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Some setvalued and multivalued contraction results in fuzzy cone metric spaces
Journal of Inequalities and Applications volume 2021, Article number: 110 (2021)
Abstract
This paper aims to present the concept of multivalued mappings in fuzzy cone metric spaces and prove some basic lemmas, a Hausdorff metric, and fixed point results for setvalued fuzzy conecontraction and for multivalued fuzzy conecontraction mappings. We prove a fixed point theorem for multivalued rational type fuzzy conecontractions in fuzzy cone metric spaces. Our results extend and improve some results given in the literature.
1 Introduction
Huang et al. [1] introduced the concept of cone metric spaces by using an ordered Banach space instead of a real number set and proved some fixed point results under cone contraction conditions. After the publication of this article, a number of researchers contributed their ideas to the problems on cone metric spaces by using different contractive type mappings and spaces (see, e.g., [2–11] and the references therein).
Kramosil et al. [12] introduced a fuzzy metric space (FMspace) by using the notion of a fuzzy set and some more notions derived from the one in ordered. These researchers have compared the fuzzy metric notion with the statistical metric space and proved that both conceptions are equivalent in some cases. Later on, the modified form of the metric fuzziness was given by George et al. in [13] by using the continuous tnorm. After that, a number of authors have studied and contributed their ideas to the problems on FMspaces. Some of their results can be found in [14–25] and the references therein.
Lopez et al. [26] introduced the Hausdorff fuzzy metric on a compact set for a given FMspace and proved some properties for a Hausdorff fuzzy metric. Kiany et al. [19] proved some fixed point results for setvalued mappings and an endpoint theorem in FMspaces by using contraction conditions. Some other properties and fixed theorems on multivalued mappings in FMspaces can be found in [27–29].
The concept of a fuzzy cone metric space (FCMspace) was given by Oner et al. in [30]. They established some properties and a fuzzy cone Banach principle theorem. Some more topological properties, fixed point theorems, and common fixed point theorems in FCMspaces can be found in [31–37].
In this paper, we introduce the concept of multivalued mappings in FCMspaces and prove some basic lemmas and a Hausdorff metric in FCMspaces. Our result extends and improves the result of Kiany et al. [19] and presents a setvalued fuzzy cone contraction theorem in FCMspaces. Moreover, we present some fixed point results via multivalued fuzzy cone contractions in FCMspaces by extending and improving the result of Ali et al. [27] and a rational type multivalued fuzzy cone contraction theorem.
2 Preliminaries
Definition 2.1
([38])
A binary operation \(\ast:[0,1]^{2}\to [0,1]\) is called a continuous tnorm if:

(i)
∗ is associative, commutative, and continuous;

(ii)
\(\forall a_{0},a_{1},b_{0},b_{1}\in [0,1]\), then \(1\ast a_{0}=a_{0}\), while \(a_{0}\ast a_{1}\le b_{0}\ast b_{1}\), whenever \(a_{0}\le b_{0}\) and \(a_{1}\le b_{1}\).
The basic continuous tnorms are minimum, the product and the Lukasiewicz tnorms are defined, respectively, as follows (see [38]):
Throughout this paper, a set of natural numbers is denoted by \(\mathbb{N}\) and a real Banach space is denoted by \(\mathbb{E}\). θ represents the zero element of \(\mathbb{E}\).
Definition 2.2
([1])
A subset \(P\subset \mathbb{E}\) is known as a cone if

(i)
\(P\neq \emptyset \), closed, and \(P\ne \{\theta \}\);

(ii)
If \(a_{0}, b_{0}\geq 0\) and \(\mu, \nu \in P\), then \(a_{0} \mu + b_{0} \nu \in P\);

(iii)
If both \(\mu,\mu \in P\), then \(\mu =\theta \).
A partial ordering “⪯” on \(P\subset \mathbb{E}\) is defined by \(\mu \preceq \nu \) if and only if \(\nu  \mu \in P\). \(\mu \prec \nu \) stands for \(\mu \preceq \nu \) and \(\mu \ne \nu \), while \(\mu \ll \nu \) stands for \(\nu  \mu \in \operatorname{int}(P)\) and all cones have nonempty interior.
Definition 2.3
([30])
A 3tuple \((U,F_{m},\ast )\) is known as an FCMspace if \(P\subset \mathbb{E}\) is a cone, U is an arbitrary set, ∗ is a continuous tnorm, and \(F_{m}\) is a fuzzy set on \(U\times U\times \operatorname{int}(P)\) satisfying the following;

(i)
\(F_{m}(\mu, \nu, t)>0\), and \(F_{m}(\mu,\nu,t)=1\) if and only if \(\mu =\nu \);

(ii)
\(F_{m}(\mu,\nu,t)=F_{m}(\nu,\mu,t)\);

(iii)
\(F_{m}(\mu,\omega,t)\ast F_{m}(\omega,\nu,s)\le F_{m}(\mu,\nu,t+s)\);

(iv)
\(F_{m}(\mu,\nu,.):\operatorname{int}(P)\to [0,1]\) is continuous
for all \(\mu,\nu,\omega \in U\) and \(t,s\in \operatorname{int}(P)\).
Definition 2.4
([30])
Let \((U,F_{m},\ast )\) be an FCMspace, \(\mu \in U\), and \((\mu _{n})\) be a sequence in U. Then

(i)
\((\mu _{n})\) is said to converge to μ if, for \(t\gg \theta \) and \(0< r<1\), there exists \(n_{1}\in \mathbb{N}\) such that
$$\begin{aligned} F_{m}(\mu _{n},\mu,t)>1r,\quad \forall n\ge n_{1}. \end{aligned}$$This can be written as \(\lim_{n\to \infty }\mu _{n}=\mu \) or \(\mu _{n}\to \mu \), as \(n\to \infty \).

(ii)
\((\mu _{n})\) is said to be a Cauchy sequence if, for \(t\gg \theta \) and \(0< r<1\), there exists \(n_{1}\in \mathbb{N}\) such that
$$\begin{aligned} F_{m}(\mu _{m},\mu _{n},t)>1r,\quad \forall m,n\ge n_{1}. \end{aligned}$$ 
(iii)
\((U,F_{m},*)\) is complete if every Cauchy sequence is convergent in U.
Lemma 2.5
([30])
Let \((U,F_{m},\ast )\) be an FCMspace. The following statements hold:

(1)
Let \(\mu \in U\) and \((\mu _{n})\) be a sequence in U. Then \(\mu _{n}\to \mu \) if and only if \(\lim_{n\to \infty }F_{m}(\mu _{n},\mu,t)=1\) for \(t\gg \theta \).

(2)
An open ball \(B (\mu _{0}, r, t)\) with center \(\mu _{0}\) and radius \(0< r <1\) can be defined as follows for \(t\gg \theta \):
$$\begin{aligned} B (\mu _{0}, r, t) = \bigl\{ \mu \in U: F_{m} (\mu _{0}, \mu, t) > 1 r\bigr\} . \end{aligned}$$Let
$$\begin{aligned} T_{fc} = \bigl\{ A \subset U: \mu _{0} \in A \textit{ iff } \exists 0< r< 1 \textit{ and } t \gg \theta \textit{ such that } B (\mu _{0}, r, t) \subset A\bigr\} . \end{aligned}$$Then \(T_{fc}\) is a topology on U.
We recall the following definitions given in [27].
Definition 2.6
Let \((U,F_{m},\ast )\) be an FCMspace;

(i)
A function \(g:U\to \mathbb{R}\) is said to be lower semicontinuous if, for any \((\mu _{i})\subset U\) and \(\mu \in U\), \(\mu _{i}\to \mu \) implies \(g(\mu )\leq \limsup_{i\to \infty }g(\mu _{i})\).

(ii)
A function \(g:U\to \mathbb{R}\) is said to be upper semicontinuous if, for any \((\mu _{i})\subset U\) and \(\mu \in U\), \(\mu _{i}\to \mu \) implies \(g(\mu )\geq \limsup_{i\to \infty }g(\mu _{i})\).

(iii)
A multivalued mapping \(G:U\to 2^{U}\) (\(2^{U}\) is the collection of all nonempty subsets of a set U) is called upper semicontinuous if, for any \(\mu \in U\) and a neighborhood B of \(G(\mu )\), there is a neighborhood A of μ such that, for any \(\nu \in A\), we have \(G(\nu )\subset B\).

(iv)
A multivalued mapping \(G:U\to 2^{U}\) is said to be lower semicontinuous if, for any \(\mu \in U\) and a neighborhood B, \(G(\mu )\cap B\neq \emptyset \), there is a neighborhood A of μ such that, for any \(\nu \in A\), we have \(G(\nu )\cap B\neq \emptyset \).
Definition 2.7
Assume that \((U,F_{m},\ast )\) is an FCMspace, \(\mu \in U\), and \((\mu _{i})_{i\in \mathbb{N}}\) is a sequence in U. Then:

(i)
a subset \(A\subseteq U\) is closed if, for every convergent sequence \((\mu _{i})\) in A such that \(\mu _{i}\to \mu \), we have \(\mu \in A\).

(ii)
a subset \(A\subseteq U\) is compact if every sequence in A has a convergent subsequence in A.
Throughout this paper, \(\mathbb{K}(U)\) represents the set of all compact subsets of a set U and \(\mathbb{P}(U)\) represents the set of all nonempty subsets of a set U.
3 Some properties and a Hausdorff fuzzy metric in FCMspaces
Proposition 3.1
Let \((U,F_{m},\ast )\) be an FCMspace. Then \(F_{m}\) is continuous on \(U^{2}\times \operatorname{int}(P)\) for every \(t\gg \theta \) (i.e., \(t\in \operatorname{int}(P)\)).
Proof
Let \(\mu,\nu \in U\), \(t\gg \theta \), and \((\mu _{i},\nu _{i},t_{i})_{i}\) be a sequence in \(X^{2}\times \operatorname{int}(P)\) converging to \((\mu,\nu,t)\). Since \((F_{m}(\mu _{i},\nu _{i},t_{i}))_{i}\) is a sequence in \((0,1]\), there is a subsequence \((\mu _{i_{n}},\nu _{i_{n}},t_{i_{n}})_{n}\) of the sequence \((\mu _{i},\nu _{i},t_{i})_{i}\) such that \((F_{m}(\mu _{i_{n}},\nu _{i_{n}},t_{i_{n}}))_{n}\) converges to a point in \([0,1]\). Fix any \(\varepsilon >0\) such that \(\varepsilon <\frac{t}{2}\), so there is \(i_{0}\in \mathbb{N}\) such that \(tt_{i}<\varepsilon \) for all \(i\ge i_{0}\). Then we have
and
Therefore, by the continuity of the function \(t\longmapsto F_{m}(\mu,\nu,t)\), we can deduce that
Thus, \(F_{m}\) is continuous on \(U^{2}\times \operatorname{int}(P)\). □
Lemma 3.2
Let \((U,F_{m},\ast )\) be an FCMspace such that
for all \(\mu,\nu \in U\), \(t\gg \theta \), and \(b>1\). Let \((\mu _{i})\) be a sequence in U such that
for all \(i\in \mathbb{N}\) and \(a\in (0,1)\). Then \((\mu _{i})\) is a Cauchy sequence in U.
Proof
For every \(i\in \mathbb{N}\) and \(t\gg \theta \), we have that
Thus, for all \(i\in \mathbb{N}\) and \(t\gg \theta \), we have
Now, we choose a constant \(b>1\) and \(l\in \mathbb{N}\) such that \(ab<1\) and \(\sum_{j=l}^{\infty }\frac{1}{b^{j}}=\frac{1/b^{l}}{1(1/b)}<1\). Hence, for \(k\geq i\) and \(t\gg \theta \), we have that
This proves that \((\mu _{i})\) is a Cauchy sequence in U. □
Lemma 3.3
Let \((U,F_{m},\ast )\) be an FCMspace. Then, for every \(\mu \in U\), \(A\in \mathbb{K}(U)\) and \(t\gg \theta \), there exists \(a_{0}\in A\) such that
Proof
Let \(\mu \in U\), \(A\in \mathbb{K}(U)\), and \(t\gg \theta \). Then, by Proposition 3.1, the function \(\nu \longmapsto F_{m}(\mu,\nu,t)\) is continuous. Thus, by the compactness of A, there exists \(a_{0}\in A\) such that
that is,
□
Lemma 3.4
Let \((U,F_{m},\ast )\) be an FCMspace. Then, for all \(\mu \in U\) and \(A\in \mathbb{K}(U)\), the function \(t\longmapsto F_{m}(\mu,A,t)\) is continuous on \(\operatorname{int}(P)\), where \(t\gg \theta \).
Proof
Since \(F_{m}(\mu,A,t)=\sup_{a_{0}\in A}F_{m}(\mu,a_{0},t)\) and for every \(a_{0}\in A\), the function \(t\longmapsto F_{m}(\mu,a_{0},t)\) is continuous on \(\operatorname{int}(P)\), it follows that \(t\longmapsto F_{m}(\mu,A,t)\) is lower semicontinuous on \(\operatorname{int}(P)\). Now, we prove that \(t\longmapsto F_{m}(\mu,A,t)\) is upper semicontinuous on \(\operatorname{int}(P)\).
Let \(t\gg \theta \) and \((t_{j})_{j}\) be a sequence in \(\operatorname{int}(P)\) which converges to t. By Lemma 3.3, there exists \(a_{j}\in A\) such that, for all \(j\in \mathbb{N}\),
Since \(A\in \mathbb{K}(U)\), there are a subsequence \((a_{j_{n}})_{n}\) of the sequence \((a_{j})_{j}\) and a point \(a^{*}\in A\) such that \(a_{j_{n}}\to a^{*}\) in \((U,F_{m},\ast )\). Hence,
for \(t\gg \theta \). Now, by Proposition 3.1, we have that
for \(t\gg \theta \). Consequently, the function \(t\longmapsto F_{m}(\mu,A,t)\) is upper semicontinuous on \(\operatorname{int}(P)\), which concludes the required proof. □
Lemma 3.5
Let \((U,F_{m},\ast )\) be an FCMspace. Then, for every \(A\in \mathbb{K}(U)\) and \(B\in \mathbb{P}(U)\), there exists \(a^{*}\in A\) such that
for \(t\gg \theta \).
Proof
By putting \(\beta =\inf_{a_{0}\in A}F_{m}(a_{0},B,t)\), there is a sequence \((a_{j})_{j}\) in A such that \(\beta +\frac{1}{j}>F_{m}(a_{j},B,t)\) for all \(j\in \mathbb{N}\). Since \(A\in \mathbb{K}(U)\), there are a subsequence \((a_{j_{n}})_{n}\) of \((a_{j})_{j}\) and a point \(a^{*}\in A\) such that \(a_{j_{n}}\to a^{*}\) in \((U,F_{m},\ast )\). Here, we choose an arbitrary point \(b_{0}\in B\). Now, by Proposition 3.1, we have
for \(t\gg \theta \). Since for all \(n\in \mathbb{N}\) and \(\beta +\frac{1}{j_{n}}>F_{m}(a_{j_{n}},b_{0},t)\). Then, by taking the limit \(n\to \infty \), we get
□
Proposition 3.6
Let \((U,F_{m},\ast )\) be an FCMspace. Then, for every \(A,B\in \mathbb{K}(U)\), \(t\longmapsto \inf_{a^{*}\in A}F_{m}(a^{*},B,t)\) is a continuous function in \(\operatorname{int}(P)\), where \(t\gg \theta \).
Proof
By Lemma 3.4, \(t\longmapsto F_{m}(a^{*},B,t)\) is a continuous function in \(\operatorname{int}(P)\). Therefore, \(t\longmapsto \inf_{a^{*}\in A}F_{m}(a^{*},B,t)\) is an upper semicontinuous function in \(\operatorname{int}(P)\).
Now, we prove that \(t\longmapsto \inf_{a^{*}\in A}F_{m}(a^{*},B,t)\) is lower semicontinuous in \(\operatorname{int}(P)\). Let \((t_{j})_{j}\) be any sequence in \(\operatorname{int}(P)\) such that \((t_{j})_{j}\to t\) in \(\operatorname{int}(P)\), where \(t\gg \theta \). By Lemma 3.5, there exists \(a_{j}\in A\) such that, for all \(j\in \mathbb{N}\),
Since \(A\in \mathbb{K}(U)\), there are a subsequence \((a_{j_{n}})_{n}\) of \((a_{j})_{j}\) and a point \(a_{1}\in A\) such that \(a_{j_{n}}\to a_{1}\) in \((U,F_{m},\ast )\). Then, by Lemma 3.3, there exists \(b_{1}\in B\) such that
Now, by Proposition 3.1,
Therefore, for given \(\delta >0\), there exists \(n_{0}\in \mathbb{N}\) such that, for all \(n\geq n_{0}\),
Hence,
for all \(n\geq n_{0}\). Consequently, \(t\longmapsto \inf_{a^{*}\in A}F_{m}(a^{*},B,t)\) is a lower semicontinuous function in \(\operatorname{int}(P)\). It completes the proof. □
Remark 3.7
Note that Proposition 3.6 showed that, for any \(A,B\in \mathbb{K}(U)\), \(t\longmapsto \inf_{b^{*}\in B}F_{m}(A,b^{*},t)\) is a continuous function in \(\operatorname{int}(P)\).
Hausdorff fuzzy cone metric on \(\mathbb{K}(U)\): Let \((U,F_{m},\ast )\) be an FCMspace. Then we define a function \(F_{H}\) on \(\mathbb{K}(U)\times \mathbb{K}(U)\times \operatorname{int}(P)\) by
for all \(A,B\in \mathbb{K}(U)\) and \(t\gg \theta \).
Lemma 3.8
Let \((U,F_{m},\ast )\) be an FCMspace, \(\mu \in U\), \(A\in \mathbb{K}(U)\), \(B\in \mathbb{P}(U)\), and \(s,t\gg \theta \). Then
where \(a_{\mu }\in A\) satisfies \(F_{m}(\mu,A,t)=F_{m}(\mu,a_{\mu },t)\).
Proof
First, we note that an element \(a_{\mu }\in A\) satisfying \(F_{m}(\mu,A,t)=F_{m}(\mu,a_{\mu },t)\) exists by Lemma 3.3. Now, for every \(b\in B\), we have that
Thus, by the continuity of ∗,
□
Theorem 3.9
Assume that \((U,F_{m},\ast )\) is an FCMspace. Then \((\mathbb{K}(U),F_{H},\ast )\) is an FCMspace.
Proof
Suppose that \(A,B,C\in \mathbb{K}(U)\) and \(s,t\gg \theta \). Then, by Lemma 3.5, there exist \(a^{*}\in A\) and \(b^{*}\in B\) such that
and
for \(t\gg \theta \). Thus, \(F_{H}(A,B,t)>0\).
In addition, we know that \(F_{H}(A,B,t)=1\) if and only if \(A=B\), and hence \(F_{H}\) is symmetric, that is,
Moreover, we note that, by Lemma 3.8 and by the continuity of ∗, we have that
for \(s,t\gg \theta \). Since \(\{b_{a_{0}}: a_{0}\in A\}\subseteq B\) such that
for \(s\gg \theta \), we have
for \(s,t\gg \theta \). Similarly, we get that
It follows that
Finally, the continuity of the function \(t\longmapsto F_{H}(A,B,t)\) on the cone is a direct consequence of Proposition 3.6 and Remark 3.7. We conclude that \((\mathbb{K}(U),F_{H},\ast )\) is an FCMspace. □
4 Setvalued mapping results in FCMspaces
In this section, we prove a fixed point theorem for setvalued mappings in FCMspaces.
Theorem 4.1
Let \((U,F_{m},\ast )\) be a complete FCMspace and \(G: U\to U\) be a setvalued mapping with nonempty compact values such that, for all \(\mu,\nu \in U\) and \(t\gg \theta \),
where \(\delta:\operatorname{int}(P)\to [0,1)\) satisfies
and \(d(\mu,\nu,t)=\frac{t}{F_{m}(\mu,\nu,t)}t\). Moreover, we suppose that \((U,F_{m},\ast )\) satisfies (3.1) for some \(\mu _{0}\in U\) and \(\mu _{1}\in G\mu _{0}\). Then G has a fixed point in U.
Proof
First, we notice that, if A and B are nonempty compact subsets of a set U and \(a\in A\), then by Lemma 3.3, there exists \(b\in B\) such that
for \(t\gg \theta \). Thus, given \(\delta \leq F_{H}(A,B,t)\), there exists a point \(b\in B\) such that \(\delta \leq F_{m}(a,b,t)\).
Now, let us fix \(\mu _{0}\) in U and \(\mu _{1}\in G\mu _{0}\). If \(G\mu _{0}=G\mu _{1}\), then \(\mu _{1}\in G\mu _{1}\) and \(\mu _{1}\) is a fixed point of G. The proof is completed. Otherwise, we may assume that \(G\mu _{0}\neq G\mu _{1}\). Then, from (4.1), we have
for \(t\gg \theta \). Since \(\mu _{1}\in G\mu _{0}\) and G is a compactvalued mapping, then again by Lemma 3.3, there exists \(\mu _{2}\in G\mu _{1}\) such that
for \(t\gg \theta \). Similarly,
By induction, we choose a sequence \((\mu _{n})_{n\geq 0}\) in U such that \(\mu _{n}\in G\mu _{n1}\). If \(G\mu _{n1}=G\mu _{n}\) for some n, then \(\mu _{n}\in G\mu _{n}\), and so \(\mu _{n}\) is a fixed point of G. The proof is completed. Otherwise, we may assume that \(G\mu _{n1}\neq G\mu _{n}\). Then from (4.1) we have
Hence, \((F_{m}(\mu _{n},\mu _{n+1},t))_{n}\) is a nondecreasing sequence. Thus, \((d(\mu _{n},\mu _{n+1},t))_{n}\) is a positive nonincreasing sequence, and so it is convergent to some constant, say \(\xi \geq 0\). Recall that
Then there are \(\beta <1\) and \(n_{0}\in \mathbb{N}\) such that
Since \(F_{m}(\mu,\nu,.)\) is nondecreasing, we have from (4.1) and (4.3) that, for \(t\gg \theta \),
Thus, we get that
Hence, by Lemma 3.2, we conclude that \((\mu _{n})\) is a Cauchy sequence in U. Since \((U,F_{m},\ast )\) is complete, there exists \(u\in U\) such that
This implies that
Therefore,
Then there exists \(\beta <\xi <1\) such that
Now, we have to show that \(u\in Gu\). Since \(\mu _{n+1}\in G\mu _{n}\), one writes
for \(t\gg \theta \). Hence, we get that
Thus, there exists a sequence \((r_{n})\) in Gu such that
Now, by Definition 2.3(iii), we have that
for each \(n\in \mathbb{N}\). By using (4.4), (4.5) together with (4.6), we can get
This implies that \(\lim_{n\to \infty }r_{n}=u\). Since \(r_{n}\to u\) and \(r_{n}\in Gu\), using the fact that Gu is closed and compact, we get \(u\in Gu\). □
Without δ mapping directly, we can get the following two corollaries from Theorem 4.1.
Corollary 4.2
Let \((U,F_{m},\ast )\) be a complete FCMspace and \(G: U\to U\) be a setvalued mapping with nonempty compact values such that, for all \(\mu,\nu \in U\) and \(t\gg \theta \), it satisfies
where \(\beta \in (0,1)\). Furthermore, we assume that \((U,F_{m},\ast )\) satisfies (3.1) for some \(\mu _{0}\in U\) and \(\mu _{1}\in G\mu _{0}\). Then G has a fixed point in U.
Corollary 4.3
Let \((U,F_{m},\ast )\) be a complete FCMspace and \(G: U\to U\) be a setvalued mapping with nonempty compact values such that, for all \(\mu,\nu \in U\) and \(t\gg \theta \), it satisfies
where \(\beta \in (0,1)\). Furthermore, we assume that \((U,F_{m},\ast )\) satisfies (3.1) for some \(\mu _{0}\in U\) and \(\mu _{1}\in G\mu _{0}\). Then G has a fixed point in U.
5 Multivalued contraction results in FCMspaces
In this section, we present some fixed point results for multivalued contractions in FCMspaces. Further, we present a fixed point theorem for rational type multivalued contractions. We present some illustrative examples.
Let \(G: U\to 2^{U}\) be a multivalued map. Consider \(g(\mu )=F_{m}(\mu,G\mu,t)\) for \(t\gg \theta \). For \(\alpha \in (0,1)\), we take the set
Theorem 5.1
Let \((U,F_{m},\ast )\) be a complete FCMspace and \(G: U\to \mathbb{K}(U)\) be a multivalued map. If there exists a constant \(\beta \in (0,1)\) such that, for any \(\mu \in U\), there is \(\nu \in j_{\alpha }^{\mu }\), so that
for \(t\gg \theta \). Suppose that \((U,M,\ast )\) verifies (3.1) for some \(\mu _{0}\in U\). Then G has a fixed point in U, provided \(\beta <\alpha \) and g is upper semicontinuous.
Proof
Since \(G(\mu )\in \mathbb{K}(U)\), by Lemma 3.3, \(J_{\alpha }^{\mu }\) is nonempty for all μ in U and \(\alpha \in (0,1)\). Let us fix \(\mu _{0}\) in U, so there exists \(\mu _{1}\in J_{\alpha }^{\mu _{0}}\), that is, \(\mu _{1}\in G\mu _{0}\) such that
for \(t\gg \theta \). Similarly, for \(\mu _{1}\) in U, there exists \(\mu _{2}\in J_{\alpha }^{\mu _{1}}\), that is, \(\mu _{2}\in G\mu _{1}\), which satisfies
for \(t\gg \theta \). By induction, we obtain a sequence \((\mu _{i})_{i\geq 0}\) in U such that there exists \(\mu _{i+1}\in J_{\alpha }^{\mu _{i}}\), that is, \(\mu _{n+1}\in G\mu _{i}\), which satisfies
for \(t\gg \theta \). On the other hand, \(\mu _{i+1}\in J_{\alpha }^{\mu _{i}}\), which gives that
From (5.3) and (5.4), we get that
i.e.,
Let \(a=\frac{\beta }{\alpha }\), then (5.5) can be expressed as follows:
for \(t\gg \theta \), \(i\in \mathbb{N}\), and \(a\in (0,1)\). Choose a constant \(b>1\) such that \(ab<1\) and \(\sum_{n=0}^{\infty }\frac{1}{b^{n}}<1\), i.e., \(\sum_{n=i}^{j1}\frac{1}{b^{n}}<1\). Then, for all \(j>i\), we get that
where \(i,j\in \mathbb{N}\). Then we have
for \(t\gg \theta \). By Lemma 3.2, it is proved that \((\mu _{i})\) is a Cauchy sequence in U. Since U is complete, there exists \(\mu \in U\) such that \(\mu _{i}\to \mu \) as \(i\to \infty \). In view of (5.3) and (5.4), it is clear that \(g(\mu _{i})=F_{m}(\mu _{i},G\mu _{i},t)\) is an increasing function and converges to 1. Since g is upper semicontinuous, we have
This implies that \(g(\mu )=1\), so that \(F_{m}(\mu,G\mu,t)=1\). Hence, by using Lemma 3.3, we get that \(\mu \in G\mu \). Hence, the proof is completed. □
Corollary 5.2
Let \((U,F_{m},\ast )\) be a complete FCMspace and \(G: U\to \mathbb{K}(U)\) be a multivalued mapping. If there exists a constant \(\beta \in (0,1)\) such that, for any \(\mu \in U\), we have \(\nu \in j_{\alpha }^{\mu }\) with
for \(t\gg \theta \). Suppose that \((U,F_{m},\ast )\) satisfies (3.1) for some \(\mu _{0}\in U\). Then G has a fixed point in U provided that \(\beta <\alpha \) and g is upper semicontinuous.
In a special case, we get the following corollary of Kiany et al. [19].
Corollary 5.3
([19])
Let \((U,F_{m},\ast )\) be a complete FCMspace and \(G: U\to \mathbb{K}(U)\) be a multivalued map. Suppose that there exists \(\beta \in (0,1)\) such that
for all \(\mu,\nu \in U\) and \(t\gg \theta \). Moreover, assume that \((U,F_{m},\ast )\) satisfies (3.1) for some \(\mu _{0}\in U\) and \(\mu _{1}\in G\mu _{0}\). Then G has a fixed point in U.
Remark 5.4
Corollary 5.2 is the generalized form of Corollary 5.3. Suppose that G satisfies the conditions of Corollary 5.3, and if g is upper semicontinuous, then from (5.10) we obtain, for any \(\mu \in U\), \(\nu \in G\mu \), and \(t\gg \theta \),
for \(t\gg \theta \). Hence, G verifies the conditions of Corollary 5.2 and the existence of a fixed point has been proved.
In the following (Example 5.5), we show that Corollary 5.2 is the generalized form of Corollary 5.3.
Example 5.5
Let \(U= \{ \frac{1}{3},\frac{1}{9},\ldots,\frac{1}{3^{i}},\ldots \} \cup \{0,1\}\) and the fuzzy metric \(F_{m}: U^{2}\times (0,\infty )\to [0,1]\) be defined as
Then \((U,F_{m},\ast )\) is a complete FCMspace, where \(\ast:[0,1]^{2}\to [0,1]\) is defined as \(a\ast b=ab\).
Let the multivalued mapping \(G:U\to \mathbb{K}(U)\) be defined as
Since
this shows that G satisfies (3.1). Moreover,
and
There does not exist any \(\beta \in (0,1)\) such that Corollary 5.3 is satisfied. If it exists, then we get
This implies that \(\beta \geq 3^{i1}\), which is a contradiction. On the other hand,
is continuous and so there exists \(\nu \in J_{\frac{3}{4}}^{\mu }\) for any μ such that
Hence, there exists \(\beta =\frac{2}{3}<\frac{3}{4}\) such that
Then, by Corollary 5.2, we can get the existence of a fixed point of G in U.
Now, we will deal with rational type multivalued contractions in FCMspaces. For this, let \(G: U\to 2^{U}\) be a multivalued map. Define \(g(\mu )=F_{m}(\mu,G\mu,t)\) for \(t\gg \theta \). For \(\alpha \in (0,1)\), define the set
Theorem 5.6
Let \((U,F_{m},\ast )\) be a complete FCMspace and \(G: U\to \mathbb{K}(U)\) be a multivalued map. If there exists a constant \(\beta \in (0,1)\) such that, for any \(\mu \in U\), there is \(\nu \in j_{\alpha }^{\mu }\) so that
for \(t\gg \theta \). Suppose that \((U,F_{m},\ast )\) satisfies (3.1) for some \(\mu _{0}\in U\). Then G has a fixed point in U provided that \(\beta <\alpha \) and g is upper semicontinuous.
Proof
Since \(G(\mu )\in \mathbb{K}(U)\), by Lemma 3.3, we have that \(J_{\alpha }^{\mu }\) is nonempty for any μ in U and \(\alpha \in (0,1)\). Let us fix \(\mu _{0}\in U\), so there exists \(\mu _{1}\in J_{\alpha }^{\mu _{0}}\). Then, by (5.12), for \(t\gg \theta \),
by using Definition 2.3(iii) and \(F_{m}(\mu _{0},\mu _{2},2t)\geq F_{m}(\mu _{0},\mu _{1},t)\ast F_{m}( \mu _{1},\mu _{2},t)\) for \(t\gg \theta \). After simplification, we get that
Again for \(\mu _{1}\in U\), there exists \(\mu _{2}\in J_{\alpha }^{\mu _{1}}\). In view of (5.12),
Again by Definition 2.3(iii), \(F_{m}(\mu _{1},\mu _{3},2t)\geq F_{m}(\mu _{1},\mu _{2},t)\ast F_{m}( \mu _{2},\mu _{3},t)\) for \(t\gg \theta \). After simplification, we get that
Similarly, by induction, we obtain a sequence \((\mu _{i})_{i\geq 0}\) in U such that there exists \(\mu _{i+1}\in J_{\alpha }^{\mu _{i}}\), then by (5.12)
On the other hand, by (5.11) and \(\mu _{i+1}\in J_{\alpha }^{\mu _{i}}\),
From (5.13) and (5.14), we can obtain
for \(t\gg \theta \), that is,
for \(t\gg \theta \). Let \(a=\frac{\beta }{\alpha }\), then (5.15) can be expressed as follows:
for all \(i\in \mathbb{N}\) and \(a\in (0,1)\). Choose a constant \(b>1\) such that \(ab<1\) and \(\sum_{n=0}^{\infty }\frac{1}{b^{n}}<1\), i.e., \(\sum_{n=i}^{j1}\frac{1}{b^{n}}<1\). Then, for all \(j>i\), we get
where \(i,j\in \mathbb{N}\). Then we have
for \(t\gg \theta \). By Lemma 3.2, we have that
It is proved that \((\mu _{i})\) is a Cauchy sequence in U. Since U is complete, there exists \(\mu \in U\) such that \(\mu _{i}\to \mu \) as \(i\to \infty \). In view of (5.13) and (5.14), it is clear that \(g(\mu _{i})=F_{m}(\mu _{i},G\mu _{i},t)\) is an increasing function and converges to 1. Since g is upper semicontinuous, we have
This implies that \(g(\mu )=1\), so that \(F_{m}(\mu,G\mu,t)=1\). Hence, by using Lemma 3.3, we get that \(\mu \in G\mu \). □
Directly from Theorem 5.6, we get the following corollary.
Corollary 5.7
Let \((U,F_{m},\ast )\) be a complete FCMspace and \(G: U\to \mathbb{K}(U)\) be a multivalued map. If there exists a constant \(\beta \in (0,1)\) such that
for all \(\mu,\nu \in U\) and \(t\gg \theta \). Moreover, assume that \((U,F_{m},\ast )\) satisfies (3.1) for some \(\mu _{0}\in U\) and \(\mu _{1}\in G\mu _{0}\). Then G has a fixed point in U.
Example 5.8
Let \(U=\{0.4, 0.4^{2},\ldots, 0.4^{i}, \ldots \}\cup \{0,1\}\). Let \(G:U\to \mathbb{K}(U)\) be defined as
Since
which shows that G satisfies (3.1). By a direct calculation as discussed in Example 5.5, we get \(\beta \geq 0.4^{i1}\) which is a contradiction to the fact that \(\beta \geq 0.4^{i1}\to 0\), as \(i\to \infty \), where \(\beta \in (0,1)\). On the other hand, we define
It is continuous and so there exists \(\nu \in J_{0.3}^{\mu }\) for any μ such that
That is,
Hence, there exists \(\beta =0.5< 0.8\), and from (5.20) we get, for \(t\gg \theta \),
Then the existence of a fixed point follows from Corollary 5.7.
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Rehman, S.U., Aydi, H., Chen, GX. et al. Some setvalued and multivalued contraction results in fuzzy cone metric spaces. J Inequal Appl 2021, 110 (2021). https://doi.org/10.1186/s13660021026463
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DOI: https://doi.org/10.1186/s13660021026463
MSC
 47H10
 54H25
Keywords
 Fixed point
 Fuzzy cone metric space
 Hausdorff metric
 Contraction conditions