Some fixed point theorems for \((\alpha,\psi)\)-rational type contractive mappings
- Hamed H Alsulami^{1, 2}Email author,
- Sumit Chandok^{3},
- Mohamed-Aziz Taoudi^{4, 5} and
- İnci M Erhan^{6}
https://doi.org/10.1186/s13663-015-0332-3
© Alsulami et al. 2015
Received: 21 January 2015
Accepted: 17 May 2015
Published: 27 June 2015
Abstract
In this paper, we introduce the concept of \((\alpha,\psi)\)-rational type contractive mappings and provide sufficient conditions for the existence and uniqueness of a fixed point for such class of generalized nonlinear contractive mappings in the setting of generalized metric spaces. We also deduce several interesting corollaries.
Keywords
MSC
1 Introduction and preliminaries
Fixed point theory has gained very large impetus due to its wide range of applications in various fields such as engineering, economics, computer science, and many others. It is well known that the contractive conditions are very indispensable in the study of fixed point theory, and Banach’s fixed point theorem [1] for contraction mappings is one of the pivotal result in analysis. This theorem has been extended and generalized by various authors (see, e.g., [2–28]) in various abstract spaces, one of which is generalized metric space.
- (T1)
A generalized metric does not need to be continuous.
- (T2)
A convergent sequence in generalized metric spaces does not need to be Cauchy.
- (T3)
A generalized metric space does not need to be Hausdorff, and hence the uniqueness of the limits cannot be guaranteed.
In this paper, we introduce the concept of \((\alpha,\psi)\)-rational type contractive mappings and provide sufficient conditions for the existence and uniqueness of fixed points for such class of generalized nonlinear contractive mappings in the framework of generalized metric spaces by caring the problems (T1)-(T3) mentioned above. We also deduce several interesting corollaries. The proved results generalize and extend various well-known results in the literature. The techniques used in this paper have been studied and improved by various authors (see [3–9, 16] and references cited therein).
To start with, we give some notations and introduce some definitions which will be used in the sequel.
Definition 1.1
[2]
- (GMS1)
\(d(x, y) = 0\) if and only if \(x = y\),
- (GMS2)
\(d(x, y) = d(y, x)\),
- (GMS3)
\(d(x, y) \le d(x,u) + d(u, v) + d(v, y)\).
Then the map d is called a generalized metric and abbreviated as GM. Here, the pair \((X, d)\) is called a generalized metric space and abbreviated as GMS.
In the above definition, if d satisfies only (GMS1) and (GMS2), then it is called a semimetric (see, e.g., [18]).
A sequence \(\{x_{n}\}\) in a GMS \((X,d)\) is GMS convergent to a limit x if and only if \(d(x_{n}, x)\to0\) as \(n\to\infty\).
A sequence \(\{x_{n}\}\) in a GMS \((X, d)\) is GMS Cauchy if and only if for every \(\epsilon> 0\) there exists a positive integer \(N(\epsilon)\) such that \(d(x_{n}, x_{m}) < \epsilon\), for all \(n > m > N(\epsilon)\).
A GMS \((X,d)\) is called complete if every GMS Cauchy sequence in X is GMS convergent.
A mapping \(T : (X, d)\to(X, d)\) is continuous if for any sequence \(\{ x_{n}\}\) in X such that \(d(x_{n}, x)\to0\) as \(n\to\infty\), we have \(d(Tx_{n}, Tx)\to0\) as \(n\to\infty\).
- (W)
For each pair of (distinct) points u, v, there is a number \(r_{u,v} > 0\) such that for every \(z \in X\), \(r_{u,v} < d(u, z) + d(z, v)\).
Proposition 1.1
[20]
In a semimetric space, the assumption (W) is equivalent to the assertion that the limits are unique.
Proposition 1.2
[20]
Suppose that \(\{x_{n}\}\) is a Cauchy sequence in a GMS \((X,d)\) with \(\lim_{n\to\infty} d(x_{n},u) = 0\), where \(u\in X\). Then \(\lim_{n\to \infty} d(x_{n}, z) = d(u, z)\), for all \(z\in X\). In particular, the sequence \(\{x_{n}\}\) does not converge to z if \(z \neq u\).
Definition 1.2
Let X be a nonempty set, \(T :X\to X\) and \(\alpha : X\times X\to[0,\infty)\) be two mappings. We say that T is an α-admissible mapping if \(\alpha(x, y)\ge1\) implies \(\alpha (Tx, Ty)\ge1\), for all \(x, y \in X\).
Definition 1.3
Let \((X,d)\) be a GMS and \(\alpha:X\times X\to[0,\infty)\). X is called α-regular GMS if, for a sequence \(\{x_{n}\}\) in X such that \(x_{n}\to x\) and \(\alpha(x_{n}, x_{n+1})\ge 1\), there exists a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \(\alpha(x_{n_{k}}, x)\ge 1\) for all \(k \in\mathbb{N}\).
Throughout the paper, \(F(T)\) denotes the set of fixed points of the mapping T.
2 Main results
- (i)
ψ is upper semi-continuous, strictly increasing;
- (ii)
\(\{\psi^{n}(t)\}_{n\in\mathbb{N}}\) converges to 0 as \(n\to\infty \), for all \(t > 0\);
- (iii)
\(\psi(t) < t\), for every \(t >0\).
Definition 2.1
Next, we state and prove an existence and uniqueness theorem for fixed point of \((\alpha,\psi)\)-rational type-I contractive mappings.
Theorem 2.1
- (i)
T is an α-admissible mapping;
- (ii)
T is an \((\alpha,\psi)\)-rational type-I contractive mapping;
- (iii)
there exists \(x_{0}\in X\) such that \(\alpha( x_{0}, T x_{0}) \ge1\), \(\alpha( x_{0}, T^{2} x_{0}) \ge1\);
- (iv)
either T is continuous, or X is α-regular.
Then T has a fixed point \(x^{*}\in X\) and \(\{T^{n}x_{0}\}\) converges to \(x^{*}\). Further, if for all \(x, y\in F(T)\), we have \(\alpha(x, y)\ge1\) then T has a unique fixed point in X.
Proof
Let \(x_{0}\in X\) satisfies \(\alpha( x_{0}, T x_{0}) \ge1\) and \(\alpha( x_{0}, T^{2} x_{0}) \ge1\). We construct the sequence \(\{ x_{n}\}\) in X as \(x_{n}= T^{n} x_{0} = T x_{ n-1}\), for \(n\in\mathbb{N}\). It is obvious that if \(x_{n_{0}}=x_{n_{0}+1}\), for some \(n_{0}\in\mathbb {N}\), then \(x_{n_{0}}\) is a fixed point of T. Consequently, we suppose that \(x_{n}\ne x_{n+1}\) for all \(n \in\mathbb{N}\).
Since T is α-admissible, \(\alpha(x_{0},Tx_{0})=\alpha(x_{0},x_{1})\ge 1 \Longrightarrow\) \(\alpha(Tx_{0},Tx_{1})=\alpha(x_{1},x_{2})\ge1 \Longrightarrow\) and thus, \(\alpha(Tx_{1},Tx_{2})=\alpha(x_{2},x_{3})\ge1 \ldots\) , and hence by induction, we get \(\alpha(x_{n},x_{n+1})\ge1\) for all \(n\ge0\).
Now, we shall prove that \(\{ x_{n}\}\) is a Cauchy sequence, that is, \(\lim_{n\to\infty}d(x_{n},x_{n+k})=0\), for all \(k\in\mathbb{N}\). We have already proved the cases for \(k=1\) and \(k=2\) in (2.6) and (2.9), respectively. Take arbitrary \(k\ge3\). We discuss two cases.
Definition 2.2
For this class of mappings we state a similar existence and uniqueness theorem.
Theorem 2.2
- (i)
T is an α-admissible mapping;
- (ii)
T is an \((\alpha,\psi)\)-rational type-II contractive mapping;
- (iii)
there exists \(x_{0}\in X\) such that \(\alpha( x_{0}, T x_{0}) \ge1\) and \(\alpha( x_{0}, T^{2} x_{0}) \ge1\);
- (iv)
either T is continuous, or X is α-regular.
Then T has a fixed point \(x^{*}\in X\) and \(\{T^{n}x_{0}\}\) converges to \(x^{*}\). Further, if for all \(x, y\in F(T)\), we have \(\alpha(x, y)\ge1\), then T has a unique fixed point in X.
Proof
The proof can be done by following the lines of the proof of Theorem 2.1. □
The following example illustrating Theorem 2.1 is inspired by [4].
Example 2.1
3 Some consequences
In this section we give some consequences of the main results presented above. Specifically, we apply our results to generalized metric spaces endowed with a partial order.
Definition 3.1
Let \((X,\preceq)\) be a partially ordered set. A mapping \(T:X\to X\) is said to be nondecreasing with respect to ⪯ if for every \(x,y\in X\) \(x\preceq y\) implies \(Tx\preceq Ty\).
Definition 3.2
Let \((X,d,\preceq)\) be a partially ordered GMS. X is called regular GMS if, whenever \(\{x_{n}\}\) is a sequence in X such that \(x_{n}\to x\) and \(x_{n}\preceq x_{n+1}\), then there exists a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that \(x_{n_{k}}\preceq x\) for all \(k \in\mathbb{N}\).
Theorem 3.1
- (i)There exists a function \(\psi\in\Psi\) for whichwhere$$ d(Tx,Ty)\le\psi\bigl(M(x,y)\bigr), $$(3.1)for all \(x,y\in X\) with \(x\preceq y\).$$M(x,y)=\max\biggl\{ d(x,y),d(x,Tx),d(y,Ty),\frac {d(x,Tx)d(y,Ty)}{1+d(x,y)},\frac{d(x,Tx)d(y,Ty)}{1+d(Tx,Ty)} \biggr\} $$
- (ii)
There exists \(x_{0}\in X\) such that \(x_{0}\preceq Tx_{0}\) and \(x_{0}\preceq T^{2}x_{0}\).
- (iii)
Either T is continuous, or X is regular.
Then T has a fixed point \(x^{*}\in X\) and \(\{T^{n}x_{0}\}\) converges to \(x^{*}\).
Proof
Theorem 3.2
- (i)There exist a function \(\psi\in\Psi\) for whichwhere$$ d(Tx,Ty)\le\psi\bigl(M(x,y)\bigr), $$(3.2)for all \(x,y\in X\) with \(x\preceq y\);$$\begin{aligned} M(x,y)={}&\max\biggl\{ d(x,y),d(x,Tx),d(y,Ty), \\ &{} \frac{d(x,Tx)d(y,Ty)}{1+d(x,y)+d(x,Ty)+d(y,Tx)}, \frac{d(x,Ty)d(x,y)}{1+d(x,Tx)+d(y,Tx)+d(y,Ty)}\biggr\} \end{aligned}$$
- (ii)
there exists \(x_{0}\in X\) such that \(x_{0}\preceq Tx_{0}\) and \(x_{0}\preceq T^{2}x_{0}\);
- (iii)
either T is continuous, or X is regular.
Then T has a fixed point \(x^{*}\in X\) and \(\{T^{n}x_{0}\}\) converges to \(x^{*}\).
Proof
Several particular cases can also be deduced from the above results.
Corollary 3.1
- (i)
T is an α-admissible mapping;
- (ii)T satisfieswhere$$ d(Tx,Ty)\le kM(x,y), $$(3.3)for some \(k\in[0,1)\);$$\begin{aligned} M(x,y)={}&\max\biggl\{ d(x,y),d(x,Tx),d(y,Ty), \\ & {} \frac{d(x,Tx)d(y,Ty)}{1+d(x,y)+d(x,Ty)+d(y,Tx)}, \frac{d(x,Ty)d(x,y)}{1+d(x,Tx)+d(y,Tx)+d(y,Ty)}\biggr\} \end{aligned}$$
- (iii)
there exists \(x_{0}\in X\) such that \(\alpha( x_{0}, T x_{0}) \ge1\), \(\alpha(x_{0}, T^{2}x_{0}) \ge1\);
- (iv)
either T is continuous, or X is α-regular.
Then T has a fixed point \(x^{*}\in X\) and \(\{T^{n}x_{0}\}\) converges to \(x^{*}\). Further, if, for all \(x, y\in F(T)\), we have \(\alpha(x, y)\ge1\), then T has a unique fixed point in X.
Proof
Define \(\psi(t)=kt\). Clearly, \(\psi\in\Psi\). By Theorem 2.2, T has a unique fixed point. □
Corollary 3.2
- (i)where$$ d(Tx,Ty)\le kM(x,y), $$(3.4)for all \(x,y\in X\) with \(x\preceq y\) and some \(k\in[0,1)\);$$\begin{aligned} M(x,y)={}&\max\biggl\{ d(x,y),d(x,Tx),d(y,Ty), \\ & {} \frac{d(x,Tx)d(y,Ty)}{1+d(x,y)+d(x,Ty)+d(y,Tx)}, \frac{d(x,Ty)d(x,y)}{1+d(x,Tx)+d(y,Tx)+d(y,Ty)}\biggr\} \end{aligned}$$
- (ii)
there exists \(x_{0}\in X\) such that \(x_{0}\preceq Tx_{0}\) and \(x_{0}\preceq T^{2}x_{0}\);
- (iii)
either T is continuous, or X is regular.
Then T has a fixed point \(x^{*}\in X\) and \(\{T^{n}x_{0}\}\) converges to \(x^{*}\).
Proof
Declarations
Acknowledgements
The authors would like to thank referees for their useful comments and suggestions for the improvement of the paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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