Abstract
In this paper, we introduce a new class of \(\alpha_{qs^{p}}\)-admissible mappings and provide some fixed point theorems involving this class of mappings satisfying some new conditions of contractivity in the setting of b-metric-like spaces. Our results extend, unify, and generalize classical and recent fixed point results for contractive mappings.
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1 Introduction
In the past years extensions of a metric fixed point theory to generalized structures have received much attention. Also in these structures the concepts of fixed point theorems and contractions have appeared with a remarkable influence on applications in the theory of differential and integral equations, and giving appropriate mathematical models for solving a variety of applied problems in the mathematical sciences and engineering. Some generalizations are b-metric spaces introduced by Bakhtin [1] (and later extensively used by Czerwik [2]), partial metric spaces by Matthews [3], b-partial metric spaces by Shukla [4], metric-like spaces by Harandi [5], and b-metric-like spaces by Alghmandi et al. [6]. Later, Hussain [7] discussed the topological structure of b-metric-like spaces.
Also these generalizations have been associated with new and generalized classes of contractive mappings. In this direction, Samet et al. [8] introduced the concept of α-admissible, α-contractive, and \(\alpha - \psi\)-contractive mappings, further extended to the \((\alpha,\beta ) \)-contractive mappings. Many papers dealing with these notions have been considered to prove fixed point results (for example, see [8–23]).
In this paper, working in this direction, we introduce the concept of an \(\alpha_{qs^{p}} \)-admissible mapping and provide some fixed point results involving \(\alpha_{qs^{p}} - \lambda\) contractions and generalized \((\alpha_{qs^{p}} - \psi,\phi )\) contractive mappings in the larger framework of b-spaces, precisely, in the setting of b-metric-like spaces. The presented theorems improve, extend, generalize, and unify a number of existing results in the literature.
2 Preliminaries
Definition 2.1
([2])
Let X be a nonempty set. A mapping \(d:X \times X \to [ 0,\infty )\) is called a b-metric if the following conditions hold for all \(x,y,z \in X\) and for some \(s \ge 1\):
The pair \(( X,d )\) is called a b-metric space with parameter s.
Definition 2.2
([3])
Let X be a nonempty set. A mapping \(p:X \times X \to [ 0,\infty )\) is called a partial metric if the following conditions hold for all \(x,y,z \in X\) and \(s \ge 1\): \(x = y \Leftrightarrow p ( x,x ) = p ( x,y ) = p ( y,y )\);
The pair \(( X,p )\) is called a partial metric space.
Definition 2.3
([4])
Let X be a nonempty set. A mapping \(p_{b}:X \times X \to [ 0,\infty )\) is called a partial b-metric if, for any real number \(s \ge 1\) and for all \(x,y,z \in X\):
The pair \(( X,p_{b} )\) is called a partial b-metric space.
Definition 2.4
([5])
Let X be a nonempty set. A mapping \(\sigma:X \times X \to [ 0,\infty )\) is called metric-like if the following conditions hold for all \(x,y,z \in X\):
The pair \(( X,\sigma )\) is called a metric-like space.
Definition 2.5
([6])
Let X be a nonempty set. A mapping \(\sigma_{b}:X \times X \to [ 0,\infty )\) is called b-metric-like if the following conditions hold for all \(x,y,z \in X\) and for some \(s \ge 1\):
The pair \(( X,\sigma_{b} )\) is called a b-metric-like space.
In a b-metric-like space \(( X,\sigma_{b} )\), if \(x,y \in X\) and \(\sigma_{b} ( x,y ) = 0\), then \(x = y\), but the converse need not be true, and \(\sigma_{b} ( x,x )\) may be positive for \(x \in X\).
Remark 2.6
The class of b-metric-like spaces is larger than either metric-like spaces or b-metric-spaces, since a b-metric-like space is a metric-like space when \(s = 1\) and since every b-metric space is a b-metric-like space with the same parameter s. However, the converse implications do not hold.
Example 2.7
([6])
Let \(X = R^{ +} \cup \{ 0\}\). Define the function \(\sigma_{b}:X^{2} \to [ 0,\infty )\) by \(\sigma_{b}(x,y) = (x + y)^{2}\) for all \(x,y \in X\). Then \(( X,\sigma_{b} )\) is a b-metric-like space with parameter \(s = 2\).
Example 2.8
([24])
Let \(X = R^{ +} \cup \{ 0\}\). Define the function \(\sigma_{b}:X^{2} \to [ 0,\infty )\) by \(\sigma_{b}(x,y) = (\max \{ x,y \})^{2}\) for all \(x,y \in X\). Then \(( X,\sigma_{b} )\) is a b-metric-like space with parameter \(s = 2\). Clearly, \(( X,\sigma_{b} )\) is not a b-metric or metric-like space.
Definition 2.9
([6])
Let \(( X,\sigma_{b} )\) be a b-metric-like space with parameter s, let \(\{ x_{n} \}\) be any sequence in X, and let \(x \in X\). Then
-
(a)
The sequence \(\{ x_{n} \}\) is said to converge to x if \(\lim_{n \to \infty} \sigma_{b} ( x_{n},x ) = \sigma_{b} ( x,x )\);
-
(b)
The sequence \(\{ x_{n} \}\) is said to be a Cauchy sequence in \(( X,\sigma_{b} )\) if \(\lim_{n,m \to \infty} \sigma_{b} ( x_{n},x_{m} )\) exists and is finite;
-
(c)
\(( X,\sigma_{b} )\) is said to be a complete b-metric-like space if, for every Cauchy sequence \(\{ x_{n} \}\) in X, there exists \(x \in X\) such that \(\lim_{n,m \to \infty} \sigma_{b} ( x_{n},x_{m} ) = \lim_{n \to \infty} \sigma_{b} ( x_{n},x ) = \sigma_{b} ( x,x )\).
The limit of a sequence in a b-metric-like space need not be unique.
Proposition 2.10
([6])
Let \(( X,\sigma_{b} )\) be a b-metric-like space with parameter s, and let \(\{ x_{n} \}\) be any sequence in X with \(x \in X\) such that \(\lim_{n \to \infty} \sigma_{b} ( x_{n},x ) = 0\).Then
-
(a)
x is unique,
-
(b)
\(\sigma_{b} ( x,y ) / s \le \lim_{n \to \infty} \sigma_{b} ( x_{n},y ) \le s\sigma_{b} ( x,y )\) for all \(y \in X\).
In 2012, Samet et al. [8] introduced the class of α-admissible mappings.
Definition 2.11
Let X be a nonempty set, \(f:X \to X\), and \(\alpha:X \times X \to R^{ +} \). We say that f is an α-admissible mapping if \(\alpha ( x,y ) \ge 1\) implies that \(\alpha ( fx,fy ) \ge 1\) for all \(x,y \in X\).
Since, in general, a b-metric-like space is not continuous, we quote the following lemmas about the convergence of sequences.
Lemma 2.12
([7])
Let \(( X,\sigma_{b} )\) be a b-metric-like space with parameter \(s \ge 1\), and suppose that \(\{ x_{n} \}\) and \(\{ y_{n} \}\) are \(\sigma_{b}\)-convergent to x and y, respectively. Then we have
In particular, if \(\sigma_{b} ( x,y ) = 0\), then we have \(\lim_{n \to \infty} \sigma_{b} ( x_{n},y_{n} ) = 0\).
Moreover, for each \(z \in X\), we have
In particular, if \(\sigma_{b} ( x,x ) = 0\), then
The following result is useful.
Lemma 2.13
Let \(( X,\sigma_{b} )\) be a b-metric-like space with parameter \(s \ge 1\). Then
-
(a)
If \(\sigma_{b}(x,y) = 0\), then \(\sigma_{b}(x,x) = \sigma_{b}(y,y) = 0\);
-
(b)
If \((x_{n})\) is a sequence such that \(\lim_{n \to \infty} \sigma_{b}(x_{n},x_{n + 1}) = 0\), then we have
$$\lim_{n \to \infty} \sigma_{b}(x_{n},x_{n}) = \lim_{n \to \infty} \sigma_{b}(x_{n + 1},x_{n + 1}) = 0; $$ -
(c)
If \(x \ne y\), then \(\sigma_{b}(x,y) > 0\).
Proof
The proof is obvious. □
Lemma 2.14
Let \(( X,\sigma_{b} )\) be a complete b-metric-like space with parameter \(s \ge 1\), and let \(\{ x_{n} \}\) be a sequence such that
If \(\{ x_{n} \}\) is not Cauchy, then there exist \(\varepsilon > 0\) and two subsequences \(\{ x_{m_{k}} \}\) and \(\{ x_{n_{k}} \}\) of \(\{ x_{n} \}\) with \(n _{k} > m _{k} >k\) (positive integers) such that \(\sigma_{b}(x_{m_{k}},x_{n_{k}}) \ge \varepsilon\), \(\sigma_{b}(x_{m_{k}},x_{n_{k} - 1}) < \varepsilon\), \(\varepsilon / s^{2} \le \lim \sup_{k \to \infty} \sigma_{b}(x_{m_{k} - 1},x_{n_{k} - 1}) \le \varepsilon s\), \(\varepsilon / s \le \lim \sup_{k \to \infty} \sigma_{b}(x_{n_{k} - 1},x_{m_{k}}) \le \varepsilon s^{2}\), and \(\varepsilon / s \le \lim \sup_{k \to \infty} \sigma_{b}(x_{m_{k} - 1},x_{n_{k}}) \le \varepsilon s^{2}\).
Proof
If \(\{ x_{n} \}\) is not a \(\sigma_{b}\)-Cauchy sequence, then there exists \(\varepsilon > 0\) for which we can find two subsequences \(\{ x_{m_{k}} \}\) and \(\{ x_{n_{k}} \}\) of \(\{ x_{n} \}\) such that \(n_{k}\) is the smallest index for which
This means that
From (2.2) and property (c) of Definition 2.4 we have
Taking the upper limit as \(k \to \infty\) in (2.4) and using (2.1), (2.2), and (2.3), we get
By the triangle inequality we have
so, taking the upper limit as \(k \to \infty\) and using (2.1), we get
Also, we have
and, taking the upper limit as \(k \to \infty\), we get
Again
Taking the upper limit as \(k \to \infty\) and using (2.1), we get
Since \(\sigma_{b} ( x_{n_{k} - 1},x_{m_{k}} ) \le s\sigma_{b} ( x_{n_{k} - 1},x_{m_{k} - 1} ) + s\sigma_{b} ( x_{m_{k} - 1},x_{m_{k}} )\), from (2.1) and (2.7) we have
Consequently,
Also,
Then from (2.7), (2.8), and (2.1) we have
Consequently,
This completes proof. □
3 Main results
We begin this section with the following definition.
Definition 3.1
Let \(( X,\sigma_{b} )\) be a b-metric-like space with parameter \(s \ge 1\), let \(\alpha:X \times X \to [ 0,\infty )\) be a function, and let \(q \ge 1\) and \(p \ge 2\) be arbitrary constants. A mapping \(f:X \to X\) is \(\alpha_{qs^{p}} \)-admissible if \(\alpha ( x,y ) \ge qs^{p}\) implies \(\alpha ( fx,fy ) \ge qs^{p}\) for all \(x,y \in X\).
Remark 3.2
-
(i)
Taking \(q = 1\) in this definition, we obtain an \(\alpha_{s^{p}} \)-admissible mapping defined in a b-metric-like space or in a b-metric space.
-
(ii)
Note that, for \(s = 1\), the definition reduces to an \(\alpha_{q} \)-admissible mapping defined in a metric space or in a metric-like space.
-
(iii)
For \(s = 1\) and \(q = 1\), the definition reduces to the definition of an α-admissible mapping in a metric space [8].
-
(iv)
The class of \(\alpha_{qs^{p}} \)-admissible mappings is strictly larger, and, more generally, because the constant \(p \ge 2\), it is not restricted to some certain values.
We further consider the following properties.
Let \(( X,\sigma_{b} )\) be a complete b-metric-like space with parameter \(s \ge 1\), and let \(\alpha:X \times X \to [ 0,\infty )\) be a function. Then:
- (\(H_{qs^{p}}\)):
-
If \(\{ x_{n} \}\) is a sequence in X such that \(x_{n} \to x \in X\) as \(n \to \infty\) and \(\alpha ( x_{n},x_{n + 1} ) \ge qs^{p}\), then there exists a subsequence \(\{ x_{n_{k}} \}\) of \(\{ x_{n} \}\) such that \(\alpha ( x_{n_{k}},x ) \ge qs^{p}\) for all \(k \in N\).
- (\(U_{qs^{p}}\)):
-
For all \(x,y \in \operatorname{Fix} ( f )\), we have \(\alpha ( x,y ) \ge qs^{p}\), where \(\operatorname{Fix} ( f )\) denotes the set of fixed points of f.
Example 3.3
Let \(X = ( 0, + \infty )\). Define \(f:X \to X\) and \(\alpha:X \times X \to [ 0, + \infty )\) by \(fx = \ln x\) for all \(x \in X\), and let
Then, f is \(\alpha_{qs^{p}} \)-admissible.
Example 3.4
Let \(X = ( 0, + \infty )\). Define \(f:X \to X\) and \(\alpha:X \times X \to [ 0,\infty )\) by \(fx = 3x\) for all \(x \in X\) and
Then f is \(\alpha_{qs^{p}} \)-admissible.
Based on the definition of quasi-contraction from Ćirić, we introduce the following definition in the setting of a b-metric-like space.
Definition 3.5
Let \(( X,\sigma_{b} )\) be a complete b-metric-like space with parameter \(s \ge 1\), and let \(f:X \to X\) be a given mapping. We say that f is a generalized \(\alpha_{qs^{p}} - \lambda\)-quasi-contraction if f is an \(\alpha_{qs^{p}} \)-admissible mapping such that
for all \(x,y \in X\) and \(\lambda \in [ 0,1 / 2 )\).
Remark 3.6
If we take \(\alpha ( x,y ) = s^{2}\) (\(p = 2\) and \(q = 1\)), then the definition reduces to the definition of an \(s - \lambda\) quasi-contraction, and if we take \(s = 1\), then the definition reduces to the λ-quasi-contraction in the setting of metric spaces.
Theorem 3.7
Let \(( X,\sigma_{b} )\) be a complete b-metric-like space with parameter \(s \ge 1\), let \(f:X \to X\) be a self-mapping, and let \(\alpha:X \times X \to R^{ +} \) be a given function. Suppose that the following conditions are satisfied:
-
(i)
f is an \(\alpha_{qs^{p}} \)-admissible mapping;
-
(ii)
f is an \(\alpha_{qs^{p}} - \lambda\) contractive mapping;
-
(iii)
there exists \(x_{0} \in X\) such that \(\alpha ( x_{0},fx_{0} ) \ge qs^{p}\);
-
(iv)
either f is continuous, or property \(H_{qs^{p}}\) is satisfied.
Then f has a fixed point. Moreover, f has a unique fixed point if property \(U_{qs^{p}}\) is satisfied.
Proof
By hypothesis (iii) there exists \(x_{0} \in X\) such that \(\alpha ( x_{0},fx_{0} ) \ge qs^{p}\). We define the sequence \(\{ x_{n} \}\) in X by \(x_{n} = fx_{n - 1}\) for all \(n \in N\). If \(x_{n} = x_{n + 1}\) for some \(n \in N\), then \(u = x_{n}\) is a fixed point for f. Consequently, we suppose that \(x_{n} \ne x_{n + 1}\) (\(\sigma_{b} ( x_{n},x_{n + 1} ) > 0\)) for all \(n \in N\).
Since f is an \(\alpha_{qs^{p}} \)-admissible mapping, we have
Hence, by induction we get
By condition (3.1) we have:
If \(\sigma_{b} ( x_{n - 1},x_{n} ) < \sigma_{b} ( x_{n},x_{n + 1} )\) for some \(n \in {N}\), then from inequality (3.2) we have \(\sigma_{b} ( x_{n}, x_{n + 1} ) \le 2\lambda / qs^{p - 1}\sigma_{b} ( x_{n},x_{n + 1} )\), a contradiction since \(2\lambda / qs^{p - 1} < 1\).
Hence, for all \(n \in N\), \(\sigma_{b} ( x_{n},x_{n + 1} ) \le \sigma_{b} ( x_{n - 1},x_{n} )\), and also by inequality (3.2) we get
Similarly, by the contractive condition of theorem we have:
Generally, from (3.3) and (3.4) we have, for all n,
where \(0 \le c = 2\lambda / qs^{p - 1} < 1\). Taking limit as \(n \to \infty\) in (3.5), we have
Now we prove that \(\{ x_{n} \}\) is a Cauchy sequence. To do this, let \(m,n > 0\) be such that \(m > n\).
Using Definition 2.4(c), we have
Taking the limit as \(n,m \to \infty\), we have \(\sigma_{b} ( x_{n},x_{m} ) \to 0\), since \(0 \le cs = 2\lambda s / qs^{p - 1} = 2\lambda / qs^{p - 2} < 1\). Therefore \(\{ x_{n} \}\) is a Cauchy sequence in the complete b-metric-like space \(( X,\sigma_{b} )\). Thus there is some \(u \in X\) such that \(\{ x_{n} \}\) converges to u.
If f is a continuous mapping, then we get:
Thus u is a fixed point of f.
On the other hand, if f is not a continuous function and property \(H_{qs^{p}}\) holds, then there exists a subsequence \(\{ x_{n_{k}} \}\) of \(\{ x_{n} \}\) such that \(\alpha ( x_{n_{k}},u ) \ge qs^{p}\) for all \(k \in {N}\).
Since \(\alpha ( x_{n_{k}},u ) \ge qs^{p}\), applying condition (3.1) with \(x = x_{n_{k}}\) and \(y = u\), we obtain
Taking the upper limit as \(k \to \infty\) in (3.7) and using (3.6), and Lemmas 2.12 and 2.13, we have
From (3.8) we get \(\sigma_{b} ( u,fu ) = 0\), which implies that \(fu = u\). Hence u is a fixed point of f.
Further, suppose that u and v are two fixed points of f, where \(fu = u\) and \(fv = v\) for some \(u \ne v\). Since property \(U_{qs^{p}}\) is satisfied, we have \(\alpha ( u,v ) \ge qs^{p}\). Hence, from (3.1) we have
So \(\sigma_{b} ( u,v ) = 0\), and since \(0 \le c = 2\lambda / qs^{p - 1} < 1\), we get \(\sigma_{b} ( u,v ) = 0\). Hence the fixed point is unique. □
The following theorem is a version of the Hardy-Rogers result.
Theorem 3.8
Let \(( X,\sigma_{b} )\) be a complete b-metric-like space with parameter \(s \ge 1\), and let \(f:X \to X\) be a given self-mapping. Suppose that there exists a function \(\alpha:X \times X \to [ 0,\infty )\) such that
for all \(x,y \in X\) and the constants \(a_{i} \ge 0\), \(i = 1,\ldots,5\), where \(a_{1} + a_{2} + a_{3} + a_{4} + a_{5} < 1 / 2\). Assume also that:
-
(i)
f is an \(\alpha_{qs^{p}} \)-admissible mapping;
-
(ii)
there exists \(x_{0} \in X\) such that \(\alpha ( x_{0},fx_{0} ) \ge qs^{p}\);
-
(iii)
either f is continuous, or property \(H_{qs^{p}}\) is satisfied.
Then f has a fixed point. Moreover, f has a unique fixed point if property \(U_{qs^{p}}\) is satisfied.
Proof
This theorem can be considered as a corollary of Theorem 3.7, since, for all \(x,y \in X\), we have
where \(0 < k = a_{1} + a_{2} + a_{3} + a_{4} + a_{5} < 1 / 2\). □
Corollary 3.9
Let \(( X,\sigma_{b} )\) be complete b-metric-like space with parameter \(s \ge 1\). If \(f:X \to X\) is a self-mapping and there exist constants \(a_{i} \ge 0\), \(i = 1,\ldots,5\), with \(a_{1} + a_{2} + a_{3} + a_{4} + a_{5} < 1 / 2\) such that
for all \(x,y \in X\) and a constant \(p \ge 2\), then f has a unique fixed point in X.
Proof
In Theorem 3.8, take the function \(\alpha ( x,y ) = qs^{p}\). □
Remark 3.10
Theorem 3.7 generalizes Theorem 18 in [7]. For \(\alpha ( x,y ) = s^{2}\) and for all \(x,y \in X\), Theorems 3.7 and 3.8 reduce to Theorems 3.2 and 3.13 of [19]. In Theorem 3.7 and Corollary 3.9, by choosing the constants \(a_{i}\) in certain manner, we obtain, as particular cases, certain classes of \(\alpha_{qs^{p}} \)-types of Kannan, Chatterjea, Reich, and Zamfirescu contractions.
The notion of \(\alpha - \psi\) contractive mappings is defined in a complete metric space in [8]. Thereafter, many authors provided various fixed point theorems for such a class of mappings. In the following definition, we extend and generalize the notions of \(\alpha - \psi\) and \((\psi - \phi )\)-contractive mappings in the context of larger spaces, such as b-metric-like spaces. The aim of this section is to extend and generalize the main classical result and other existing results in the literature on b-metric and metric-like spaces.
Let \(( X,\sigma_{b} )\) be a b-metric-like space with parameter \(s \ge 1\). For a self-mapping \(f:X \to X\), we define \(N ( x,y )\) by
for all \(x,y \in X\).
The families Ψ, Φ with altering distance functions are defined as follows:
Let \(\mathbb{S}\) be the set of all mappings \(\beta: [ 0,\infty ) \to [ 0,1 )\) satisfying the condition
Definition 3.11
Let \(( X,\sigma_{b} )\) be a b-metric-like space with parameter \(s \ge 1\), and let \(f:X \to X\) be a self-mapping. Also, let \(\alpha:X \times X \to [ 0,\infty )\) and \(q \ge 1\), \(p \ge 2\). We say that f is an \(( \alpha_{qs^{p}} - \psi,\phi )\) generalized contractive mapping if there exist \(\psi \in \Psi\), \(\phi \in \Phi\) such that
for all \(x,y \in X\) with \(\alpha ( x,y ) \ge qs^{p}\), where \(N ( x,y )\) is defined by (3.10).
Remark 3.12
-
(i)
Taking \(q = 1\) in the definition, we obtain \(\alpha_{s} - (\psi,\phi )\) admissible mappings defined in a b-metric-like space or in a b-metric space.
-
(ii)
Note that, for \(\alpha ( x,y ) = q\), the definition reduces to an \(\alpha_{q} \)-admissible mapping defined in a metric space or in a metric-like space.
-
(iii)
For \(s = 1\) and \(q = 1\), the definition reduces to the definition of an α-admissible mapping in a metric space.
-
(iv)
The definition reduces to a \((\psi,\phi ) \)-contractive mapping if we take \(\alpha ( x,y ) = 1\).
-
(v)
The definition reduces to an \(\alpha_{qs^{p}} - \phi\) contractive mapping if we take \(\psi ( t ) = t\).
-
(vi)
The definition reduces to an \(\alpha_{qs^{p}} - \lambda\) contractive mapping if we take \(\psi ( t ) = t\) and \(\phi ( t ) = \lambda t\) for \(\lambda \in ( 0,1 )\).
We now present the following theorem.
Theorem 3.13
Let \(( X,\sigma_{b} )\) be a complete b-metric-like space with parameter \(s \ge 1\), and let \(f:X \to X\) be an \(( \alpha_{qs^{p}} - \psi,\phi )\) generalized contractive mapping. Suppose that the following conditions are satisfied:
-
(i)
f is an \(\alpha_{qs^{p}} \)-admissible mapping;
-
(ii)
there exists \(x_{0} \in X\) such that \(\alpha ( x_{0},fx_{0} ) \ge qs^{p}\);
-
(iii)
either f is continuous, or property \(H_{qs^{p}}\) is satisfied.
Then f has a fixed point \(x \in X\). Moreover, f has a unique fixed point if property \(U_{qs^{p}}\) is satisfied.
Proof
By assumption (ii) there exists a point \(x_{0} \in X\) such that \(\alpha ( x_{0},fx_{0} ) \ge qs^{p}\). We construct a sequence \(\{ x_{n} \}\) in X by \(x_{n} = f^{n}x_{0} = f ( x_{n - 1} )\) for all \(n \in {N}\). If we suppose that \(\sigma_{b} ( x_{n},x_{n + 1} ) = 0\) for some n, then \(x_{n + 1} = x_{n}\), and the proof is completed, since \(u = x_{n} = x_{n + 1} = f ( x_{n} ) = fu\). Consequently, throughout the proof, we assume that
Since f is an \(\alpha_{qs^{p}} \)-admissible mapping, we observe that
In general, by induction we derive that
By (3.13) and condition (3.11) we have:
where
If we assume that, for some \(n \in {N}\),
then from inequality (3.15) we get
Again, by (3.13) and condition (3.11) we have:
By the property ψ inequality (3.17) implies that
From (3.16) and (3.18) we have
From (3.17), using (3.19), we obtain
which gives a contradiction, since we have assumed that \(\sigma_{b} ( x_{n},x_{n + 1} ) > 0\) and \(\phi ( t ) < \psi ( t )\) for all \(t > 0\). Hence, for all \(n \in {N}\), \(\sigma_{b} ( x_{n},x_{n + 1} ) \le \sigma_{b} ( x_{n - 1},x_{n} )\), and the sequence \(\{ \sigma_{b} ( x_{n},x_{n + 1} ) \}\) is decreasing and bounded below. Hence there exists \(l \ge 0\) such that \(\sigma_{b} ( x_{n},x_{n + 1} ) \to l\). Also,
We shall prove that \(l = 0\).
Consider
If we assume that \(l > 0\), taking the limit in (3.21), we have
which is a contradiction since \(\psi ( t ) > \phi ( t )\) for \(t > 0\). Hence \(l = 0\), and
Next, we shall show that \(\{ x_{n} \}\) is a Cauchy sequence in X. Suppose, on the contrary, that \(\{ x_{n} \}\) is not a Cauchy sequence. Then by Lemma 2.14 there exist \(\varepsilon > 0\) and two subsequences \(\{ x_{m_{k}} \}\) and \(\{ x_{n_{k}} \}\) of \(\{ x_{n} \}\), with \(n_{k} > m_{k} > k\), such that
From the definition of \(N ( x,y )\) we have
Taking the upper limit as \(k \to \infty\) in (3.24) and using (3.22), (3.23), we get
Using the \(\alpha_{qs^{p}}\)-weak contractive condition, we have
Taking the upper limit in (3.26), using (3.23) and (3.25), we obtain
which is a contradiction, since \(\varepsilon > 0\). Therefore \(\{ x_{n} \}\) is a Cauchy sequence in the complete b-metric-like space \(( X,\sigma_{b} )\). Thus, there is some \(u \in X\) such that \(\{ x_{n} \}\) converges to u. If f is a continuous mapping, we get:
and u is a fixed point of f.
If the self-map f is not continuous, then from (3.13) and condition \(H_{qs^{p}}\), there exists a subsequence \(\{ x_{n_{k}} \}\) of \(\{ x_{n} \}\) such that \(\alpha ( x_{n_{k}},u ) \ge qs^{p}\) for all \(k \in {N}\). Since \(\alpha ( x_{n_{k}},u ) \ge qs^{p}\), applying contractive condition (3.11), with \(x = x_{n_{k}}\) and \(y = u\), we obtain
where
Taking the upper limit in (3.28) and using Lemma 2.13 and result (3.22), we obtain
Taking the upper limit as \(k \to \infty\) in (3.27) and using (3.29) and Lemma 2.13, we obtain
From (3.30) we get \(\sigma_{b} ( u,fu ) = 0\), which implies that \(fu = u\). Hence u is a fixed point of f.
Suppose that u and v are two fixed points of f, where \(fu = u\) and \(fv = v\) are such that \(u \ne v\). Then, by hypothesis \(U_{qs^{p}}\), \(\alpha ( u,v ) \ge qs^{p}\), and applying (3.11), we have
where
From inequality (3.31) it follows that \(\sigma_{b} ( u,u ) = 0\) (also \(\sigma_{b} ( v,v ) = 0\)).
Again we have
where \(N ( u,v ) = \sigma_{b} ( u,v )\).
Inequality (3.32) implies that \(\sigma_{b} ( u,v ) = 0\). Therefore \(u = v\), and the fixed point is unique. □
Remark 3.14
Our theorem extends Theorems 2.1, 2.2, and 2.7 of Aydi et al. [9].
By taking \(\phi ( t ) = \psi ( t ) - \varphi ( t )\), where \(\varphi \in \Psi\), in Theorem 3.13 we obtain the following result.
Corollary 3.15
Let \(( X,\sigma_{b} )\) be a complete b-metric-like space with parameter \(s \ge 1\), let \(f:X \to X\) be a self-mapping, and let \(\alpha:X \times X \to [ 0,\infty )\). Suppose that the following conditions are satisfied:
-
(i)
f is an \(\alpha_{qs^{p}} \)-admissible mapping;
-
(ii)
there exist functions \(\psi,\varphi \in \Psi\) such that
$$\psi \bigl( \alpha ( x,y )\sigma_{b}(fx,fy) \bigr) \le \psi \bigl( N(x,y) \bigr) - \varphi \bigl( N(x,y) \bigr); $$ -
(iii)
there exists \(x_{0} \in X\) such that \(\alpha ( x_{0},fx_{0} ) \ge qs^{p}\);
-
(iv)
either f is continuous, or property \(H_{qs^{p}}\) is satisfied.
Then f has a fixed point \(x \in X\). Moreover, f has a unique fixed point if property \(U_{qs^{p}}\) is satisfied.
Remark 3.16
This corollary extends Theorems 3 and 4 of Roshan et al. [25].
By taking \(\psi ( t ) = t\) and \(\phi ( t ) = \beta ( t )t\) where \(\beta \in \mathbb{S}\) is as in Theorem 3.13, we obtain the following result.
Corollary 3.17
Let \(( X,\sigma_{b} )\) be a complete b-metric-like space with parameter \(s \ge 1\), let \(f:X \to X\) be a self-mapping, and let \(\alpha:X \times X \to [ 0,\infty )\). Suppose that the following conditions are satisfied:
-
(i)
f is an \(\alpha_{qs^{p}} \)-admissible mapping;
-
(ii)
there exist functions \(\psi,\varphi \in \Psi\) such that
$$\psi \bigl( \alpha ( x,y )\sigma_{b}(fx,fy) \bigr) \le \beta \bigl( N(x,y) \bigr) \bigl( N(x,y) \bigr); $$ -
(iii)
there exists \(x_{0} \in X\) such that \(\alpha ( x_{0},fx_{0} ) \ge qs^{p}\);
-
(iv)
either f is continuous, or property \(H_{qs^{p}}\) is satisfied.
Then f has a fixed point \(x \in X\). Moreover, f has a unique fixed point if property \(U_{qs^{p}}\) is satisfied.
If we take \(\psi ( t ) = t\) in Theorem 3.13, then we get the following result.
Corollary 3.18
Let \(( X,\sigma_{b} )\) be a complete b-metric-like space with parameter \(s \ge 1\), let \(f:X \to X\) be a self-mapping, and let \(\alpha:X \times X \to [ 0,\infty )\). Suppose that the following conditions are satisfied:
-
(i)
f is an \(\alpha_{qs^{p}} \)-admissible mapping;
-
(ii)
there exist functions \(\varphi \in \Psi\) such that
$$\alpha ( x,y )\sigma_{b}(fx,fy) \le \varphi \bigl( N(x,y) \bigr); $$ -
(iii)
there exists \(x_{0} \in X\) such that \(\alpha ( x_{0},fx_{0} ) \ge qs^{p}\);
-
(iv)
either f is continuous, or property \(H_{qs^{p}}\) is satisfied.
Then f has a fixed point \(x \in X\). If property \(U_{qs^{p}}\) is satisfied, then f has a unique fixed point.
Remark 3.19
Corollary 3.18 generalizes and extends Theorem 2.7 of Samet et al. [8].
Corollary 3.20
Let \(( X,\sigma_{b} )\) be a complete b-metric-like space with parameter \(s \ge 1\), let \(f:X \to X\) be a self-mapping, and let \(\alpha:X \times X \to [ 0,\infty )\). Suppose that the following conditions are satisfied:
-
(i)
f is an \(\alpha_{qs^{p}} \)-admissible mapping;
-
(ii)
there exists a function \(\varphi \in \Psi\) such that
$$\alpha ( x,y )\sigma_{b}(fx,fy) \le N(x,y) - \varphi \bigl( N(x,y) \bigr); $$ -
(iii)
there exists \(x_{0} \in X\) such that \(\alpha ( x_{0},fx_{0} ) \ge qs^{p}\);
-
(iv)
either f is continuous, or property \(H_{qs^{p}}\) is satisfied.
Then f has a fixed point \(x \in X\). Moreover, f has a unique fixed point if property \(U_{qs^{p}}\) is satisfied.
Proof
It follows from Corollary 3.15 by taking \(\psi ( t ) = t\). □
Remark 3.21
Corollary 3.20 generalizes Theorem 2.7 of Harandi [5].
Corollary 3.22
Let \(( X,\sigma_{b} )\) be a complete b-metric like space with parameter \(s \ge 1\), and let \(f,g\) be two selfmaps of X with \(\psi \in \Psi\), \(\varphi \in \Phi\) satisfying the condition
for all \(x,y \in X\), where \(M ( x,y )\) is defined in (3.15), and \(q > 1\). Then f and g have a unique common fixed point in X.
Proof
In Theorem 3.13, take \(\varphi (t) = \lambda \psi ( t )\) where \(0 < \lambda < 1\). □
Corollary 3.23
Let \(( X,\sigma_{b} )\) be a complete b-metric-like space with parameter \(s \ge 1\), and let \(f:X \to X\) be a self-mapping such that, for all \(x,y \in X\) and any arbitrary coefficient \(p \ge 1\),
where \(k \in (0,1)\). Then f has a unique fixed point.
Proof
It follows from Corollary 3.15 by taking \(\alpha ( x,y ) = qs^{p}\), \(\psi ( t ) = t\), and \(\varphi (t) = ( 1 - k )t\) for all \(t \in [0,\infty )\) and \(k \in (0,1)\). □
Remark 3.24
It is clear that we can derive several corresponding results by replacing the b-metric-like space with some other spaces such as a b-metric space, a metric space, a metric-like space, and a partial metric space. Conditions (3.1) and (3.12) are more general than the analogues in the previous literature, and theorems related to those conditions have a more general character because of the parameter s and arbitrary coefficients q, p.
3.1 Application
In this section, we will use Corollary 3.23 to show that there is a solution to the following integral equation:
Let \(X = C ( [ 0,T ] )\) be the set of real continuous functions defined on \([ 0,T ]\) for \(T > 0\).
We endow X with
It is evident that \(( X,\sigma_{b} )\) is a complete b-metric-like space with parameter \(s = 2^{m - 1}\) with \(m > 1\).
Consider the mapping \(f:X \to X\) defined by \(fx ( t ) = \int_{0}^{T} G ( t,r,x ( r ) )\,dr\).
Theorem 3.25
Consider equation (3.33) and suppose that
-
(a)
\(G: [ 0,T ] \times [ 0,T ] \times R \to R^{ +} = [ 0,\infty )\) (that is, \(G ( t,r,x ( r ) ) \ge 0\)) is continuous;
-
(b)
there exists a continuous \(\gamma: [ 0,T ] \times [ 0,T ] \to R\);
-
(c)
\(\sup_{t \in [ 0,T ]}\int_{0}^{T} \gamma ( t,r ) \,dr \le 1\);
-
(d)
there exists a constant \(L \in ( 0,1 )\) such that, for all \(( t,r ) \in [ 0,T ]^{2}\) and \(x,y \in R\),
$$\bigl\vert G \bigl( t,r,x ( r ) \bigr) + G \bigl( t,r,y ( r ) \bigr) \bigr\vert \le \biggl( \frac{L}{s^{3}} \biggr)^{\frac{1}{m}}\gamma ( t,r ) \bigl( \bigl\vert x ( r ) \bigr\vert + \bigl\vert y ( r ) \bigr\vert \bigr). $$
Then the integral equation (3.33) has a unique solution in \(x \in X\).
Proof
For \(x,y \in X\), from conditions (c) and (d), for all t, we have
which implies that
where \(k = L / s^{2} \in ( 0,1 )\).
Therefore, all of the conditions of Corollary 3.23 are satisfied, and, as a result, the mapping f has a unique fixed point in X, which is a solution of the integral equation (3.33). □
4 Conclusions
In this paper, the class of \(\alpha_{qs^{p}} \)-admissible mappings is introduced in a larger structure such as a b-metric-like space. Some fixed point results dealing with \((\alpha - \psi,\phi )\) contractions are obtained, and they cover and unify a huge number of published results in the related literature.
Change history
25 January 2018
In the publication of this article [Fixed Point Theory Appl. 2017:26, 2017], there is an error in Section 3.
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Zoto, K., Rhoades, B.E. & Radenović, S. Some generalizations for \((\alpha - \psi,\phi )\)-contractions in b-metric-like spaces and an application. Fixed Point Theory Appl 2017, 26 (2017). https://doi.org/10.1186/s13663-017-0620-1
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DOI: https://doi.org/10.1186/s13663-017-0620-1