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Fuzzy fixed point theorems on the complete fuzzy spaces under supremum metric
Fixed Point Theory and Applications volume 2015, Article number: 167 (2015)
Abstract
By using the concept of a class of functions, the \(\mathcal {R}\)-functions, we provide some fuzzy fixed point theorems on a space of fuzzy sets equipped with the supremum metric. By presenting a technique of constructing a sequence of successive approximations, we obtain some interesting results that improve many existing results. The related cases are also shown and discussed.
1 Introduction
By using a natural generalization of the concept of a set, the fuzzy set, which was introduced initially by Zadeh [1], considering mathematical programming problems which are expressed as optimizing some goal function given certain constraints, this be relaxed by means of a subjective gradation. In 1981, Heilpern [2] used the concept of fuzzy sets and introduced a class of fuzzy mappings, which is a generalization of the set-valued mapping, and proved a fixed point theorem for fuzzy contraction mappings in metric linear spaces. It is worth noting that the result announced by Heilpern [2] is a fuzzy extension of the Banach contraction principle. Subsequently, several other authors have studied the existence of fixed points of fuzzy mappings; for example, Estruch and Vidal [3] proved a fixed point theorem for fuzzy contraction mappings over a complete metric space, which is a generalization of the given Heilpern fixed point theorem, and Sedghi et al. [4] gave an extended version of the Estruch and Vidal [3] theorem (for more examples, see [5–14]).
Although many kinds of fixed point theorems for fuzzy contraction mappings in complete metric spaces have been studied extensively in recent years, we have to point out that one has given most attention to the class of fuzzy sets with nonempty compact α-cut sets in the metric space X, but little attention to the class of fuzzy sets with nonempty bounded or closed, or even bounded closed, α-cut sets. However, it is well known that all compact sets are bounded closed sets in a general metric space and the converse is not always true.
In 2008, Qui and Shu [12] established the completeness of \(\mathcal{CB}(X)\) with respect to the completeness of the metric space X, where \(\mathcal{CB}(X)\) denotes the class of fuzzy sets with nonempty bounded closed α-cut sets equipped with the generalized Hausdorff metric \(d_{\infty}\), which takes the supremum on the Hausdorff distances between the corresponding α-cut sets. Also they proved the following common fixed point theorem for a family of fuzzy mappings.
Theorem 1.1
[12]
Let \((X,d)\) be a complete metric space and let \(\{F_{i}\}\) be a sequence of self-mappings of \(\mathcal{CB}(X)\). If there exists a constant \(q\in(0, 1)\) such that for each \(\mu_{1}, \mu_{2} \in \mathcal{CB}(X)\), and for arbitrary positive integers i and j, \(i \neq j\),
where
Then there exists a \(\mu_{*} \in\mathcal{CB}(X)\) such that \(\mu_{*} \subseteq F_{i}(\mu_{*})\) for all \(i \in\mathbb{N}\).
Later, by using the concept of \(d_{\infty}\) metric, Qiu et al. [15] proved the following common fixed point theorem, but under the assumption of a compact cut set \(\mathcal{C}(X)\) instead of a closed bounded cut set \(\mathcal{CB}(X)\).
Theorem 1.2
[15]
Let \((X,d)\) be a compact metric space and let \(\{F_{i}\}_{i = 1}^{\infty }\) be a sequence of self-mappings of \(\mathcal{C}(X)\). Let \(\Phi: [0, \infty)\rightarrow[0, \infty)\) be a non-decreasing function satisfying the following condition: Φ is continuous from the right and
where \(\Phi^{n}\) denotes the nth iterative function of Φ. Suppose that for arbitrary positive integers i and j, \(i \neq j\),
where \(M_{(i, j)}(\mu_{1}, \mu_{2})\) is defined as in Theorem 1.1. Then there exists a \(\mu_{*} \in\mathcal{C}(X)\) such that \(\mu_{*} \subseteq F_{i}(\mu_{*})\) for all \(i \in\mathbb{N}\).
Notice that Theorem 1.2 can be used to apply to a larger class of mappings than that of Theorem 1.1. However, after careful consideration, one may see that Theorem 1.2 is relevant when the considered space is a compact metric space instead of a complete metric space, which has been considered in Theorem 1.1.
Based on the above remarks, here we will present an extension of Theorem 1.1, but in a complete metric space setting. In fact, we will use the concept of a class of functions, so-called \(\mathcal {R}\)-functions, to show some fixed point theorems for self-mappings of \(\mathcal{CB}(X)\) with the supremum metric for fuzzy sets. Of course, our results improve and extend those results which have been presented in [12] and [15].
2 Preliminaries
In this section, we will provide some important basic concepts and useful results. Let \((X,d)\) be a metric space, and let \(CB(X)\) be the set of all nonempty bounded closed subsets of X. Recall that the Hausdorff metric is a function H on \(CB(X)\) defined by
where \(\rho(A,B) =\sup_{x \in A}d(x,B)\) is the Hausdorff separation of A from B.
2.1 Fuzzy sets and fuzzy mappings
Let \(I = [0, 1]\). A fuzzy set μ of a metric space X is defined by its membership function \(\mu(x)\), which is a mapping from X into I. We denote by \(\mathfrak{F}(X)\) the set of all fuzzy mappings on X, that is, \(\mathfrak{F}(X):=\{\mu| \mu: X\rightarrow[0, 1]\}\). For any \(\alpha\in(0, 1]\), the α-cut of the fuzzy set μ is defined by
where \(\alpha\in(0, 1]\), and we separately specify the support \([\mu]_{0}\) of μ to be the closure of the union of \([\mu]_{\alpha}\) for \(0 < \alpha\le1\). We denote by \(\mathcal{CB}(X)\) the totality of fuzzy sets \(\mu: X\rightarrow I\) for which, for each \(\alpha\in I\), the α-cut of μ is a nonempty closed bounded subset of X.
Let \(\mu_{1}, \mu_{2} \in\mathfrak{F}(X)\). Then \(\mu_{1}\) is said to be included in \(\mu_{2}\), denoted by \(\mu_{1} \subseteq\mu_{2}\), if and only if \(\mu_{1}(x) \leq\mu_{2}(x)\) for each \(x \in X\). Thus we have \(\mu_{1} \subseteq\mu_{2}\) if and only if \([\mu_{1}]^{\alpha} \subseteq[\mu_{2}]^{\alpha}\) for all \(\alpha\in I\). Let X, Y be any underling sets and \(\mathfrak{U}\), \(\mathfrak{V}\) are subsets of \(\mathfrak{F}(X)\) and \(\mathfrak{F}(Y)\), respectively. A mapping \(F:\mathfrak{U}\rightarrow\mathfrak{V}\) is said to be a fuzzy mapping, i.e., \(F(\mu) \in\mathfrak{V}\) for each \(\mu\in\mathfrak {U}\). An element \(\mu_{*} \in\mathfrak{U}\) is said to be a fixed point of a fuzzy self-mapping F on \(\mathfrak{U}\) if and only if \(\mu_{*} \subseteq F(\mu_{*})\).
The \(d_{\infty}\)-metric (called supremum or generalized Hausdorff metric) is a metric on \(\mathcal{CB}(X)\) which is defined as follows:
where \(\mu_{1}, \mu_{2} \in\mathcal{CB}(X)\), and
is the Hausdorff separation of \(\mu_{1}\) from \(\mu_{2}\). Notice that the supremum in (2.1) may be not attained, and so it cannot be replaced by a maximum. To clarify this, we include the following example, which can be found in [16].
Example 2.1
Let X be a set of real numbers and \(\mu, \nu\in\mathfrak{F}{(X)}\) be fuzzy subsets of X such that the corresponding level sets are
and
It follows that
Then \(d_{\infty}(\mu, \nu) =\sup_{0\leq\alpha\leq1}H([\mu ]^{\alpha},[\nu]^{\alpha}) = 1\), but this is not attained.
Note that if \(\{\mu_{n}\}\) be a sequence in \(\mathcal{CB}(X)\), then it follows from the definition of \(d_{\infty}\) that \(\{\mu_{n}\}\) converges with respect to the \(d_{\infty}\)-metric if and only if \([\mu_{n}]^{\alpha}\) converges uniformly in \(\alpha\in I\) with respect to the Hausdorff metric. Further, we know that the metric space \((\mathcal{CB}(X),d_{\infty})\) is complete provided \((X,d)\) is complete (see [12]). Here, we collect some useful properties of the \(d_{\infty}\)-metric, which will be used in order to obtain our results.
Lemma 2.2
[12]
Let \(\mu_{1}, \mu_{2}, \mu_{3} \in\mathcal{CB}(X)\). The following items are true:
-
(i)
\(\rho_{\infty}(\mu_{1},\mu_{2}) = 0 \) if and only if \(\mu _{1} \subseteq\mu_{2}\),
-
(ii)
if \(\mu_{1} \subseteq\mu_{2}\), then \(\rho_{\infty }(\mu _{1},\mu_{3}) \leq d_{\infty}(\mu_{2},\mu_{3})\),
-
(iii)
\(\rho_{\infty}(\mu_{1},\mu_{3}) \leq d_{\infty}(\mu _{1},\mu_{2}) + \rho_{\infty}(\mu_{2},\mu_{3})\).
Theorem 2.3
[12]
Let \((X,d)\) be a metric space and \(\mu_{1}, \mu_{2} \in\mathcal{CB}(X)\). Then for any \(\beta> 1\) and any \(\mu_{3} \in\mathcal{CB}(X)\) satisfying \(\mu_{3} \subseteq\mu_{1}\), there exists a \(\mu_{4} \in\mathcal {CB}(X)\) such that \(\mu_{4} \subseteq\mu_{2}\) and \(d_{\infty}(\mu_{3}, \mu_{4}) \leq\beta d_{\infty}(\mu_{1}, \mu_{2})\).
2.2 \(\mathcal{R}\)-Functions
In this subsection, we will recall an important tool related to our considered class of mappings. A function \(\varphi: [0,\infty) \rightarrow[0,1)\) is said to be an \(\mathcal{R}\)-function if
Note that if \(\varphi:[0,\infty)\rightarrow[0,1)\) is a non-decreasing function or a non-increasing function, then φ is an \(\mathcal{R}\)-function. This means the set of \(\mathcal{R}\)-functions is a rich class. In [17], Du proved some of the following characterizations for the class of \(\mathcal{R}\)-functions.
Theorem 2.4
[17]
Let \(\varphi: [0,\infty) \rightarrow[0,1)\) be a function. Then the following statements are equivalent.
-
(a)
φ is an \(\mathcal{R}\)-function.
-
(b)
For any nonincreasing sequence \(\{x_{n}\}_{n\in\mathbb{N}}\) in \([0,\infty)\), we have \(0 \leq\sup_{n\in\mathbb{N}}\varphi (x_{n}) < 1\).
3 Fixed point theorems for fuzzy mappings induced by \(\mathcal {R}\)-functions
Now, we are in a position to present our main results.
Theorem 3.1
Let \((X,d)\) be a complete metric space and let \(\{F_{i}\}_{i = 1}^{\infty}\) be a sequence of fuzzy self-mappings of \(\mathcal{CB}(X)\). Assume that there exists an \(\mathcal{R}\)-function \(\varphi: [0,\infty ) \rightarrow[0,1)\) such that for each \(\mu_{1}, \mu_{2} \in \mathcal {CB}(X)\), and for arbitrary positive integers i and j, \(i \neq j\),
where \(M_{(i, j)}(\mu_{1}, \mu_{2})\) is defined as in Theorem 1.1. Then there exists a \(\mu_{*} \in\mathcal{CB}(X)\) such that \(\mu_{*} \subseteq F_{i}(\mu_{*})\) for all positive integers i.
Proof
Let us define a function \(k : [0,\infty) \rightarrow[0,1)\) by
Note that we have \(0 \leq\varphi(t) < k(t) < 1\) for all \(t \in [0,\infty)\).
We will start by picking a fuzzy set \(\mu_{0} \in\mathcal{CB}(X)\). We subsequently choose \(\mu_{1} \subseteq F_{1}(\mu_{0})\) and a positive real number \(\varepsilon_{0}\) such that \(\varepsilon_{0} \in (\frac{1-k(d_{\infty}(\mu_{0},\mu _{1}))}{2},1-k(d_{\infty}(\mu_{0},\mu_{1})) )\). Next, by using this \(\varepsilon_{0}\), we can find a positive real number \(\beta_{0}\) such that \(\beta_{0}\in (1, \frac{1-\varepsilon _{0}}{k(d_{\infty }(\mu_{0},\mu_{1}))} )\). Now, by Theorem 2.3, there exists \(\mu_{2} \in\mathcal{CB}(X)\) such that \(\mu_{2} \subseteq F_{2}(\mu _{1})\) and
Next, let us to choose a positive real number \(\varepsilon_{1}\) such that \(\varepsilon_{1} \in (\frac{1-k(d_{\infty}(\mu_{1},\mu _{2}))}{2},1-k(d_{\infty}(\mu_{1},\mu_{2})) )\), and then pick a positive real number \(\beta_{1}\) such that \(\beta_{1}\in (1, \frac{1-\varepsilon_{1}}{k(d_{\infty}(\mu_{1},\mu_{2}))} )\). Similarly to the above, by Theorem 2.3, we can find \(\mu_{3} \in\mathcal{CB}(X)\) such that \(\mu_{3} \subseteq F_{3}(\mu_{2})\) and
By continuing this process, we obtain two sequences of positive real numbers \(\{\varepsilon_{n}\}\), \(\{\beta_{n}\}\) and a sequence \(\{\mu _{n}\}\) in \(\mathcal{CB}(X)\) such that
for each \(n\in\Bbb{N}\).
In order to complete the proof, we will divide it into three steps.
Step 1. We show that \(\sup_{n \in\mathbb {N}}\beta _{n}k(d_{\infty}(\mu_{n},\mu_{n+1})) < 1\).
Note that, by Lemma 2.2(iii), we have
for each \(n\in\Bbb{N}\). Subsequently, since \(\mu_{n} \subseteq F_{n}(\mu_{n-1})\), in view of Lemma 2.2(i) we obtain
for each \(n\in\Bbb{N}\).
Using this fact, we now derive
Using this one, in view of the inequalities (3.1) and (3.2), we obtain
Then, by inequality (3.3), we must have
Indeed, if (3.4) is not true, we would have
by inequality (3.3). This leads to a contradiction, since \(0 <\beta_{n}k(d_{\infty}(\mu _{n},\mu_{n+1})) < 1-\varepsilon_{n} < 1 \) for all \(n \in\mathbb{N}\). This shows that (3.4) holds. Subsequently, since \(\beta_{n}k(d_{\infty}(\mu_{n},\mu_{n+1})) \in (0,1)\), we have
This means the sequence \(\{d_{\infty}(\mu_{n},\mu_{n+1})\}\) is strictly decreasing in \([0,\infty)\). Thus, by applying Theorem 2.4, we know that
This implies
On the other hand, we observe that
Using this together with (3.6), we can conclude that
as required.
Step 2. We show that \(\{\mu_{n}\}\) is a Cauchy sequence in \(\mathcal{CB}(X)\).
According to step 1, let us put \(c:=\sup_{n \in\mathbb{N}}\beta _{n}k(d_{\infty}(\mu_{n},\mu_{n+1})) \in(0,1)\). Then, by (3.5), we have
for each \(n \in\mathbb{N}\). By using this relation, we deduce that
for each \(n \in\mathbb{N}\).
So, for arbitrary positive integers m and k, we see that
Since \(c\in(0, 1)\), we can conclude that \(\{\mu_{n}\}\) is a Cauchy sequence in \(\mathcal{CB}(X)\), as required.
Step 3. We show that there is \(\mu_{*}\in\mathcal {CB}(X)\) such that \(\mu_{*}\subset F_{i}(\mu_{*})\), for all \(i\in \Bbb{N}\).
Since \(\{\mu_{n}\}\) is a Cauchy sequence in a complete metric space \((\mathcal{CB}(X), d_{\infty})\), there is \(\mu_{*} \in\mathcal{CB}(X)\) such that \(\mu_{n} \rightarrow\mu_{*} \) as \(n \rightarrow\infty\). We now show that \(\mu_{*} \subseteq F_{i}(\mu_{*})\) for all \(i\in \mathbb{N}\).
Let \(i\in\mathbb{N}\) be arbitrary. Firstly, by Lemma 2.2(iii) and (iv), let us notice that
since \(\mu_{j}\subset F_{j}(\mu_{j-1}) \) for arbitrary natural numbers j such that \(i \neq j\).
Subsequently, by using (3.7) and Lemma 2.2(iii), we derive
We now consider the following two possible cases:
Case 1. If
then, by inequality (3.8), we have
This is equivalent to
Using, the previous argument together with inequality (3.7), we have
Note that, since \(d_{\infty}(\mu_{j},\mu_{*}) \rightarrow0\) as \(j \rightarrow\infty\), without loss of generality (passing to a subsequence if necessary), we may assume that \(\{d_{\infty}(\mu _{j},\mu _{*})\}_{j=1}^{\infty}\) is a nonincreasing sequence. Subsequently, by Theorem 2.4, we have
Subsequently, by inequality (3.9), we obtain
Letting \(j \rightarrow\infty\) in inequality (3.10), we have
This implies, by Lemma 2.2(i), that \(\mu_{*} \subseteq F_{i}(\mu_{*})\).
Case 2. Assume
In this situation, by inequality (3.8), we have
So, it follows by inequality (3.7) that
Letting \(j \rightarrow\infty\) in inequality (3.11), we obtain
which implies that \(\rho_{\infty}(\mu_{*},F_{i}(\mu_{*})) = 0\). Again, by Lemma 2.2(i), it follows that \(\mu_{*} \subseteq F_{i}(\mu _{*})\). Hence, by Cases 1 and 2, the proof is completed. □
Remark 3.2
Theorem 3.1 recovers Theorem 1.1 as a special case. Meanwhile, it improves Theorem 1.2 since we are considering a larger class of metric space settings.
Using Theorem 3.1, we also obtain the following result.
Corollary 3.3
Let \((X,d)\) be a complete metric space and let \(\{F_{i}\}_{i = 1}^{\infty}\) be a sequence of fuzzy self-mappings of \(\mathcal{CB}(X)\). If there exist nonnegative constants a, b, c, \(a+2b+2c < 1\), such that for each \(\mu_{1}, \mu_{2} \in\mathcal{CB}(X)\), and for arbitrary positive integers i and j, \(i\neq j\),
then there exists a \(\mu_{*} \in\mathcal{CB}(X)\) such that \(\mu_{*} \subseteq F_{i}(\mu_{*})\) for all \(i \in\Bbb{N}\).
Proof
Let us define an \(\mathcal{R}\) function \(\varphi: [0,\infty ) \rightarrow[0,1)\) by
Then one can derive that the relation (3.12) is transformed to (3.1), and so the required result follows immediately from Theorem 3.1. □
Next, motivated by the idea of Berinde [18], we now present another fuzzy fixed point theorem.
Theorem 3.4
Let \((X,d)\) be a complete metric space and let \(\{F_{i}\}_{i = 1}^{\infty}\) be a sequence of fuzzy self-mappings of \(\mathcal{CB}(X)\). If there exist an \(\mathcal{R}\)-function \(\varphi: [0,\infty) \rightarrow[0,1)\) and \(L \geq0\) such that for each \(\mu_{1}, \mu_{2} \in\mathcal{CB}(X)\), and for arbitrary positive integers i and j, \(i \neq j\),
then there exists a \(\mu_{*} \in\mathcal{CB}(X)\) such that \(\mu_{*} \subseteq F_{i}(\mu_{*})\) for all \(i \in\Bbb{N}\).
Proof
Let us construct and consider again a function k and the sequences which we have defined in (3.2). Recall that, by our constructive method, we also know that
for each \(n\in\Bbb{N}\). Using this together with (3.13), we derive
for each \(n\in\Bbb{N}\). This is equivalent to
for each \(n\in\Bbb{N}\). Observe that, since \(\beta_{n}k(d_{\infty}(\mu_{n},\mu_{n+1})) \in(0,1)\), we also have
for all \(n \in\mathbb{N}\). Thus, by (3.14), we see that \(\{ d_{\infty}(\mu_{n},\mu_{n+1})\}\) is a strictly decreasing sequence in \([0,\infty)\). Since φ is an \(\mathcal{R}\)-function, by applying Theorem 2.4, we have
Next, as we have done in order to prove Theorem 3.1, we write
Let us take \(c := \frac{\lambda}{2-\lambda}\). It follows that \(c \in (0,1)\) and, by (3.14), we also have
Now, again by reasoning along the lines of proving Theorem 3.1, we can show that there is a \(\mu^{*}\in\mathcal{CB}(X)\) which is the limit point of the considered sequence \(\{\mu_{n}\}\), under the supremum metric \(d_{\infty}\), and it satisfies
Finally, we will show that \(\mu_{*} \subseteq F_{i}(\mu_{*})\) for all \(i \in\Bbb{N}\). Let \(i\in\Bbb{N}\) be arbitrary. Note again that
for each \(j\in\Bbb{N}\) such that \(j\neq i\). Consider
where \(j\in\Bbb{N}\) with \(j\neq i\). Subsequently, from (3.17) together with (3.16), we obtain
By letting \(j \rightarrow\infty\) on the right of inequality (3.18), we have
which implies that \(\rho_{\infty}(\mu_{*},F_{i}(\mu_{*})) = 0\). Then, by Lemma 2.2(i), we have \(\mu_{*} \subseteq F_{i}(\mu_{*})\). Since \(i\in\Bbb{N}\) is arbitrary, we complete the proof. □
The following result can be deduced from Theorem 3.4.
Corollary 3.5
Let \((X,d)\) be a complete metric space and let \(\{F_{i}\}_{i = 1}^{\infty}\) be a sequence of fuzzy self-mappings of \(\mathcal{CB}(X)\). If there exist an \(\mathcal{R}\)-function \(\varphi: [0,\infty) \rightarrow[0,1)\) and \(L \geq0\) such that for each \(\mu_{1}, \mu_{2} \in\mathcal{CB}(X)\), and for arbitrary positive integers i and j, \(i \neq j\),
then there exists a \(\mu_{*} \in\mathcal{CB}(X)\) such that \(\mu_{*} \subseteq F_{i}(\mu_{*})\) for all \(i \in\Bbb{N}\).
Proof
Since for each \(a,b \in\mathbb{R}^{+}\), we have \(\sqrt{ab} \leq\frac{a+b}{2}\), the required result follows immediately from our Theorem 3.4. □
Remark 3.6
Corollary 3.5 recovers a result which has presented in [12], when \(L = 0\).
4 Conclusions
In this paper we have presented fuzzy fixed point theorems on the space of fuzzy sets under a kind of supremum metric setting. We would like to point out that this kind of space is very general and interesting. Moreover, as one can observe, a fixed point in this situation is, in fact, a fixed (fuzzy) set. This means that our presented results are very general and we recover many existing results on fixed point theory as regards both single-valued and set-valued mappings.
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Acknowledgements
This paper is partially supported by the Thailand Research Fund under the project RTA5780007. N Petrot is partially supported by Naresuan University Project R2558C098. W Saksirikun is supported by the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program (Grant No. PHD/0248/2553).
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Suantai, S., Petrot, N. & Saksirikun, W. Fuzzy fixed point theorems on the complete fuzzy spaces under supremum metric. Fixed Point Theory Appl 2015, 167 (2015). https://doi.org/10.1186/s13663-015-0418-y
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DOI: https://doi.org/10.1186/s13663-015-0418-y