) and for some positive integer p, there exists a finite sequence \(\{\xi_{i}\}_{i=0}^{p}\subset X\) such that
Then \(\{T^{n}x_{0}\}\) is a Cauchy sequence in \((X,d)\).
Proof
Let \(\varphi=\sigma\psi\). By the definition of \(\Sigma_{\psi}^{b}\), we have \(\varphi\in\Psi_{b}\). Let \(\{\xi_{i}\}_{i=0}^{p}\) be a finite sequence in X satisfying (2.1). Consider the sequence \(\{x_{n}\}_{n\in\mathbb{N}}\) in X defined by \(x_{n+1}=Tx_{n}\), \(n\in\mathbb{N}\). We claim that
Let \(i\in\{0,1,\ldots,p-1\}\). From (2.1), we have
which implies that
Again, we have
which implies that
Since φ is an increasing function (from Lemma 1.6), from (2.3) and (2.4), we obtain
Continuing this process, by induction we obtain (2.2).
Now, using the property (iii) of a b-metric and (2.2), for every \(n\in\mathbb{N}\), we have
Thus we proved that
which implies that for \(q\geq1\),
Since \(b\geq1\), using Lemma 1.5(i), we obtain
This proves that \(\{x_{n}\}\) is a Cauchy sequence in the b-metric space \((X,d)\). □
Our first main result is the following fixed point theorem which requires the continuity of the mapping T.
Theorem 2.2
Let \((X,d)\) be a complete b-metric space with constant \(b\geq1\), and let \(T: X\to X\) be a given mapping. Suppose that there exist \(\alpha: X\times X \to\mathbb{R}\) and \(\psi \in\Psi_{b}\) such that T is an α-ψ contraction. Suppose also that (2.1) is satisfied. Then \(\{T^{n}x_{0}\}\) converges to some \(x^{*}\in X\). Moreover, if T is continuous, then \(x^{*}\in\operatorname{Fix}(T)\).
Proof
From Proposition 2.1, we know that \(\{ T^{n}x_{0}\}\) is a Cauchy sequence. Since \((X,d)\) is a complete b-metric space, there exists \(x^{*}\in X\) such that
The continuity of T yields
By the uniqueness of the limit, we obtain \(x^{*}=Tx^{*}\), that is, \(x^{*}\in \operatorname{Fix}(T)\). □
In the next theorem, we omit the continuity assumption of T.
Theorem 2.3
Let \((X,d)\) be a complete b-metric space with constant \(b\geq1\), and let \(T: X\to X\) be a given mapping. Suppose that there exist \(\alpha: X\times X \to\mathbb{R}\) and \(\psi \in\Psi_{b}\) such that T is an α-ψ contraction. Suppose also that (2.1) is satisfied. Then \(\{T^{n}x_{0}\}\) converges to some \(x^{*}\in X\). Moreover, if there exists a subsequence \(\{T^{\gamma(n)}x_{0}\}\) of \(\{T^{n}x_{0}\}\) such that
then \(x^{*}\in\operatorname{Fix}(T)\).
Proof
From Proposition 2.1 and the completeness of the b-metric space \((X,d)\), we know that \(\{T^{n}x_{0}\}\) converges to some \(x^{*}\in X\).
Suppose now that there exists a subsequence \(\{T^{\gamma(n)}x_{0}\}\) of \(\{T^{n}x_{0}\}\) such that
Since T is an α-ψ contraction, we have
and
Thus we have
From (2.5), we get
On the other hand, using the property (iii) of a b-metric, we get
Letting \(n\to\infty\) in the above inequality, using Lemma 1.6 and Lemma 1.3(ii) and (iv), we obtain
which implies that \(d(x^{*},Tx^{*})=0\), that is, \(x^{*}\in\operatorname {Fix}(T)\). □
We provide now a sufficient condition for the uniqueness of the fixed point.
Theorem 2.4
Let \((X,d)\) be a b-metric space with constant \(b\geq1\), and let \(T: X\to X\) be a given mapping. Suppose that there exist \(\alpha: X\times X \to\mathbb{R}\) and \(\psi \in\Psi_{b}\) such that T is an α-ψ contraction. Suppose also that
-
(i)
\(\operatorname{Fix}(T)\neq\emptyset\);
-
(ii)
for every pair \((x,y)\in\operatorname{Fix}(T)\times \operatorname{Fix}(T)\) with \(x\neq y\), if \(\alpha(x,y)<1\), then there exists \(\eta\in\Sigma_{\psi}^{b}\) and for some positive integer q, there is a finite sequence \(\{\zeta_{i}(x,y)\}_{i=0}^{q}\subset X\) such that
$$\zeta_{0}(x,y)=x, \qquad \zeta_{q}(x,y)=y,\qquad \alpha \bigl(T^{n}\zeta_{i}(x,y),T^{n} \zeta_{i+1}(x,y)\bigr)\geq\eta^{-1} $$for \(n\in\mathbb{N}\) and \(i=0,\ldots,q-1\).
Then T has a unique fixed point.
Proof
Let \(\varphi=\eta\psi\in\Psi_{b}\). Suppose that \(u,v\in X\) are two fixed points of T such that \(d(u,v)>0\). We consider two cases.
Case 1: \(\alpha(u,v)\geq1\). Since T is an α-ψ contraction, we have
On the other hand, from Lemma 1.6 and Lemma 1.3(iii), we have
The two above inequalities yield a contradiction.
Case 2: \(\alpha(u,v)<1\). By assumption, there exists a finite sequence \(\{\zeta_{i}(u,v)\} _{i=0}^{q}\) in X such that
for \(n\in\mathbb{N}\) and \(i=0,\ldots,q-1\). As in the proof of Proposition 2.1, we can establish that
On the other hand, we have
Then \(u=v\), which is a contradiction. □
3 Particular cases
In this section, we deduce from our main theorems several fixed point theorems in b-metric spaces.
3.1 The class of ψ-type contractions in b-metric spaces
Definition 3.1
Let \((X,d)\) be a b-metric space with constant \(b\geq1\). A mapping \(T: X\to X\) is said to be a ψ-contraction if there exists \(\psi\in\Psi_{b}\) such that
Theorem 3.2
Let \((X,d)\) be a b-metric space with constant \(b\geq1\), and let \(T: X\to X\) be a given mapping. Suppose that there exists \(\psi\in\Psi_{b}\) such that T is a ψ-contraction. Then there exists \(\alpha: X\times X \to\mathbb{R}\) such that T is an α-ψ contraction.
Proof
Consider the function \(\alpha: X\times X \to\mathbb{R}\) defined by
Clearly, from (3.1), T is an α-ψ contraction. □
Corollary 3.3
([17])
Let \((X,d)\) be a complete b-metric space with constant \(b\geq1\), and let \(T: X\to X\) be a given mapping. If T is a ψ-contraction for some \(\psi\in\Psi_{b}\), then T has a unique fixed point. Moreover, for any \(x_{0}\in X\), the Picard sequence \(\{T^{n}x_{0}\}\) converges to this fixed point.
Proof
From Lemma 1.6, we have
which implies that T is a continuous mapping. From Theorem 3.2, T is an α-ψ contraction, where α is defined by (3.2). Clearly, for any \(x_{0}\in X\), (2.1) is satisfied with \(p=1\) and \(\sigma=1\). By Theorem 2.2, \(\{T^{n}x_{0}\}\) converges to a fixed point of T. The uniqueness follows immediately from (3.2) and Theorem 2.4. □
Corollary 3.4
Let \((X,d)\) be a complete b-metric space with constant \(b\geq1\), and let \(T: X\to X\) be a given mapping. Suppose that
for some constant \(k\in(0,1/b)\). Then T has a unique fixed point. Moreover, for any \(x_{0}\in X\), the Picard sequence \(\{T^{n}x_{0}\}\) converges to this fixed point.
Proof
It is an immediate consequence of Corollary 3.3 with \(\psi(t)=kt\). □
3.2 The class of rational-type contractions in b-metric spaces
3.2.1 Dass-Gupta-type contraction in b-metric spaces
Definition 3.5
Let \((X,d)\) be a b-metric space with constant \(b\geq1\). A mapping \(T: X\to X\) is said to be a Dass-Gupta contraction if there exist constants \(\lambda,\mu\geq0\) with \(\lambda b+\mu<1\) such that
Theorem 3.6
Let \((X,d)\) be a b-metric space with constant \(b\geq1\), and let \(T: X\to X\) be a given mapping. Suppose that T is a Dass-Gupta contraction. Then there exist \(\psi\in \Psi_{b}\) and \(\alpha: X\times X \to\mathbb{R}\) such that T is an α-ψ contraction.
Proof
From (3.3), for all \(x,y\in X\), we have
which yields
Consider the functions \(\psi:[0,\infty)\to[0,\infty)\) and \(\alpha:X\times X\to\mathbb{R}\) defined by
and
Since \(0\leq\lambda b<1\), then \(\psi\in\Psi_{b}\). On the other hand, from (3.4) we have
Then T is an α-ψ contraction. □
Corollary 3.7
Let \((X,d)\) be a complete b-metric space with constant \(b\geq1\), and let \(T: X\to X\) be a given mapping. If T is a Dass-Gupta contraction with parameters \(\lambda,\mu\geq0\) such that \(\lambda b+\mu<1\), then T has a unique fixed point. Moreover, for any \(x_{0}\in X\), the Picard sequence \(\{ T^{n}x_{0}\}\) converges to this fixed point.
Proof
Let \(x_{0}\) be an arbitrary point in X. If for some \(r\in\mathbb{N}\), \(T^{r}x_{0}=T^{r+1}x_{0}\), then \(T^{r}x_{0}\) will be a fixed point of T. So we can suppose that \(T^{r}x_{0}\neq T^{r+1}x_{0}\) for all \(r\in\mathbb{N}\). From (3.6), for all \(n\in\mathbb{N}\), we have
On the other hand, from (3.5) we have
From the condition \(\lambda b+\mu<1\), clearly we have \((1-\mu)^{-1}\psi \in\Psi_{b}\), which is equivalent to \((1-\mu)^{-1}\in\Sigma_{\psi}^{b}\). Then (2.1) is satisfied with \(p=1\) and \(\sigma=(1-\mu)^{-1}\). From the first part of Theorem 2.3, the sequence \(\{T^{n}x_{0}\}\) converges to some \(x^{*}\in X\).
Suppose that \(x^{*}\) is not a fixed point of T, that is, \(d(x^{*},Tx^{*})>0\). Then
From (3.6), we have
On the other hand, using the property (iii) of a b-metric, we have
Thus we have
Since
we have
By Theorem 2.3, we deduce that \(x^{*}\in \operatorname{Fix}(T)\), which is a contradiction. Thus \(\operatorname {Fix}(T)\neq\emptyset\).
For the uniqueness, observe that for every pair \((x,y)\in\operatorname {Fix}(T)\times\operatorname{Fix}(T)\) with \(x\neq y\), we have \(\alpha(x,y)=1\). By Theorem 2.4, \(x^{*}\) is the unique fixed point of T. □
If \(b=1\), Corollary 3.7 recovers the Dass-Gupta fixed point theorem [18].
3.2.2 Jaggi-type contraction in b-metric spaces
Definition 3.8
Let \((X,d)\) be a b-metric space with constant \(b\geq1\). A mapping \(T: X\to X\) is said to be a Jaggi contraction if there exist constants \(\lambda,\mu\geq0\) with \(\lambda b+\mu<1\) such that
Theorem 3.9
Let \((X,d)\) be a b-metric space with constant \(b\geq1\), and let \(T: X\to X\) be a given mapping. Suppose that T is a Jaggi contraction. Then there exist \(\psi\in\Psi_{b}\) and \(\alpha: X\times X \to\mathbb{R}\) such that T is an α-ψ contraction.
Proof
From (3.7), for all \(x,y\in X\) with \(x\neq y\), we have
which yields
Consider the functions \(\psi:[0,\infty)\to[0,\infty)\) and \(\alpha:X\times X\to\mathbb{R}\) defined by
and
Since \(\lambda b<1\), we have \(\psi\in\Psi_{b}\). From (3.8), we have
Then T is an α-ψ contraction. □
Corollary 3.10
Let \((X,d)\) be a complete b-metric space with constant \(b\geq1\), and let \(T: X\to X\) be a continuous mapping. If T is a Jaggi contraction with parameters \(\lambda,\mu\geq0\) such that \(\lambda b+\mu<1\), then T has a unique fixed point. Moreover, for any \(x_{0}\in X\), the Picard sequence \(\{T^{n}x_{0}\}\) converges to this fixed point.
Proof
Let \(x_{0}\) be an arbitrary point in X. Without loss of generality, we can suppose that \(T^{r}x_{0}\neq T^{r+1}x_{0}\) for all \(r\in\mathbb{N}\). From (3.10), for all \(n\in\mathbb{N}\), we have
On the other hand, from (3.9), for all \(t\geq0\), we have
Since \(\lambda b+\mu<1\), we have \((1-\mu)^{-1}\psi\in\Psi_{b}\), that is, \((1-\mu)^{-1}\in\Sigma_{\psi}^{b}\). Then (2.1) is satisfied with \(p=1\) and \(\sigma=(1-\mu)^{-1}\). By the first part of Theorem 2.2, \(\{T^{n}x_{0}\}\) converges to some \(x^{*}\in X\). Since T is continuous, by the second part of Theorem 2.2, \(x^{*}\) is a fixed point of T. Moreover, for every pair \((x,y)\in\operatorname{Fix}(T)\times\operatorname{Fix}(T)\) with \(x\neq y\), we have \(\alpha(x,y)=1\). Then, by Theorem 2.4, \(x^{*}\) is the unique fixed point of T. □
If \(b=1\), Corollary 3.10 recovers the Jaggi fixed point theorem [19].
3.3 The class of Berinde-type mappings in b-metric spaces
Definition 3.11
Let \((X,d)\) be a b-metric space with constant \(b\geq1\). A mapping \(T: X\to X\) is said to be a Berinde-type contraction if there exist \(\lambda\in(0,1/b)\) and \(L\geq0\) such that
Theorem 3.12
Let \((X,d)\) be a b-metric space with constant \(b\geq1\), and let \(T: X\to X\) be a given mapping. If T is a Berinde-type contraction, then there exist \(\alpha: X\times X \to\mathbb{R}\) and \(\psi\in\Psi_{b}\) such that T is an α-ψ contraction.
Proof
From (3.11), we have
which yields
Consider the functions \(\psi:[0,\infty)\to[0,\infty)\) and \(\alpha:X\times X\to\mathbb{R}\) defined by
and
Since \(\lambda b<1\), then \(\psi\in\Psi_{b}\). From (3.12), we have
Then T is an α-ψ contraction. □
Corollary 3.13
Let \((X,d)\) be a complete b-metric space with constant \(b\geq1\), and let \(T: X\to X\) be a given mapping. If T is a Berinde-type contraction with parameters \(\lambda,L \geq0\) such that \(0<\lambda b<1\), then for any \(x_{0}\in X\), the Picard sequence \(\{T^{n}x_{0}\}\) converges to a fixed point of T.
Proof
Let \(x_{0}\) be an arbitrary point in X. Without loss of generality, we can suppose that \(T^{r}x_{0}\neq T^{r+1}x_{0}\) for all \(r\in\mathbb{N}\). From (3.13), for all \(n\in\mathbb{N}\), we have
Then (2.1) holds with \(\sigma=1\) and \(p=1\). From the first part of Theorem 2.3, the sequence \(\{T^{n}x_{0}\}\) converges to some \(x^{*}\in X\).
Suppose now that \(x^{*}\) is not a fixed point of T, that is, \(d(x^{*},Tx^{*})>0\). Then
From (3.13), we have
Using the property (iii) of a b-metric, we have
Thus we have
Since
then
By Theorem 2.3, we deduce that \(x^{*}\in \operatorname{Fix}(T)\), which is a contradiction.
Thus \(x^{*}\) is a fixed point of T. □
If \(b=1\), Corollary 3.13 recovers the Berinde fixed point theorem [20].
Note that a Berinde mapping need not have a unique fixed point (see [21], Example 2.11).
Corollary 3.14
Let \((X,d)\) be a complete b-metric space with constant \(b\geq1\), and let \(T: X\to X\) be a given mapping. Suppose that there exists a constant \(k\in(0,1/b(b+1))\) such that
Then, for any \(x_{0}\in X\), the Picard sequence \(\{T^{n}x_{0}\}\) converges to a fixed point of T.
Proof
At first, observe that from (3.14), for all \(x,y\in X\), we have
where
With the condition \(k\in(0,1/b(b+1))\), we have \(0<\lambda<1/b\) and \(L\geq0\). Then T is a Berinde-type contraction. From Corollary 3.13, if \(x_{0}\in X\), then \(\{T^{n}x_{0}\}\) converges to a fixed point of T. □
If \(b=1\), Corollary 3.14 recovers the Kannan fixed point theorem [22].
Corollary 3.15
Let \((X,d)\) be a complete b-metric space with constant \(b\geq1\), and let \(T: X\to X\) be a given mapping. Suppose that there exists a constant \(k\in(0,1/2b^{2})\) such that
Then, for any \(x_{0}\in X\), the Picard sequence \(\{T^{n}x_{0}\}\) converges to a fixed point of T.
Proof
From (3.15), we have
where
With the condition \(k\in(0,1/2b^{2})\), we have \(0<\lambda<1/b\) and \(L\geq0\). Then T is a Berinde-type contraction. From Corollary 3.13, if \(x_{0}\in X\), then \(\{T^{n}x_{0}\}\) converges to a fixed point of T. □
If \(b=1\), Corollary 3.15 recovers the Chatterjee fixed point theorem [23].
3.4 Ćirić-type mappings in b-metric spaces
Definition 3.16
Let \((X,d)\) be a b-metric space with constant \(b\geq1\). A mapping \(T: X\to X\) is said to be a Ćirić-type mapping if there exists \(\lambda\in(0,1/b)\) such that for all \(x,y\in X\), we have
Theorem 3.17
Let \((X,d)\) be a b-metric space with constant \(b\geq1\), and let \(T: X\to X\) be a given mapping. If T is a Ćirić-type mapping with parameter \(\lambda\in(0,1/b)\), then there exist \(\alpha: X\times X \to\mathbb{R}\) and \(\psi\in\Psi _{b}\) such that T is an α-ψ contraction.
Proof
Consider the functions \(\psi:[0,\infty)\to[0,\infty)\) and \(\alpha:X\times X\to\mathbb{R}\) defined by
and
From (3.16), we have
which implies that T is an α-ψ contraction. □
Corollary 3.18
Let \((X,d)\) be a complete b-metric space with constant \(b\geq1\), and let \(T: X\to X\) be a continuous mapping. If T is a Ćirić-type mapping with parameter \(\lambda\in(0,1/b)\), then for any \(x_{0}\in X\), the Picard sequence \(\{T^{n}x_{0}\}\) converges to a fixed point of T.
Proof
Let \(x_{0}\in X\) be an arbitrary point. Without loss of generality, we can suppose that \(T^{r}x_{0}\neq T^{r+1}x_{0}\) for all \(r\in\mathbb{N}\). From (3.18), for all \(n\in\mathbb{N}\), we have
Suppose that for some \(n\in\mathbb{N}\), we have
In this case, from (3.17) and (3.19), we have
This implies (from the assumption \(T^{r}x_{0}\neq T^{r+1}x_{0}\) for all \(r\in\mathbb{N}\)) that \(\lambda\geq1\), which is a contradiction. Then
Then (2.1) is satisfied with \(p=1\) and \(\sigma=1\). By Theorem 2.3, we deduce that the sequence \(\{T^{n}x_{0}\}\) converges to a fixed point of T. □
If \(b=1\), Corollary 3.18 recovers Ćirić’s fixed point theorem [24].
3.5 Edelstein fixed point theorem in b-metric spaces
Another consequence of our main results is the following generalized version of Edelstein fixed point theorem [25] in b-metric spaces.
Corollary 3.19
Let \((X,d)\) be a complete b-metric space with constant \(b\geq1\), and ε-chainable for some \(\varepsilon>0\); i.e., given \(x,y\in X\), there exist a positive integer N and a sequence \(\{x_{i}\}_{i=0}^{N}\subset X\) such that
Let \(T: X\to X\) be a given mapping such that
for some \(\psi\in\Psi_{b}\). Then T has a unique fixed point.
Proof
It is clear from (3.21) that the mapping T is continuous. Now, consider the function \(\alpha:X\times X\to\mathbb{R}\) defined by
From (3.21), we have
Let \(x_{0}\in X\). For \(x=x_{0}\) and \(y=Tx_{0}\), from (3.20) and (3.22), for some positive integer p, there exists a finite sequence \(\{\xi_{i}\}_{i=0}^{p}\subset X\) such that
Now, let \(i\in\{0,\ldots,p-1\}\) be fixed. From (3.22) and (3.21), we have
Again,
By induction, we obtain
Then (2.1) is satisfied with \(\sigma=1\). From Theorem 2.2, the sequence \(\{T^{n}x_{0}\}\) converges to a fixed point of T. Using a similar argument, we can see that condition (ii) of Theorem 2.4 is satisfied, which implies that T has a unique fixed point. □
3.6 Contractive mapping theorems in b-metric spaces with a partial order
Let \((X,d)\) be a b-metric space with constant \(b\geq1\), and let ⪯ be a partial order on X. We denote
Corollary 3.20
Let \(T: X\to X\) be a given mapping. Suppose that there exists \(\psi\in\Psi_{b}\) such that
Suppose also that
-
(i)
T is continuous;
-
(ii)
for some positive integer p, there exists a finite sequence \(\{\xi_{i}\}_{i=0}^{p}\subset X\) such that
$$ \xi_{0}=x_{0},\qquad \xi_{p}=Tx_{0}, \qquad \bigl(T^{n}\xi_{i},T^{n}\xi_{i+1} \bigr)\in\Delta, \quad n\in\mathbb{N}, i=0,\ldots,p-1. $$(3.24)
Then \(\{T^{n}x_{0}\}\) converges to a fixed point of T.
Proof
Consider the function \(\alpha:X\times X\to\mathbb {R}\) defined by
From (3.23), we have
Then the result follows from Theorem 2.2 with \(\sigma=1\). □
Corollary 3.21
Let \(T: X\to X\) be a given mapping. Suppose that
Then \(\{T^{n}x_{0}\}\) converges to some \(x^{*}\in X\). Moreover, if
-
(iii)
there exist a subsequence \(\{T^{\gamma(n)}x_{0}\}\) of \(\{ T^{n}x_{0}\}\) and \(N\in\mathbb{N}\) such that
$$\bigl(T^{\gamma(n)}x_{0},x^{*}\bigr)\in\Delta, \quad n\geq N, $$
then \(x^{*}\) is a fixed point of T.
Proof
We continue to use the same function α defined by (3.25). From the first part of Theorem 2.3, the sequence \(\{T^{n}x_{0}\}\) converges to some \(x^{*}\in X\). From (iii) and (3.25), we have
By the second part of Theorem 2.3 (with \(\ell=1\)), we deduce that \(x^{*}\) is a fixed point of T. □
The next result follows from Theorem 2.4 with \(\eta=1\).
Corollary 3.22
Let \(T: X\to X\) be a given mapping. Suppose that
-
(i)
there exists \(\psi\in\Psi_{b}\) such that (3.23) holds;
-
(ii)
\(\operatorname{Fix}(T)\neq\emptyset\);
-
(iii)
for every pair \((x,y)\in\operatorname{Fix}(T) \times\operatorname{Fix}(T)\) with \(x\neq y\), if \((x,y)\notin\Delta\), there exist a positive integer q and a finite sequence \(\{\zeta_{i}(x,y)\}_{i=0}^{q}\subset X\) such that
$$\zeta_{0}(x,y)=x,\qquad \zeta_{q}(x,y)=y,\qquad \bigl(T^{n}\zeta_{i}(x,y),T^{n} \zeta_{i+1}(x,y)\bigr)\in\Delta $$for \(n\in\mathbb{N}\) and \(i=0,\ldots,q-1\).
Then T has a unique fixed point.
Observe that in our results we do not suppose that T is monotone or T preserves order as it is supposed in many papers (see [26–28] and others).
References
Samet, B: Fixed points for α-ψ contractive mappings with an application to quadratic integral equations. Electron. J. Differ. Equ. 2014, 152 (2014)
Amiri, P, Rezapour, S, Shahzad, N: Fixed points of generalized α-ψ-contractions. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 108, 519-526 (2014)
Bota, MF, Karapinar, E, Mleşniţe, O: Ulam-Hyers stability results for fixed point problems via α-ψ-contractive mapping in \((b)\)-metric space. Abstr. Appl. Anal. 2013, Article ID 825293 (2013)
Karapinar, E, Samet, B: Generalized α-ψ contractive type mappings and related fixed point theorems with applications. Abstr. Appl. Anal. 2012, Article ID 793486 (2012)
Karapinar, E, Shahi, P, Tas, K: Generalized α-ψ-contractive type mappings of integral type and related fixed point theorems. J. Inequal. Appl. 2014, 16 (2014)
Bakhtin, IA: The contraction principle in quasimetric spaces. In: Functional Analysis, vol. 30, pp. 26-37. Gos. Ped. Ins., Unianowsk (1989) (Russian)
Bourbaki, N: Topologie générale. Hermann, Paris (1961)
Czerwik, S: Nonlinear set-valued contraction mappings in b-metric spaces. Atti Semin. Mat. Fis. Univ. Modena 46, 263-276 (1998)
Amini-Harandi, A: Fixed point theory for quasi-contraction maps in b-metric spaces. Fixed Point Theory 15(2), 351-358 (2014)
Aydi, H, Bota, MF, Mitrovic, S, Karapinar, E: A fixed point theorem for set-valued quasi-contractions in b-metric spaces. Fixed Point Theory Appl. 2012, 88 (2012)
Bota, MF, Karapinar, E: A note on some results on multi-valued weakly Jungck mappings in b-metric space. Cent. Eur. J. Math. 11(9), 1711-1712 (2013)
Kadelburg, Z, Radenović, S: Pata-type common fixed point results in b-metric and b-rectangular metric spaces. J. Nonlinear Sci. Appl. (in press)
Rus, IA: Generalized Contractions and Applications. Cluj University Press, Cluj-Napoca (2001)
Berinde, V: Contracţii generalizate şi aplicaţii. Editura Cub Press 22, Baia Mare (1997)
Berinde, V: Sequences of operators and fixed points in quasimetric spaces. Stud. Univ. Babeş-Bolyai, Math. 16(4), 23-27 (1996)
Berinde, V: Une généralization de critère du d’Alembert pour les séries positives. Bul. Ştiinţ. - Univ. Baia Mare 7, 21-26 (1991)
Pacurar, M: A fixed point result for ϕ-contractions on b-metric spaces without the boundedness assumption. Fasc. Math. 43, 127-137 (2010)
Dass, BK, Gupta, S: An extension of Banach contraction principle through rational expressions. Indian J. Pure Appl. Math. 6, 1455-1458 (1975)
Jaggi, DS: Some unique fixed point theorems. Indian J. Pure Appl. Math. 8, 223-230 (1977)
Berinde, V: Approximating fixed points of weak contractions using the Picard iteration. Nonlinear Anal. Forum 9, 43-53 (2004)
Berinde, V: Iterative Approximation of Fixed Points. Lecture Notes in Mathematics (2007)
Kannan, R: Some results on fixed points. Bull. Calcutta Math. Soc. 60, 71-76 (1968)
Chatterjee, SK: Fixed point theorems. C. R. Acad. Bulgare Sci. 25, 727-730 (1972)
Cirić, L: On some maps with a nonunique fixed point. Publ. Inst. Math. (Belgr.) 17, 52-58 (1974)
Edelstein, M: An extension of Banach’s contraction principle. Proc. Am. Math. Soc. 12, 7-10 (1961)
Jachymski, J: The contraction principle for mappings on a metric space with a graph. Proc. Am. Math. Soc. 136(4), 1359-1373 (2008)
Nieto, JJ, Rodríguez-López, R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22(3), 223-239 (2005)
Ran, ACM, Reurings, MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 132, 1435-1443 (2004)
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This project was funded by the National Plan for Science, Technology and Innovation (MAARIFAH), King Abdulaziz City for Science and Technology, Kingdom of Saudi Arabia, Award Number (12-MAT 2895-02).
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Samet, B. The class of \((\alpha,\psi)\)-type contractions in b-metric spaces and fixed point theorems. Fixed Point Theory Appl 2015, 92 (2015). https://doi.org/10.1186/s13663-015-0344-z
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DOI: https://doi.org/10.1186/s13663-015-0344-z