Fixed point theorems for Meir-Keeler type mappings in M-metric spaces with applications
- Mehdi Asadi^{1}Email author
https://doi.org/10.1186/s13663-015-0460-9
© Asadi 2015
Received: 7 July 2015
Accepted: 5 November 2015
Published: 17 November 2015
Abstract
In this paper, we establish some fixed point theorems for a Meir-Keeler type contraction in M-metric spaces via Gupta-Saxena type contraction. Also, we extend and improve very recent results in fixed point theory.
Keywords
MSC
1 Introduction and preliminaries
Ekeland formulated a variational principle that is the foundation of modern variational calculus, having applications in many branches of mathematics, including optimization and fixed point theory [1] and applications in nonlinear analysis, since it entails the existence of approximate solutions of minimization problems for a lower semi-continuous function that is bounded from below on complete metric spaces. Also, Ekeland’s variational principle is a fruitful tool in simplifying and unifying the proofs of already known theorems and has many generalizations; see Borwein and Zhu [2].
Matthews in 1994 [3] introduced a partial metric space and proved the contraction principle of Banach in this new framework. Afterward, by several mathematicians many fixed point theorems were founded in partial metric spaces. Recently Haghi et al. [4] published a paper which stated that we should ‘be careful on partial metric fixed point results’ along with very some results therein. They showed that fixed point generalizations to partial metric spaces can be obtained from the corresponding results in metric spaces.
In 2014, Asadi et al. [5] introduced the M-metric space, which extends the p-metric space and certain fixed point theorems obtained therein.
In this paper, we establish some of the fixed point theorem for a Meir-Keeler type contraction in M-metric spaces via a Gupta-Saxena type contraction. Also, we extend and improve very recent results in fixed point theory.
Definition 1.1
- (p1)
\(p(x,x)=p(y,y)=p(x,y)\iff x=y\),
- (p2)
\(p(x,x)\leq p(x,y)\),
- (p3)
\(p(x, y)=p(y,x)\),
- (p4)
\(p(x, y)\leq p(x,z)+p(z,y)-p(z,z)\).
Notation
- (i)
\(m_{xy}:=\min\{m(x,x), m(y,y)\}=m(x,x)\vee m(y,y)\),
- (ii)
\(M_{xy}:=\max\{m(x,x), m(y,y)\}=m(x,x)\wedge m(y,y)\).
Now we want to extend Definition 1.1 as follows.
Definition 1.2
- (m1)
\(m(x,x)=m(y,y)=m(x,y)\iff x=y\),
- (m2)
\(m_{xy}\leq m(x,y)\),
- (m3)
\(m(x, y)=m(y,x)\),
- (m4)
\((m(x, y)-m_{xy} )\leq (m(x,z)-m_{xz} )+ (m(z,y)-m_{zy} )\).
According to the above, our definition of the condition (p1) in the definition [3] changes to (m1) and (p2) for \(p(x,x)\) is expressed by just \(p(y,y)=0\); we may have \(p(y,y)\neq0\), so we improved that condition by replacing it by \(\min\{p(x,x), p(y,y)\}\leq p(x,y)\), and also we improved the condition (p4) to the form (m4). In the sequel we present an example that holds for the m-metric, but not for the p-metric.
Remark 1.1
- (i)
\(0\leq M_{xy}+m_{xy} =m(x,x)+m(y,y)\),
- (ii)
\(0\leq M_{xy}-m_{xy} = |m(x,x)-m(y,y)|\),
- (iii)
\(M_{xy}-m_{xy} \leq(M_{xz}-m_{xz}) +(M_{zy}-m_{zy})\).
The next examples state that \(m^{s}\) and \(m^{w}\) are ordinary metrics.
Example 1.1
- (i)
\(m^{w}(x,y)=m(x,y)-2m_{xy}+M_{xy}\),
- (ii)
\(m^{s}(x,y)=m(x,y)-m_{xy}\) when \(x\neq y\) and \(m^{s}(x,y)=0\) if \(x=y\).
Proof
In the following example, we present an example of a m-metric which is not a p-metric.
Remark 1.2
- (i)
\(m(x,y)-M_{xy}\leq m^{w}(x,y)\leq m(x,y)+M_{xy}\),
- (ii)
\((m(x,y)-M_{xy})\leq m^{s}(x,y)\leq m(x,y)\).
Example 1.2
Example 1.3
([5])
Example 1.4
Let \((X,d)\) be a metric space. Then \(m(x,y)=ad(x,y)+b\) where \(a,b>0\) is an m-metric, because we can put \(\phi(t)=at+b\).
Remark 1.3
2 Topology for M-metric space
Definition 2.1
- (1)A sequence \(\{x_{n}\}\) in an M-metric space \((X, m)\) converges to a point \(x \in X\) if and only if$$ \lim_{n\to\infty} \bigl(m(x_{n}, x)-m_{x_{n},x}\bigr)=0. $$(2)
- (2)A sequence \(\{x_{n}\}\) in an M-metric space \((X, m)\) is called an m-Cauchy sequence ifin this space exist (and are finite).$$ \lim_{n,m\to \infty}\bigl(m(x_{n}, x_{m})-m_{x_{n},x_{m}}\bigr)\quad \mbox{and}\quad \lim _{n,m\to\infty}(M_{x_{n}, x_{m}}-m_{x_{n},x_{m}}) $$(3)
- (3)An M-metric space \((X, m)\) is said to be complete if every m-Cauchy sequence \(\{x_{n}\}\) in X converges, with respect to \(\tau _{m}\), to a point \(x\in X\) such that$$\Bigl(\lim_{n\to\infty}\bigl(m(x_{n}, x)-m_{x_{n},x} \bigr)=0 \mbox{ and } \lim_{n\to\infty}(M_{x_{n}, x}-m_{x_{n},x})=0 \Bigr). $$
Lemma 2.1
- (1)
\(\{x_{n}\}\) is a m-Cauchy sequence in \((X, m)\) if and only if it is a Cauchy sequence in the metric space \((X, m^{w})\).
- (2)An M-metric space \((X, m)\) is complete if and only if the metric space \((X, m^{w})\) is complete. Furthermore,$$\lim_{n\to\infty}m^{w}(x_{n}, x) = 0\ \iff\ \Bigl(\lim_{n\to\infty}\bigl(m(x_{n}, x)-m_{x_{n},x}\bigr)=0 \textit{ and } \lim_{n\to\infty}(M_{x_{n}, x}-m_{x_{n},x})=0 \Bigr). $$
Likewise the above definition holds also for \(m^{s}\).
Lemma 2.2
Proof
From Lemma 2.2 we can deduce the following lemma.
Lemma 2.3
Lemma 2.4
Assume that \(x_{n}\to x\) and \(x_{n}\to y\) as \(n\to\infty\) in an M-metric space \((X, m)\). Then \(m(x,y)=m_{xy}\). Further if \(m(x,x)=m(y,y)\), then \(x=y\).
Proof
Lemma 2.5
- (A)
\(\lim_{n\to\infty}m(x_{n},x_{n-1})=0\),
- (B)
\(\lim_{n\to\infty}m(x_{n},x_{n})=0\),
- (C)
\(\lim_{m,n\to\infty}m_{x_{m}x_{n}}=0\),
- (D)
\(\{x_{n}\}\) is an m-Cauchy sequence.
Proof
Clearly, (C) holds, since \(\lim_{n\to\infty}m(x_{n},x_{n})=0\). □
Theorem 2.1
The topology \(\tau_{m}\) is not Hausdorff.
Theorem 2.2
Theorem 2.3
3 Main result and fixed point theorems
The following definition is new version of the definition in [7] for an M-metric space.
Definition 3.1
Theorem 3.1
Proof
Equation (10) implies that \(m(x_{n},Tx_{n})\to0\). Since \(m_{x_{n},Tx_{n}}\to0\), by Lemma 2.2, we get \(m(x^{*},Tx^{*})=m_{x^{*},Tx^{*}}\).
Theorem 3.2
Proof
Corollary 3.1
(Gupta and Saxena [8])
4 Applications
In this section, after an idea of Samet et al. [9], we shall state an integral version of the Gupta-Saxena result.
Theorem 4.1
- (1)
\(\varphi(0)=0\) and \(t>0\Rightarrow\varphi(t)>0\);
- (2)
φ is nondecreasing and right continuous;
- (3)for every \(\varepsilon>0\), there exists \(\delta>0\) such thatfor some \(0< k<\frac{1}{3}\) and for all \(x,y\in X\) with \(x\neq y\).$$ \varepsilon\leq\varphi\bigl(kC(x,y)\bigr)< \varepsilon+\delta \Rightarrow\varphi \bigl(m(Tx,Ty)\bigr)< \varepsilon, $$(17)
Proof
Corollary 4.1
- (1)
\(t>0\Rightarrow\int_{0}^{t} h(s)\,ds>0\);
- (2)for every \(\varepsilon>0\), there exists \(\delta>0\) such thatfor some \(0< k<\frac{1}{3}\) and for all \(x,y\in X\) with \(x\neq y\).$$\frac{1}{k}\varepsilon\leq\int_{0}^{C(x,y)}h(s)\,ds< \frac {1}{k}\varepsilon+\delta\Rightarrow\int_{0}^{ \frac {1}{k}m(Tx,Ty)}h(s)\,ds< \frac{1}{k}\varepsilon, $$(20)
Declarations
Acknowledgements
This research was supported by the Zanjan Branch, Islamic Azad University, Zanjan, Iran. The author would like to acknowledge this support. The author expresses deep gratitude to the referee for his/her valuable comments and suggestions.
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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