New result on fixed point theorems for φ-contractions in Menger spaces
- Huichun Hua^{1},
- Minjiang Chen^{2} and
- Shenghua Wang^{1}Email author
https://doi.org/10.1186/s13663-015-0417-z
© Hua et al. 2015
Received: 10 April 2015
Accepted: 4 September 2015
Published: 6 November 2015
Abstract
Very recently, Fang (Fuzzy Sets Syst. 267:86-99, 2015) gave some fixed point theorems for probabilistic φ-contractions in Menger spaces. Fang’s results improve the one of Jachymski (Nonlinear Anal. 73:2199-2203, 2010) by relaxing the restriction on the gauge function φ. In this paper, inspired by the results of Fang, we prove a new fixed point theorem for a probabilistic φ-contraction in Menger spaces in which a weaker condition on the function φ is required. Our result improves the corresponding one of Fang and some others. Finally, an example is given to illustrate our result.
Keywords
MSC
1 Introduction
Theorem 1.1
([13])
By using the condition (1.1), Fang gave the following result.
Theorem 1.2
([14])
Let \((X,F,{\varDelta })\) be a complete Menger space with a t-norm Δ of H-type. If \(T: X\to X\) is a probabilistic φ-contraction, where \(\varphi\in\boldsymbol{\Phi}_{\mathbf{w}}\), then T has a unique fixed point \(x^{*}\in X\), and \(\{T^{n}x_{0}\}\) converges to \(x^{*}\) for each \(x_{0}\in X\).
Can the condition (1.1) in Theorem 1.2 be replaced by a more weak condition?
In this paper, we give a positive answer to the question of Fang by proving a new fixed point theorem for a probabilistic φ-contraction in Menger spaces. In our result, the function φ is required to satisfy a more weak condition than (1.1) and the t-norm is not required to be of H-type. Our result improves the corresponding one of Fang [14] and some others. Finally, an example is given to illustrate our result.
2 Preliminaries
In the rest of this paper, let \(\mathbb{R}=(-\infty,+\infty)\), \(\mathbb {R}^{+}=[0,+\infty)\) and \(\mathbb{N}\) denote the set of all natural numbers.
A mapping \(F: \mathbb{R}\to[0,1]\) is called a distribution function if it is non-decreasing and left-continuous with \(\inf_{t\in\mathbb {R}}F(t)=0\). If in addition \(F(0)=0\), then F is called a distance distribution function. A distance distribution function F satisfying \(\lim_{t\to\infty }F(t)=1\) is called a Menger distance distribution function.
Definition 2.1
([15])
- (1)
Δ is associative and commutative;
- (2)
\({\varDelta } (a,1)=a\) for all \(a\in{}[0,1]\);
- (3)
\({\varDelta }(a, b)\leq {\varDelta } (c,d)\) whenever \(a\leq c\) and \(b\leq d\) for all \(a,b,c,d\in{}[0,1]\).
Two typical examples of the continuous t-norm are \({\varDelta } _{P}(a,b)=ab\) and \({\varDelta } _{M}(a,b)=\min\{a,b\}\) for all \(a,b\in{}[0,1]\).
Definition 2.2
([16])
It is easy to see that \({\varDelta }_{M}\) is a t-norm of H-type but \({\varDelta }_{P}\) is not of H-type. Here we give a new t-norm of H-type by \({\varDelta }_{M}\) and \({\varDelta }_{P}\).
Example 2.1
Let \({\varDelta }(x,1)={\varDelta }(1,x)=x\) for all \(x\in[0,1]\), \({\varDelta }(x,y)={\varDelta }_{P}(x,y)\) for all \(x,y\in[0,1]\) with \(\max\{x,y\}\in[0,\frac{1}{2}]\) and \({\varDelta }(x,y)={\varDelta }_{M}(x,y)\) for all \(x,y\in[0,1]\) with \(\max\{x,y\} \in(\frac{1}{2},1]\). It is easy to check that Δ is a t-norm. Now we show that it is of H-type. For any given \(\epsilon\in (0,\frac{1}{2})\), set \(\delta=\epsilon\). Then \(1-\delta=1-\epsilon> \frac{1}{2}\). Thus, for all \(t\in(1-\delta,1)\), one has \({\varDelta }^{n}(t)=t>1-\delta=1-\epsilon\) for all \(n\in\mathbb{N}\). For \(\epsilon\in [\frac{1}{2},1)\), taking \(\delta\in(0,\frac{1}{2})\) arbitrarily, then we have \(1-\delta>\frac{1}{2}\geq1-\epsilon\). Thus for all \(t\in (1-\delta,1)\), \({\varDelta }^{n}(t)=t>1-\delta>\frac{1}{2}\geq 1-\epsilon\) for all \(n\in\mathbb{N}\). Therefore, Δ is a t-norm of H-type.
Example 2.2
Let \(\delta\in(0,1]\) and let Δ be a t-norm. Define \({\varDelta }_{\delta}\) by \({\varDelta }_{\delta}(x,y)={\varDelta }(x,y)\), if \(\max\{x,y\}\leq1-\delta\), and \({\varDelta }_{\delta}(x,y)=\min\{x,y\}\), if \(\max\{x,y\}>1-\delta\). then \({\varDelta }_{\delta}\) is a t-norm of H-type; see [17]. However, if \({\varDelta }_{\delta}(x,1)={\varDelta }_{\delta}(1,x)=x\) for all \(x\in[0,1]\), \({\varDelta }_{\delta}(x,y)=\delta\) for all \(x,y\in[\delta,1)\) and \({\varDelta }_{\delta}(x,y)=0\) for all \(x,y\in[0,1]\) with \(\min\{x,y\}\in[0,\delta)\), then \({\varDelta }_{\delta}\) is a t-norm but not of H-type.
For other t-norms of H-type, the reader may refer to [16].
Definition 2.3
([18])
- (PM-1)
\(F_{x,y}(t)=\epsilon_{0}(t)\) for all \(t\in\mathbb{R}\) if and only if \(x=y\);
- (PM-2)
\(F_{x,y}(t)=F_{y,x}(t)\) for all \(t\in\mathbb{R}\);
- (PM-3)
\(F_{x,y}(t+s)\geq{ \varDelta }(F_{x,z}(t), F_{z,y}(s))\) for all \(x,y,z\in X\) and \(t,s>0\).
Definition 2.4
([15])
Let \((X,F,{\varDelta })\) be a Menger space and \(\{x_{n}\}\) be a sequence in X. The sequence \(\{x_{n}\}\) is said to be convergent to \(x\in X\) if \(\lim_{n\to\infty}F_{x_{n},x}(t)=1\) for all \(t>0\); the sequence \(\{x_{n}\}\) is said to be a Cauchy sequence if for any given \(t>0\) and \(\epsilon\in(0,1)\), there exists \(N_{\epsilon,t}\in\mathbb {N}\) such that \(F_{x_{n},x_{m}}(t)>1-\epsilon\) whenever \(m,n>N_{t,\epsilon }\); the Menger space \((X,F,{\varDelta })\) is said to be complete, if each Cauchy sequence in X is convergent to some point in X.
3 Main results
It is easy to see that for each \(\varphi\in\boldsymbol{\Phi}_{\mathbf{w}}\), \(\varphi\in \boldsymbol{\Phi}_{\mathbf{w}^{*}}\). In fact, if \(\varphi\in\boldsymbol{\Phi}_{\mathbf{w}}\), then for each \(t_{1},t_{2}>0\), there exist \(r_{1}\geq t_{1}\) and \(r_{2}\geq t_{2}\) such that \(\lim_{n\to\infty}\varphi^{n}(r_{1})=\lim_{n\to\infty}\varphi^{n}(r_{2})=0\). Assume that \(t_{1}\leq t_{2}\). Then there exists \(N\in\mathbb{N}\) such that \(\varphi^{n}(r_{2})< t_{1}\) for all \(n>N\). Thus \(\varphi\in\boldsymbol{\Phi}_{\mathbf{w}^{*}}\).
However, if \(\varphi\in\boldsymbol{\Phi}_{\mathbf{w}^{*}}\), then it is unnecessary that \(\varphi\in\boldsymbol{\Phi}_{\mathbf{w}}\).
Example 3.1
Let \(\varphi:\mathbb{R}^{+}\to\mathbb{R}^{+}\) by \(\varphi (t)=t\) for all \(t\in[0,1]\), \(\varphi(t)=t-1\) for all \(t\in(1,\infty)\). Then \(\varphi\in\boldsymbol{\Phi}_{\mathbf{w}^{*}}\). In fact, for each \(t_{1},t_{2}\in (0,\infty)\), there exists \(N\in\mathbb{N}\) such that \(r=1+N+\epsilon >\max\{t_{1},t_{2}\}\), where \(\epsilon\in(0,\min\{t_{1},t_{2},1\})\). Then we have \(\varphi^{n}(r)=\epsilon< \min\{t_{1},t_{2}\}\) for all \(n> N+1\). So \(\varphi\in\boldsymbol{\Phi}_{\mathbf{w}^{*}}\). However, since \(\lim_{n\to\infty }\varphi^{n}(r)\neq0\) for all \(r>0\), \(\varphi\notin\boldsymbol{\Phi}_{\mathbf{w}}\).
From Example 3.1 we see that \(\boldsymbol{\Phi}_{\mathbf{w}^{*}}\) is a proper subclass of \(\boldsymbol{\Phi}_{\mathbf{w}}\). On \(\boldsymbol{\Phi}_{\mathbf{w}^{*}}\), \(\boldsymbol{\Phi}_{\mathbf{w}}\), and Φ, we have \(\boldsymbol{\Phi} \subset \boldsymbol{\Phi} _{\mathbf{w}}\subset\boldsymbol{\Phi}_{\mathbf{w}^{*}}\).
Lemma 3.1
Let \(\varphi\in\boldsymbol{\Phi}_{\mathbf{w}^{*}}\). Then for each \(t>0\), there exists \(r\geq t\) such that \(\varphi(r)< t\).
Proof
Suppose that there is \(t_{0}>0\) such that \(\varphi(r)\geq t_{0}\) for all \(r\geq t_{0}\). By induction, we obtain \(\varphi^{n}(r)\geq t_{0}\) for all \(n\in\mathbb{N}\). From (3.2) it follows that there exist \(r\geq t_{0}\) and \(N\in\mathbb{N}\) such that \(\varphi^{n}(r)< t_{0}\) for all \(n>N\), which contradicts \(\varphi^{n}(r)\geq t_{0}\) for all \(r\geq t_{0}\) and \(n\in\mathbb{N}\). Thus for each \(t>0\), there exists \(r\geq t\) such that \(\varphi(r)\leq t\). This completes the proof. □
Lemma 3.2
Proof
Lemma 3.3
- (1)
\(\varphi(t)>0\) for all \(t>0\);
- (2)
\(F_{x_{n},x_{m}}(\varphi(t))\geq F_{x_{n-1},x_{m-1}}(t)\) for all \(n,m\in\mathbb{N}\) and \(t>0\).
Proof
Lemma 3.4
Let \((X,F,{\varDelta })\) be a Menger space where Δ is of H-type and continuous at \((1,1)\) and let \(\{x_{n}\}\) be a sequence in X. Suppose that there exists a function \(\varphi\in\boldsymbol {\Phi}_{\mathbf{w}^{*}}\) satisfying the conditions (1) and (2) in Lemma 3.3. Then \(\{x_{n}\}\) is a Cauchy sequence.
Proof
Theorem 3.1
Let \((X,F,{\varDelta })\) be a complete Menger space where Δ is of H-type and continuous at \((1,1)\). Let \(T: X\to X\) be a probabilistic φ-contraction, where \(\varphi\in\boldsymbol{\Phi} _{\mathbf{w}^{*}}\) satisfies \(\varphi(t)>0\) for all \(t>0\). Then T has a unique fixed point \(x^{*}\in X\), and \(\{T^{n}x_{0}\}\) converges to \(x^{*}\) for each \(x_{0}\in X\).
Proof
Corollary 3.1
Proof
Let \(T=T_{0}T_{1}\). Then (3.11) implies that T is a probabilistic φ-contraction. From Theorem 3.1 it follows that T has a unique fixed point \(x^{*}\in X\). Since \(T_{0}\) commutes with \(T_{1}\), we have \(T_{0}T_{1}x^{*}=T_{1}T_{0}x^{*}\). Further we have \(T(T_{0}x^{*})=(T_{0}T_{1})(T_{0}x^{*})=T_{0}(T_{0}T_{1}x^{*})=T_{0}(Tx^{*})=T_{0}x^{*}\), which implies that \(T_{0}x^{*}\) is a fixed point of T. Since T has a unique fixed point \(x^{*}\), one has \(T_{0}x^{*}=x^{*}\). Similarly, we have \(T_{1}x^{*}=x^{*}\). Thus \(x^{*}\) is the common fixed point of \(T_{0}\) and \(T_{1}\). Assume that \(x'\in X\) is another common fixed point of \(T_{0}\) and \(T_{1}\). Since \(T_{0}\) commutes with \(T_{1}\), we have \(T(T_{0}x')=(T_{0} T_{1})(T_{0}x')=T_{0}(T_{0}T_{1}x')=T_{0}(T_{1}T_{0}x')=T_{0}x'\), which implies that \(T_{0}x'\) is the fixed point of T. Since \(x^{*}\) is a unique fixed point of T, one has \(x'=T_{0}x'=x^{*}\). Thus \(x^{*}\) is the unique common fixed point of \(T_{0}\) and \(T_{1}\). This completes the proof. □
Finally, we give an example to illustrate Theorem 3.1.
Example 3.2
- (a)For \((x,y)\in\{ (0,3^{n+3}),(3,3^{n+3}), (27,3^{n+3}):n\in\mathbb {N}\}\), it is easy to conclude that \(\varphi(t)\leq|Tx-Ty|\) implies that \(t\leq|x-y|\) for all \(t>0\). Thus if \(\varphi(t)\leq|Tx-Ty|\), thenTherefore (3.12) holds.$$F_{Tx,Ty}\bigl(\varphi(t)\bigr)=\frac{1}{2}=F_{x,y}(t) \quad \mbox{for all } t>0. $$
- (b)For \((x,y)\in\{(3^{n+3},3^{m+3}): m,n\in\mathbb{N} \mbox{ with } m>n\}\), we have \(\varphi(t)\leq |Tx-Ty|=3^{m+2}-3^{n+2}<3(3^{m+2}-3^{n+2})=|y-x|\) for \(t\in(0,1]\). For \(t>1\), from \(\varphi(t)=t-1\leq|Tx-Ty|=3^{m+2}-3^{n+2}\), we have \(t\leq3^{m+2}-3^{n+2}+1< 3^{m+3}-3^{n+3}= |x-y|\) since \(3^{m+3}-3^{n+3}-3^{m+2}+3^{n+2}=2(3^{m+2}-3^{n+2})>1\). So \(\varphi (t)\leq|Tx-Ty|\) implies that \(t\leq|x-y|\) for all \(t>0\). Thus if \(\varphi(t)\leq|Tx-Ty|\), thenTherefore (3.12) holds.$$F_{Tx,Ty}\bigl(\varphi(t)\bigr)=\frac{1}{2}=F_{x,y}(t) \quad \mbox{for all } t>0. $$
By the discussion above, (3.12) holds for all \(x,y\in X\) and \(t>0\). Therefore, T is a probabilistic φ-contraction. All the conditions of Theorem 3.1 are satisfied. By Theorem 3.1, T has a unique fixed point \(x^{*}\in X\). Obviously, \(x^{*}=0\) is the unique fixed point of T. However, since \(\varphi\notin\boldsymbol{\Phi}_{\mathbf{w}}\), Theorem 1.2, i.e., Theorem 3.1 of [14] cannot be applied to this example.
4 Conclusion
In this paper, we prove a new fixed point theorems for a probabilistic φ-contraction in Menger spaces. In the theorem, a more weak condition on the gauge function φ is required. Thus our result improves Theorem 1.2 of Fang [14] and some others, such as Jachymski [13], Ćirić [2], and Xiao et al. [19]. By using Theorem 3.1, it is easy to prove some fixed point theorems for φ-contraction in fuzzy metric spaces like Theorems 4.1-4.4 in [14]. For shortening the length of this paper, we omit the proofs of these theorems.
Declarations
Acknowledgements
This work is supported by the Fundamental Research Funds for the Central Universities (Grant numbers: 13MS109, 2014MS164, 2014ZD44, 2015MS78). The authors thank the editor and reviewers.
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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