On extended, and extended rectangular, Menger probabilistic b-metric spaces: applications to the existence of solutions of integral, and fractional differential, equations

Abstract

We introduce the notions of extended Menger probabilistic b-metric space and extended rectangular Menger probabilistic b-metric space. Thereafter, we obtain new fixed-point results for single- and multi-valued mappings in both spaces. Additionally, we present two applications, involving integral, and fractional differential, equations, proving the existence of solutions.

Appendix

Herein, non-trivial examples are given for Theorems 1 and 2. Additionally, examples concerning eMPbM and eRMPbM spaces are provided.

Example 1

Let us consider ${\mathcal {Q}} = \{1,2,3\}$, ${\mathcal {T}}(\zeta ,\varsigma )=\min \{\zeta ,\varsigma \}$, and $\epsilon (\zeta ,\varsigma )=\frac{1}{\zeta +\varsigma }$. Also, let us introduce ${\mathbb {E}}^{e}: {\mathcal {Q}} \times {\mathcal {Q}} \rightarrow D^{+}$ as

$$\begin{aligned} {\mathbb {E}}^{e}_{ \zeta ,\varsigma } (t) ={\left\{ \begin{array}{ll} \frac{t}{t+\vert \zeta -\varsigma \vert ^{2}}, \quad \text{ if } t>0, \\ 0,\quad \text{ otherwise }, \end{array}\right. } \end{aligned}$$

where $\alpha \ge \beta >1$. We can show that ${\mathbb {E}}^{e}$ is an eMPbM.

Conditions (Ee1) and (Ee2) are trivial. Let $t=1$ and $s=2$. For (Ee3), we get

$$\begin{aligned} {\mathbb {E}}^{e}_{1,3}(t+s)=\frac{1+2}{1+2+\vert 1-3\vert ^{2}} \ge \min \{{\mathbb {E}}^{e}_{1,2}(\epsilon (1,3)t), {\mathbb {E}}^{e}_{2,3}(\epsilon (1,3)s)\}=\frac{(1+3)^{-1}}{(1+3)^{-1}+1}. \end{aligned}$$

For other cases, we can proceed in a similar way. Therefore, for every $\zeta ,\varsigma \in {\mathcal {Q}}$ with distinct $u,v \in {\mathcal {Q}} {\setminus } \{\zeta ,\varsigma \}$, we obtain

$$\begin{aligned} {\mathbb {E}}^{e}_{ \zeta ,\varsigma }(t+s)\ge {\mathcal {T}}({\mathbb {E}}^{e}_{ \zeta ,u}(\epsilon (\zeta ,\varsigma ) t),{\mathbb {E}}^{e}_{ u,v}(\epsilon (\zeta ,\varsigma )s)). \end{aligned}$$

Thus, $({\mathcal {Q}},{\mathbb {E}}^{e},{\mathcal {T}})$ is an eMPbM-space. Nevertheless, it is not an MPM-space, because

$$\begin{aligned} {\mathbb {E}}^{e}_{1,3}(t+s)=\frac{1+2}{1+2+\vert 1-3\vert ^{2}} \le \min \{{\mathbb {E}}^{e}_{1,2}(t),{\mathbb {E}}^{e}_{2,3}(s)\}=\frac{1}{1+\vert 1-2\vert ^{2}}. \end{aligned}$$

Example 2

Let us suppose that ${\mathcal {Q}} ={\mathbb {R}}^{+}$, ${\mathcal {T}}(\zeta ,\varsigma )=\min \{\zeta ,\varsigma \}$ and ${\mathbb {E}}^{e}: {\mathcal {Q}} \times {\mathcal {Q}} \rightarrow D^{+}$ is

$$\begin{aligned} {\mathbb {E}}^{e}_{ \zeta ,\varsigma } (t) ={\left\{ \begin{array}{ll} \frac{t}{t+\vert \zeta -\varsigma \vert ^{2}}, \quad \text{ if } t>0 \\ 0\quad \text{ otherwise }, \end{array}\right. } \end{aligned}$$

for every $\zeta ,\varsigma \in {\mathcal {Q}}$ and $\epsilon (\zeta ,\varsigma )=\frac{1}{\zeta +\varsigma }$ (Hasanvand and Khanehgir 2015). Let us introduce the function $\varpi : {\mathcal {Q}} \rightarrow {\mathcal {Q}}$ with $\varpi \zeta =\frac{\zeta }{4}$ and $\varphi : {\mathbb {R}}^{ +}\rightarrow {\mathbb {R}}^{ +}$ with $\varphi (t) = t$. Also, let us set $c=\frac{1}{2}$ and $\lambda =1$. From

$$\begin{aligned} {\mathbb {E}}^{e}_{\varpi \zeta ,\varpi \varsigma }(\epsilon (\zeta ,\varsigma )^{k}\varphi (t))&\ge \lambda \min \lbrace {\mathbb {E}}^{e}_{\zeta ,\varsigma }(\epsilon (\zeta ,\varsigma )^{k-1}\varphi (\frac{t}{c}),\\&\quad {\mathbb {E}}^{e}_{\zeta ,\varpi \zeta }(\epsilon (\zeta ,\varsigma )^{k-1}\varphi (\frac{t}{c}), {\mathbb {E}}^{e}_{\varpi \varsigma ,\varsigma }(\epsilon (\zeta ,\varsigma )^{k-1}\varphi (\frac{t}{c})\rbrace , \end{aligned}$$

we obtain

$$\begin{aligned} {\mathbb {E}}^{e}_{\varpi \zeta ,\varpi \varsigma }(\epsilon (\zeta ,\varsigma )^{k}\varphi (t))&=\frac{\frac{1}{(\zeta +\varsigma )^{k}}t}{\frac{1}{(\zeta +\varsigma )^{k}}t +\vert \frac{\zeta }{4}-\frac{\varsigma }{4}\vert ^{2}} =\frac{t}{t+(\zeta +\varsigma )^{k-4}\vert \zeta -\varsigma \vert ^{2}}\\&\ge \frac{t}{t+(\zeta +\varsigma )^{k-2}\vert \zeta -\varsigma \vert ^{2}}\\&={\mathbb {E}}^{e}_{\zeta ,\varsigma } (\epsilon (\zeta ,\varsigma )^{k-1}\varphi (\frac{t}{c}))\\&\ge \min \lbrace {\mathbb {E}}^{e}_{\zeta ,\varsigma } (\epsilon (\zeta ,\varsigma )^{k-1}\varphi (\frac{t}{c}), {\mathbb {E}}^{e}_{\zeta ,\varpi \zeta }(\epsilon (\zeta ,\varsigma )^{k-1}\varphi (\frac{t}{c}),\\&\quad {\mathbb {E}}^{e}_{\varpi \varsigma ,\varsigma } (\epsilon (\zeta ,\varsigma )^{k-1}\varphi (\frac{t}{c})\rbrace \\&=\lambda \min \lbrace {\mathbb {E}}^{e}_{\zeta ,\varsigma } (\epsilon (\zeta ,\varsigma )^{k-1}\varphi (\frac{t}{c}),{\mathbb {E}}^{e}_{\zeta ,\varpi \zeta } (\epsilon (\zeta ,\varsigma )^{k-1}\varphi (\frac{t}{c}),\\ {}&\qquad {\mathbb {E}}^{e}_{\varpi \varsigma ,\varsigma } (\epsilon (\zeta ,\varsigma )^{k-1}\varphi (\frac{t}{c})\rbrace . \end{aligned}$$

Thus, Theorem 1 implies that $\varpi$ has a FP.

Example 3

Let us consider ${\mathcal {Q}} = \{1,2,3,4\}$, ${\mathcal {T}}(\zeta ,\varsigma )=\min \{\zeta ,\varsigma \}$, and $\epsilon (\zeta ,\varsigma )=\frac{1}{\zeta +\varsigma }$. Also, let us introduce ${\mathbb {E}}^{\zeta }: {\mathcal {Q}} \times {\mathcal {Q}} \rightarrow D^{+}$ as

$$\begin{aligned} {\mathbb {E}}^{\zeta }_{ \zeta ,\varsigma } (t) = 1 \ if\ \zeta =\varsigma ,\\ {\mathbb {E}}^{\zeta }_{\zeta ,\varsigma }(t) ={\mathbb {E}}^{\zeta }_{\varsigma ,\zeta }(t)\ for \ all\ \zeta ,\varsigma \in {\mathcal {Q}},\\ {\mathbb {E}}^{\zeta }_{1,2}(t)=\frac{t}{t+\alpha },\\ {\mathbb {E}}^{\zeta }_{1,4}(t)={\mathbb {E}}^{\zeta }_{2,4}(t) ={\mathbb {E}}^{\zeta }_{3,4}(t)=\frac{t}{t+4\alpha },\\ {\mathbb {E}}^{\zeta }_{1,3}(t)={\mathbb {E}}^{\zeta }_{2,3}(t) =\frac{3t}{3t+\beta }, \end{aligned}$$

where $\alpha \ge \beta >1$. We can show that ${\mathbb {E}}^{\zeta }$ is an eRMPbM.

Conditions (Ex1) and (Ex2) are trivial. For (Ex3), we get

$$\begin{aligned} {\mathbb {E}}^{\zeta }_{1,2}(t+s+z)=\frac{t}{t+\alpha } \ge \min \{{\mathbb {E}}^{\zeta }_{1,3}(\epsilon (1,2)t), {\mathbb {E}}^{\zeta }_{2,4}(\epsilon (1,2)s),{\mathbb {E}}^{\zeta }_{3,4}(\epsilon (1,2)z)\} =\frac{\frac{t}{1+2}}{\frac{t}{1+2}+4\alpha }. \end{aligned}$$

For other cases, we can proceed in a similar way. Therefore, for every $\zeta ,\varsigma \in {\mathcal {Q}}$ with distinct $u,v \in {\mathcal {Q}} {\setminus } \{\zeta ,\varsigma \}$, we obtain

$$\begin{aligned} {\mathbb {E}}^{\zeta }_{ \zeta ,\varsigma }(t+s+z) \ge {\mathcal {T}}({\mathcal {T}}({\mathbb {E}}^{\zeta }_{ \zeta ,u}(\epsilon (\zeta ,\varsigma ) t), {\mathbb {E}}^{\zeta }_{ u,v}(\epsilon (\zeta ,\varsigma )s)){\mathbb {E}}^{\zeta }_{ v,\varsigma }(\epsilon (\zeta ,\varsigma )z).). \end{aligned}$$

Thus, $({\mathcal {Q}},{\mathbb {E}}^{\zeta },{\mathcal {T}})$ is an eRMPbM-space, but not an eMPbM-space, because

$$\begin{aligned} {\mathbb {E}}^{\zeta }_{1,2}(t+s)=\frac{t}{t+\alpha }\ngeq \min \{{\mathbb {E}}^{\zeta }_{1,3} (\epsilon (1,2)t),{\mathbb {E}}^{\zeta }_{3,2}(\epsilon (1,2)s)\}=\frac{t}{t+\beta }. \end{aligned}$$

Example 4

Let us assume that ${\mathcal {Q}} =[0,1]$, ${\mathbb {E}}^{\zeta }$ and ${\mathcal {T}}$ are as in Example 3. Therefore, $({\mathcal {Q}},{\mathbb {E}}^{\zeta },{\mathcal {T}})$ is a complete eRMPbM-space with $\epsilon (\zeta ,\varsigma ) =\frac{\zeta +\varsigma }{2}$. Let us introduce the function $\varpi : {\mathcal {Q}} \rightarrow CB({\mathcal {Q}})$ as $\varpi \zeta =[0,\frac{\zeta }{4}]$ and $\varphi : {\mathbb {R}}^{ +}\rightarrow {\mathbb {R}}^{ +}$ by $\varphi (t) = t$. Using the definition of the probabilistic Hausdorff metric in Definition 5, we get

$$\begin{aligned} a_{1}H_{\varpi \zeta ,\varpi \varsigma }(\alpha ^{k}\varphi (t))=a_{1}\frac{\frac{1}{2^{k}}t}{\frac{1}{2^{k}}t+\vert \frac{\zeta }{4}-\frac{\varsigma }{4}\vert ^{2}}&=a_{1}\frac{t}{t+2^{k-4}\vert \zeta -\varsigma \vert ^{2}}\\&\ge a_{1}\frac{t}{t+2^{k-2}\vert \zeta -\varsigma \vert ^{2}}\\&=a_{1}{\mathbb {E}}^{\zeta }_{\zeta ,\varsigma }(\alpha ^{k-1}\varphi (\frac{t}{c}))\\&\ge -a_{4}{\mathbb {E}}^{\zeta }_{\zeta ,\varsigma }(\alpha ^{k-1}\varphi (\frac{t}{c}))-a_{5} {\mathbb {E}}^{\zeta }_{\zeta ,u}(\alpha ^{k-1}\varphi (\frac{t}{c})), \end{aligned}$$

where, $c=\frac{1}{2},a_{1}=1,a_{4}=-2, a_{5}=-1$ and $a_{2}=a_{3}=0$. Since $\varpi$ has compact values, it has ${\mathbb {E}}$-APV and closed mapping. On the other hand, by the definition of $\varpi$, all conditions of Theorem 2 are satisfied. Hence, Theorem 2 implies that $\varpi$ has an FP.