Stochastic prey-predator model with small random immigration

1.   Introduction

Fixed point theory has a vital role in mathematics and applied sciences. Also, this theory has lot of applications in differential equations and integral equations to guarantee the existence and uniqueness of the solutions ^[1,2]. The Banach contraction principle ^[3] has an imperative role in fixed point theory. Since after the appearance of this principle, it has become very popular and there has been a lot of activity in this area. On the other hand, to establish Banach contraction principle in a more general structure, the notion of a metric space was generalized by Bakhtin ^[4] in 1989 by introducing the idea of a $b$-metric space. Afterwards, the same idea was further investigated by Czerwik ^[5] to establish different results in $b$-metric spaces. The study of $b$-metric spaces holds a prominent place in fixed point theory. Banach contraction principle is generalized in many ways by changing the main platform of the metric space ^[6,7,8,9].

Zadeh ^[10] introduced the notion of a fuzzy set theory to deal with the uncertain states in daily life. Motivated by the concept, Kramosil and Michálek ^[11] defined the idea of fuzzy metric spaces. Grabiec ^[12] gave contractive mappings on a fuzzy metric space and extended fixed point theorems of Banach and Edelstein in such spaces. Successively, George and Veeramani ^[13] slightly altered the concept of a fuzzy metric space introduced by Kramosil and Michálek ^[11] and then attained a Hausdorff topology and a first countable topology on it. Numerous fixed point theorems have been constructed in fuzzy metric spaces. For instance, see ^{[14,15,16,17,18,19,20]}.

Nǎdǎban ^[21] studied the notion of a fuzzy $b$-metric space and proved some results. Rakić et al. ^[22] (see also ^[23]) proved some new fixed point results in $b$-fuzzy metric spaces. The notion of a Hausdorff fuzzy metric on compact sets is introduced in ^[24] and recently studied by Shahzad et al. ^[25] to establish fixed point theorems for multivalued mappings in complete fuzzy metric spaces. In this paper, we use the idea of a fuzzy $b$-metric space and establish some fixed point results for multivalued mappings in Hausdorff fuzzy $b$-metric spaces. Some fixed point theorems are also derived from these results. Finally, we investigate the applicability of the obtained results to integral equations.

Throughout the article, fuzzy metric space and fuzzy $b$-metric space are denoted by FMS and FBMS, respectively.

2.   Preliminaries

Bakhtin ^[4] defined the notion of a $b$-metric space as follows:

Definition 2.1. ^[4] Let $\Omega$ be a non-empty set. For any real number $b\geq 1$, a function $d_{b}\colon \Omega \times \Omega \longrightarrow \mathbb{R}$ is called a $b$-metric if it satisfies the following properties for all $\zeta_{1}, \zeta_{2}, \zeta_{3} \in \Omega$:

$BM1$ : $d_{b}(\zeta_{1}, \zeta_{2})\geq 0$;

$BM2$ : $d_{b}(\zeta_{1}, \zeta_{2}) = 0$ if and only if $\zeta_{1} = \zeta_{2}$;

$BM3$ : $d_{b}(\zeta_{1}, \zeta_{2}) = d_{b}(\zeta_{2}, \zeta_{1})$ for all $\zeta_{1}, \zeta_{2} \in \Omega$;

$BM4$ : $d_{b}(\zeta_{1}, \zeta_{3}) \leq b\Bigl[d_{b}(\zeta_{1}, \zeta_{2})+d_{b}(\zeta_{2}, \zeta_{3})\Bigr]$.

The pair $(\Omega, d_{b})$ is called a $b$-metric space.

Definition 2.2. ^[26] A binary operation $* : [0, 1] \times [0, 1] \rightarrow [0, 1]$ is called a continuous $t$-norm if it satisfies the following conditions:

(1) $*$ is associative and commutative;

(2) $*$ is continuous;

(3) $\zeta * 1 = \zeta$ for all $\zeta \in [0, 1]$;

(4) $\zeta_{1} * \zeta_{2} \leq \zeta_{3}* \zeta_{4} \; \; \rm{wherever}\; \; \zeta_{1} \leq \zeta_{3} \; \; \rm{and}\; \; \zeta_{2} \leq \zeta_{4}$, for all $\zeta_{1}, \zeta_{2}, \zeta_{3}, \zeta_{4}\in [0, 1]$.

Example 2.1. Define a mapping $* : [0, 1] \times [0, 1] \rightarrow [0, 1]$ by

It is then obvious that $*$ is a continuous $t$-norm, known as the product norm.

George and Veermani ^[13] defined a fuzzy metric space as follows:

Definition 2.3. ^[13] Consider a nonempty set $\Omega,$ then $(\Omega, F, *)$ is a fuzzy metric space if $*$ is a continuous $t$-norm and $F$ is a fuzzy set on $\Omega \times \Omega \times[0, +\infty)$ satisfying the following for all $\zeta_{1}, \zeta_{2}, \zeta_{3} \in \Omega$ and $\alpha, \beta > 0$:

$[F1]:$ $F(\zeta_{1}, \zeta_{2}, \alpha) > 0 \; ;$

$[F2]:$ $F(\zeta_{1}, \zeta_{2}, \alpha) = 1$ if and only if $\zeta_{1} = \zeta_{2} \; ;$

$[F3]:$ $F(\zeta_{1}, \zeta_{2}, \alpha) = F(\zeta_{2}, \zeta_{1}, \alpha)\; ;$

$[F4]:$ $F(\zeta_{1}, \zeta_{3}, \alpha+\beta)\geq F(\zeta_{1}, \zeta_{2}, \alpha)*F(\zeta_{2}, \zeta_{3}, \beta)\; ;$

$[F5]:$ $F(\zeta_{1}, \zeta_{2}, .)\colon [0, +\infty) \rightarrow [0, 1]$ is continuous.

In ^[27], the idea of a fuzzy $b$-metric space is given as:

Definition 2.4. ^[27] Let $\Omega \neq \phi$ be a set, $b\geq 1$ be a real number and $*$ be a continuous $t$-norm. A fuzzy set $F_{b}$ on $\Omega\times \Omega \times [0, +\infty)$ is called a fuzzy $b$-metric on $\Omega$ if for all $\zeta_{1}, \zeta_{2}, \zeta_{3} \in \Omega$, the following conditions hold:

$[Fb1]:$ $F_{b}(\zeta_{1}, \zeta_{2}, \alpha) > 0 \; ;$

$[Fb2]:$ $F_{b}(\zeta_{1}, \zeta_{2}, \alpha) = 1, {\rm{ for \; all}} \; \alpha > 0$ if and only if $\zeta_{1} = \zeta_{2}\; ;$

$[Fb3]:$ $F_{b}(\zeta_{1}, \zeta_{2}, \alpha) = F_{b}(\zeta_{2}, \zeta_{1}, \alpha)\; ;$

$[Fb4]:$ $F_{b}(\zeta_{1}, \zeta_{3}, b(\alpha+\beta))\geq F_{b}(\zeta_{1}, \zeta_{2}, \alpha)* F_{b}(\zeta_{2}, \zeta_{3}, \beta)$ ${\rm{ for \; all}} \; \alpha, \beta \geq 0 \; ;$

$[Fb5]:$ $F_{b}(\zeta_{1}, \zeta_{2}, .)\colon (0, +\infty) \rightarrow [0, 1]$ is left continuous.

Example 2.2. Let $(\Omega, d_ b)$ be a $b$-metric space. Define a mapping $F_{b} \colon \Omega\times \Omega \times [0, +\infty)\rightarrow [0, 1]$ by

Then $(\Omega, F_{b}, \wedge)$ is a fuzzy $b$-metric space, where

$\wedge$ is a $t$-norm, known as the minimum $t$-norm.

Following Grabiec ^[12], we extend the idea of a $G$-Cauchy sequence and the notion of completeness in the FBMS as follows:

Definition 2.5. Let $(\Omega, F_{b}, *)$ be a FBMS.

$1)$ A sequence $\lbrace \zeta_{n}\rbrace$ in $\Omega$ is said to be a $G$-Cauchy sequence if $\lim\limits_{n\rightarrow +\infty}F_b(\zeta_n, \zeta_{n+q}, \alpha) = 1$ for $\alpha > 0$ and $q > 0$.

$2)$ A FBMS in which every $G$-Cauchy sequence is convergent is called a $G$-complete FBMS.

Similarly, for a FBMS $(\Omega, F_{b}, *)$, a sequence $\lbrace \zeta_{n}\rbrace$ in $\Omega$ is said to be convergent if there exits $\zeta \in \Omega$ such that for all $\alpha > 0$,

Definition 2.6. ^[25] Let $\mho\neq \phi$ be a subset of a FBMS $(\Omega, F, *)$ and $\alpha > 0$, then the fuzzy distance $\mathcal{F}$ of an element $\varrho_{1} \in \Omega$ and the subset $\mho\subset \Omega$ is

Note that $\mathcal{F}(\varrho_{1}, \mho, \alpha) = \mathcal{F}(\mho, \varrho_{1}, \alpha)$.

Lemma 2.1. ^[28] If $\Lambda \in CB(\Omega)$, then $\zeta_{1} \in \Lambda$ if and only if $\mathcal{F}(\Lambda, \zeta_{1}, \alpha) = 1$ for all $\alpha > 0$, where $CB(\Omega)$ is the collection of closed bounded subsets of $\Omega$.

Definition 2.7. ^[25] Let $(\Omega, F, *)$ be a FMS. Define a function $\Theta_{\mathcal{F}}$ on $\hat{C_{0}} (\Omega)\times \hat{C_{0}} (\Omega)\times (0, +\infty)$ by

for all $\Lambda, \mho\in \hat{C_{0}} (\Omega)$ and $\alpha > 0$, where $\hat{C_{0}} (\Omega)$ is the collection of all nonemty compact subsets of $\Omega$.

Lemma 2.2. ^[29] Let $(\Omega, F, *)$ be a complete FMS and $F(\zeta_{1}, \zeta_{2}, k\alpha)\geq F\left(\zeta_{1}, \zeta_{2}, \alpha\right)$ for all $\zeta_{1}, \zeta_{2} \in \Omega, \; k \in (0, 1)$ and $\alpha > 0$ then $\zeta_{1} = \zeta_{2}$.

Lemma 2.3. ^[25] Let $(\Omega, F, *)$ be a complete FMS such that $(\hat{C_{0}}, \Theta_{\mathcal{F}}, *)$ is a Hausdorff FMS on $\hat{C_{0}}$. Then for all $\Lambda, \mho\in\hat{C_{0}}$, for each $\zeta\in \Lambda$ and for $\alpha > 0$, there exists $\varrho_{\zeta}\in \mho$ so that $\mathcal{F}(\zeta, \mho, \alpha) = F(\zeta, \varrho_{\zeta}, \alpha)$ then

The notion of a Hausdorff FMS in Definition 2.6 of ^[25] can be extended naturally for a Hausdorff FBMS on $\hat{C_{0}}$ as follows:

Definition 2.8. Let $(\Omega, F_{b}, *)$ be a FBMS. Define a function $\Theta_{\mathcal{F}_{b}}$ on $\hat{C_{0}} (\Omega)\times \hat{C_{0}} (\Omega)\times (0, +\infty)$ by

for all $\Lambda, \mho\in \hat{C_{0}} (\Omega)$ and $\alpha > 0$.

3.   Main results

This section deals with the idea of Hausdorff FBMS and certain new fixed point results in a fuzzy FBMS. Note that, one can easily extend Lemma 2.1 to 2.3 in the setting of fuzzy $b$-metric spaces.

Lemma 3.1. If $\Lambda \in C\mho(\Omega),$ then $\zeta \in \Lambda$ if and only if $\mathcal{F}_{b}(\Lambda, \zeta, \alpha) = 1 \; {{for \; all}}\; \alpha > 0$.

Proof. Since

there exists a sequence $\lbrace \varrho_{n}\rbrace\subset \Lambda$ such that $F_b(\zeta, \varrho_{n}, \alpha) > 1-\dfrac{1}{n}.$ Letting $n\rightarrow +\infty,$ we get $\varrho_{n}\rightarrow \zeta.$ From $\Lambda \in C\mho(\Omega),$ it follows that $\zeta \in \Lambda.$ Conversely, if $\zeta \in \Lambda$, we have

Again, due to ^[17], the following fact follows from $[Fb5]$.

Lemma 3.2. Let $(\Omega, F_{b}, *)$ be a $G$-complete FBMS. If for two elements $\zeta, \varrho\in\Omega$ and for a number$k < 1$

then $\zeta = \varrho$.

Lemma 3.3. Let $(\Omega, F_{b}, *)$ be a $G$-complete FBMS, such that $(\hat{C_{0}}, \Theta_{\mathcal{F}_{b}}, *)$ is a Hausdorff FBMS on $\hat{C_{0}}$. Then for all $\Lambda, \mho\in\hat{C_{0}}$, foreach $\zeta\in \Lambda$ and for $\alpha > 0$ there exists $\varrho_{\zeta}\in \mho$, satisfying $\mathcal{F}_{b}(\zeta, \mho, \alpha) = F_{b}(\zeta, \varrho_{\zeta}, \alpha)$ also

Proof. If

then

Since for each $\zeta\in \Lambda$, there exists $\varrho_{\zeta}\in\mho$ satisfying

Hence,

Now, if

this implies

for some $\varrho_{\zeta}\in \mho.$ Hence, in both cases, the result is proved.

Theorem 3.1. Let $(\Omega, F_{b}, *)$ be a $G$-complete FBMS with $b\geqslant 1$ and $\Theta_{\mathcal{F}_{b}}$ be aHausdorff FBMS. Let $S \colon \Omega \rightarrow \hat{C_{0}} (\Omega)$ be a multivalued mapping satisfying

for all $\zeta, \varrho \in \Omega$, where $bk < 1,$then $S$ has a fixed point.

Proof. For $a_{0} \in\Omega$, we choose a sequence $\lbrace \zeta_{n}\rbrace$ in $\Omega$ as follows: Let $a_{1} \in \Omega$ such that $a_{1} \in Sa_{0}$. By using Lemma 6, we can choose $a_{2} \in Sa_{1}$ such that

By induction, we have $a_{n+1} \in Sa_{n}$ satisfying

Now, by (3.1) together with Lemma 3.3, we have

For any $q \in \mathbb{N}$, writing $\alpha = \frac{\alpha}{qb}+\frac{\alpha(q-1)}{qb}$ and using $[Fb4]$ to get

Again, writing $\frac{(q-1)\alpha}{qb} = \frac{\alpha}{qb}+\frac{\alpha(q-2)}{qb}$ together with $[Fb4]$, we have

Continuing in the same way and using $[Fb4]$ repeatedly for $(q-2)$ more steps, we obtain

Using (3.2) and $[Fb5]$, we get

Consequently,

Since for all $n, q\in \mathbb{N}$, we have $bk < 1$, taking limit as $n \rightarrow +\infty$, we get

Hence, $\lbrace a_{n}\rbrace$ is a $G$-Cauchy sequence. Then, the $G$-completeness of $\Omega$ implies that there exists $z \in \Omega$ such that

By Lemma 3.1, we have $z \in Sz$. Hence, $z$ is a fixed point for $S$.

Example 3.1. Let $\Omega = \left[0, 1\right]$ and $F_{b}(\zeta, \varrho, \alpha) = \dfrac{\alpha}{\alpha+(\zeta-\varrho)^{2}}$.

It is easy to verify that $(\Omega, F_{b}, *)$ is a $G$-complete FBMS with $b\geq1.$

For $k\in(0, 1)$, define a mapping $S \colon\Omega \rightarrow \hat{C_{0}} (\Omega)$ by

In the case $\zeta = \varrho$, we have

For $\zeta\neq \varrho$, we have the following cases:

If $\zeta = 0$ and $\varrho\in(0, 1]$, we have

It follows that $\Theta_{\mathcal{F}_{b}}(S(0), S(\varrho), k\alpha) > F_{b}(0, \varrho, \alpha) = \dfrac{\alpha}{\alpha+\varrho^{2}}.$

If $\zeta$ and $\varrho\in(0, 1]$, an easy calculation with either possibility of supremum and infimum, yield that

Thus, for all cases, we have

Hence, all the conditions of Theorem 3.1 are satisfied and 0 is a fixed point of $S$.

Theorem 3.2. Let $(\Omega, F_{b}, *)$ be a $G$-complete FBMS with $b\geqslant 1$ and $\Theta_{\mathcal{F}_{b}}$ be a Hausdorff FBMS. Let $S \colon \Omega \rightarrow \hat{C_{0}} (\Omega)$ be a multivalued mapping which satisfies

for all $\zeta, \varrho \in \Omega$, where $bk < 1,$then $S$ has a fixed point.

Proof. In the same way as in Theorem 3.1 for $a_{0} \in\Omega$, we choose a sequence $\lbrace a_{n}\rbrace$ in $\Omega$ as follows: Let $a_{1} \in \Omega$ such that $a_{1} \in Sa_{0}.$ By Lemma 2.3, we can choose $a_{2} \in Sa_{1}$ such that

By induction, we have $a_{n+1} \in Sa_{n}$ satisfying

Now, by (3.3) together with Lemma 3.3, we have

If

then (3.4) implies

Then nothing to prove by Lemma 3.2. If

then from (3.4) we have

By adopting the same procedure as in Theorem 3.1 after inequality (3.2), we can complete the proof.

Remark 3.1. By taking $b = 1$ in Theorem 3.2, Theorem 2.1 of ^[25] can be obtained.

Theorem 3.3. Let $(\Omega, F_{b}, *)$ be a $G$-complete FBMS (with $b\geqslant 1$) and $\Theta_{\mathcal{F}_{b}}$ be a Hausdorff FBMS. Let $S \colon \Omega \rightarrow \hat{C}_{0}(\Omega)$ be a multivalued map which satisfies

for all $\zeta, \varrho \in\Omega$, where $bk < 1,$then $S$ has a fixed point.

Proof. Starting same way as in Theorem 3.1, we have

By induction, we have $a_{n+1} \in Sa_{n}$ satisfying

Now, by (3.5) together with Lemma 3.3, we have

If

then (3.6) implies

Then nothing to prove by Lemma 3.2.

If

then from (3.6), we have

By adopting the same procedure as in Theorem 3.1 after inequality (3.2), we can complete the proof.

Next, a corollary of Theorem 3.3 is given.

Corollary 3.1. Let $(\Omega, F, *)$ be a $G$-complete FMS and $\Theta_{\mathcal{F}}$ be a Hausdorff FMS. Let $S \colon\Omega \rightarrow \hat{C}_{0}(\Omega)$ be a multivalued mapping satisfying

for all $\zeta, \varrho \in\Omega$, where $0 < k < 1,$ then $S$ has a fixed point.

Proof. Taking $b = 1$ in Theorem 3.3, one can complete the proof.

Theorem 3.4. Let $(\Omega, F_{b}, *)$ be a $G$-complete FBMS with $b\geqslant 1$ and $\Theta_{\mathcal{F}_{b}}$ be a Hausdorff FBMS. Let $S \colon \Omega \rightarrow \hat{C}_{0}(\Omega)$ be a multivalued map which satisfies

for all $\zeta, \varrho \in\Omega$, where $bk < 1,$then $S$ has a fixed point.

Proof. For $a_{0} \in \Omega$, we choose a sequence $\lbrace x_{n}\rbrace$ in $\Omega$ as follows: Let $a_{1} \in \Omega$ such that $a_{1} \in Sa_{0}.$ By using Lemma 2.3, we can choose $a_{2} \in Sa_{1}$ such that

By induction, we have $a_{n+1} \in Sa_{n}$ satisfying

Now, by (3.7) together with 3.3, we have

If

so (3.8) implies

Then nothing to prove by Lemma 3.2. While, if

then from (3.6) we have

By adopting the same procedure as in Theorem 3.1 after inequality (3.2) we can complete the proof.

The same result in fuzzy metric spaces is stated as follows.

Corollary 3.2. Let $(\Omega, F, *)$ be a $G$-complete FMS and $\Theta_{\mathcal{F}}$ be a Hausdorff FMS. Let $S \colon \Omega\rightarrow \hat{C_{0}}(\Omega)$ be a multivalued mapping satisfying

for all $\zeta, \varrho \in \Omega,$then $S$ has a fixed point.

Proof. Taking $b = 1$ in Theorem 3.4, one can complete the proof.

4.   Consequences

This section is about the construction of some fixed point results involving integral inequalities as consequences of our results. Define a function $\tau \colon [0, +\infty) \rightarrow [0, +\infty)$ by

where $\tau(\alpha)$ is a non-decreasing and continuous function. Moreover, $\psi(\alpha) > 0$ for $\alpha > 0$ and $\psi(\alpha) = 0$ if and only if $\alpha = 0.$

Theorem 4.1. Let $(\Omega, F_{b}, *)$ be a complete fuzzy $b$-metric space and $\Theta_{\mathcal{F}_{b}}$ be a Hausdorff fuzzy $b$-metric space. Let $S \colon \Omega \rightarrow \hat{C}_{0}(\Omega)$ be a multivalued mapping satisfying

for all $\zeta, \varrho \in\Omega$, where $bk < 1,$ then $S$ has a fixed point.

Proof. Taking (4.1) in account, (4.2) implies that

Since $\tau$ is continuous and non-decreasing, we have

The rest of the proof follows immediately from Theorem 3.1.

A more general form of Theorem 4.1 can be stated as an immediate consequence of Theorem 3.2.

Theorem 4.2. Let $(\Omega, F_{b}, *)$ be a complete FBMS and $\Theta_{\mathcal{F}_{b}}$ be a Hausdorff FBMS. Let $S \colon \Omega\rightarrow \hat{C}_{0}(\Omega)$ be a multivalued mapping satisfying

where

for all $\zeta, \varrho \in\Omega$, where $bk < 1,$ then $S$ has a fixed point.

Proof. Taking (4.1) in account, (4.3) implies that

Since $\tau$ is continuous and non-decreasing, we have

The rest of the proof follows immediately from Theorem 3.2.

Remark 4.1. By taking $b = 1$ in Theorem 4.2, Theorem 3.1 of ^[25] can be obtained.

Theorem 4.3. Let $(\Omega, F_{b}, *)$ be a complete FBMS and $\Theta_{\mathcal{F}_{b}}$ be a Hausdorff FBMS. Let $S \colon \Omega\rightarrow \hat{C}_{0}(\Omega)$ be a multivalued mapping satisfying

where

for all $\zeta, \varrho \in\Omega$, where $bk < 1,$ then $S$ has a fixed point.

Note that if $\beta(\zeta, \varrho, \alpha) = F_{b}(\zeta, \varrho, \alpha)$, then the above result follows from Theorem 4.1. Similar results on integral inequalities can be obtained as a consequence of Theorem 3.4.

Theorem 4.4. Let $(\Omega, F_{b}, *)$ be a complete FBMS and $\Theta_{\mathcal{F}_{b}}$ be a Hausdorff FBMS. Let $S \colon \Omega \rightarrow \hat{C}_{0}(\Omega)$ be a multivalued mapping satisfying

where

for all $\zeta, \varrho \in \Omega$, where $bk < 1$, then $S$ has a fixed point.

5.   An application

Nonlinear integral equations arise in a variety of fields of physical science, engineering, biology, and applied mathematics ^[31,32]. This theory in abstract spaces is a rapidly growing field with lot of applications in analysis as well as other branches of mathematics ^[33].

Fixed point theory is a valuable tool for the existence of a solution of different kinds of integral as well as differential inclusions, such as ^[33,34,35]. Many authors provided a solution of different integral inclusions in this context, for instance see ^{[30,36,37,38,39,40]}. In this section, a Volterra-type integral inclusion as an application of Theorem 3.1 is studied.

Consider $\Omega = C([0, 1], \mathbb{R})$ as the space of all continuous functions defined on $[0, 1]$ and define the $G$-complete fuzzy $b$-metric on $\Omega$ by

for all $\alpha > 0$ and $\xi, \varrho \in \Omega.$

Consider the integral inclusion:

where $G \colon [0, 1]\times[0, 1] \times \mathbb{R}\rightarrow P_{cv}(\mathbb{R})$ is a multivalued continuous function.

For the above integral inclusion, we define a multivalued operator $S: \Omega\rightarrow \hat{C_{0}} (\Omega)$ by

The next result proves the existence of a solution of the integral inclusion (5.1).

Theorem 5.1. Let $S: \Omega \rightarrow \hat{C_{0}} (\Omega)$ be the multivalued integral operator given by

Suppose the following conditions are satisfied:

1) $G \colon [0, 1]\times[0, 1] \times \mathbb{R}\rightarrow P_{cv}(\mathbb{R})$ is such that $G(\varepsilon, \sigma, \xi(\sigma))$ is lower semi-continuous in $[0, 1]\times[0, 1]$;

2) For all $\varepsilon, \sigma \in[0, 1]$, $f(\varepsilon, \sigma) \in \Omega$ and for all $\xi, \varrho \in\Omega$, we have

where $f \colon [0, 1]\rightarrow [0, +\infty)$ is continuous;

3) There exists $0 < k < 1$ such that

Then the integral inclusion (5.1) has the solution in $\Omega.$

Proof. For $G \colon [0, 1]\times[0, 1] \times \mathbb{R}\rightarrow P_{cv}(\mathbb{R})$, it follows from Michael's selection theorem that there exists a continuous operator

such that $G_{i}(\varepsilon, \sigma, \xi(\sigma))\in G(\varepsilon, \sigma, \xi(\sigma))$ for all $\varepsilon, \sigma \in[0, 1].$ It follows that

hence $S(\xi(\varepsilon))\neq \emptyset$ and closed. Moreover, since $h(\varepsilon)$ is continuous on $[0, 1]$, and $G$ is continuous, their ranges are bounded. This means that $S(\xi(\varepsilon))$ is bounded and $S(\xi(\varepsilon))\in\hat{C_{0}} (\Omega)$. For $q, r \in \Omega$, there exist $q(\varepsilon)\in S(\xi(\varepsilon))$ and $r(\varepsilon)\in S(\varrho(\varepsilon))$ such that

and

It follows from item 5.1 that

Now,

So, we have

By interchanging the roll of $\xi$ and $\varrho$, we reach to

Hence, $S$ has a fixed point in $\Omega$, which is a solution of the integral inclusion (5.1).

6.   Conclusions

In this article we proved certain fixed point results for Hausdorff fuzzy $b$-metric spaces. The main results are validated by an example. Theorem 3.2 generalizes the result of ^[25]. These results extend the theory of fixed points for multivalued mappings in a more general class of fuzzy $b$-metric spaces. For instance, some fixed point results can be obtained by taking $b = 1$ (corresponding to $G$-complete FMSs). An application for the existence of a solution for a Volterra type integral inclusion is also provided.

Acknowledgments

The author Aiman Mukheimer would like to thank Prince Sultan University for paying APC and for the support through the TAS research LAB.

Conflict of interest

The authors declare no conflict of interest.