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On ϕ-contractions in partially ordered fuzzy metric spaces
- Shuang Wang^{1}Email author
Received: 18 August 2015
Accepted: 9 October 2015
Published: 23 December 2015
Abstract
Under some weaker conditions, some coincidence point and common fixed point theorems are established in partially ordered fuzzy metric spaces using weakly compatible mappings. By using the theorems, we obtain some coupled and multidimensional fixed point results, which are generalization and improvement of very recent theorems in the corresponding literature. In order to illustrate our main results, we give three examples.
Keywords
MSC
1 Introduction
In 1987, the notion of coupled fixed point was first introduced by Guo and Lakshmikantham [1]. Recently, Gnana-Bhaskar and Lakshmikantham [2] established some coupled fixed point theorems in partially ordered metric space. The fuzzy version of the results of Gnana-Bhaskar and Lakshmikantham [2] was studied by Sedghi et al. [3]. After that, common coupled fixed point results in fuzzy metric spaces were established by Hu [4] and Hu et al. [5]. Very recently, Choudhury et al. [6] established coupled coincidence point and fixed point results for compatible mappings in partially ordered fuzzy metric spaces. Later, Roldán et al. [7] obtained multidimensional coincidence point theorems for nonlinear mappings in any number of variables in partially ordered fuzzy metric spaces. Their results generalize, clarify and unify several classical and very recent related results in the literature in the setting of metric spaces.
But many results (see, e.g., [4–7]) are obtained under the assumptions: (a) \(\phi(t)=kt\) for all \(t>0\), where \(k\in(0,1)\); or (b) \(\sum^{\infty}_{n=1}\phi^{n}(t)<\infty\) for all \(t>0\). It is obvious that the condition (a) is special. In [8], Ćirić [8] has pointed out, the condition (b) is very strong and difficult for testing in practice. Then Ćirić introduced the condition (CBW): \(\phi(0)=0\), \(\phi(t)< t\) and \(\liminf_{r\rightarrow t^{+}}\phi(t)< t\) for all \(t>0\). Later, Jachymski [9] presented the condition (c): \(0<\phi(t)<t\) and \(\lim_{n\rightarrow\infty}\phi^{n}(t)=0\) for all \(t>0\). In order to weaken the condition (c) further, Fang [10] introduced the condition (d): for each \(t>0\) there exists \(r\geq t\) such that \(\lim_{n\rightarrow \infty}\phi^{n}(r)=0\) in the context of Menger probabilistic metric spaces and fuzzy metric spaces. In this paper, under the condition (d), we present some coincidence point and common fixed point results for weakly compatible mappings in partially ordered fuzzy metric spaces. By using the theorems, we obtain some coupled and multidimensional fixed point results, which are generalization and improvement of very recent theorems in the corresponding literature. In addition, we illustrate our main results with three examples.
2 Preliminaries
In order to fix the framework needed to state our main results, we recall the following notions. Let \(n\in \mathbb {N}\), X be a non-empty set and \(X^{n}\) be the Cartesian product of n copies of X. For brevity, \(g(x)\), \((y_{1},y_{2},\ldots,y_{n})\), \((y^{1}_{m},y^{2}_{m},\ldots,y^{n}_{m})\), \((z^{1}_{m},z^{2}_{m},\ldots,z^{n}_{m})\), \((z_{1},z_{2},\ldots,z_{n})\), \((v_{1},v_{2},\ldots,v_{n})\) and \((x^{1}_{0},x^{2}_{0},\ldots,x^{n}_{0})\) will be denoted by gx, Y, \(Y_{m}\), \(Z_{m}\), Z, V, and \(X_{0}\), respectively.
Proposition 2.1
([12])
If \(Y\preceq_{n} V\), it follows that \((y_{\sigma(1)},y_{\sigma (2)},\ldots,y_{\sigma(n)})\preceq_{n}(v_{\sigma(1)},v_{\sigma (2)},\ldots, v_{\sigma(n)})\) if \(\sigma\in\Omega_{AB}\), \((y_{\sigma (1)},y_{\sigma(2)},\ldots,y_{\sigma(n)})\succeq _{n}(v_{\sigma (1)},v_{\sigma(2)},\ldots,v_{\sigma(n)})\) if \(\sigma\in \Omega'_{AB}\).
Definition 2.2
([13])
Definition 2.3
([14])
Definition 2.4
([11])
Definition 2.5
([13])
An element \(Y\in X^{n}\) is called a coincidence point of the mappings \(T:X^{n}\rightarrow X^{n}\) and \(G:X^{n}\rightarrow X^{n}\) if \(T(Y)=G(Y)\). Furthermore, if \(T(Y)=G(Y)=Y\), then we say that Y is a common fixed point of T and G.
Definition 2.6
- (i)
A coupled coincidence point ([14]) if \(n=2\), \(F(x_{1},x_{2})=g(x_{1})\), and \(F(x_{2},x_{1})=g(x_{2})\). If g is the identity mapping on X, then \((x_{1},x_{2})\in X^{2}\) is called a coupled fixed point of the mapping F ([2]). A coupled common fixed point of F and g ([15]) if \(n=2\), \(F(x_{1},x_{2})=g(x_{1})=x_{1}\), and \(F(x_{2},x_{1})=g(x_{2})=x_{2}\).
- (ii)A ϒ-coincidence point ([16]) of F and g iffor \(i\in\Lambda_{n}\). If g is the identity mapping on X, then \((x_{1},x_{2},\ldots,x_{n})\in X^{n}\) is called a ϒ-fixed point of the mapping F.$$F(x_{\sigma_{i}(1)},x_{\sigma_{i}(2)},\ldots,x_{\sigma_{i}(n)})=gx_{i} $$
Definition 2.7
([17])
- (i)
If \(\{x_{m}\}\) is a non-decreasing sequence and \(\{x_{m}\}\rightarrow x\), then \(gx_{m}\preceq gx\) for all m.
- (ii)
If \(\{y_{m}\}\) is a non-increasing sequence and \(\{y_{m}\}\rightarrow y\), then \(gy_{m}\succeq gy\) for all m.
Definition 2.8
([18])
Definition 2.9
([19])
A t-norm is said to be of H-type if the sequence \(\{\ast^{m}a\}^{\infty}_{m=1}\) is equicontinuous at \(a=1\), i.e., for all \(\varepsilon\in(0,1)\), there exists \(\eta\in (0,1)\) such that if \(a\in(1-\eta,1]\), then \(\ast^{m}a>1-\varepsilon\) for all \(m\in \mathbb {N}\).
Definition 2.10
([20])
- (FM-1)
\(M(x,y,0)=0\);
- (FM-2)
\(M(x,y,t)=1\), for all \(t>0\) if and only if \(x=y\);
- (FM-3)
\(M(x,y,t)=M(y,x,t)\);
- (FM-4)
\(M(x,y,t)\ast M(y,z,s)\leq M(x,z,t+s)\);
- (FM-5)
\(M(x,y,\cdot):\mathbb{R}^{+}\rightarrow\mathbb{I}\) is left continuous.
Remark 2.11
Note that ∗ is continuous in the original definition in [20].
Definition 2.12
([21])
A triple \((X,M,\ast)\) is called a fuzzy metric space (in the sense of George and Veeramani) if X is an arbitrary non-empty set, ∗ is a continuous t-norm and \(M:X\times X\times\mathbb{R}^{+}\rightarrow\mathbb{I}\) is a fuzzy set satisfying, for each \(x,y,z\in X\) and \(t,s>0\), conditions (FM-2), (FM-3), (FM-4), (GV-5): \(M(x,y,\cdot):(0,\infty)\rightarrow\mathbb{I}\) is continuous, and (GV-1): \(M(x,y,t)>0\).
Definition 2.13
([21])
Let \((X,M,\ast)\) be a FMS. A sequence \(\{ x_{n}\}\) in X is said to be convergent to \(x\in X\) if \(\lim_{n\rightarrow\infty}M(x_{n},x,t)=1\) for all \(t>0\). A sequence \(\{x_{n}\}\) in X is said to an M-Cauchy sequence, if for each \(\varepsilon\in (0,1)\) and \(t>0\) there exists \(n_{0}\in \mathbb {N}\) such that \(M(x_{n},x_{m},t)>1-\varepsilon\) for all \(m,n\geq n_{0}\). A fuzzy metric space is called complete if every M-Cauchy sequence is convergent in X.
Lemma 2.14
([22])
If \((X,M)\) is a FMS under some t-norm and \(x,y\in X\), then \(M(x,y,\cdot)\) is a non-decreasing function on \((0,\infty)\).
Definition 2.15
([7])
A partially ordered fuzzy metric space (for short, poFMS) is a quadruple \((X,M,\ast,\preceq)\) such that \((X,M,\ast)\) is a FMS and ⪯ is a partial order on X.
Definition 2.16
([7])
Let \(p\in \mathbb {N}\) and let \((X,M,\ast)\) be a FMS. A mapping \(G:X^{p}\rightarrow X\) is said to be continuous at a point \(Y_{0}\in X^{p}\) if, for any sequence \(\{Y_{m}\}_{m\geq0}\) in \(X^{p}\) converging to \(Y_{0}\), the sequence \(\{G(Y_{m})\}_{m\geq0}\) converges to \(G(Y_{0})\). If G is continuous at each \(Y_{0}\in X^{p}\), then G is said continuous on \(X^{p}\).
Definition 2.17
([4])
Definition 2.18
([7])
Remark 2.19
If \(n=1\) in Definition 2.18, then \(F,g:X\rightarrow X\) are compatible w.r.t. \((X,M, \ast,\preceq)\).
Definition 2.20
([23])
We will say that the maps \(f,g:X\rightarrow X\) are weakly compatible (or the pair \((f,g)\) is w-compatible) if \(fgx=gfx\) for all \(x\in X\) such that \(fx=gx\).
Let \(\Phi'\) denote the family of all functions \(\phi:\mathbb {R}^{+}\rightarrow\mathbb{R}^{+}\) such that \(\lim_{n\rightarrow \infty }\phi^{n}(t)=0\) for all \(t>0\), and let \(\Phi_{w}\) denote the family of all functions \(\phi:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}\) verifying the condition (d), that is, for each \(t>0\) there exists \(r\geq t\) such that \(\lim_{n\rightarrow\infty}\phi^{n}(r)=0\).
It is evident that the condition \(\lim_{n\rightarrow\infty}\phi ^{n}(t)=0\) for all \(t>0\) implies the condition (d). However, the following example shows that the reverse is not true in general. Hence \(\Phi'\subseteq\Phi_{w}\).
Example 2.21
([10])
Lemma 2.22
([10])
Let \(\phi\in\Phi_{w}\), then for each \(t>0\) there exists \(r\geq t\) such that \(\phi(r)< t\).
3 Main results
In this section we establish our main results and use them to obtain some coupled and multidimensional fixed point theorems.
Lemma 3.1
If \((X,M,\ast)\) is a FMS with \(M(x,y,\cdot):\mathbb {R}^{+}\rightarrow \mathbb{I}\) is continuous, then M is a continuous mapping on \(X^{2}\times(0,\infty)\).
Proof
The proof is the same as that for a fuzzy metric space in the sense of George and Veeramani (see Rodríguez-López and Romaguera [24], Proposition 1). □
Lemma 3.2
- (i)
\(\phi(t)>0\) for all \(t>0\);
- (ii)
\(M(x_{n},x_{m},\phi(t))\geq M(x_{n-1},x_{m-1},t)\) for all \(n,m\in \mathbb {N}\) and \(t>0\);
- (iii)
\(\lim_{t\rightarrow\infty}M(x_{0},x_{1},t)=1\),
Proof
We proceed with the following steps:
Now, we state and prove some fixed point results for weakly compatible mappings in partially ordered fuzzy metric spaces.
Theorem 3.3
- (C1)
T and G are continuous and compatible and \(M(x,y,\cdot ):\mathbb{R}^{+}\rightarrow\mathbb{I}\) is continuous or
- (C2)
\((X,\tau_{M},\preceq)\) has the sequential monotone property and \(G(X)\) is closed.
If there exists \(y_{0}\in X\) such that \(G(y_{0})\asymp T(y_{0})\) and \(\lim_{t\rightarrow\infty}M(G(y_{0}),T(y_{0}),t)=1\). Then T and G have a coincidence point.
Proof
Let \(y_{0}\in X\) such that \(G(y_{0})\asymp T(y_{0})\) and \(\lim_{t\rightarrow \infty}M(G(y_{0}),T(y_{0}),t)=1\). Since \(T(X)\subseteq G(X)\), there exists \(y_{1}\in X\) such that \(G(y_{1})=T(y_{0})\). Recursively, we see that, for every \(m\in \mathbb {N}_{0}\), there exists \(y_{m+1}\in X\) such that \(G(y_{m+1})=T(y_{m})\). Set \(z_{0}=G(y_{0})\) and \(z_{m+1}=G(y_{m+1})=T(y_{m})\) for every \(m\in \mathbb {N}_{0}\).
Theorem 3.4
- (C3)
\(G(u)\) is comparable to \(G(y)\) and \(G(v)\);
- (C4)
\(\lim_{t\rightarrow\infty}M(G(u),G(y),t)=\lim_{t\rightarrow \infty}M(G(u),G(v),t)=1\).
Proof
Denote \(w=T(y)=G(y)\). Since T and G are weakly compatible mappings, we have \(T(w)=TG(y)=GT(y)=G(w)\). So, w is also a coincidence point of T and G. Therefore, \(G(w)=G(y)=w\) and w is a common fixed point of T and G. In order to prove the uniqueness, assume that \(w^{*}\) is another common fixed point of T and G. Then we have \(w^{*}=G(w^{*})=G(w)=w\). This completes the proof. □
Example 3.5
- (i)
\(T(X)\subseteq G(X)\) and T is a G-isotone mapping.
- (ii)
The condition (C1) holds.
- (iii)
There exists \(y_{0}=0\) such that \(G(y_{0})=0\leq\frac{2}{3}=T(y_{0})\).
By Theorems 3.3 and 3.4, T and G have a unique common fixed point, which is \(z=1\). In this example, computing according to \(z_{0}=G(y_{0})\) and \(z_{m+1}=G(y_{m+1})=T(y_{m})\) for every \(m\in \mathbb {N}_{0}\), we obtain \(\{ z_{0}=0,z_{1}=\frac{2}{3},z_{2}=\frac{22}{27},z_{3}=\frac{1942}{2187},\ldots\}\). Thus the sequence \(\{z_{n}\}\) is a non-trivial sequence.
Example 3.6
- (i)
\(T(X)\subseteq G(X)\).
- (ii)
The condition (C2) holds (since \(\tau_{M}\) is the discrete topology on X).
- (iii)
There exists \(y_{0}=0\) such that \(G(y_{0})\preceq T(y_{0})\) and \(\lim_{t\rightarrow\infty}M(G(y_{0}),T(y_{0}),t)=1\).
- (iv)
All conditions of Theorem 3.4 hold. In fact, \(y=0\) and \(v=0.5\) are all coincidence points of T and G. Since \(TG(0)=GT(0)\) and \(TG(0.5)=GT(0.5)\), by Definition 2.20, G is weakly compatible with T. In addition, there exists \(u=1.5\) such that \(G(y)\preceq G(u)\) and \(G(v)\preceq G(u)\). It follows from (13) that (C4) holds.
- (v)T is a G-isotone mapping. Indeed, let \(y,v\in X\) such that \(G(y)\preceq G(v)\).
- (a)
If \(G(y)=G(v)\) then \(y=v\) or \(y,v\in\{0,0.5\}\) or \(y,v\in\{ 0.25,1.75,2\}\). Thus, \(T(y)=T(v)\).
- (b)
If \((G(y),G(v))=(0,0.25)\), then \(y\in\{0,0.5\}\) and \(v\in\{ 0.25,1.75,2\}\). Thus, \(T(y)=T(v)\).
- (c)
If \((G(y),G(v))=(0,0.5)\), then \(y\in\{0,0.5\}\) and \(v=1.5\). Thus, \((T(y),T(v))=(0,0.25)\).
- (a)
If G is the identity mapping on X in Theorems 3.3 and 3.4, then the following corollary is obtained immediately.
Corollary 3.7
Furthermore, suppose that for all fixed points \(y,v\in X\) of T, there exists \(u\in X\) such that u is comparable to y and v and \(\lim_{t\rightarrow\infty}M(u,y,t)=\lim_{t\rightarrow \infty }M(u,v,t)=1\). Then T has a unique fixed point.
Example 3.9
Next, we give some basic concepts and results that we will need to obtain some coupled and multidimensional fixed point results.
Definition 3.10
Let \(F:X^{n}\rightarrow X\) and \(g:X\rightarrow X\) be two mappings. A point \((x_{1},x_{2},\ldots,x_{n})\in X^{n}\) is a common fixed point of F and g if \(F(x_{\sigma_{i}(1)},x_{\sigma _{i}(2)},\ldots,x_{\sigma_{i}(n)})=gx_{i}=x_{i}\) for \(i\in\Lambda_{n}\).
Definition 3.11
Lemma 3.12
- (i)
\((X^{n},M^{n},\ast)\) is also a FMS.
- (ii)
Let \(\{A_{m}=(a^{1}_{m},a^{2}_{m},\ldots,a^{n}_{m})\}\) be a sequence on \(X^{n}\) and let \(A=(a_{1},a_{2},\ldots,a_{n})\in X^{n}\). Then \(\{A_{m}\} \rightarrow A\) if, and only if, \(\{a^{i}_{m}\}\rightarrow a_{i}\) for all \(i\in\{ 1,2,\ldots,n\}\).
- (iii)
If \((X,M,\ast)\) is complete, then \((X^{n},M^{n},\ast)\) is complete.
Proof
The following multidimensional fixed point theorem is an immediate consequence of Theorems 3.3 and 3.4.
Theorem 3.13
- (C5)
F and g are continuous and Φ-compatible and \(M(x,y,\cdot ):\mathbb{R}^{+}\rightarrow\mathbb{I}\) is continuous, or
- (C6)
\((X,\tau_{M},\preceq)\) has the sequential monotone property and \(g(X)\) is closed.
- (C7)
\((gu_{1},gu_{2},\ldots,gu_{n})\) is comparable to \((gx_{1},gx_{2},\ldots,gx_{n})\) and \((gy_{1},gy_{2},\ldots,gy_{n})\);
- (C8)
\(\lim_{t\rightarrow\infty}M(gu_{i},gx_{i},t)=\lim_{t\rightarrow \infty}M(gu_{i},gy_{i},t)=1\) for \(i\in\Lambda_{n}\).
Proof
The conditions (C7) and (C8) imply that (C3) and (C4) hold w.r.t. \((X^{n},M^{n},\ast,\preceq_{n})\). It is easy to deduce that T and G are weakly compatible if assumption (C2) holds w.r.t. \((X^{n},M^{n},\ast ,\preceq _{n})\). If F and g are continuous, then T and G are continuous.
Next we shall prove that the condition (C2) of Theorem 3.3 holds w.r.t. \((X^{n},M^{n},\ast,\preceq_{n})\). Since \(g(X)\) is closed, so is \(G(X)\). Suppose that \(\{Z_{m}\}\) is a non-decreasing sequence in \(X^{n}\) such that \(Z_{m}\rightarrow Z\) (\(m\rightarrow\infty\)). Using Lemma 3.12, we have \(z^{i}_{m}\rightarrow z^{i}\) (\(m\rightarrow\infty\)) for all \(i\in\Lambda_{n}\). Since \(Z_{m}\preceq_{n} Z_{m+1}\) for all \(m\in \mathbb {N}_{0}\), then \((z^{i}_{m})_{m\in \mathbb {N}_{0}}\) is a non-decreasing sequence when \(i\in A\) and \((z^{i}_{m})_{m\in \mathbb {N}_{0}}\) is a non-increasing sequence when \(i\in B\). If \(i\in A\), as \((X,\tau_{M},\preceq)\) has the sequential monotone property, then we have \(z^{i}_{m}\preceq z^{i}\) for all \(m\in \mathbb {N}_{0}\). Similarly, if \(i\in B\), then \(z^{i}_{m}\succeq z^{i}\) for all \(m\in \mathbb {N}_{0}\). That is, \(Z_{m}\preceq_{n} Z\) for every \(m\in \mathbb {N}_{0}\). The other case is treated similarly.
Therefore, all conditions of Theorems 3.3 and 3.4 hold w.r.t. \((X^{n},M^{n},\ast,\preceq_{n})\). Theorem 3.3 implies that T and G have a coincidence point, which is a ϒ-coincidence point of F and g. Moreover, it follows from Theorem 3.4 that T and G have a unique common fixed point, which is a unique common fixed point of F and g. □
Remark 3.14
- (i)
We use \(\phi\in\Phi_{w}\), and \(\Phi_{w}\) is a class of more general functions than \(\phi(t)=kt\), \(k\in(0,1)\).
- (ii)
The continuity of g and Φ-compatible of F and g are omitted if assumption (C6) holds. The continuity of γ is not necessary.
- (iii)
The condition \(\lim_{t\rightarrow\infty}M(x,y,t)=1\) for all \(x,y\in X\) is weakened by conditions (17) and (C8).
- (iv)
Our result is valid for fuzzy metric spaces in the sense of Kramosil and Michálek, so it is also valid for fuzzy metric spaces in the sense of George and Veeramani. The completeness is a weaker kind of completeness in Theorem 3.13 (see [26]).
Taking \(n=2\), \(A=\{1\}\), and \(B=\{2\}\) in Theorem 3.13, we deduce the following coupled fixed point theorem improving Theorem 3.1 in [6].
Corollary 3.15
Furthermore, assume that for all pairs of coupled coincidence points \((x_{1},x_{2})\), \((y_{1},y_{2})\in X^{2}\) of F and g there exists \((u_{1},u_{2})\in X^{2}\) such that \((gu_{1},gu_{2})\) is comparable to \((gx_{1},gx_{2})\) and \((gy_{1},gy_{2})\), \(\lim_{t\rightarrow\infty}M(gu_{i},gx_{i},t)=\lim_{t\rightarrow\infty }M(gu_{i},gy_{i},t)=1\) for \(i\in\Lambda_{2}\). Also, assume that F are weakly compatible with g if assumption (C6) holds. Then F and g have a unique common coupled fixed point.
Remark 3.16
- (i)
In Corollary 3.15, we use \(\phi\in\Phi_{w}\), and \(\Phi _{w}\) is a class of more general functions than Φ in Theorem 3.2 of [5].
- (ii)
Corollary 3.15 is valid for partially ordered fuzzy metric spaces in the sense of Kramosil and Michálek, so it is also valid for fuzzy metric spaces in the sense of George and Veeramani.
Declarations
Acknowledgements
The author thanks the editor and the referees for their useful comments and suggestions. Supported by the Natural Science Foundation of Jiangsu Province under grant (13KJB110028).
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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