Best Proximity Point Results for Fuzzy Proximal Quasi Contractions with Applications

Abstract

In this work, we introduce a new type of multivalued fuzzy proximal quasi-contraction. These are generalized contractions which are a hybrid of $\mathcal{H}$-contractive mappings and quasi-contractions. Furthermore, we establish the best proximity point results for newly introduced fuzzy contractions in the context of fuzzy b-metric spaces. Fuzzy b-metric spaces are more general than fuzzy metric spaces and are linked with the cosine distance, which is used in various contexts of artificial intelligence to measure the similarity between elements of a vector space.

1. Introduction and Preliminaries

Let $\left(Z,d\right)$ be a metric space and $T:Z\to Z$ be a self-mapping. The fixed point problem of the mapping T is finding a point z in Z such that $Tz=z$, and a solution to the fixed point problem of T is called a fixed point of T. Banach established the foundations of metric fixed point theory by presenting an instrumental tool in nonlinear analysis, known as the Banach contraction principle (BCP) [1]. The BCP guarantees the existence of a unique solution to a fixed point problem associated with Banach contraction in a complete metric space $(Z,d).$ Many problems in mathematics and related disciplines can be transformed into corresponding fixed point problems of certain mappings. If L and M are two non-empty subsets of a metric space $(Z,d)$ and $T:L\to M$ is a non-self-mapping, then the necessary condition for the solution to the equation $T\left(\xi \right)=\xi$ is $T\left(L\right)\cap L\ne \varnothing .$ Obviously, if $L\cap M=\varnothing ,$ then the equation $T\left(\xi \right)=\xi$ has no solution. In this case, we try to minimize $d\left(\xi ,T\xi \right)$. To be precise, we try to find a point $\xi$ in L such that

$$d\left(\xi ,T\xi \right)=d\left(L,M\right),$$

where

$$d\left(L,M\right)=\underset{l\in L,m\in M}{inf}d(l,m).$$

If there exists such a point, then it is known as a best proximity point. The best proximity point theory has been discussed in the context of metric spaces (see for details [2,3,4,5,6] and the references therein).

Zadeh [7] introduced the fuzzy set theory in 1965 and opened a new horizon of research in many areas of engineering and mathematics. Kramosil and Michalek defined fuzzy metric spaces (FMSs) by embedding the idea of probabilistic metric spaces by Menger [8] on the fuzzy sets. Some results related to probabilistic Menger spaces can be seen in [9]. George and Veeramani [10] redefined the concept of FMS given in [11] such that the topology induced by FMSs is a Hausdorff space and completely metrizable [12]. The notion of b-metric spaces was initiated by Bakhtin [13] and Czerwik [14]. Sedghi and Shobe [15] generalized b-metric spaces to b-FMSs by applying the idea given in [15]. Interested readers can find some work related to Menger $PbM$-spaces in [16]. Furthermore, Grabiec [17] launched the fuzzy fixed point theory by proving the BCP in FMSs. Heilpern [18] introduced fuzzy contraction mappings and proved a fixed point theorem for fuzzy contraction mappings. Moreover, Gregori and Sapena [19] introduced the notion of fuzzy contractive mappings and applied the BCP in different classes of complete FMSs in the sense of George and Veeramani, Kramosil and Michalek, and Grabiec. Nadler’s fixed point theorem [20] generalized the BCP for multivalued mappings. Gopal and Vetro [21] introduced the notions of (α-ϕ)-fuzzy contractive mappings and (β-ψ)-fuzzy contractive mappings and proved a couple of theorems about the existence and uniqueness of a fixed point for the above-mentioned mappings. The theory of best proximity points has also been considered in the context of non-Archimedean FMSs (see for details [22] and the references therein).

Wardowski [23] coined a new term, fuzzy $\mathcal{H}$-contraction, which is a generalization of the fuzzy contractive mappings in the sense of Gregori and Sapena. Furthermore, they proved some fixed point results for $\mathcal{H}$-contractive mappings. Beg et al. [24] introduced the idea of $\alpha$-fuzzy $\mathcal{H}$-contractive mapping, which is essentially weaker than the class of fuzzy contractive mappings and is stronger than the concept of (α-ϕ)-fuzzy contractive mappings.

In this article, we have extended the idea of $\alpha$-fuzzy $\mathcal{H}$-contractive mapping by introducing multivalued fuzzy (α-ξ-φ)-proximal contraction of type I and type II, and we also consider multivalued fuzzy (β-ψ)-proximal contraction in the b-FMS.

We highlight some basic notions and results that will be used in a follow-up work to obtain the main results presented in this paper. Throughout this article, I represents the interval $\left[0,1\right]$, Z is a non-empty set, and L and M are non-empty subsets of Z. Moreover, we denote $2^M\setminus \left\{\varphi \right\}$ and $2^{M^{\ast }}$ as the set of all non-empty subsets and the set of all non-empty closed subsets of M, respectively.

Definition 1 

([25]). A continuous v-norm is a binary operation $★:I\times I\to I$ such that the following are true:

$\left({\mathcal{T}}_1\right)$

★ is associative and commutative;

$\left({\mathcal{T}}_2\right)$

★ is continuous;

$\left({\mathcal{T}}_3\right)$

$1★p=p$ for every p in I;

$\left({\mathcal{T}}_4\right)$

$p★r\le q★s$ whenever $p\le q$ and $r\le s$ for all $p,q,r,$ and s in $I.$

The three important v-norms, namely the minimum, product, and Lukasiewicz norms, are defined as follows:

$$\begin{array}{c}p{★}_{min}q=min\left\{p,q\right\},\hfill \\ p{★}_{prod}q=pq,\hfill \\ p{★}_{L}q=max\left\{0,p+q-1\right\},\hfill \end{array}$$

One can easily check that ${★}_L\le {★}_{prod}\le {★}_{min}.$ In fact, ${★}_{min}$ is the largest v-norm. For $l_1,l_2,···,l_m\in \left[0,1\right]$ and $m\in \mathbb{N},$ the product $l_1★l_2★···★l_n$ will be denoted by

$${\displaystyle \prod _{j=1}^m}{l}_j.$$

Definition 2. 

Let Z be a non-empty set, ★ be a continuous v-norm ($b\ge 1$), and a fuzzy set $\mathcal{F}$ be defined on $Z\times Z\times (0,\infty )$, satisfying the following conditions for all $\nu ,\varrho ,\eta \in Z$ and $v,w\in (0,\infty )$:

$\left(G_1\right)$

$\mathcal{F}(\varrho ,\nu ,v)>0;$

$\left(G_2\right)$

$\mathcal{F}(\varrho ,\nu ,v)=1$ if and only if $\varrho =\nu ;$

$\left(G_3\right)$

$\mathcal{F}(\varrho ,\nu ,v)=\mathcal{F}(\nu ,\varrho ,v);$

$\left(G_4\right)$

$\mathcal{F}(\varrho ,\nu ,·):(0,\infty )\to [0,1]$ is continuous;

$\left(G_5\right)$

$\mathcal{F}(\varrho ,\nu ,v)★\mathcal{F}(\nu ,\eta ,w)\le \mathcal{F}(\varrho ,\eta ,v+w);$

$\left(G_6\right)$

$\mathcal{F}\left(\varrho ,\nu ,{\displaystyle \frac{v}{b}}\right)★\mathcal{F}\left(\nu ,\eta ,{\displaystyle \frac{w}{b}}\right)\le \mathcal{F}(\varrho ,\eta ,v+w).$

If $\mathcal{F}$ satisfies $\left(G_1\right)$–$\left(G_5\right)$, then the triplet $(Z,\mathcal{F},★)$ is called an FMS in the sense of George and Veeramani [10]. If $\mathcal{F}$ satisfies $\left(G_1\right)$–$\left(G_4\right)$ and $\left(G_6\right)$, then the quadruple $(Z,\mathcal{F},★,b)$ is called a b-FMS [15]. Note that a b-fuzzy metric is a fuzzy metric if we have $b=1$, but the converse does not hold in general.

Remark 1 

([26]). A b-FMS is not continuous in general.

Definition 3 

([15]). Let $(Z,\mathcal{F},★,b)$ be a b-FMS. For $v>0$, the open ball $D(\varrho ,r,v)$ with a center $\varrho \in Z$ and radius $0<r<1$ is defined by

$$D(\varrho ,r,v)=\{\nu \in Z:\mathcal{F}(\varrho ,\nu ,v)>1-r\}$$

Definition 4 

([15]). Let $\left(Z,\mathcal{F},★,b\right)$ be a b-fuzzy metric space. Then, the following are true:

(i)

A sequence $\left\{{\varrho }_n\right\}$ in Z is said to converge to ϱ in Z if and only if $\mathcal{F}\left({\varrho }_n,\varrho ,v\right)\to 1$ as $n\to \infty$ for each $v>0,$ or equivalently, if

$$\underset{n\to \infty }{lim}\mathcal{F}\left({\varrho }_n,\varrho ,v\right)=1$$

for all $v>0$, denoted as ${\varrho }_n\stackrel{\mathcal{F}}{\to }\varrho$ as $n\to \infty .$

(ii)

A sequence $\left\{{\varrho }_n\right\}$$\subset Z$ is an M-Cauchy sequence if and only if for all $\epsilon \in \left(0,1\right)$ and $v>0$, there exists $n_0$ such that

$$\mathcal{F}\left({\varrho }_m,{\varrho }_n,v\right)>1-\epsilon$$

for all $m,n\ge n_0.$

(iii)

The b-fuzzy metric space is M-complete if every M-Cauchy sequence converges to some $\varrho \in Z.$

Lemma 1 

([15]). In a b-FMS $(Z,\mathcal{F},★,b)$, the following assertions hold:

(i)

The limit of a convergent sequence $\left\{{\nu }_n\right\}$ in Z is unique.

(ii)

Every convergent sequence $\left\{{\nu }_n\right\}$ in Z is Cauchy sequence.

Definition 5 

(Compare with [3]). Let $\left(Z,\mathcal{F},★,b\right)$ be a b-FMS. A set M is said to be fuzzy approximatively compact (FAC) with respect to L if every sequence $\left\{{\nu }_n\right\}$ in M satisfying the condition

$$\mathcal{F}\left(\varrho ,{\nu }_n,v\right)\to \mathcal{F}\left(\varrho ,M,v\right)$$

as $n\to \infty$ and for some $\varrho \in L$ has a convergent subsequence.

Definition 6. 

Let $\left(Z,\mathcal{F},★,b\right)$ be a b-FMS and $T:L\to 2^M\setminus \left\{\varphi \right\}$ be a multivalued mapping. Then, $\mathcal{F}$ has the property Q if for any sequences ${\varrho }_n\in L, {\nu }_n\in T{\varrho }_{n-1}$ such that

$${\varrho }_n\stackrel{\mathcal{F}}{\to }{\varrho }^{\ast }\mathit{as}n\to \infty ,\mathit{and}\mathcal{F}({\varrho }_n,{\nu }_n,v)=\mathcal{F}\left(L,M,v\right)$$

for all $n\in \mathbb{N}$ implies

$$\underset{n\to \infty }{lim}\mathcal{F}({\varrho }^{\ast },{\nu }_n,v)=\mathcal{F}\left(L,M,v\right)$$

for every $v>0$.

Now, we consider some classes of functions which will be used in the follow-up work.

Definition 7. 

Let $\mathcal{H}$ denote the family of mappings $\xi :(0,1]\to [0,\infty )$ with the following two conditions:

$\left(1_{\xi }\right)$

$\xi :(0,1]\stackrel{\mathit{onto}}{\to }[0,\infty )$;

$\left(2_{\xi }\right)$

ξ is strictly decreasing.

Note that $\left(1_{\xi }\right)$ and $\left(2_{\xi }\right)$ imply $\xi \left(1\right)=0$ (compare with [23]).

Definition 8. 

Let Ψ denote the family of mappings $\psi :I\to I,$ with the following two conditions:

$\left(1_{\psi }\right)$

ψ is continuous and non-decreasing;

$\left(2_{\psi }\right)$

$\psi \left(r\right)>r$ for every $r\in (0,1)$.

It is easy to show that if $\psi \in \Psi ,$ then $\psi \left(1\right)=1$ and $\underset{n\to \infty }{lim}{\psi }^n\left(r\right)=1$ for all $r\in \left(0,1\right)$ (compare with [21]).

Definition 9 

([27]). Suppose that Φ denotes the class of all functions $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ satisfying the following conditions:

$\left(1_{\phi }\right)$

φ is monotonically increasing;

$\left(2_{\phi }\right)$

$\phi \left(r\right)<r$ for all $r>0$;

$\left(3_{\phi }\right)$

φ is continuous;

$\left(4_{\phi }\right)$

${\displaystyle \sum _{n=0}^{\infty }}{\phi }^n\left(r\right)$$<\infty$ for all $r>0$, where ${\phi }^n$ is the $n^{th}$ iteration of φ.

Let $\left(Z,\mathcal{F},★,b\right)$ be a b-FMS. We define

$$\begin{array}{cc}\hfill L_0\left(v\right)& =\left\{\varrho \in L:\mathcal{F}\left(\varrho ,\nu ,v\right)=\mathcal{F}\left(L,M,v\right)\mathrm{for}\mathrm{some}\nu \in M\right\},\hfill \\ \hfill M_0\left(v\right)& =\left\{\nu \in M:\mathcal{F}\left(\varrho ,\nu ,v\right)=\mathcal{F}\left(L,M,v\right)\mathrm{for}\mathrm{some}\varrho \in L\right\},\hfill \end{array}$$

where

$$\mathcal{F}\left(L,M,v\right)=\underset{\varrho \in L,\nu \in M}{sup}\mathcal{F}\left(\varrho ,\nu ,v\right)$$

and the distance of a point $\varrho \in Z$ from a non-empty set M for $v>0$ is

$$\mathcal{F}(\varrho ,M,v)=\underset{\nu \in M}{sup}\mathcal{F}(\varrho ,\nu ,v).$$

Definition 10. 

Let $\left(Z,\mathcal{F},★,b\right)$ be a b-FMS. An element ${\varrho }^{\ast }\in L$ is said to be a best proximity point (BPP) of a multivalued mapping $T:L\to 2^M\setminus \left\{\varphi \right\}$ if

$$\mathcal{F}\left({\varrho }^{\ast },T{\varrho }^{\ast },v\right)=\mathcal{F}\left(L,M,v\right).$$

Definition 11. 

Let $(Z,\mathcal{F},★)$ be an FMS. A mapping $T:Z\to Z$ is fuzzy $\mathcal{H}$-quasi-contractive for $\xi \in \mathcal{H}$ if there is $k\in (0,1)$, satisfying

$$\xi \left(\mathcal{F}\left(T\varrho ,T\nu ,v\right)\right)\le kmax\left\{\begin{array}{c}\xi \left(\mathcal{F}\right(\varrho ,\nu ,v\left)\right),\xi \left(\mathcal{F}\right(\varrho ,T\varrho ,v\left)\right),\xi \left(\mathcal{F}\right(\nu ,T\nu ,v\left)\right),\hfill \\ \xi \left(\mathcal{F}\right(\varrho ,T\nu ,v\left)\right),\xi \left(\mathcal{F}\right(\nu ,T\varrho ,v\left)\right)\hfill \end{array}\right\}$$

for all $\nu ,\varrho \in Z$ and for any $v>0$ (see [28] for details).

Definition 12 

(Compare with [5]). Let $\left(Z,\mathcal{F},★,b\right)$ be a b-FMS. A mapping $T:L\to 2^{M^{\ast }}$ is multivalued fuzzy ${\alpha }_p$-proximal admissible if there exist mappings $\alpha :L\times L\times \left(0,\infty \right)\to [0,+\infty )$ and $p:L\times L\times \left(0,\infty \right)\to [1,\infty )$ such that for any ${\varrho }_0,{\varrho }_1,u_1,u_2\in L$ and ${\nu }_1\in T{\varrho }_0,{\nu }_2\in T{\varrho }_1,$ we have

$$\left\{\begin{array}{c}\alpha \left({\varrho }_0,{\varrho }_1,v\right)\ge p\left({\varrho }_0,{\varrho }_1,v\right)\hfill \\ \mathcal{F}(u_1,{\nu }_1,v)=\mathcal{F}\left(L,M,v\right)\hfill \\ \mathcal{F}(u_2,{\nu }_2,v)=\mathcal{F}\left(L,M,v\right)\hfill \end{array}\right.\mathit{implies}\alpha \left(u_1,u_2,v\right)\ge p\left(u_1,u_2,v\right)\mathit{for}\mathit{all}v>0.$$

Definition 13. 

Let $\left(Z,\mathcal{F},★,b\right)$ be a b-FMS. A mapping $T:L\to 2^{M^{\ast }}$ is said to be a multivalued fuzzy ${\beta }_q$-proximal admissible if there exist mappings $\beta :L\times L\times \left(0,\infty \right)\to [0,+\infty )$ and $q:L\times L\times \left(0,\infty \right)\to (0,1]$ such that for any ${\varrho }_0,{\varrho }_1,u_1,u_2\in L$ and ${\nu }_1\in T{\varrho }_0,{\nu }_2\in T{\varrho }_1,$ we have

$$\left\{\begin{array}{c}\beta \left({\varrho }_0,{\varrho }_1,v\right)\le q\left({\varrho }_0,{\varrho }_1,v\right)\\ \mathcal{F}(u_1,{\nu }_1,v)=\mathcal{F}\left(L,M,v\right)\\ \mathcal{F}(u_2,{\nu }_2,v)=\mathcal{F}\left(L,M,v\right)\end{array}\right.\mathit{implies}\beta \left(u_1,u_2,v\right)\le q\left(u_1,u_2,v\right)\mathit{for}\mathit{all}v>0.$$

Definition 14. 

Let L and M be non-empty closed subsets of b-FMS $\left(Z,\mathcal{F},★,b\right)$ and $\alpha :L\times L\times \left(0,\infty \right)\to [0,\infty )$ be a mapping. Suppose that $L_0\left(v\right)$ is non-empty for every $v>0.$ A mapping $T:L\to 2^{M^{\ast }}$ is said to be a fuzzy (α-ξ-φ)-proximal contraction of type I if there exist $\xi \in \mathcal{H}$ and $\phi \in \Phi$ such that

$$\left.\begin{array}{c}\mathcal{F}(u_1,T{\varrho }_0,v)=\mathcal{F}\left(L,M,v\right)\\ \mathcal{F}(u_2,T{\varrho }_1,v)=\mathcal{F}\left(L,M,v\right)\end{array}\right\}\mathit{implies}\alpha \left({\varrho }_0,{\varrho }_1,v\right)\xi \left(\mathcal{F}\left(u_1,u_2,v\right)\right)\le \phi \left(L_{\xi }({\varrho }_0,{\varrho }_1,v)\right),$$

for all ${\varrho }_0,{\varrho }_1,u_1,u_2\in L,$ where

$$L_{\xi }({\varrho }_0,{\varrho }_1,v)=max\left\{\xi \left(\mathcal{F}({\varrho }_0,{\varrho }_1,v)\right),\xi \left(\mathcal{F}\left({\varrho }_0,u_1,v\right)\right),\xi \left(\mathcal{F}\left({\varrho }_1,u_1,v\right)\right)\right\}.$$

Definition 15. 

Let L and M be non-empty closed subsets of b-FMS $\left(Z,\mathcal{F},★,b\right)$. Suppose that $L_0\left(v\right)$ is non-empty for every $v>0$ and $\alpha :L\times L\times \left(0,\infty \right)\to [0,\infty )$ is a mapping. A mapping $T:L\to 2^{M^{\ast }}$ is said to be a fuzzy (α-ξ-φ)-proximal contraction of type II if there exist $\xi \in \mathcal{H}$ and $\phi \in \Phi$ such that

$$\left.\begin{array}{c}\mathcal{F}(u_1,T{\varrho }_0,v)=\mathcal{F}\left(L,M,v\right)\\ \mathcal{F}(u_2,T{\varrho }_1,v)=\mathcal{F}\left(L,M,v\right)\end{array}\right\}\mathit{implies}\alpha \left({\varrho }_0,{\varrho }_1,v\right)\xi \left(\mathcal{F}\left(u_1,u_2,v\right)\right)\le \phi \left(M_{\xi }({\varrho }_0,{\varrho }_1,v)\right),$$

for all ${\varrho }_0,{\varrho }_1,u_1,u_2\in L,$ where

$$M_{\xi }({\varrho }_0,{\varrho }_1,v)=max\left\{\xi \left(\mathcal{F}({\varrho }_0,{\varrho }_1,v)\right),\xi \left(\mathcal{F}\left({\varrho }_0,u_1,v\right)\right),\xi \left(\mathcal{F}\left({\varrho }_1,u_1,v\right)\right),\xi \left(\mathcal{F}\left({\varrho }_1,u_2,v\right)\right)\right\}.$$

Remark 2. 

Note that every fuzzy (α-ξ-φ)-proximal contraction of type I is fuzzy (α-ξ-φ)-proximal contraction of type II.

Definition 16 

(Compare with [21]). Let L and M be non-empty closed subsets of b-FMS $\left(Z,\mathcal{F},★,b\right)$. Suppose that $L_0\left(v\right)$ is non-empty for every $v>0$ and $\beta :L\times L\times \left(0,\infty \right)\to [0,\infty )$ is a mapping. A mapping $T:L\to 2^{M^{\ast }}$ is said to be a fuzzy (β-ψ)-proximal contraction if there exists $\psi \in \Psi$ such that

$$\left.\begin{array}{c}\mathcal{F}(u_1,T{\varrho }_0,v)=\mathcal{F}\left(L,M,v\right)\\ \mathcal{F}(u_2,T{\varrho }_1,v)=\mathcal{F}\left(L,M,v\right)\end{array}\right\}\mathit{implies}\beta \left({\varrho }_0,{\varrho }_1,v\right)\mathcal{F}\left(u_1,u_2,v\right)\ge \psi \left(L_{\psi }({\varrho }_0,{\varrho }_1,v)\right)$$

where

$$L_{\psi }({\varrho }_0,{\varrho }_1,v)=min\left\{\mathcal{F}({\varrho }_0,{\varrho }_1,v),\mathcal{F}\left({\varrho }_0,u_1,v\right),\mathcal{F}\left({\varrho }_1,u_1,v\right),\mathcal{F}\left({\varrho }_1,u_2,v\right)\right\}.$$

In addition, $\psi \in \Psi ,{\varrho }_0,{\varrho }_1,u_1,u_2\in L,$ and $v>0.$

Lemma 2 

(Compare with [23]). Let $\left(Z,\mathcal{F},★,b\right)$ be a b-FMS and $\xi \in \mathcal{H}$. A sequence $\left\{{\varrho }_n\right\}\subset Z$ is an M-Cauchy sequence if and only if for every $\epsilon >0$ and $v>0$, there exists $n_0\in \mathbb{N}$ such that

$$\xi \left(\mathcal{F}\left({\varrho }_m,{\varrho }_n,v\right)\right)<\epsilon ,m,n\ge n_0.$$

(1)

Lemma 3 

(Compare with [23]). Let $\left(Z,\mathcal{F},★,b\right)$ be a b-FMS and $\xi \in \mathcal{H}$. A sequence $\left\{{\varrho }_n\right\}\subset Z$ is convergent to ϱ if and only if

$$\underset{n\to \infty }{lim}\xi \left(\mathcal{F}\left({\varrho }_n,\varrho ,v\right)\right)=0,\mathrm{for}\mathrm{all}v>0.$$

2. Main Results

In this section, we prove the following result for the existence of the BPP theorem for fuzzy (α-ξ-φ)-proximal contractions of type $I.$

Theorem 1. 

Let $(Z,\mathcal{F},★,b)$ be an M-complete b-FMS having the property Q, L and M be two non-empty and closed subsets of Z, M be FAC with respect to L, and $T:L\to 2^{M^{\ast }}$ be a fuzzy (α-ξ-φ)-proximal contraction of type $I.$ Moreover, the following conditions hold:

(i)

$\xi \left(r★w\right)\le \xi \left(r\right)+\xi \left(w\right)$ for all $r,w\in (0,1];$

(ii)

$L_0\left(v\right)$ is non-empty for all $v>0$, and for every $\varrho \in L_0\left(v\right),$$T\left(\varrho \right)\subseteq M_0\left(v\right)$ for all $v>0$;

(iii)

T is ${\alpha }_p$-proximal admissible, and there exist ${\varrho }_0,{\varrho }_1$ in $L_0\left(v\right)$ and ${\nu }_1\in T{\varrho }_0$ such that $\mathcal{F}({\varrho }_1,{\nu }_1,v)=\mathcal{F}\left(L,M,v\right)$ and $\alpha \left({\varrho }_0,{\varrho }_1,v\right)\ge p\left({\varrho }_0,{\varrho }_1,v\right)$ for all $v>0;$

(iv)

$\left\{{\varrho }_n\right\}$ is a sequence in $L_0\left(v\right)$ such that $\alpha \left({\varrho }_n,{\varrho }_{n+1},v\right)\ge p\left({\varrho }_n,{\varrho }_{n+1},v\right),$$n\in \mathbb{N}$, and ${\varrho }_n\stackrel{\mathcal{F}}{\to }\varrho$ as $n\to \infty$ implies $\alpha \left({\varrho }_n,\varrho ,v\right)\ge p\left({\varrho }_n,\varrho ,v\right)$ for all $n\in \mathbb{N}$ and $v>0.$

Then, T has a BPP in $L_0\left(v\right)$.

Proof. 

From the given assumption, there exist ${\varrho }_0,{\varrho }_1$ in $L_0\left(v\right)$, and ${\nu }_1\in T{\varrho }_0$ such that

$$\mathcal{F}({\varrho }_1,{\nu }_1,v)=\mathcal{F}\left(L,M,v\right).$$

Hence, we have

$$\mathcal{F}\left(L,M,v\right)\ge \mathcal{F}\left({\varrho }_1,T{\varrho }_0,v\right)\ge \mathcal{F}({\varrho }_1,{\nu }_1,v)=\mathcal{F}\left(L,M,v\right).$$

Consequently, we obtain

$$\mathcal{F}\left({\varrho }_1,T{\varrho }_0,v\right)=\mathcal{F}\left(L,M,v\right)\mathrm{and}\alpha \left({\varrho }_1,{\varrho }_0,v\right)\ge p\left({\varrho }_1,{\varrho }_0,v\right).$$

As $T{\varrho }_1\subseteq M_0\left(v\right)$, and $T{\varrho }_1$ is non-empty, then for some ${\nu }_2\in T{\varrho }_1\subseteq M_0\left(v\right)$, there exists ${\varrho }_2\in L_0\left(v\right)$ such that

$$\mathcal{F}({\varrho }_2,{\nu }_2,v)=\mathcal{F}\left(L,M,v\right),$$

In other words, we have

$$\mathcal{F}\left(L,M,v\right)\ge \mathcal{F}({\varrho }_2,T{\varrho }_1,v)\ge \mathcal{F}({\varrho }_2,{\nu }_2,v)=\mathcal{F}\left(L,M,v\right).$$

This implies that

$$\left\{\begin{array}{c}\alpha \left({\varrho }_0,{\varrho }_1,v\right)\ge p\left({\varrho }_0,{\varrho }_1,v\right),\hfill \\ \mathcal{F}({\varrho }_1,T{\varrho }_0,v)=\mathcal{F}\left(L,M,v\right),\hfill \\ \mathcal{F}({\varrho }_2,T{\varrho }_1,v)=\mathcal{F}\left(L,M,v\right).\hfill \end{array}\right.$$

As T is ${\alpha }_p$-proximal admissible, we then have

$$\alpha \left({\varrho }_1,{\varrho }_2,v\right)\ge p\left({\varrho }_1,{\varrho }_2,v\right)$$

By continuing in this way, we obtain the sequences ${\varrho }_n\in L_0\left(v\right)$ and ${\nu }_n\in T{\varrho }_{n-1}$ such that

$$\mathcal{F}({\varrho }_n,{\nu }_n,v)=\mathcal{F}\left(L,M,v\right)$$

and

$$\left\{\begin{array}{c}\alpha \left({\varrho }_{n-1},{\varrho }_n,v\right)\ge p\left({\varrho }_{n-1},{\varrho }_n,v\right),\hfill \\ \mathcal{F}({\varrho }_n,T{\varrho }_{n-1},v)=\mathcal{F}\left(L,M,v\right),\hfill \\ \mathcal{F}({\varrho }_{n+1},T{\varrho }_n,v)=\mathcal{F}\left(L,M,v\right).\hfill \end{array}\right.$$

(2)

Note that ${\varrho }_n\ne {\varrho }_{n+1}$ for all $n.$ If ${\varrho }_{n_0}={\varrho }_{n_0+1}$ for some $n_0,$ then

$$\mathcal{F}({\varrho }_{n_0+1},T{\varrho }_{n_0},v)=\mathcal{F}({\varrho }_{n_0+1},T{\varrho }_{n_0+1},v)=\mathcal{F}\left(L,M,v\right)$$

which implies that ${\varrho }_{n_0}$ is the BPP of T. Since T is a fuzzy (α-ξ-φ)-proximal contraction of type I, we therefore have

$$\begin{array}{cc}\hfill \xi \left(\mathcal{F}\left({\varrho }_n,{\varrho }_{n+1},v\right)\right)& \le p\left({\varrho }_{n-1},{\varrho }_n,v\right)\xi \left(\mathcal{F}\left({\varrho }_n,{\varrho }_{n+1},v\right)\right)\hfill \\ \hfill & \le \alpha \left({\varrho }_{n-1},{\varrho }_n,v\right)\xi \left(\mathcal{F}\left({\varrho }_n,{\varrho }_{n+1},v\right)\right)\hfill \\ \hfill & \le \phi \left(L_{\xi }({\varrho }_{n-1},{\varrho }_n,v)\right)\hfill \\ \hfill & \le \phi \left(max\left\{\begin{array}{c}\xi \left(\mathcal{F}({\varrho }_{n-1},{\varrho }_n,v)\right),\hfill \\ \xi \left(\mathcal{F}({\varrho }_{n-1},{\varrho }_n,v)\right),\xi \left(\mathcal{F}({\varrho }_n,{\varrho }_n,v)\right)\hfill \end{array}\right\}\right)\hfill \\ \hfill & =\phi \left(\xi \left(\mathcal{F}({\varrho }_{n-1},{\varrho }_n,v)\right)\right),\hfill \end{array}$$

implying

$$\xi \left(\mathcal{F}\left({\varrho }_n,{\varrho }_{n+1},v\right)\right)\le \phi \left(\xi \left(\mathcal{F}({\varrho }_{n-1},{\varrho }_n,v)\right)\right).$$

(3)

Now, from Equation (3), we obtain

$$\begin{array}{cc}\hfill \xi \left(\mathcal{F}\left({\varrho }_n,{\varrho }_{n+1},v\right)\right)& \le \phi \left(\xi \left(\mathcal{F}\left({\varrho }_{n-1},{\varrho }_n,v\right)\right)\right)\hfill \\ \hfill & \le {\phi }^2\left(\xi \left(\mathcal{F}({\varrho }_{n-2},{\varrho }_{n-1},v)\right)\right)\hfill \\ \hfill & ⋮\hfill \\ \hfill & \le {\phi }^n\left(\xi \left(\mathcal{F}({\varrho }_0,{\varrho }_1,v)\right)\right),\hfill \end{array}$$

In other words, we have

$$\xi \left(\mathcal{F}\left({\varrho }_n,{\varrho }_{n+1},v\right)\right)\le {\phi }^n\left(\xi \left(\mathcal{F}({\varrho }_0,{\varrho }_1,v)\right)\right)$$

for all $v>0$ and $n\in \mathbb{N}.$ Let $m,n$$\in \mathbb{N}$ with $m<n$. Suppose that $\left\{x_k\right\}$ is a strictly decreasing sequence of positive numbers such that

$${\displaystyle \sum _{k=1}^{\infty }}{x}_k=1.$$

For $b\ge 1$, we have

$$\begin{array}{cc}\hfill \mathcal{F}({\varrho }_n,{\varrho }_m,v)& \ge \mathcal{F}\left({\varrho }_n,{\varrho }_n,{\displaystyle \frac{v}{b}}-\underset{k=n}{{\displaystyle \sum }^{m-1}}{\displaystyle \frac{x_{k}v}{b}}\right)★\mathcal{F}\left({\varrho }_n,{\varrho }_m,\underset{k=n}{{\displaystyle \sum }^{m-1}}{\displaystyle \frac{x_{k}v}{b}}\right)\hfill \\ \hfill & \ge \mathcal{F}\left({\varrho }_n,{\varrho }_{n+1},{\displaystyle \frac{x_{n}v}{b^2}}\right)★\mathcal{F}\left({\varrho }_{n+1},{\varrho }_{n+2},{\displaystyle \frac{x_{n+1}v}{b^3}}\right)★...\hfill \\ \hfill & ★\mathcal{F}\left({\varrho }_{m-1},{\varrho }_m,{\displaystyle \frac{x_{m-1}v}{b^{m-n+1}}}\right).\hfill \end{array}$$

Thus, we obtain

$$\begin{array}{cc}\hfill \xi \left(\mathcal{F}({\varrho }_n,{\varrho }_m,v)\right)& \le \xi \left(\underset{k=n}{{\displaystyle \prod }^{m-1}}\mathcal{F}\left({\varrho }_k,{\varrho }_{k+1},{\displaystyle \frac{x_{k}v}{b^{k-n+2}}}\right)\right)\hfill \\ \hfill & \le \underset{k=n}{{\displaystyle \sum }^{m-1}}\xi \left(\mathcal{F}\left({\varrho }_k,{\varrho }_{k+1},{\displaystyle \frac{x_{k}v}{b^{k-n+2}}}\right)\right)\hfill \\ \hfill & \le \underset{k=n}{{\displaystyle \sum }^{m-1}}{\phi }^k\left(\xi \left(\mathcal{F}\left({\varrho }_0,{\varrho }_1,{\displaystyle \frac{x_{k}v}{b^{k-n+2}}}\right)\right)\right)\hfill \end{array}$$

for all $n\in \mathbb{N}$ and $v>0.$ Let $\epsilon >0$ be given. Since

$$\underset{k=1}{{\displaystyle \sum }^{\infty }}{\phi }^k\left(\xi \left(\mathcal{F}\left({\varrho }_0,{\varrho }_1,{\displaystyle \frac{x_{k}v}{b^{k-n+2}}}\right)\right)\right)<\infty ,$$

we therefore have

$$\underset{k\ge h}{{\displaystyle \sum }^{\infty }}{\phi }^k\left(\xi \left(\mathcal{F}\left({\varrho }_0,{\varrho }_1,{\displaystyle \frac{x_{k}v}{b^{k-n+2}}}\right)\right)\right)<\epsilon$$

for some $h\in \mathbb{N}.$ Hence, we obtain

$$\xi \left(\mathcal{F}({\varrho }_n,{\varrho }_m,v)\right)\le \underset{k\ge h}{{\displaystyle \sum }^{\infty }}{\phi }^k\left(\xi \left(\mathcal{F}\left({\varrho }_0,{\varrho }_1,{\displaystyle \frac{x_{k}v}{b^{k-n+2}}}\right)\right)\right)<\epsilon$$

for all $m>n>h$ and $v>0.$ Thus, under Lemma 2, it follows that $\left\{{\varrho }_n\right\}$ is an M-Cauchy sequence in a closed subset L of a complete b-FMS Z. Therefore, there exists some ${\varrho }^{\ast }\in$L such that ${\varrho }_n\stackrel{\mathcal{F}}{\to }{\varrho }^{\ast }$ as $n\to +\infty$. We show that T has a BPP. As ${\varrho }_n\in L_0\left(v\right)$, ${\nu }_n\in T{\varrho }_{n-1}\subset M_0\left(v\right)$ and $\mathcal{F}({\varrho }_n,{\nu }_n,v)=\mathcal{F}\left(L,M,v\right),$ then under property Q, we have

$$\underset{n\to \infty }{lim}\mathcal{F}\left({\varrho }^{\ast },{\nu }_n,v\right)=\mathcal{F}\left(L,M,v\right),$$

which implies

$$\underset{n\to \infty }{lim}\mathcal{F}\left({\varrho }^{\ast },{\nu }_n,v\right)=\mathcal{F}\left(L,M,v\right)\ge \mathcal{F}\left({\varrho }^{\ast },M,v\right)\ge \underset{n\to \infty }{lim}\mathcal{F}\left({\varrho }^{\ast },{\nu }_n,v\right).$$

Consequently, we have

$$\underset{n\to \infty }{lim}\mathcal{F}\left({\varrho }^{\ast },{\nu }_n,v\right)=\mathcal{F}\left({\varrho }^{\ast },M,v\right).$$

Since M is FAC with respect to L, there exists a convergent subsequence $\left\{{\nu }_{n_k}\right\}$ of $\left\{{\nu }_n\right\}$ such that ${\nu }_{n_k}\stackrel{\mathcal{F}}{\to }{\nu }^{\ast }$ as $k\to \infty$. Since M is closed, ${\nu }^{\ast }\in M.$ Because

$${\nu }_n\in T{\varrho }_n\mathrm{and}{\nu }_{n_k}\in T{\varrho }_{n_k-1}\mathrm{imply}\mathcal{F}\left({\varrho }_{n_k},{\nu }_{n_k},v\right)=\mathcal{F}\left(L,M,v\right)$$

we thus have

$$\mathcal{F}\left({\varrho }^{\ast },{\nu }^{\ast },v\right)=\underset{n\to \infty }{lim}\mathcal{F}\left({\varrho }_{n_k},{\nu }_{n_k},v\right)=\mathcal{F}\left(L,M,v\right)$$

which implies ${\varrho }^{\ast }\in L_0\left(v\right).$ Since $T{\varrho }^{\ast }\subseteq M_0\left(v\right)$ ensures that for every $w^{\ast }\in T{\varrho }^{\ast },$ there is an $\eta \in L_0\left(v\right)$ such that

$$\mathcal{F}\left(\eta ,w^{\ast },v\right)=\mathcal{F}\left(L,M,v\right).$$

we thus have

$$\mathcal{F}\left(L,M,v\right)\ge \mathcal{F}\left(\eta ,T{\varrho }^{\ast },v\right)\ge \mathcal{F}\left(\eta ,w^{\ast },v\right)=\mathcal{F}\left(L,M,v\right).$$

Consequently, we obtain

$$\mathcal{F}(\eta ,T{\varrho }^{\ast },v)=\mathcal{F}\left(L,M,v\right).$$

(4)

Now, we show that $\eta ={\varrho }^{\ast }$. On the contrary, assume that $\eta \ne {\varrho }^{\ast }.$ Now, using Equations (2) and (4), and from a given assumption, we have

$$\left\{\begin{array}{c}\alpha \left({\varrho }_n,{\varrho }^{\ast },v\right)\ge p\left({\varrho }_n,{\varrho }^{\ast },v\right),\hfill \\ \mathcal{F}({\varrho }_{n+1},T{\varrho }_n,v)=\mathcal{F}\left(L,M,v\right),\hfill \\ \mathcal{F}(\eta ,T{\varrho }^{\ast },v)=\mathcal{F}\left(L,M,v\right).\hfill \end{array}\right.$$

(5)

As T is an (α-ξ-φ)-proximal contraction of type $I,$ we thus have

$$\begin{array}{c}\xi \left(\mathcal{F}\left({\varrho }_{n+1},\eta ,v\right)\right)\hfill \\ \le p\left({\varrho }_n,{\varrho }^{\ast },v\right)\xi \left(\mathcal{F}\left({\varrho }_{n+1},\eta ,v\right)\right)\hfill \\ \le \alpha \left({\varrho }_n,{\varrho }^{\ast },v\right)\xi \left(\mathcal{F}\left({\varrho }_{n+1},\eta ,v\right)\right)\hfill \\ \le \phi (L_{\xi }\left(\mathcal{F}({\varrho }_n,{\varrho }^{\ast },v)\right)\hfill \\ =\phi \left(max\left\{\begin{array}{c}\xi \left(\mathcal{F}({\varrho }_n,{\varrho }^{\ast },v)\right),\xi \left(\mathcal{F}({\varrho }_n,{\varrho }_{n+1},v)\right),\hfill \\ \xi \left(\mathcal{F}({\varrho }^{\ast },{\varrho }_{n+1},v)\right)\hfill \end{array}\right\}\right)\hfill \\ \le \phi \left(max\left\{\begin{array}{c}\xi \left(\mathcal{F}({\varrho }_n,{\varrho }^{\ast },v)\right),\hfill \\ \xi \left(\mathcal{F}({\varrho }_n,{\varrho }^{\ast },{\displaystyle \frac{v}{2b}})★\mathcal{F}({\varrho }^{\ast },{\varrho }_{n+1},{\displaystyle \frac{v}{2b}})\right),\hfill \\ \xi \left(\mathcal{F}({\varrho }^{\ast },{\varrho }_{n+1},v)\right)\hfill \end{array}\right\}\right)\hfill \\ \le \phi \left(max\left\{\begin{array}{c}\xi \left(\mathcal{F}({\varrho }_n,{\varrho }^{\ast },v)\right),\hfill \\ \xi \left(\mathcal{F}({\varrho }_n,{\varrho }^{\ast },{\displaystyle \frac{v}{2b}})\right)+\xi \left(\mathcal{F}({\varrho }^{\ast },{\varrho }_{n+1},{\displaystyle \frac{v}{2b}})\right),\hfill \\ \xi \left(\mathcal{F}({\varrho }^{\ast },{\varrho }_{n+1},v)\right)\hfill \end{array}\right\}\right).\hfill \end{array}$$

(6)

By applying the limit as n→∞ in Equation (6), we obtain

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}\xi \left(\mathcal{F}\left({\varrho }_{n+1},\eta ,v\right)\right)& \le \underset{n\to \infty }{lim}\phi \left(max\left\{\begin{array}{c}\xi \left(\mathcal{F}({\varrho }_n,{\varrho }^{\ast },v)\right),\hfill \\ \xi \left(\mathcal{F}({\varrho }_n,{\varrho }^{\ast },{\displaystyle \frac{v}{2b}})\right)+\xi \left(\mathcal{F}({\varrho }^{\ast },{\varrho }_{n+1},{\displaystyle \frac{v}{2b}})\right),\hfill \\ \xi \left(\mathcal{F}({\varrho }^{\ast },{\varrho }_{n+1},v)\right)\hfill \end{array}\right\}\right)\hfill \\ \hfill & =0.\hfill \end{array}$$

Hence, according to Lemma 3, we find

$$\underset{n\to \infty }{lim}\xi \left(\mathcal{F}\left({\varrho }_{n+1},\eta ,v\right)\right)=0\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}\eta =\underset{n\to \infty }{lim}{\varrho }_{n+1}={\varrho }^{\ast }.$$

Consequently, using $\eta ={\varrho }^{\ast }$ in Equation (4), we obtain

$$\mathcal{F}({\varrho }^{\ast },T{\varrho }^{\ast },v)=\mathcal{F}\left(L,M,v\right),$$

which implies that ${\varrho }^{\ast }$ is the BPP of $T.$ □

Now, we prove the following theorem for the fuzzy (α-ξ-φ)-proximal contraction mapping of type II via an assumption of continuity on the function $\xi .$

Theorem 2. 

Let $(Z,\mathcal{F},★,b)$ be an M-complete b-FMS having the property Q, L and M be two non-empty and closed subsets of Z, M be FAC with respect to L, and $T:L\to 2^{M^{\ast }}$ be a fuzzy (α-ξ-φ)-proximal contraction of type II. Moreover, the following conditions hold:

(i)

$\xi \left(r★w\right)\le \xi \left(r\right)+\xi \left(w\right)$ for all $r,w\in (0,1]$, and ξ is continuous;

(ii)

$L_0\left(v\right)$ is non-empty for all $v>0$, and for every $\varrho \in L_0\left(v\right),$$T\left(\varrho \right)\subseteq M_0\left(v\right)$ for all $v>0$;

(iii)

T is ${\alpha }_p$-proximal admissible, and there exist ${\varrho }_0,{\varrho }_1$ in $L_0\left(v\right)$ and ${\nu }_1\in T{\varrho }_0$ such that $\mathcal{F}({\varrho }_1,{\nu }_1,v)=\mathcal{F}\left(L,M,v\right)$ and $\alpha \left({\varrho }_0,{\varrho }_1,v\right)\ge p\left({\varrho }_0,{\varrho }_1,v\right)$ for all $v>0$;

(iv)

$\left\{{\varrho }_n\right\}$ is a sequence in $L_0\left(v\right)$ such that $\alpha \left({\varrho }_n,{\varrho }_{n+1},v\right)\ge p\left({\varrho }_n,{\varrho }_{n+1},v\right),$$n\in \mathbb{N}$, and ${\varrho }_n\stackrel{\mathcal{F}}{\to }\varrho$, as $n\to \infty$ implies $\alpha \left({\varrho }_n,\varrho ,v\right)\ge p\left({\varrho }_n,\varrho ,v\right)$ for all $n\in \mathbb{N}$ and $v>0.$

Then, T has a BPP in $L_0\left(v\right)$.

Proof. 

Following similar lines to those in the proof of Theorem 1, we find the sequences $\left\{{\varrho }_n\right\}$ and $\left\{{\nu }_n\right\}$ in $L_0\left(v\right)$ and $M_0\left(v\right)$, respectively, such that $\left\{{\varrho }_n\right\}$ is an M-Cauchy sequence. As L is a closed subset of an M-complete b-FMS Z, then there exists some ${\varrho }^{\ast }\in$L such that ${\varrho }_n\stackrel{\mathcal{F}}{\to }{\varrho }^{\ast }$ as $n\to +\infty$. We show that T has a BPP. As ${\varrho }_n\in L_0\left(v\right)$, ${\nu }_n\in T{\varrho }_{n-1}\subset M_0\left(v\right)$, and $\mathcal{F}({\varrho }_n,{\nu }_n,v)=\mathcal{F}\left(L,M,v\right),$ then under property $Q,$ we have

$$\underset{n\to \infty }{lim}\mathcal{F}\left({\varrho }^{\ast },{\nu }_n,v\right)=\mathcal{F}\left(L,M,v\right),$$

which implies

$$\underset{n\to \infty }{lim}\mathcal{F}\left({\varrho }^{\ast },{\nu }_n,v\right)=\mathcal{F}\left(L,M,v\right)\ge \mathcal{F}\left({\varrho }^{\ast },M,v\right)\ge \underset{n\to \infty }{lim}\mathcal{F}\left({\varrho }^{\ast },{\nu }_n,v\right).$$

Consequently, we have

$$\underset{n\to \infty }{lim}\mathcal{F}\left({\varrho }^{\ast },{\nu }_n,v\right)=\mathcal{F}\left({\varrho }^{\ast },M,v\right).$$

Since M is FAC with respect to L, there exists a convergent subsequence $\left\{{\nu }_{n_k}\right\}$ of $\left\{{\nu }_n\right\}$ such that ${\nu }_{n_k}\stackrel{\mathcal{F}}{\to }{\nu }^{\ast }$ as $k\to \infty$. Since M is closed, ${\nu }^{\ast }\in M.$ Because

$${\nu }_n\in T{\varrho }_n\mathrm{and}{\nu }_{n_k}\in T{\varrho }_{n_k-1}\mathrm{imply}\mathcal{F}\left({\varrho }_{n_k},{\nu }_{n_k},v\right)=\mathcal{F}\left(L,M,v\right)$$

we thus have

$$\mathcal{F}\left({\varrho }^{\ast },{\nu }^{\ast },v\right)=\underset{n\to \infty }{lim}\mathcal{F}\left({\varrho }_{n_k},{\nu }_{n_k},v\right)=\mathcal{F}\left(L,M,v\right)$$

which implies ${\varrho }^{\ast }\in L_0\left(v\right).$ Here, $T{\varrho }^{\ast }\subseteq M_0\left(v\right)$ ensures that for every $w^{\ast }\in T{\varrho }^{\ast },$ there is an $\eta \in L_0\left(v\right)$ such that

$$\mathcal{F}\left(\eta ,w^{\ast },v\right)=\mathcal{F}\left(L,M,v\right).$$

Hence, we have

$$\mathcal{F}\left(L,M,v\right)\ge \mathcal{F}\left(\eta ,T{\varrho }^{\ast },v\right)\ge \mathcal{F}\left(\eta ,w^{\ast },v\right)=\mathcal{F}\left(L,M,v\right).$$

Consequently, we obtain

$$\mathcal{F}\left(\eta ,T{\varrho }^{\ast },v\right)=\mathcal{F}\left(L,M,v\right).$$

(7)

Now, we show that $\eta ={\varrho }^{\ast }$. On the contrary, assume that $\eta \ne {\varrho }^{\ast }.$ Now, using Equations (2) and (7), and from a given assumption, we have

$$\left\{\begin{array}{c}\alpha \left({\varrho }_n,{\varrho }^{\ast },v\right)\ge p\left({\varrho }_n,{\varrho }^{\ast },v\right),\hfill \\ \mathcal{F}({\varrho }_{n+1},T{\varrho }_n,v)=\mathcal{F}\left(L,M,v\right),\hfill \\ \mathcal{F}(\eta ,T{\varrho }^{\ast },v)=\mathcal{F}\left(L,M,v\right).\hfill \end{array}\right.$$

(8)

As T is an (α-ξ-φ)-proximal contraction of type II, we therefore have

$$\begin{array}{c}\xi \left(\mathcal{F}\left({\varrho }_{n+1},\eta ,v\right)\right)\hfill \\ \le p\left({\varrho }_n,{\varrho }^{\ast },v\right)\xi \left(\mathcal{F}\left({\varrho }_{n+1},\eta ,v\right)\right)\hfill \\ \le \alpha \left({\varrho }_n,{\varrho }^{\ast },v\right)\xi \left(\mathcal{F}\left({\varrho }_{n+1},\eta ,v\right)\right)\hfill \\ \le \phi ({\mathcal{M}}_{\xi }\left(\mathcal{F}({\varrho }_n,{\varrho }^{\ast },v)\right)\hfill \\ =\phi \left(max\left\{\begin{array}{c}\xi \left(\mathcal{F}({\varrho }_n,{\varrho }^{\ast },v)\right),\xi \left(\mathcal{F}({\varrho }_n,{\varrho }_{n+1},v)\right),\\ \xi \left(\mathcal{F}({\varrho }^{\ast },{\varrho }_{n+1},v)\right),\xi \left(\mathcal{F}({\varrho }^{\ast },\eta ,v)\right)\end{array}\right\}\right).\hfill \end{array}$$

(9)

Now, if

$$max\left\{\begin{array}{c}\xi \left(\mathcal{F}({\varrho }_n,{\varrho }^{\ast },v)\right),\xi \left(\mathcal{F}({\varrho }_n,{\varrho }_{n+1},v)\right),\\ \xi \left(\mathcal{F}({\varrho }^{\ast },{\varrho }_{n+1},v)\right),\xi \left(\mathcal{F}({\varrho }^{\ast },\eta ,v)\right)\end{array}\right\}=\xi \left(\mathcal{F}({\varrho }^{\ast },\eta ,v)\right)$$

then with the continuity of $\xi$, we have

$$\xi \left(\mathcal{F}({\varrho }^{\ast },\eta ,v)\right)=\underset{n\to \infty }{lim}\xi \left(\mathcal{F}\left({\varrho }_{n+1},\eta ,v\right)\right)\le \phi \left(\xi \left(\mathcal{F}({\varrho }^{\ast },\eta ,v)\right)\right)<\xi \left(\mathcal{F}({\varrho }^{\ast },\eta ,v)\right)$$

which implies that either ${\varrho }^{\ast }=\eta$ or there is a contradiction. Hence, we have

$$\xi \left(\mathcal{F}\left({\varrho }_{n+1},\eta ,v\right)\right)\le \phi \left(max\left\{\begin{array}{c}\xi \left(\mathcal{F}({\varrho }_n,{\varrho }^{\ast },v)\right),\xi \left(\mathcal{F}({\varrho }_n,{\varrho }_{n+1},v)\right),\hfill \\ \xi \left(\mathcal{F}({\varrho }^{\ast },{\varrho }_{n+1},v)\right)\hfill \end{array}\right\}\right)$$

By applying the limit as $n\to$∞ in the above inequality, we obtain

$$\underset{n\to \infty }{lim}\xi \left(\mathcal{F}\left({\varrho }_{n+1},\eta ,v\right)\right)\le 0$$

implying

$$\underset{n\to \infty }{lim}\xi \left(\mathcal{F}\left({\varrho }_{n+1},\eta ,v\right)\right)=0\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}\eta =\underset{n\to \infty }{lim}{\varrho }_{n+1}={\varrho }^{\ast }.$$

Hence, $\eta ={\varrho }^{\ast }.$ Consequently, by using $\eta ={\varrho }^{\ast }$ in Equation (7), we find that

$$\mathcal{F}({\varrho }^{\ast },T{\varrho }^{\ast },v)=\mathcal{F}\left(L,M,v\right),$$

which implies that ${\varrho }^{\ast }$ is the BPP of $T.$ □

The following result concerns the existence of the best proximity points of multivalued fuzzy $\left(\beta \text{-}\psi \right)$-proximal contractions of b-FMS.

Theorem 3. 

Let $(Z,\mathcal{F},★,b)$ be an M-complete b-FMS having the property Q, L and M be two non-empty and closed subsets of Z, M be FAC with respect to L, and $T:L\to 2^{M^{\ast }}$ be a fuzzy $\left(\beta \text{-}\psi \right)$-proximal contraction. Moreover, let the following conditions hold:

(i)

$L_0\left(v\right)$ is non-empty for all $v>0$, and for every $\varrho \in L_0\left(v\right),$$T\left(\varrho \right)\subseteq M_0\left(v\right)$ for all $v>0$;

(ii)

T is ${\beta }_q$-proximal admissible, and there exist ${\varrho }_0,{\varrho }_1$ in $L_0\left(v\right)$ and ${\nu }_1\in T{\varrho }_0$ such that $\mathcal{F}({\varrho }_1,{\nu }_1,v)=\mathcal{F}\left(L,M,v\right)$ and $\beta \left({\varrho }_0,{\varrho }_1,v\right)\le q\left({\varrho }_0,{\varrho }_1,v\right)$ for all $v>0;$

(iii)

$\left\{{\varrho }_n\right\}$ is a sequence in $L_0\left(v\right)$ such that $\beta \left({\varrho }_n,{\varrho }_{n+1},v\right)\le q\left({\varrho }_n,{\varrho }_{n+1},v\right),$$n\in \mathbb{N}$, and ${\varrho }_n\stackrel{\mathcal{F}}{\to }\varrho$ as $n\to \infty$, which implies $\beta \left({\varrho }_n,\varrho ,v\right)\le q\left({\varrho }_n,\varrho ,v\right)$ for all $n\in \mathbb{N}$ and $v>0.$

Then, T has a BPP in $L_0\left(v\right).$

Proof. 

From the given assumption, there exist ${\varrho }_0,{\varrho }_1$ in $L_0\left(v\right)$ and ${\nu }_1\in T{\varrho }_0$ such that

$$\mathcal{F}({\varrho }_1,{\nu }_1,v)=\mathcal{F}\left(L,M,v\right).$$

Hence, we have

$$\mathcal{F}\left(L,M,v\right)\ge \mathcal{F}\left({\varrho }_1,T{\varrho }_0,v\right)\ge \mathcal{F}({\varrho }_1,{\nu }_1,v)=\mathcal{F}\left(L,M,v\right).$$

Consequently, we obtain

$$\mathcal{F}({\varrho }_1,T{\varrho }_0,v)=\mathcal{F}\left(L,M,v\right)\mathrm{and}\beta \left({\varrho }_0,{\varrho }_1,v\right)\le q\left({\varrho }_0,{\varrho }_1,v\right).$$

As $T{\varrho }_1\subseteq M_0\left(v\right)$, and $T{\varrho }_1$ is non-empty, then for some ${\nu }_2\in T{\varrho }_1\subseteq M_0\left(v\right)$, there exists ${\varrho }_2\in L_0\left(v\right)$ such that

$$\mathcal{F}({\varrho }_2,{\nu }_2,v)=\mathcal{F}\left(L,M,v\right),$$

In other words, we have

$$\mathcal{F}\left(L,M,v\right)\ge \mathcal{F}\left({\varrho }_2,T{\varrho }_1,v\right)\ge \mathcal{F}({\varrho }_2,{\nu }_2,v)=\mathcal{F}\left(L,M,v\right).$$

This implies that

$$\left\{\begin{array}{c}\beta \left({\varrho }_0,{\varrho }_1,v\right)\le q\left({\varrho }_0,{\varrho }_1,v\right),\hfill \\ \mathcal{F}({\varrho }_1,T{\varrho }_0,v)=\mathcal{F}\left(L,M,v\right),\hfill \\ \mathcal{F}({\varrho }_2,T{\varrho }_1,v)=\mathcal{F}\left(L,M,v\right).\hfill \end{array}\right.$$

As T is ${\beta }_q$-proximal admissible, thus

$$\beta \left({\varrho }_1,{\varrho }_2,v\right)\le q\left({\varrho }_1,{\varrho }_2,v\right).$$

By continuing this way, we find the sequences ${\varrho }_n\in L_0\left(v\right)$ and ${\nu }_n\in T{\varrho }_{n-1}$ such that

$$\mathcal{F}({\varrho }_n,{\nu }_n,v)=\mathcal{F}\left(L,M,v\right)$$

and

$$\left\{\begin{array}{c}\beta \left({\varrho }_{n-1},{\varrho }_n,v\right)\le q\left({\varrho }_{n-1},{\varrho }_n,v\right),\\ \mathcal{F}({\varrho }_n,T{\varrho }_{n-1},v)=\mathcal{F}\left(L,M,v\right),\\ \mathcal{F}({\varrho }_{n+1},T{\varrho }_n,v)=\mathcal{F}\left(L,M,v\right).\end{array}\right.$$

(10)

Note that ${\varrho }_n\ne {\varrho }_{n+1}$ for all $n.$ If ${\varrho }_{n_0}={\varrho }_{n_0+1}$ for some $n_0,$ then

$$\mathcal{F}({\varrho }_{n_0+1},T{\varrho }_{n_0},v)=\mathcal{F}({\varrho }_{n_0+1},T{\varrho }_{n_0+1},v)=\mathcal{F}\left(L,M,v\right)$$

which implies that ${\varrho }_{n_0}$ is the BPP of T. Since T is a $\left(\beta \text{-}\psi \right)$-proximal contraction, therefore

$$\begin{array}{cc}\hfill \mathcal{F}\left({\varrho }_n,{\varrho }_{n+1},v\right)& \ge q\left({\varrho }_{n-1},{\varrho }_n,v\right)\mathcal{F}\left({\varrho }_n,{\varrho }_{n+1},v\right)\hfill \\ \hfill & \ge \beta \left({\varrho }_{n-1},{\varrho }_n,v\right)\mathcal{F}\left({\varrho }_n,{\varrho }_{n+1},v\right)\hfill \\ \hfill & \ge \psi \left(L_{\psi }\left({\varrho }_{n-1},{\varrho }_n,v\right)\right)\hfill \\ \hfill & =\psi \left(min\left\{\begin{array}{c}\mathcal{F}({\varrho }_{n-1},{\varrho }_n,v),\mathcal{F}({\varrho }_{n-1},{\varrho }_n,v)\hfill \\ \mathcal{F}({\varrho }_n,{\varrho }_n,v),\mathcal{F}({\varrho }_n,{\varrho }_{n+1},v)\hfill \end{array}\right\}\right)\hfill \\ \hfill & =\psi \left(min\left\{\mathcal{F}({\varrho }_{n-1},{\varrho }_n,v),\mathcal{F}({\varrho }_n,{\varrho }_{n+1},v)\right\}\right),\hfill \end{array}$$

which implies that

$$\mathcal{F}\left({\varrho }_n,{\varrho }_{n+1},v\right)\ge \psi \left(min\left\{\left(\mathcal{F}({\varrho }_{n-1},{\varrho }_n,v)\right),\left(\mathcal{F}({\varrho }_n,{\varrho }_{n+1},v)\right)\right\}\right).$$

(11)

If

$$min\left\{\mathcal{F}({\varrho }_{n-1},{\varrho }_n,v),\mathcal{F}({\varrho }_n,{\varrho }_{n+1},v)\right\}=\mathcal{F}({\varrho }_n,{\varrho }_{n+1},v)$$

then Equation (11) implies

$$\mathcal{F}\left({\varrho }_n,{\varrho }_{n+1},v\right)\ge \psi \left(\mathcal{F}\left({\varrho }_n,{\varrho }_{n+1},v\right)\right)>\mathcal{F}\left({\varrho }_n,{\varrho }_{n+1},v\right)$$

which is a contradiction. This implies that

$$min\left\{\mathcal{F}({\varrho }_{n-1},{\varrho }_n,v),\mathcal{F}({\varrho }_n,{\varrho }_{n+1},v)\right\}=\mathcal{F}({\varrho }_{n-1},{\varrho }_n,v).$$

Now, from Equation (11), we obtain

$$\begin{array}{cc}\hfill \mathcal{F}\left({\varrho }_n,{\varrho }_{n+1},v\right)& \ge \psi \left(\mathcal{F}\left({\varrho }_{n-1},{\varrho }_n,v\right)\right)\hfill \\ \hfill & \ge {\psi }^2\left(\mathcal{F}\left({\varrho }_{n-2},{\varrho }_{n-1},v\right)\right)\hfill \\ \hfill & ⋮\hfill \\ \hfill & \ge {\psi }^n\left(\mathcal{F}\left({\varrho }_0,{\varrho }_1,v\right)\right),\hfill \end{array}$$

In other words, we have

$$\mathcal{F}\left({\varrho }_n,{\varrho }_{n+1},v\right)\ge {\psi }^n\left(\mathcal{F}\left({\varrho }_0,{\varrho }_1,v\right)\right)$$

for all $v>0$ and $n\in \mathbb{N}$. Since $\underset{n\to \infty }{lim}{\psi }^n\left(v\right)=1$ for all $v\in \left(0,1\right),$ we obtain

$$\underset{n\to \infty }{lim}\mathcal{F}\left({\varrho }_n,{\varrho }_{n+1},v\right)\ge \underset{n\to \infty }{lim}{\psi }^n\left(\mathcal{F}\left({\varrho }_0,{\varrho }_1,v\right)\right)=1$$

which implies that

$$\underset{n\to \infty }{lim}\mathcal{F}\left({\varrho }_n,{\varrho }_{n+1},v\right)=1,\mathrm{for}\mathrm{all}v>0.$$

(12)

Now, we prove that $\left\{{\varrho }_n\right\}$ is an M-Cauchy sequence. Assume, on the contrary, that $\left\{{\varrho }_n\right\}$ is not an M-Cauchy sequence, that is, there is an $\epsilon \in \left(0,1\right)$ and $v_0>0$ such that for every $k\in \mathbb{N},$ there are $n_k,m_k\in \mathbb{N}$ with $m_k>n_k\ge k$ and

$$\mathcal{F}\left({\varrho }_{m_k},{\varrho }_{n_k},v_0\right)\le 1-\epsilon .$$

(13)

Let $m_k$ be the lowest integer greater than $n_k$ satisfying Equation (13), that is, let

$$\mathcal{F}\left({\varrho }_{m_k-1},{\varrho }_{n_k},v_0\right)>1-\epsilon ,$$

which implies that for every k, we obtain

$$\begin{array}{cc}1-\epsilon \hfill & \ge \mathcal{F}\left({\varrho }_{m_k},{\varrho }_{n_k},v_0\right)\ge \mathcal{F}\left({\varrho }_{m_k-1},{\varrho }_{m_k},{\displaystyle \frac{v_0}{2b}}\right)★\mathcal{F}\left({\varrho }_{m_k-1},{\varrho }_{n_k},{\displaystyle \frac{v_0}{2b}}\right)\hfill \\ \hfill & >\mathcal{F}\left({\varrho }_{m_k-1},{\varrho }_{m_k},{\displaystyle \frac{v_0}{2b}}\right)★\left(1-\epsilon \right).\hfill \end{array}$$

(14)

By applying the limit as $k\to \infty$ in Equation (14) and using Equation (12), we have

$$\underset{k\to \infty }{lim}\mathcal{F}\left({\varrho }_{m_k},{\varrho }_{n_k},v_0\right)=1-\epsilon .$$

Similarly, we can find that

$$\underset{k\to \infty }{lim}\mathcal{F}\left({\varrho }_{m_k+1},{\varrho }_{n_k+1},v_0\right)=1-\epsilon .$$

Furthermore, we have

$$\left\{\begin{array}{c}\beta \left({\varrho }_{m_k},{\varrho }_{n_k},{\displaystyle \frac{v_0}{3b}}\right)\le q\left({\varrho }_{m_k},{\varrho }_{n_k},{\displaystyle \frac{v_0}{3b}}\right)\hfill \\ \mathcal{F}\left({\varrho }_{m_{k+1}},T{\varrho }_{m_k},{\displaystyle \frac{v_0}{3b}}\right)=\mathcal{F}\left(L,M,v\right)\hfill \\ \mathcal{F}\left({\varrho }_{n_{k+1}},T{\varrho }_{n_k},{\displaystyle \frac{v_0}{3b}}\right)=\mathcal{F}\left(L,M,v\right)\hfill \end{array}\right.$$

Since T is a $\left(\beta \text{-}\psi \right)$-proximal contraction, therefore

$$\begin{array}{c}\mathcal{F}\left({\varrho }_{m_k},{\varrho }_{n_k},v_0\right)\ge \mathcal{F}\left({\varrho }_{m_k},{\varrho }_{m_k+1},{\displaystyle \frac{v_0}{3b}}\right)★\mathcal{F}\left({\varrho }_{m_k+1},{\varrho }_{n_k+1},{\displaystyle \frac{v_0}{3b}}\right)★\hfill \\ \mathcal{F}\left({\varrho }_{n_k+1},{\varrho }_{n_k},{\displaystyle \frac{v_0}{3b}}\right)\hfill \\ \ge \mathcal{F}\left({\varrho }_{m_k},{\varrho }_{m_k+1},{\displaystyle \frac{v_0}{3b}}\right)★\mathcal{F}\left({\varrho }_{n_k+1},{\varrho }_{n_k},{\displaystyle \frac{v_0}{3b}}\right)★\hfill \\ q\left({\varrho }_{m_k},{\varrho }_{n_k},{\displaystyle \frac{v_0}{3b}}\right)\mathcal{F}\left({\varrho }_{m_k+1},{\varrho }_{n_k+1},{\displaystyle \frac{v_0}{3b}}\right)\hfill \\ \ge \mathcal{F}\left({\varrho }_{m_k},{\varrho }_{m_k+1},{\displaystyle \frac{v_0}{3b}}\right)★\mathcal{F}\left({\varrho }_{n_k+1},{\varrho }_{n_k},{\displaystyle \frac{v_0}{3b}}\right)★\hfill \\ \beta \left({\varrho }_{m_k},{\varrho }_{n_k},{\displaystyle \frac{v_0}{3b}}\right)\mathcal{F}\left({\varrho }_{m_k+1},{\varrho }_{n_k+1},{\displaystyle \frac{v_0}{3b}}\right).\hfill \end{array}$$

(15)

Equation (15) implies that

$$\begin{array}{c}\mathcal{F}\left({\varrho }_{m_k},{\varrho }_{n_k},v_0\right)\ge \mathcal{F}\left({\varrho }_{m_k},{\varrho }_{m_k+1},{\displaystyle \frac{v_0}{3b}}\right)★\mathcal{F}\left({\varrho }_{n_k+1},{\varrho }_{n_k},{\displaystyle \frac{v_0}{3b}}\right)★\\ \psi \left(L_{\psi }\left({\varrho }_{m_k},{\varrho }_{n_k},{\displaystyle \frac{v_0}{3b}}\right)\right)\end{array}$$

(16)

where

$$L_{\psi }\left({\varrho }_{m_k},{\varrho }_{n_k},{\displaystyle \frac{v_0}{3b}}\right)=min\left\{\begin{array}{c}\mathcal{F}\left({\varrho }_{m_k},{\varrho }_{n_k},{\displaystyle \frac{v_0}{3b}}\right),\mathcal{F}\left({\varrho }_{m_k},{\varrho }_{m_k+1},{\displaystyle \frac{v_0}{3b}}\right),\\ \mathcal{F}\left({\varrho }_{n_k},{\varrho }_{m_k+1},{\displaystyle \frac{v_0}{3b}}\right),\mathcal{F}\left({\varrho }_{n_k},{\varrho }_{n_k+1},{\displaystyle \frac{v_0}{3b}}\right)\end{array}\right\}.$$

Now, if

$$min\left\{\begin{array}{c}\mathcal{F}\left({\varrho }_{m_k},{\varrho }_{n_k},{\displaystyle \frac{v_0}{3b}}\right),\mathcal{F}\left({\varrho }_{m_k},{\varrho }_{m_k+1},{\displaystyle \frac{v_0}{3b}}\right),\\ \mathcal{F}\left({\varrho }_{n_k},{\varrho }_{m_k+1},{\displaystyle \frac{v_0}{3b}}\right),\mathcal{F}\left({\varrho }_{n_k},{\varrho }_{n_k+1},{\displaystyle \frac{v_0}{3b}}\right)\end{array}\right\}=\mathcal{F}\left({\varrho }_{m_k},{\varrho }_{n_k},{\displaystyle \frac{v_0}{3b}}\right)$$

then Equation (16) implies

$$\begin{array}{cc}\hfill \mathcal{F}\left({\varrho }_{m_k},{\varrho }_{n_k},v_0\right)& \ge \mathcal{F}\left({\varrho }_{m_k},{\varrho }_{m_k+1},{\displaystyle \frac{v_0}{3b}}\right)★\mathcal{F}\left({\varrho }_{n_k+1},{\varrho }_{n_k},{\displaystyle \frac{v_0}{3b}}\right)★\hfill \\ \hfill & \psi \left(\mathcal{F}\left({\varrho }_{m_k},{\varrho }_{n_k},{\displaystyle \frac{v_0}{3b}}\right)\right)\hfill \end{array}$$

while taking the limit as $k\to \infty$ implies

$$1-\epsilon \ge 1★1★\psi (1-\epsilon )=\psi (1-\epsilon )>1-\epsilon$$

which is a contradiction. If

$$min\left\{\begin{array}{c}\mathcal{F}\left({\varrho }_{m_k},{\varrho }_{n_k},{\displaystyle \frac{v_0}{3b}}\right),\mathcal{F}\left({\varrho }_{m_k},{\varrho }_{m_k+1},{\displaystyle \frac{v_0}{3b}}\right),\\ \mathcal{F}\left({\varrho }_{n_k},{\varrho }_{m_k+1},{\displaystyle \frac{v_0}{3b}}\right),\mathcal{F}\left({\varrho }_{n_k},{\varrho }_{n_k+1},{\displaystyle \frac{v_0}{3b}}\right)\end{array}\right\}=\mathcal{F}\left({\varrho }_{m_k},{\varrho }_{m_k+1},{\displaystyle \frac{v_0}{3b}}\right)$$

then Equation (16) implies

$$\begin{array}{cc}\hfill \mathcal{F}\left({\varrho }_{m_k},{\varrho }_{n_k},v_0\right)& \ge \mathcal{F}\left({\varrho }_{m_k},{\varrho }_{m_k+1},{\displaystyle \frac{v_0}{3b}}\right)★\mathcal{F}\left({\varrho }_{n_k+1},{\varrho }_{n_k},{\displaystyle \frac{v_0}{3b}}\right)★\hfill \\ \hfill & \psi \left(\mathcal{F}\left({\varrho }_{m_k},{\varrho }_{m_k+1},{\displaystyle \frac{v_0}{3b}}\right)\right)\hfill \end{array}$$

while taking the limit as $n\to \infty$ implies

$$1-\epsilon \ge 1★1★\psi \left(1\right)=1\mathrm{implies}\epsilon <0,$$

which is a contradiction. If

$$min\left\{\begin{array}{c}\mathcal{F}\left({\varrho }_{m_k},{\varrho }_{n_k},{\displaystyle \frac{v_0}{3b}}\right),\mathcal{F}\left({\varrho }_{m_k},{\varrho }_{m_k+1},{\displaystyle \frac{v_0}{3b}}\right),\\ \mathcal{F}\left({\varrho }_{n_k},{\varrho }_{m_k+1},{\displaystyle \frac{v_0}{3b}}\right),\mathcal{F}\left({\varrho }_{n_k},{\varrho }_{n_k+1},{\displaystyle \frac{v_0}{3b}}\right)\end{array}\right\}=\mathcal{F}\left({\varrho }_{n_k},{\varrho }_{m_k+1},{\displaystyle \frac{v_0}{3b}}\right)$$

then Equation (16) implies

$$\begin{array}{cc}\hfill \mathcal{F}\left({\varrho }_{m_k},{\varrho }_{n_k},v_0\right)& \ge \mathcal{F}\left({\varrho }_{m_k},{\varrho }_{m_k+1},{\displaystyle \frac{v_0}{3b}}\right)★\mathcal{F}\left({\varrho }_{n_k+1},{\varrho }_{n_k},{\displaystyle \frac{v_0}{3b}}\right)★\hfill \\ \hfill & \psi \left(\mathcal{F}\left({\varrho }_{n_k},{\varrho }_{m_k+1},{\displaystyle \frac{v_0}{3b}}\right)\right)\hfill \end{array}$$

while taking the limit as $n\to \infty$ implies

$$1-\epsilon \ge 1★1★\psi \left(1-\epsilon \right)=\psi \left(1-\epsilon \right)>1-\epsilon$$

which is a contradiction. Hence, $\left\{{\varrho }_n\right\}$ is an M-Cauchy sequence in a closed subset L of a complete b-FMS Z. Thus, there exists some ${\varrho }^{\ast }\in$L such that ${\varrho }_n\stackrel{\mathcal{F}}{\to }{\varrho }^{\ast }$ as $n\to +\infty$. We show that T has a BPP. As ${\varrho }_n\in L_0\left(v\right)$, ${\nu }_n\in T{\varrho }_{n-1}\subset M_0\left(v\right)$, and $\mathcal{F}({\varrho }_n,{\nu }_n,v)=\mathcal{F}\left(L,M,v\right),$ then under the property Q, we have

$$\underset{n\to \infty }{lim}\mathcal{F}\left({\varrho }^{\ast },{\nu }_n,v\right)=\mathcal{F}\left(L,M,v\right),$$

which implies

$$\underset{n\to \infty }{lim}\mathcal{F}\left({\varrho }^{\ast },{\nu }_n,v\right)=\mathcal{F}\left(L,M,v\right)\ge \mathcal{F}\left({\varrho }^{\ast },M,v\right)\ge \underset{n\to \infty }{lim}\mathcal{F}\left({\varrho }^{\ast },{\nu }_n,v\right).$$

Consequently, we have

$$\underset{n\to \infty }{lim}\mathcal{F}\left({\varrho }^{\ast },{\nu }_n,v\right)=\mathcal{F}\left({\varrho }^{\ast },M,v\right).$$

Since M is FAC with respect to L, there exists a convergent subsequence $\left\{{\nu }_{n_k}\right\}$ of $\left\{{\nu }_n\right\}$ such that ${\nu }_{n_k}\stackrel{\mathcal{F}}{\to }{\nu }^{\ast }$ as $k\to \infty$. Since M is closed, then ${\nu }^{\ast }\in M.$ Because

$${\nu }_n\in T{\varrho }_n\mathrm{and}{\nu }_{n_k}\in T{\varrho }_{n_k-1}\mathrm{imply}\mathcal{F}\left({\varrho }_{n_k},{\nu }_{n_k},v\right)=\mathcal{F}\left(L,M,v\right)$$

thus

$$\mathcal{F}\left({\varrho }^{\ast },{\nu }^{\ast },v\right)=\underset{n\to \infty }{lim}\mathcal{F}\left({\varrho }_{n_k},{\nu }_{n_k},v\right)=\mathcal{F}\left(L,M,v\right)$$

which implies ${\varrho }^{\ast }\in L_0\left(v\right).$ Here, $T{\varrho }^{\ast }\subseteq M_0\left(v\right)$ ensures that for every $w^{\ast }\in T{\varrho }^{\ast },$ there is an $\eta \in L_0\left(v\right)$ such that

$$\mathcal{F}\left(\eta ,w^{\ast },v\right)=\mathcal{F}\left(L,M,v\right).$$

Hence, we have

$$\mathcal{F}\left(L,M,v\right)\ge \mathcal{F}\left(\eta ,T{\varrho }^{\ast },v\right)\ge \mathcal{F}\left(\eta ,w^{\ast },v\right)=\mathcal{F}\left(L,M,v\right).$$

Consequently, we obtain

$$\mathcal{F}(\eta ,T{\varrho }^{\ast },v)=\mathcal{F}\left(L,M,v\right).$$

(17)

Now, we show that $\eta ={\varrho }^{\ast }$. On the contrary, assume that $\eta \ne {\varrho }^{\ast }.$ Then, by using Equations (10) and (17), and from a given assumption, we have

$$\left\{\begin{array}{c}\beta \left({\varrho }_n,{\varrho }^{\ast },v\right)\le q\left({\varrho }_n,{\varrho }^{\ast },v\right)\hfill \\ \mathcal{F}({\varrho }_{n+1},T{\varrho }_n,v)=\mathcal{F}\left(L,M,v\right)\hfill \\ \mathcal{F}(\eta ,T{\varrho }^{\ast },v)=\mathcal{F}\left(L,M,v\right)\hfill \end{array}\right.$$

Since T is a $\left(\beta \text{-}\psi \right)$-proximal contraction, therefore

$$\begin{array}{c}\mathcal{F}\left({\varrho }_{n+1},\eta ,v\right)\hfill \\ \ge q\left({\varrho }_n,{\varrho }^{\ast },v\right)\mathcal{F}\left({\varrho }_{n+1},\eta ,v\right)\hfill \\ \ge \beta \left({\varrho }_n,{\varrho }^{\ast },v\right)\mathcal{F}\left({\varrho }_{n+1},\eta ,v\right)\hfill \\ \ge \psi (L_{\psi }\left(\mathcal{F}({\varrho }_n,{\varrho }^{\ast },v)\right)\hfill \\ =\psi \left(min\left\{\begin{array}{c}\mathcal{F}({\varrho }_n,{\varrho }^{\ast },v),\mathcal{F}({\varrho }_n,{\varrho }_{n+1},v),\hfill \\ \mathcal{F}({\varrho }^{\ast },{\varrho }_{n+1},v),\mathcal{F}({\varrho }^{\ast },\eta ,v)\hfill \end{array}\right\}\right)\hfill \end{array}$$

(18)

By applying the limit as n→∞ in Equation (18), we obtain

$$\begin{array}{c}\mathcal{F}\left({\varrho }^{\ast },\eta ,v\right)=\underset{n\to \infty }{lim}\mathcal{F}\left({\varrho }_{n+1},\eta ,v\right)\hfill \\ \ge \underset{n\to \infty }{lim}\psi \left(min\left\{\begin{array}{c}\mathcal{F}({\varrho }_n,{\varrho }^{\ast },v),\mathcal{F}({\varrho }_n,{\varrho }_{n+1},v),\hfill \\ \mathcal{F}({\varrho }^{\ast },{\varrho }_{n+1},v),\mathcal{F}({\varrho }^{\ast },\eta ,v)\hfill \end{array}\right\}\right)\hfill \\ =\psi \left(min\left\{1,1,1,\mathcal{F}({\varrho }^{\ast },\eta ,v)\right\}\right)\hfill \\ =\psi \left(\mathcal{F}({\varrho }^{\ast },\eta ,v)\right)>\mathcal{F}({\varrho }^{\ast },\eta ,v),\hfill \end{array}$$

which is a contradiction. Consequently, $\eta ={\varrho }^{\ast }$. Hence, with Equation (17), we find that

$$\mathcal{F}({\varrho }^{\ast },T{\varrho }^{\ast },v)=\mathcal{F}\left(L,M,v\right),$$

which implies that ${\varrho }^{\ast }$ is the BPP of $T.$ □

Example 1. 

Let $Z=\left\{0,1,2,3\right\},$$L=\left\{2,3\right\}$, and $M=\left\{0,1\right\}.$ We define $d:Z\times Z\to {\mathbb{R}}^{+}\cup \left\{0\right\}$ as follows:

$$d\left({\varrho }_1,{\varrho }_2\right)=\left|{\varrho }_1-{\varrho }_2\right|$$

$$\mathcal{F}\left({\varrho }_1,{\varrho }_2,v\right)={\displaystyle \frac{v}{v+\rho \left({\varrho }_1,{\varrho }_2\right)}}\mathit{where}\rho \left({\varrho }_1,{\varrho }_2\right)=d^2\left({\varrho }_1,{\varrho }_2\right).$$

It is easy to check that

$$\mathcal{F}\left(L,M,v\right)={\displaystyle \frac{v}{v+1}}.$$

Clearly, $L_0\left(v\right)=\left\{2\right\}$ and $M_0\left(v\right)=\left\{1\right\}.$ We define the mapping $T:L\to M\setminus \left\{\varphi \right\}$ as follows:

$$T\varrho =\left\{\begin{array}{c}\left\{1\right\},\varrho =2,\hfill \\ \left\{0,1\right\},\varrho =3.\hfill \end{array}\right.$$

Also, $T\left(\varrho \right)\subseteq M_0\left(v\right)$ for all $\varrho \in L_0\left(v\right).$ We define

$$\beta \left({\varrho }_0,{\varrho }_1,v\right)=\left\{\begin{array}{c}1,\mathit{for}{\varrho }_0,{\varrho }_1\in L,\hfill \\ 2,\mathit{otherwise}.\hfill \end{array}\right.$$

Then, ${\beta }_q$-proximal admissibility implies the following cases:

(i)

$u_1=2,{\varrho }_0=2$, and $u_2=2,{\varrho }_1=3;$

(ii)

$u_1=2,{\varrho }_0=2$, and $u_2=2,{\varrho }_1=2;$

(iii)

$u_1=2,{\varrho }_0=3$, and $u_2=2,{\varrho }_1=3.$

If we have $\psi \left(v\right)=\sqrt{v},$ then we can use

$$\beta ({\varrho }_0,{\varrho }_1,v)\mathcal{F}(u_1,u_2,v)\ge \psi \left(L_{\psi }({\varrho }_0,{\varrho }_1,v)\right),$$

where

$$L_{\psi }({\varrho }_0,{\varrho }_1,v)=min\left\{\mathcal{F}\left({\varrho }_0,{\varrho }_1,v\right),\mathcal{F}\left({\varrho }_0,u_1,v\right),\mathcal{F}\left({\varrho }_1,u_1,v\right),\mathcal{F}\left({\varrho }_1,u_2,v\right)\right\}.$$

Now, if

$$min\left\{\mathcal{F}\left({\varrho }_0,{\varrho }_1,v\right),\mathcal{F}\left({\varrho }_0,u_1,v\right),\mathcal{F}\left({\varrho }_1,u_1,v\right),\mathcal{F}\left({\varrho }_1,u_2,v\right)\right\}=\mathcal{F}\left({\varrho }_0,{\varrho }_1,v\right),$$

then for $u_1=2,{\varrho }_0=2$, and $u_2=2,{\varrho }_1=3$, we have

$$\mathcal{F}\left(u_1,T{\varrho }_0,v\right)=\mathcal{F}\left(L,M,v\right)=\mathcal{F}\left(u_2,T{\varrho }_1,v\right)$$

(19)

and

$$\mathcal{F}\left(u_1,u_2,v\right)=1\mathit{and}\mathcal{F}\left({\varrho }_0,{\varrho }_1,v\right)={\displaystyle \frac{v}{v+1}}.$$

(20)

Consequently, we obtain

$$\beta \left({\varrho }_0,{\varrho }_1,v\right)\mathcal{F}\left(u_1,u_2,v\right)=1\ge \sqrt{{\displaystyle \frac{v}{v+1}}}=\psi \left(\mathcal{F}\left({\varrho }_0,{\varrho }_1,v\right)\right)$$

Now, for the case where $u_1=2,{\varrho }_0=2$, and $u_2=2,{\varrho }_1=2$, we have

$$\mathcal{F}\left(u_1,T{\varrho }_0,v\right)=\mathcal{F}\left(L,M,v\right)=\mathcal{F}\left(u_2,T{\varrho }_1,v\right)$$

(21)

and

$$\mathcal{F}\left(u_1,u_2,v\right)=1=\mathcal{F}\left({\varrho }_0,{\varrho }_1,v\right).$$

(22)

Consequently, we obtain

$$\beta \left({\varrho }_0,{\varrho }_1,v\right)\mathcal{F}\left(u_1,u_2,v\right)=1\ge 1=\psi \left(1\right)=\psi \left(\mathcal{F}\left({\varrho }_0,{\varrho }_1,v\right)\right).$$

The same result is present for the last case, where $u_1=2,{\varrho }_0=3$ and $u_2=2,{\varrho }_1=3.$

Thus, all the conditions for Theorem 3 are satisfied, and ${\varrho }^{\ast }=2$ is the BPP of T in $L_0\left(v\right)$.

Corollary 1. 

Let $(Z,\mathcal{F},★)$ be a complete FMS and L be a non-empty closed subset of Z. Then, the mapping $T:L\to L$ satisfies

$$\alpha \left({\varrho }_0,{\varrho }_1,v\right)\xi \left(\mathcal{F}\left(T{\varrho }_0,T{\varrho }_1,v\right)\right)\le \phi \left(L_{\xi }({\varrho }_0,{\varrho }_1,v)\right)$$

for all $u_0,u_1,{\varrho }_0,{\varrho }_1\in L$, where

$$L_{\xi }({\varrho }_0,{\varrho }_1,v)=max\left\{\xi \left(\mathcal{F}({\varrho }_0,{\varrho }_1,v)\right),\xi \left(\mathcal{F}\left({\varrho }_0,u_1,v\right)\right),\xi \left(\mathcal{F}\left({\varrho }_1,u_1,v\right)\right)\right\}.$$

Moreover, the following conditions hold:

(i)

$\xi \left(r★w\right)\le \xi \left(r\right)+\xi \left(w\right)$ for all $r,w\in (0,1];$

(ii)

T is $\alpha$-admissible, and there exists ${\varrho }_0$ in L such that $\alpha \left({\varrho }_0,T{\varrho }_0,v\right)\ge 1$ for all $v>0;$

(iii)

$\left\{{\varrho }_n\right\}$ is a sequence in L such that $\alpha \left({\varrho }_n,{\varrho }_{n+1},v\right)\ge 1,$$n\in \mathbb{N}$, and ${\varrho }_n\stackrel{\mathcal{F}}{\to }\varrho$ as $n\to \infty$, which implies $\alpha \left({\varrho }_n,\varrho ,v\right)\ge 1$ for all $n\in \mathbb{N}$ and $v>0.$

Then, T has a fixed point in $L.$

Proof. 

Let $L=M$, and consider that $b=1$. Also, insert $p\left({\varrho }_n,{\varrho }_{n+1},v\right)=1=p\left({\varrho }_n,\varrho ,v\right)$ into Theorem 1. □

Corollary 2. 

Let $(Z,\mathcal{F},★)$ be a complete FMS and L be a non-empty closed subset of Z. The mapping $T:L\to L$ satisfies

$$\alpha \left({\varrho }_0,{\varrho }_1,v\right)\xi \left(\mathcal{F}\left(T{\varrho }_0,T{\varrho }_1,v\right)\right)\le k\left(L_{\xi }({\varrho }_0,{\varrho }_1,v)\right)$$

for all $u_0,u_1,{\varrho }_0,{\varrho }_1\in L$, where

$$L_{\xi }({\varrho }_0,{\varrho }_1,v)=max\left\{\xi \left(\mathcal{F}({\varrho }_0,{\varrho }_1,v)\right),\xi \left(\mathcal{F}\left({\varrho }_0,u_1,v\right)\right),\xi \left(\mathcal{F}\left({\varrho }_1,u_1,v\right)\right)\right\}.$$

Moreover, the following conditions hold:

(i)

$\xi \left(r★w\right)\le \xi \left(r\right)+\xi \left(w\right)$ for all $r,w\in (0,1];$

(ii)

T is $\alpha$-admissible, and there exists ${\varrho }_0$ in L such that $\alpha \left({\varrho }_0,T{\varrho }_0,v\right)\ge 1$ for all $v>0;$

(iii)

$\left\{{\varrho }_n\right\}$ is a sequence in L such that $\alpha \left({\varrho }_n,{\varrho }_{n+1},v\right)\ge 1,$$n\in \mathbb{N}$, and ${\varrho }_n\stackrel{\mathcal{F}}{\to }\varrho$ as $n\to \infty$, which implies $\alpha \left({\varrho }_n,\varrho ,v\right)\ge 1$ for all $n\in \mathbb{N}$ and $v>0.$

Then, T has a fixed point in $L.$

Proof. 

Let $L=M$, and consider that $\phi \left(t\right)=kt$ in Corollary 1. □

Remark 3. 

If we take $L_{\xi }({\varrho }_0,{\varrho }_1,v)=\xi \left(\mathcal{F}({\varrho }_0,{\varrho }_1,v)\right)$ in Corollary 2, then we obtain Theorem 3.4 in [24].

3. Conclusions

In this paper, we considered the problem of the existence of best proximity points of multivalued fuzzy proximal contractions of various types in the set-up of fuzzy b-metric spaces. Notably, many distance functions in applications do not satisfy the triangle inequality like fuzzy metrics do, rather, they satisfy a relaxed triangle inequality, like the inequality of (strong) fuzzy b-metrics. For instance, compare [29,30,31,32]. One example of such a function is the cosine distance function, which has been used in artificial intelligence to measure the similarity between different objects of vector spaces (for details, see [33]). Moreover, one can prove that the fuzzy distance induced by the cosine distance function is not fuzzy metric, rather, it is fuzzy b-metric. In this way, as a future direction, one can consider establishing a link between the best proximity point problems in the context of fuzzy b-metrics induced by functions like cosine distance functions and optimization problems in artificial intelligence.