Fixed Point Results in Modified Intuitionistic Fuzzy Soft Metric Spaces with Application

Abstract

This paper establishes a new type of space, modified intuitionistic fuzzy soft metric space (MIFSMS). Basic properties and topological structures are defined in the setting of this new notion with valid examples. Moreover, we have given some new results along with suitable examples to show their validity. An application for finding the solution of an integral equation is also given by utilizing our newly developed results.

1. Introduction

The concept of vagueness came into existence when it was difficult for the interval mathematics to solve problems with different uncertainties. To tackle with such issues, Zadeh [1] gave fuzzy sets (FS) that were very well defined by using the indicator function. Many properties and important results are given in [1] that build a new field for the researchers to explore. Thereafter, K.T. Atanassov [2] presented an intuitionistic fuzzy set (IFS) consisting of membership and non membership functions as well.

Experiments solving issues in economics, engineering, environmental science, sociology, medical science and many other fields deal with the complex problems of modeling uncertain data. While some mathematical theories such as “probability theory, fuzzy set theory, rough set theory, vague set theory and the interval mathematics” are practical in defining uncertainty, each of these concepts has their own drawbacks. Further, in case of data containing parameters, Molodtsov [3] gave the concept of soft sets to deal with the uncertainties. Soft sets have applications in several fields including the “smoothness of functions, game theory, operations research, Riemann integration, Perron integration, probability theory and measurement theory”. Especially, it has been very well applied to soft decision making problems. Das and Samanta [4,5,6] applied the concept of soft sets to metric spaces (MS) and hence presented Soft Metric Spaces (SMS) using soft points of soft sets. Maji et al. [7] in 2001 introduced Fuzzy Soft Sets. Beaula and Gunaseli [8] applied the MS concept to Fuzzy Soft Sets and hence introduced Fuzzy Soft Metric Spaces (FSMS) using a fuzzy soft point and defined some of its characteristics. See also [9,10,11]. Saadiat et al. [12] gave a vital concept of Modified Intuitionistic Fuzzy Metric Spaces (MIFMS) by availing the continuous t-representable norm.

2. Preliminaries

In the given section, $\mathit{\chi }$ denotes the universe, $\Omega$ depicts the parameter set, $\overline{\Omega }$ represents the absolute soft set, and $SP(\overline{\mathit{\chi }})$ denotes the set consisting of all the soft points of $\overline{\mathit{\chi }}$.

Definition 1 

([3]). A 2-tuple $(\Delta ,\Omega )$ depicts a soft set on a universe χ where Ω represents the parameter set and Δ denotes the map from Ω to power set of χ, i.e., $\Delta :\Omega \to P(\mathit{\chi })$.

Definition 2 

([7]). A soft set $(\Delta ,\Omega )$ is an absolute soft set if $\Delta (\mathit{\iota })=\mathit{\chi }$ for every $\mathit{\iota }\in \Omega$.

Definition 3 

([7]). A soft point is a soft set $(\Delta ,\Omega )$ if $\Delta (\mathit{\iota })=\{\mathit{\kappa }\}$ for any $\mathit{\kappa }\in \mathit{\chi }$ and $\Delta (\mathit{\lambda })=\mathit{\varphi }$ for $\mathit{\lambda }\in \mathit{\chi }/\{\mathit{\iota }\}$.

Definition 4 

([4,5]). A 3-tuple $(\overline{\mathit{\chi }},\overline{\mathit{\rho }},\Omega )$ denotes a SMS, where $\overline{\mathit{\rho }}:SP(\overline{\mathit{\chi }})\times SP(\overline{\mathit{\chi }})\to R(\Omega )$ is the soft metric and $R(\Omega )$ is the set having non-negative soft real numbers with $\overline{\mathit{\rho }}$ satisfying the given assertions for all ${\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},{\overline{\mathit{\lambda }}}_{e_3}\in SP(\overline{\mathit{\chi }})$:

(i) 

$\overline{\mathit{\rho }}({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2})>\overline{0}$;

(ii) 

$\overline{\mathit{\rho }}({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2})=\overline{0}$ iff ${\overline{\mathit{\iota }}}_{e_1}={\overline{\mathit{\kappa }}}_{e_2}$;

(iii) 

$\overline{\mathit{\rho }}({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2})=\overline{\mathit{\rho }}({\overline{\mathit{\kappa }}}_{e_2},{\overline{\mathit{\iota }}}_{e_1})$;

(iv) 

$\overline{\mathit{\rho }}({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2})\le \overline{\mathit{\rho }}({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\lambda }}}_{e_3})+\overline{\mathit{\rho }}({\overline{\mathit{\lambda }}}_{e_3},{\overline{\mathit{\kappa }}}_{e_2})$.

Definition 5 

([7]). A 2-tuple $(\Delta ,\Omega )$ over a universe χ is a fuzzy soft set, where Ω represents the parameter set and Δ is the map from Ω to $F(\mathit{\chi })$ which is the set having fuzzy subsets in universe χ, i.e., $\Delta :\Omega \to F(\mathit{\chi })$.

Definition 6 

([8]). A 3-tuple  $(\overline{\mathit{\chi }},\Delta ,\ast )$ is a soft fuzzy metric space (SFMS), where SFM on $\overline{\mathit{\chi }}$ is given by map $\Delta :SP(\overline{\mathit{\chi }})\times SP(\overline{\mathit{\chi }})\times (0,\infty )\to [0,1]$ satisfying the below assertions for all ${\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},{\overline{\mathit{\lambda }}}_{e_3}\in SP(\overline{\mathit{\chi }})$ and $\mathit{\rho },\mathit{\varrho }>0$:

(i) 

$\Delta ({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},\mathit{\varrho })>0$;

(ii) 

$\Delta ({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},\mathit{\varrho })=1$ iff ${\overline{\mathit{\iota }}}_{e_1}={\overline{\mathit{\kappa }}}_{e_2}$;

(iii) 

$\Delta ({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},\mathit{\varrho })=\Delta ({\overline{\mathit{\kappa }}}_{e_2},{\overline{\mathit{\iota }}}_{e_1},\mathit{\varrho })$;

(iv) 

$\Delta ({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},\mathit{\varrho }+\mathit{\rho })\ge \Delta ({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\lambda }}}_{e_3},\mathit{\varrho })\ast \Delta ({\overline{\mathit{\lambda }}}_{e_3},{\overline{\mathit{\kappa }}}_{e_2},\mathit{\rho })$;

(v) 

$\Delta ({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},.):(0,\infty )\to [0,1]$ is continuous.

Definition 7 

([12]). A 3-tuple  $(\mathit{\chi },M_{\mathit{\omega },\mathit{\psi }},\tau )$ is a MIFMS, where χ is an arbitrary set, $\mathit{\omega },\mathit{\psi }$ depicts the fuzzy sets from ${\mathit{\chi }}^2\times (0,\infty )$ to $[0,1]$ asserting $\mathit{\omega }(\mathit{\iota },\mathit{\kappa },\mathit{\varrho })+\mathit{\psi }(\mathit{\iota },\mathit{\kappa },\mathit{\varrho })\le 1$ for every $\mathit{\iota },\mathit{\kappa }\in \mathit{\chi }$ and $\mathit{\varrho }>0$, continuous t-representable norm is denoted by τ and $M_{\mathit{\omega },\mathit{\psi }}$ depicts a map ${\mathit{\chi }}^2\times (0,\infty )\to L^{*}$ that satisfies the below assertions for every $\mathit{\iota },\mathit{\kappa },\mathit{\lambda }\in \mathit{\chi }$ and $\mathit{\rho },\mathit{\varrho }>0$:

(i) 

$M_{\mathit{\omega },\mathit{\psi }}(\mathit{\iota },\mathit{\kappa },\mathit{\varrho }){>}_{L^{*}}{0}_{L^{*}}$;

(ii) 

$M_{\mathit{\omega },\mathit{\psi }}(\mathit{\iota },\mathit{\kappa },\mathit{\varrho })=1_{L^{*}}$ iff $\mathit{\iota }=\mathit{\kappa }$;

(iii) 

$M_{\mathit{\omega },\mathit{\psi }}(\mathit{\iota },\mathit{\kappa },\mathit{\varrho })=M_{\mathit{\omega },\mathit{\psi }}(\mathit{\kappa },\mathit{\iota },\mathit{\varrho })$;

(iv) 

$M_{\mathit{\omega },\mathit{\psi }}(\mathit{\iota },\mathit{\kappa },\mathit{\varrho }+\mathit{\rho }){\ge }_{L^{*}}\tau (M_{\mathit{\omega },\mathit{\psi }}(\mathit{\iota },\mathit{\lambda },\mathit{\varrho }),M_{\mathit{\omega },\mathit{\psi }}(\mathit{\lambda },\mathit{\kappa },\mathit{\rho }))$;

(v) 

$M_{\mathit{\omega },\mathit{\psi }}(\mathit{\iota },\mathit{\kappa },.):(0,\infty )\to L^{*}$ is continuous.

Here, MIFM $M_{\mathit{\omega },\mathit{\psi }}$ is given as

$$\begin{array}{c}\hfill M_{\mathit{\omega },\mathit{\psi }}(\mathit{\iota },\mathit{\kappa },\mathit{\varrho })=(\mathit{\omega }(\mathit{\iota },\mathit{\kappa },\mathit{\varrho }),\mathit{\psi }(\mathit{\iota },\mathit{\kappa },\mathit{\varrho })).\end{array}$$

3. Modified Intuitionistic Fuzzy Soft Metric Space

Definition 8 

([13]). A map Θ on $L^{*}$, $\Theta :{(L^{*})}^2\to L^{*}$ represents a triangular norm $(t-norm)$ if it satisfies the below assertions:

(i) 

$\Theta (\mathit{\iota },1_{L^{*}})=\mathit{\iota }$, for all $\mathit{\iota }\in L^{*}$;

(ii) 

$\Theta (\mathit{\iota },\mathit{\kappa })=\Theta (\mathit{\kappa },\mathit{\iota })$, for all $(\mathit{\iota },\mathit{\kappa })\in {(L^{*})}^2$;

(iii) 

$\Theta (\mathit{\iota },\Theta (\mathit{\kappa },\mathit{\lambda }))=\Theta (\Theta (\mathit{\iota },\mathit{\kappa }),\mathit{\lambda })$, for all $(\mathit{\iota },\mathit{\kappa },\mathit{\lambda })\in {(L^{*})}^3$;

(iv) 

$\mathit{\iota }\le {\mathit{\iota }}^{\prime }$ and $\mathit{\kappa }\le {\mathit{\kappa }}^{\prime }$⇒$\Theta (\mathit{\iota },\mathit{\kappa }){\le }_{L^{*}}\Theta ({\mathit{\iota }}^{\prime },{\mathit{\kappa }}^{\prime })$ , for all $(\mathit{\iota },{\mathit{\iota }}^{\prime },\mathit{\kappa },{\mathit{\kappa }}^{\prime })\in {(L^{*})}^4$.

Definition 9 

([13]). A continuous t-representable norm is a continuous $t-norm$ Θ on $L^{*}$ if it implies the existence of a $t-conorm$ ⋄ on $[0,1]$ which is continuous so that

$$\begin{array}{c}\hfill \Theta (\mathit{\iota },\mathit{\kappa })=({\mathit{\iota }}_1\ast {\mathit{\kappa }}_1,{\mathit{\iota }}_2\diamond {\mathit{\kappa }}_2)\end{array}$$

for every $\mathit{\iota }=({\mathit{\iota }}_1,{\mathit{\iota }}_2)$, $\mathit{\kappa }=({\mathit{\kappa }}_1,{\mathit{\kappa }}_2)\in L^{*}$.

Definition 10. 

A 3-tuple $(\overline{\mathit{\chi }},M_{\mathit{\omega },\mathit{\psi }},\Theta )$ is a MIFSMS, where $\overline{\mathit{\chi }}$ is any set, ω and ψ are SFS, Θ denotes a t-representable norm possessing continuity and $M_{\mathit{\omega },\mathit{\psi }}$ represents a map from $SP(\overline{\mathit{\chi }})\times SP(\overline{\mathit{\chi }})\times (0,\infty )$ to $L^{*}$ fulfilling the below assertions for all ${\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},{\overline{\mathit{\lambda }}}_{e_3}\in SP(\overline{\mathit{\chi }})$ and $\mathit{\rho },\mathit{\varrho }>0$:

(i) 

$M_{\mathit{\omega },\mathit{\psi }}({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},\mathit{\varrho }){>}_{L^{*}}{0}_{L^{*}}$;

(ii) 

$M_{\mathit{\omega },\mathit{\psi }}({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},\mathit{\varrho })=1_{L^{*}}$ iff ${\overline{\mathit{\iota }}}_{e_1}={\overline{\mathit{\kappa }}}_{e_2}$;

(iii) 

$M_{\mathit{\omega },\mathit{\psi }}({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},\mathit{\varrho })=M_{\mathit{\omega },\mathit{\psi }}({\overline{\mathit{\kappa }}}_{e_2},{\overline{\mathit{\iota }}}_{e_1},\mathit{\varrho })$;

(iv) 

$M_{\mathit{\omega },\mathit{\psi }}({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},\mathit{\varrho }+\mathit{\rho }){\ge }_{L^{*}}\Theta (M_{\mathit{\omega },\mathit{\psi }}({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\lambda }}}_{e_3},\mathit{\varrho }),M_{\mathit{\omega },\mathit{\psi }}({\overline{\mathit{\lambda }}}_{e_3},{\overline{\mathit{\kappa }}}_{e_2},\mathit{\rho }))$;

(v) 

$M_{\mathit{\omega },\mathit{\psi }}({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},.):(0,\infty )\to L^{*}$ is continuous.

Then, $M_{\mathit{\omega },\mathit{\psi }}$ is MIFSM on $\overline{\mathit{\chi }}$ where level of closeness and non closeness between ${\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2}$ w.r.t. ϱ is depicted by the functions $\mathit{\omega }({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},\mathit{\varrho })$ and $\mathit{\psi }({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},\mathit{\varrho })$ respectively and metric $M_{\mathit{\omega },\mathit{\psi }}$ is given as

$$\begin{array}{c}\hfill M_{\mathit{\omega },\mathit{\psi }}({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},\mathit{\varrho })=(\mathit{\omega }({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},\mathit{\varrho }),\mathit{\psi }({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},\mathit{\varrho })).\end{array}$$

Remark 1. 

The function $\mathit{\omega }({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},\mathit{\varrho })$ is increasing and the function $\mathit{\psi }({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},\mathit{\varrho })$ is decreasing in a MIFSMS, for every ${\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2}\in SP(\overline{\mathit{\chi }})$.

Example 1. 

Take $(\overline{\mathit{\chi }},d)$ a SMS and $\mathit{\omega },\mathit{\psi }$ soft fuzzy sets on $SP(\overline{\mathit{\chi }})\times SP(\overline{\mathit{\chi }})\times (0,\infty )$ given as below:

$$\begin{array}{cc}\hfill M_{\mathit{\omega },\mathit{\psi }}({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},\mathit{\varrho })& =(\mathit{\omega }({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},\mathit{\varrho }),\mathit{\psi }({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},\mathit{\varrho }))\hfill \\ \hfill & =({\displaystyle \frac{h{\mathit{\varrho }}^n}{h{\mathit{\varrho }}^n+md({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2})}},{\displaystyle \frac{md({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2})}{h{\mathit{\varrho }}^n+md({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2})}})\hfill \end{array}$$

for all $h,\mathit{\varrho },m,n\in R^{+};\Theta (\overline{r},\overline{s})=({\overline{r}}_1{\overline{s}}_1,min({\overline{r}}_2+{\overline{s}}_2,1))$, where $\overline{r}=({\overline{r}}_1,{\overline{r}}_2)$ and $\overline{s}=({\overline{s}}_1,{\overline{s}}_2)\in L^{*}$. Then, 3-tuple $(\overline{\mathit{\chi }},M_{\mathit{\omega },\mathit{\psi }},\Theta )$ is a MIFSMS.

Example 2. 

Consider $\mathit{\chi }=N$. Define $\Theta (\overline{\mathcal{ȷ}},\overline{\mathcal{\ell }})=(max\{0,\overline{{\mathcal{ȷ}}_1}+\overline{{\mathcal{\ell }}_1}-1\},\overline{{\mathcal{ȷ}}_2}+\overline{{\mathcal{\ell }}_2}-\overline{{\mathcal{ȷ}}_2}\overline{{\mathcal{\ell }}_2})$, where $\overline{\mathcal{ȷ}}=(\overline{{\mathcal{ȷ}}_1},\overline{{\mathcal{ȷ}}_2})$ and $\overline{\mathcal{\ell }}=(\overline{{\mathcal{\ell }}_1},\overline{{\mathcal{\ell }}_2})\in L^{*}$. Take $\mathit{\omega },\mathit{\psi }$ be soft fuzzy sets on $SP(\overline{\mathit{\chi }})\times SP(\overline{\mathit{\chi }})\times (0,\infty )$ given as $M_{\mathit{\omega },\mathit{\psi }}({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},\mathit{\varrho })=(\mathit{\omega }({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},\mathit{\varrho }),\mathit{\psi }({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},\mathit{\varrho }))$

$$\begin{array}{c}\hfill M_{\mathit{\omega },\mathit{\psi }}({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},\mathit{\varrho })=\left\{\begin{array}{c}(\frac{{\overline{\mathit{\iota }}}_{e_1}}{{\overline{\mathit{\kappa }}}_{e_2}},\frac{{\overline{\mathit{\kappa }}}_{e_2}-{\overline{\mathit{\iota }}}_{e_1}}{{\overline{\mathit{\kappa }}}_{e_2}})\mathit{if}{\overline{\mathit{\iota }}}_{e_1}\le {\overline{\mathit{\kappa }}}_{e_2}\hfill \\ (\frac{{\overline{\mathit{\kappa }}}_{e_2}}{{\overline{\mathit{\iota }}}_{e_1}},\frac{{\overline{\mathit{\iota }}}_{e_1}-{\overline{\mathit{\kappa }}}_{e_2}}{{\overline{\mathit{\iota }}}_{e_1}})\mathit{if}{\overline{\mathit{\kappa }}}_{e_2}\le {\overline{\mathit{\iota }}}_{e_1}\hfill \end{array}\right.\end{array}$$

for all ${\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2}\in SP(\overline{\mathit{\chi }})$ and $\mathit{\varrho }>0$. Then, 3-tuple $(\overline{\mathit{\chi }},M_{\mathit{\omega },\mathit{\psi }},\Theta )$ is also a MIFSMS.

Lemma 1. 

Let $(\overline{\mathit{\chi }},M_{\mathit{\omega },\mathit{\psi }},\tau )$ be MIFSMS. Then, $M_{\mathit{\omega },\mathit{\psi }}({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},\mathit{\varrho })$ is increasing with respect to $\mathit{\varrho }>0$ in $(L^{*},{\le }_{L^{*}})$ for every ${\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2}\in SP(\overline{\mathit{\chi }})$.

Definition 11. 

Let $(\overline{\mathit{\chi }},M_{\mathit{\omega },\mathit{\psi }},\tau )$ be MIFSMS. $M_{\mathit{\omega },\mathit{\psi }}$ possesses continuity on $SP(\overline{\mathit{\chi }})\times SP(\overline{\mathit{\chi }})\times (0,\infty )$ if

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{M}_{\mathit{\omega },\mathit{\psi }}({\overline{\mathit{\iota }}}_{e_n},{\overline{\mathit{\kappa }}}_{e_n},{\mathit{\varrho }}_n)=M_{\mathit{\omega },\mathit{\psi }}({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},\mathit{\varrho })\end{array}$$

where sequence $\{({\overline{\mathit{\iota }}}_{e_n},{\overline{\mathit{\kappa }}}_{e_n},{\mathit{\varrho }}_n)\}$ converges to $({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},\mathit{\varrho })$.

Lemma 2. 

For $(\overline{\mathit{\chi }},M_{\mathit{\omega },\mathit{\psi }},\tau )$ be a MIFSMS, then $M_{\mathit{\omega },\mathit{\psi }}$ possesses continuity on $SP(\overline{\mathit{\chi }})\times SP(\overline{\mathit{\chi }})\times (0,\infty )$.

4. Results and Discussion

Definition 12. 

An open ball in a MIFSMS $(\overline{\mathit{\chi }},M_{\mathit{\omega },\mathit{\psi }},\tau )$ is depicted by $B({\overline{\mathit{\iota }}}_{e_1},\mathit{ϵ},\mathit{\varrho })$ having center ${\overline{\mathit{\iota }}}_{e_1}\in \overline{\mathit{\chi }}$ and radius $0<\mathit{ϵ}<1$ for any $\mathit{\varrho }>0$ and is given as,

$$\begin{array}{c}\hfill B({\overline{\mathit{\iota }}}_{e_1},\mathit{ϵ},\mathit{\varrho })=\{{\overline{\mathit{\kappa }}}_{e_2}\in \overline{\mathit{\chi }}:M_{\mathit{\omega },\mathit{\psi }}({\overline{\mathit{\iota }}}_{e_1},{\overline{\mathit{\kappa }}}_{e_2},\mathit{\varrho }){>}_{L^{*}}(N_s(\mathit{ϵ}),\mathit{ϵ})\}.\end{array}$$

Theorem 3. 

Every open ball $B({\overline{u}}_{e_1},\mathit{ϵ},\mathit{\varrho })$ in MIFSMS is an open set.

Proof. 

Take ${\overline{v}}_{e_2}\in B({\overline{u}}_{e_1},\mathit{ϵ},\mathit{\varrho })$, then $M_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_1},{\overline{v}}_{e_2},\mathit{\varrho }){>}_{L^{*}}(N_s(\mathit{ϵ}),\mathit{ϵ})$, so

$$\begin{array}{c}\hfill \mathit{\omega }({\overline{u}}_{e_1},{\overline{v}}_{e_2},\mathit{\varrho })>1-\mathit{ϵ},\mathit{\psi }({\overline{u}}_{e_1},{\overline{v}}_{e_2},\mathit{\varrho })<\mathit{ϵ}.\end{array}$$

Since $\mathit{\omega }({\overline{u}}_{e_1},{\overline{v}}_{e_2},\mathit{\varrho })>1-\mathit{ϵ}$, there exists ${\mathit{\varrho }}_o\in (0,\mathit{\varrho })$ so that $\mathit{\omega }({\overline{u}}_{e_1},{\overline{v}}_{e_2},{\mathit{\varrho }}_o)>1-\mathit{ϵ}$ and $\mathit{\psi }({\overline{u}}_{e_1},{\overline{v}}_{e_2},{\mathit{\varrho }}_o)<\mathit{ϵ}$.

Put ${\mathit{ϵ}}_o=\mathit{\omega }({\overline{u}}_{e_1},{\overline{v}}_{e_2},{\mathit{\varrho }}_o)$.

Since ${\mathit{ϵ}}_o>1-\mathit{ϵ}$, then there exists $\eta \in (0,1)$ so that ${\mathit{ϵ}}_0>1-\eta >1-\mathit{ϵ}$.

Now, for given $\mathit{ϵ}$ and $\eta$ such that ${\mathit{ϵ}}_o>1-\eta$, there exist ${\mathit{ϵ}}_1$,${\mathit{ϵ}}_2\in (0,1)$ such that $\tau ({\mathit{ϵ}}_o^{\prime },{\mathit{ϵ}}_1^{\prime }){>}_{L^{*}}(N_s(\eta ),\eta )$, where ${\mathit{ϵ}}_o^{\prime }=({\mathit{ϵ}}_o,1-{\mathit{ϵ}}_o)$ and ${\mathit{ϵ}}_1^{\prime }=({\mathit{ϵ}}_1,1-{\mathit{ϵ}}_2)$.

Let ${\mathit{ϵ}}_3=$max$\{{\mathit{ϵ}}_1,{\mathit{ϵ}}_2\}$ and open ball $B({\overline{v}}_{e_2},1-{\mathit{ϵ}}_3,\mathit{\varrho }-{\mathit{\varrho }}_o)$.

Claim that $B({\overline{v}}_{e_2},1-{\mathit{ϵ}}_3,\mathit{\varrho }-{\mathit{\varrho }}_o)\subset B({\overline{u}}_{e_1},\mathit{ϵ},\mathit{\varrho })$.

Consider ${\overline{w}}_{e_3}\in B({\overline{v}}_{e_2},1-{\mathit{ϵ}}_3,\mathit{\varrho }-{\mathit{\varrho }}_o)$, then $M_{\mathit{\omega },\mathit{\psi }}({\overline{v}}_{e_2},{\overline{w}}_{e_3},\mathit{\varrho }-{\mathit{\varrho }}_o){>}_{L^{*}}(N_s(1-{\mathit{ϵ}}_3),{\mathit{ϵ}}_3)$, so

$$\begin{array}{c}\hfill \mathit{\omega }({\overline{v}}_{e_2},{\overline{w}}_{e_3},\mathit{\varrho }-{\mathit{\varrho }}_o)>{\mathit{ϵ}}_3,\mathit{\psi }({\overline{v}}_{e_2},{\overline{w}}_{e_3},\mathit{\varrho }-{\mathit{\varrho }}_o)<1-{\mathit{ϵ}}_3.\end{array}$$

Now, $M_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_1},{\overline{w}}_{e_3},\mathit{\varrho }){\ge }_{L^{*}}\tau (M_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_1},{\overline{v}}_{e_2},{\mathit{\varrho }}_o),M_{\mathit{\omega },\mathit{\psi }}({\overline{v}}_{e_2},{\overline{w}}_{e_3},\mathit{\varrho }-{\mathit{\varrho }}_o)){\ge }_{L^{*}}(N_s(\mathit{ϵ}),\mathit{ϵ})$.

Thus, ${\overline{w}}_{e_3}\in B({\overline{u}}_{e_1},\mathit{ϵ},\mathit{\varrho })$. Hence, $B({\overline{v}}_{e_2},1-{\mathit{ϵ}}_3,\mathit{\varrho }-{\mathit{\varrho }}_o)\subset B({\overline{u}}_{e_1},\mathit{ϵ},\mathit{\varrho })$. □

Remark 2. 

The topology generated by $M_{\mathit{\omega },\mathit{\psi }}$ on $\overline{\mathit{\chi }}$ in MIFSMS $(\overline{\mathit{\chi }},M_{\mathit{\omega },\mathit{\psi }},\tau )$ is given as

${\tau }_{\mathit{\omega },\mathit{\psi }}$=$\{Y\subseteq \overline{\mathit{\chi }}$: for every ${\overline{u}}_{e_1}\in Y$, there exist $\mathit{\varrho }>0$ and $\mathit{ϵ}\in (0,1)$ so that $B({\overline{u}}_{e_1},\mathit{ϵ},\mathit{\varrho })\subseteq Y\}$.

Theorem 4. 

If $(\overline{\mathit{\chi }},M_{\mathit{\omega },\mathit{\psi }},\tau )$ is a MIFSMS, then it is a Hausdorff space.

Proof. 

Given that $(\overline{\mathit{\chi }},M_{\mathit{\omega },\mathit{\psi }},\tau )$ is MIFSMS, consider ${\overline{u}}_{e_1},{\overline{v}}_{e_2}\in \overline{\mathit{\chi }}$ be two distinct points, then

$$\begin{array}{c}\hfill 0_{L^{*}}{<}_{L^{*}}{M}_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_1},{\overline{v}}_{e_2},\mathit{\varrho }){<}_{L^{*}}{1}_{L^{*}}.\end{array}$$

Let $s_1=\mathit{\omega }({\overline{u}}_{e_1},{\overline{v}}_{e_2},\mathit{\varrho })$, $s_2=\mathit{\psi }({\overline{u}}_{e_1},{\overline{v}}_{e_2},\mathit{\varrho })$ and $s=$max$\{s_1,1-s_2\}$.

For each $s_0\in (s,1)$, there exist $s_3,s_4$ such that $\tau ((s_3,1-s_4),(s_3,1-s_4)){\ge }_{L^{*}}(N_s(1-s_0),s_0)$.

Let $s_5=$max$\{s_3,s_4\}$.

Consider two MIFSOBs $B({\overline{u}}_{e_1},1-s_5,\frac{\mathit{\varrho }}{2})$ and $B({\overline{v}}_{e_2},1-s_5,\frac{\mathit{\varrho }}{2})$.

Claim that $B({\overline{u}}_{e_1},1-s_5,\frac{t}{2})\cap B({\overline{v}}_{e_2},1-s_5,\frac{\mathit{\varrho }}{2})$ = $\mathit{\varphi }$.

Let ${\overline{w}}_{e_3}\in B({\overline{u}}_{e_1},1-s_5,\frac{\mathit{\varrho }}{2})\cap B({\overline{v}}_{e_2},1-s_5,\frac{\mathit{\varrho }}{2})$.

$$\begin{array}{cc}\hfill (s_1,s_2)=& M_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_1},{\overline{v}}_{e_2},\mathit{\varrho })\hfill \\ \hfill & {\ge }_{L^{*}}\tau (M_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_1},{\overline{w}}_{e_3},\frac{\mathit{\varrho }}{2}),M_{\mathit{\omega },\mathit{\psi }}({\overline{w}}_{e_3},{\overline{v}}_{e_2},\frac{\mathit{\varrho }}{2}))\hfill \\ \hfill & {\ge }_{L^{*}}\tau ((s_5,1-s_5),(s_5,1-s_5)){\ge }_{L^{*}}(N_s(1-s_0),s_0)\hfill \\ \hfill & {>}_{L^{*}}(s_1,s_2),\hfill \end{array}$$

which is contrary. So $(\overline{\mathit{\chi }},M_{\mathit{\omega },\mathit{\psi }},\tau )$ is a Hausdorff space. □

Definition 13. 

A subset Y of $\overline{\mathit{\chi }}$ in a MIFSMS $(\overline{X},M_{\mathit{\omega },\mathit{\psi }},\tau )$ is called IF-bounded if it implies the existence of $\mathit{\varrho }>0$ and $0<\mathit{ϵ}<1$ so that $M_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_1},{\overline{v}}_{e_2},t){>}_{L^{*}}(N_s(\mathit{ϵ}),\mathit{ϵ})$ for each ${\overline{u}}_{e_1},{\overline{v}}_{e_2}$∈ Y.

Theorem 5. 

If $(\overline{\mathit{\chi }},M_{\mathit{\omega },\mathit{\psi }},\tau )$ is MIFSMS and $Y\subseteq \overline{\mathit{\chi }}$ is compact, then Y is IF-bounded.

Proof. 

Consider $\mathit{\varrho }>0$ and $0<\mathit{ϵ}<1$.

Let $\{B({\overline{u}}_{e_1},\mathit{ϵ},t):\overline{u_{e_1}}\in Y\}$ be an open cover of Y.

As Y is given to be compact, there exist ${\overline{u}}_{e_1}$, ${\overline{u}}_{e_2}$,…,${\overline{u}}_{e_n}$∈ Y such that $Y \subseteq$ ${\cup }_{i=1}^n$$B({\overline{u}}_{e_i},\mathit{ϵ},t)$.

Let ${\overline{u}}_{e_1}$, ${\overline{v}}_{e_2}$∈ Y, then ${\overline{u}}_{e_1}$∈ $B({\overline{u}}_{e_i},\mathit{ϵ},\mathit{\varrho })$ and ${\overline{v}}_{e_2}$ ∈ $B({\overline{u}}_{e_j},\mathit{ϵ},\mathit{\varrho })$ for some i,j; then

$$\begin{array}{c}\hfill M_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_1},{\overline{u}}_{e_i},\mathit{\varrho }){>}_{L^{*}}(N_s(\mathit{ϵ}),\mathit{ϵ}),M_{\mathit{\omega },\mathit{\psi }}({\overline{v}}_{e_2},{\overline{u}}_{e_j},\mathit{\varrho }){>}_{L^{*}}(N_s(\mathit{ϵ}),\mathit{ϵ}).\end{array}$$

Let $\alpha =max\{\mathit{\omega }({\overline{u}}_{e_i},{\overline{u}}_{e_j},\mathit{\varrho }):1\le i,j\le p\}$ and $\beta =max\{\mathit{\psi }({\overline{u}}_{e_i},{\overline{u}}_{e_j},\mathit{\varrho }):1\le i,j\le p\}$.

Then, $\alpha ,\beta >0$.

Now,

$$\begin{array}{cc}\hfill M_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_1},{\overline{v}}_{e_2},3\mathit{\varrho })& {>}_{L^{*}}{\tau }^2(M_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_1},{\overline{u}}_{e_i},\mathit{\varrho }),M_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_i},{\overline{u}}_{e_j},\mathit{\varrho }),M_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_j},{\overline{v}}_{e_2},\mathit{\varrho }))\hfill \\ \hfill & {\ge }_{L^{*}}{\tau }^2((1-\mathit{ϵ},\mathit{ϵ}),(\alpha ,\beta ),(1-\mathit{ϵ},\mathit{ϵ}))\hfill \\ & {>}_{L^{*}}(N_s({\eta }_1),N_s({\eta }_2))\hfill \end{array}$$

for some $0<{\eta }_1,{\eta }_2<1$.

Let $\eta =max\{{\eta }_1,{\eta }_2\}$ and ${\mathit{\varrho }}^{\prime }=3\mathit{\varrho }$, then $M_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_1},{\overline{v}}_{e_2},{\mathit{\varrho }}^{\prime }){>}_{L^{*}}(N_s(\eta ),\eta )$ for all ${\overline{u}}_{e_1},{\overline{v}}_{e_2}\in Y$.

Hence, Y is IF-bounded. □

Theorem 6. 

If $(\overline{\mathit{\chi }},M_{\mathit{\omega },\mathit{\psi }},\tau )$ is a MIFSMS and ${\tau }_{\mathit{\omega },\mathit{\psi }}$ is a topology on $\overline{\mathit{\chi }}$, then ${\overline{u}}_{e_n}\to {\overline{u}}_{e_1}$ if

$$\begin{array}{c}\hfill l{t}_{n\to \infty }{M}_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_n},{\overline{u}}_{e_1},\mathit{\varrho })=1_{L^{*}}\end{array}$$

for ${\overline{u}}_{e_n}$ in $\overline{\mathit{\chi }}$.

Proof. 

Take $\mathit{\varrho }>0$.

Consider ${\overline{u}}_{e_n}\to {\overline{u}}_{e_1}$, then there exists $n_0\in N$ so that ${\overline{u}}_{e_n}\in B({\overline{u}}_{e_1},\mathit{ϵ},\mathit{\varrho })$ for all $n\ge n_0,\mathit{ϵ}\in (0,1)$; then, $M_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_n},{\overline{u}}_{e_1},\mathit{\varrho }){>}_{L^{*}}(N_s(\mathit{ϵ}),\mathit{ϵ})$.

Hence, $l{t}_{n\to \infty }{M}_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_n},{\overline{u}}_{e_1},\mathit{\varrho })=1_{L^{*}}$.

Conversely, let $l{t}_{n\to \infty }{M}_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_n},{\overline{u}}_{e_1},\mathit{\varrho })=1_{L^{*}}$, thus for $\mathit{ϵ}\in (0,1)$, there exists $n_0\in N$ satisfying $M_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_n},{\overline{u}}_{e_1},\mathit{\varrho }){>}_{L^{*}}(N_s(\mathit{ϵ}),\mathit{ϵ})$ for each $n\ge n_0$.

Thus, ${\overline{u}}_{e_n}\in B({\overline{u}}_{e_1},\mathit{ϵ},\mathit{\varrho })$ where $n\ge n_0$.

Hence, ${\overline{u}}_{e_n}\to {\overline{u}}_{e_1}$. □

Definition 14. 

Consider $(\overline{\mathit{\chi }},M_{\mathit{\omega },\mathit{\psi }},\tau )$ to be a MIFSMS and $\{{\overline{u}}_{e_n}\}$ a sequence in $\overline{\mathit{\chi }}$, then

1.

The sequence is Cauchy iff for every $\mathit{\varrho }>0$, which implies the existence of ${\delta }_o\in N$ that satisfies

$$\begin{array}{c}\hfill l{t}_{n_o\to \infty }{M}_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_n},{\overline{u}}_{e_{n+m}},t)=1_{L^{*}}\end{array}$$

where $n,m\ge {\delta }_o$.

2.

The sequence converges to $\overline{u}$ iff for every $\mathit{\varrho }>0$

$$\begin{array}{c}\hfill l{t}_{n\to \infty }{M}_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_n},\overline{u},\mathit{\varrho })=1_{L^{*}}\end{array}$$

Definition 15. 

A MIFSMS $(\overline{\mathit{\chi }},M_{\mathit{\omega },\mathit{\psi }},\tau )$ is complete iff every Cauchy sequence converges in it.

Theorem 7. 

If any Cauchy sequence in MIFSMS $(\overline{\mathit{\chi }},M_{\mathit{\omega },\mathit{\psi }},\tau )$ has a subsequence that converges in it, then it is a complete space.

Proof. 

Consider $\{{\overline{u}}_{e_n}\}$ be any sequence which is Cauchy and $\{{\overline{u}}_{e_{n_i}}\}$ be any of its subsequence converging to $\overline{u}$.

Claim that ${\overline{u}}_{e_n}\to \overline{u}$.

Take $\mathit{\varrho }>0$ and $\mathit{ϵ}\in (0,1)$.

Consider $s\in (0,1)$ such that

$$\begin{array}{c}\hfill \tau ((1-s,s),(1-s,s)){\ge }_{L^{*}}(N_s(\mathit{ϵ}),\mathit{ϵ}).\end{array}$$

Since sequence $\{u_{e_n}\}$ is given as Cauchy, there exist $e_{n_o}\in N$ such that

$$\begin{array}{c}\hfill M_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_m},{\overline{u}}_{e_n},\frac{\mathit{\varrho }}{2}){>}_{L^{*}}(N_s(s),s)\end{array}$$

for all $e_m,e_n\ge e_{n_o}$.

Since ${\overline{u}}_{e_{n_i}}\to \overline{u}$, there exist positive integer $e_{i_p}$ such that $e_{i_p}>e_{n_o}$,

$$\begin{array}{c}\hfill M_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_i},\overline{u},\frac{\mathit{\varrho }}{2}){>}_{L^{*}}(N_s(s),s).\end{array}$$

For, if $e_n\ge e_{n_o}$, we have

$$\begin{array}{cc}\hfill M_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_n},\overline{u},\mathit{\varrho })& {\ge }_{L^{*}}\tau (M_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_n},{\overline{u}}_{e_{i_p}},\frac{\mathit{\varrho }}{2}),M_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_{i_p}},\overline{u},\frac{\mathit{\varrho }}{2}))\hfill \\ \hfill & {>}_{L^{*}}\tau ((1-s,s),(1-s,s))\hfill \\ \hfill & {\ge }_{L^{*}}(N_s(\mathit{ϵ}),\mathit{ϵ}).\hfill \end{array}$$

Thus, ${\overline{u}}_{e_{n_i}}\to \overline{u}$ and $(\overline{\mathit{\chi }},M_{\mathit{\omega },\mathit{\psi }},\tau )$ is a complete space. □

5. Fixed Point Theorems

In this section, we have extended Gregori-Sapena’s [14] and Zikic’s fixed point Theorem [15] to MIFSMS.

Definition 16. 

A sequence $\{t_n\}$ is known as s-increasing if there exists $n_o\in N$ such that

$$\begin{array}{c}\hfill t_n+1\le t_{n+1},\end{array}$$

for all $n\ge n_o$.

Theorem 8. 

Consider $(\overline{\mathit{\chi }},M_{\mathit{\omega },\mathit{\psi }},\tau )$ be complete MIFSMS, so that for every s-increasing sequence $\{{\mathit{\varrho }}_n\}$ and arbitrary ${\overline{u}}_{e_1},{\overline{v}}_{e_2}\in SP(\overline{\mathit{\chi }})$, (1) holds

$$l{t}_{n\to \infty }\prod _{i=1}^n{M}_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_1},{\overline{v}}_{e_2},{\mathit{\varrho }}_i)=1_{L^{*}}.$$

(1)

Consider $h\in (0,1)$ and $S:SP(\overline{\mathit{\chi }})\to SP(\overline{\mathit{\chi }})$ be any map that satisfies

$$\begin{array}{c}\hfill M_{\mathit{\omega },\mathit{\psi }}(S{\overline{u}}_{e_1},S{\overline{v}}_{e_2},h\mathit{\varrho }){\ge }_{L^{*}}{M}_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_1},{\overline{v}}_{e_2},\mathit{\varrho })\end{array}$$

for each ${\overline{u}}_{e_1},{\overline{v}}_{e_2}\in SP(\overline{\mathit{\chi }})$. Then, S possesses a fixed point which is unique as well.

Proof. 

Consider $\overline{u}\in SP(\overline{\mathit{\chi }})$ and let ${\overline{u}}_{e_n}=S^n(\overline{u})$, $n\in N$. Thus,

$$M_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_1},{\overline{u}}_{e_2},\mathit{\varrho })=M_{\mathit{\omega },\mathit{\psi }}(S\overline{u},S^2\overline{u},\mathit{\varrho }){\ge }_{L^{*}}{M}_{\mathit{\omega },\mathit{\psi }}(\overline{u},S\overline{u},{\displaystyle \frac{\mathit{\varrho }}{h}})=M_{\mathit{\omega },\mathit{\psi }}(\overline{u},{\overline{u}}_{e_1},{\displaystyle \frac{\mathit{\varrho }}{h}}).$$

(2)

By induction, we have

$$M_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_n},{\overline{u}}_{e_{n+1}},\mathit{\varrho }){\ge }_{L^{*}}{M}_{\mathit{\omega },\mathit{\psi }}(\overline{u},{\overline{u}}_{e_1},{\displaystyle \frac{\mathit{\varrho }}{h^n}}).$$

(3)

Consider $\mathit{\varrho }>0$. Now, for $m,n\in N$, take $n<m$; considering $r_i>0$, $i=n,\dots ,m-1$ that satisfies $r_n+r_{n+1}+\cdots +r_{m-1}\le 1$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill M_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_n},{\overline{u}}_{e_m},\mathit{\varrho })& {\ge }_{L^{*}}{\tau }^{m-n-2}(M_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_n},{\overline{u}}_{e_{n+1}},r_n\mathit{\varrho })\hfill \\ \hfill & ,\dots ,M_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_{m-1}},{\overline{u}}_{e_m},r_{m-1}\mathit{\varrho }))\hfill \\ \hfill & {\ge }_{L^{*}}{\tau }^{m-n-2}(M_{\mathit{\omega },\mathit{\psi }}(\overline{u},{\overline{u}}_{e_1},\frac{r_n\mathit{\varrho }}{h^n}),\hfill \\ \hfill & \dots ,M_{\mathit{\omega },\mathit{\psi }}(\overline{u},{\overline{u}}_{e_1},\frac{r_{m-1}\mathit{\varrho }}{h^{m-1}})).\hfill \end{array}\end{array}$$

(4)

Considering $r_p={\displaystyle \frac{1}{p(p+1)}}$, $p=n,\dots ,m-1$, we get

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill M_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_n},{\overline{u}}_{e_m},\mathit{\varrho })& {\ge }_{L^{*}}{\tau }^{m-n-2}(M_{\mathit{\omega },\mathit{\psi }}(\overline{u},{\overline{u}}_{e_1},\frac{\mathit{\varrho }}{n(n+1)h^n}),\dots ,\hfill \\ \hfill & M_{\mathit{\omega },\mathit{\psi }}(\overline{u},{\overline{u}}_{e_1},\frac{\mathit{\varrho }}{(m-1)m{h}^{m-1}})).\hfill \end{array}\end{array}$$

(5)

Now, define ${\mathit{\varrho }}_s={\displaystyle \frac{\mathit{\varrho }}{s(s+1)h^s}}$. It is trivial that ${\mathit{\varrho }}_{s+1}-{\mathit{\varrho }}_s\to \infty$ as $s\to \infty$, thus $\{{\mathit{\varrho }}_s\}$ is an s-increasing sequence. So, we have

$$l{t}_{n\to \infty }\prod _{n=m}^{\infty }{M}_{\mathit{\omega },\mathit{\psi }}(\overline{u},{\overline{u}}_{e_1},{\displaystyle \frac{\mathit{\varrho }}{n(n+1)h^n}})=1_{L^{*}}.$$

(6)

Equations (5) and (6) implies $l{t}_{n\to \infty }{M}_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_n},{\overline{u}}_{e_m},\mathit{\varrho })=1_{L^{*}}$ for $n<m$. Thus, sequence $\{{\overline{u}}_{e_n}\}$ is Cauchy. As $\overline{\mathit{\chi }}$ is complete, there exist $\overline{v}\in SP(\overline{\mathit{\chi }})$ so that ${\overline{u}}_{e_n}\to \overline{v}$. Claim that S possesses $\overline{v}$ as its fixed point.

$$\begin{array}{cc}\hfill M_{\mathit{\omega },\mathit{\psi }}(S\overline{v},\overline{v},\mathit{\varrho })& {\ge }_{L^{*}}\tau (l{t}_{n\to \infty }{M}_{\mathit{\omega },\mathit{\psi }}(S\overline{v},S{\overline{u}}_{e_n},{\displaystyle \frac{\mathit{\varrho }}{2}}),l{t}_{n\to \infty }{M}_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_{n+1}},\overline{v},{\displaystyle \frac{\mathit{\varrho }}{2}}))\hfill \\ \hfill & {\ge }_{L^{*}}\tau (l{t}_{n\to \infty }{M}_{\mathit{\omega },\mathit{\psi }}(\overline{v},{\overline{u}}_{e_n},{\displaystyle \frac{\mathit{\varrho }}{2}}),l{t}_{n\to \infty }{M}_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_{n+1}},\overline{v},{\displaystyle \frac{\mathit{\varrho }}{2}}))=1_{L^{*}}.\hfill \end{array}$$

(7)

Thus, $M_{\mathit{\omega },\mathit{\psi }}(S\overline{v},\overline{v},\mathit{\varrho })=1_{L^{*}}$, that implies $S(\overline{v})=\overline{v}$.

Uniqueness: Consider $\overline{w}\in SP(\overline{X})$ to be any other fixed point of S so that $S(\overline{w})=\overline{w}$. Thus, we have

$$\begin{array}{cc}\hfill 1_{L^{*}}& {\ge }_{L^{*}}{M}_{\mathit{\omega },\mathit{\psi }}(\overline{v},\overline{w},t)=M_{\mathit{\omega },\mathit{\psi }}(S(\overline{v}),S(\overline{w}),\mathit{\varrho })\hfill \\ \hfill & {\ge }_{L^{*}}{M}_{\mathit{\omega },\mathit{\psi }}(\overline{v},\overline{w},{\displaystyle \frac{\mathit{\varrho }}{h}})=M_{\mathit{\omega },\mathit{\psi }}(S(\overline{v}),S(\overline{w}),{\displaystyle \frac{\mathit{\varrho }}{h}})\hfill \\ \hfill & {\ge }_{L^{*}}{M}_{\mathit{\omega },\mathit{\psi }}(\overline{v},\overline{w},{\displaystyle \frac{\mathit{\varrho }}{h^2}})=M_{\mathit{\omega },\mathit{\psi }}(S(\overline{v}),S(\overline{w}),{\displaystyle \frac{\mathit{\varrho }}{h^2}})\hfill \\ \hfill & \cdots \hfill \\ \hfill & {\ge }_{L^{*}}l{t}_{n\to \infty }{M}_{\mathit{\omega },\mathit{\psi }}(\overline{v},\overline{w},{\displaystyle \frac{\mathit{\varrho }}{h^n}})\hfill \\ \hfill & =1_{L^{*}}.\hfill \end{array}$$

(8)

Hence, $M_{\mathit{\omega },\mathit{\psi }}(\overline{v},\overline{w},\mathit{\varrho })=1_{L^{*}}$, that implies $\overline{v}=\overline{w}$. □

Example 3. 

Take $\overline{\mathit{\chi }}=[0,2]$ and define $\Theta (\overline{j},\overline{\mathcal{\ell }})=({\overline{j}}_1{\overline{\mathcal{\ell }}}_1,max\{{\overline{j}}_2,{\overline{\mathcal{\ell }}}_2\})$. Take $\mathit{\omega },\mathit{\psi }$ soft fuzzy sets on $SP(\overline{\mathit{\chi }})\times SP(\overline{\mathit{\chi }})\times (0,\infty )$ given as $M_{\mathit{\omega },\mathit{\psi }}(\overline{v},\overline{w},\mathit{\varrho })=(e^{-(\frac{max\{\overline{v},\overline{w}\}}{\mathit{\varrho }})},1-e^{-(\frac{max\{\overline{v},\overline{w}\}}{\mathit{\varrho }})})$, for all $\overline{v},\overline{w}\in SP(\overline{\mathit{\chi }})$, $\mathit{\varrho }>0$. Then, $(\overline{\mathit{\chi }},M_{\mathit{\omega },\mathit{\psi }},\Theta )$ is a complete MIFSMS. Define a self map $T:SP(\overline{\mathit{\chi }})\to SP(\overline{\mathit{\chi }})$ so that

$$\begin{array}{c}\hfill T(\overline{\mathit{\iota }})=\left\{\begin{array}{c}0\mathit{if}\overline{\mathit{\iota }}=1\hfill \\ \frac{\overline{\mathit{\iota }}}{2}\mathit{if}\overline{\mathit{\iota }}\in [0,1)\hfill \\ \frac{\overline{\mathit{\iota }}}{4}\mathit{if}\overline{\mathit{\iota }}\in (1,2].\hfill \end{array}\right.\end{array}$$

Then, T satisfies Theorem 8 and possesses 0 as its unique fixed point.

Lemma 9. 

If $G:(0,\infty )\to [0,1]$ is an increasing function, then it satisfies

$$l{t}_{n\to \infty }\prod _{i=n}^{\infty }G({\mathit{\rho }}_o^i)=0\Rightarrow l{t}_{n\to \infty }\prod _{i=n}^{\infty }G({\mathit{\rho }}^i)=0$$

(9)

for all $\mathit{\rho }\in (0,1)$ and ∏ is taken as co-norm ⋄.

Proof. 

Case I Consider $\mathit{\rho }<{\mathit{\rho }}_o$. Now, ${\mathit{\rho }}^i<{\mathit{\rho }}_o^i$ for $i\in N$, in view of the fact that G is increasing $G(\mathit{\rho })\le G({\mathit{\rho }}_o)$. So, ${\prod }_{i=n}^{\infty }G({\mathit{\rho }}^i)\le {\prod }_{i=n}^{\infty }G({\mathit{\rho }}_o^i)$, where $n\in N$. Hence, the proof is complete.

Case II Consider $\mathit{\rho }\ge {\mathit{\rho }}_o$. Suppose $\mathit{\rho }=\sqrt{{\mathit{\rho }}_o}$, then

$$\prod _{i=2p^{\prime }}^{\infty }G({\mathit{\rho }}^i)=[\prod _{i=p^{\prime }}^{\infty }G({\mathit{\rho }}^{2i})]\diamond [\prod _{i=p^{\prime }}^{\infty }G({\mathit{\rho }}^{2i+1})]\le [\prod _{i=p^{\prime }}^{\infty }G({\mathit{\rho }}_o^i)]\diamond [\prod _{i=p^{\prime }}^{\infty }G({\mathit{\rho }}_o^i)].$$

(10)

Thus, we have $l{t}_{p^{\prime }\to \infty }{\prod }_{i=2p^{\prime }}^{\infty }G({\mathit{\rho }}^i)\le 0\diamond 0=0$. Furthermore, $l{t}_{p^{\prime }\to \infty }{\prod }_{i=2p^{\prime }+1}^{\infty }G({\mathit{\rho }}^i)\le l{t}_{p^{\prime }\to \infty }{\prod }_{i=2p^{\prime }+2}^{\infty }G({\mathit{\rho }}^i)=0$ that implies $l{t}_{p^{\prime }\to \infty }{\prod }_{i=p^{\prime }}^{\infty }G({\mathit{\rho }}^i)=0$ if $\mathit{\rho }=\sqrt{{\mathit{\rho }}_o}$. As G is increasing it can be verified easily that $l{t}_{p^{\prime }\to \infty }{\prod }_{i=p^{\prime }}^{\infty }G({\mathit{\rho }}^i)=0$ if $\mathit{\rho }<\sqrt{{\mathit{\rho }}_o}$.

Now, if $\mathit{\rho }>{\mathit{\rho }}_o$, that implies the existence of $p^{\prime }\in N$ so that $\mathit{\rho }<{\mathit{\rho }}_o^{{(1/2)}^{p^{\prime }}}$ and on repeating the above process $p^{\prime }$-times, we obtain $l{t}_{p^{\prime }\to \infty }{\prod }_{i=p^{\prime }}^{\infty }G({\mathit{\rho }}^i)=0$. □

Lemma 10. 

If $H:(0,\infty )\to [0,1]$ is a decreasing function, then it satisfies

$$l{t}_{n\to \infty }\prod _{i=n}^{\infty }H({\mathit{\rho }}_o^i)=1\Rightarrow l{t}_{n\to \infty }\prod _{i=n}^{\infty }H({\mathit{\rho }}^i)=1$$

(11)

for all $\mathit{\rho }\in (0,1)$ and ∏ is taken as norm ∗.

Proof. 

The above can be easily proved on the similar lines of Lemma 9. □

Lemma 11. 

The sequence ${\overline{u}}_{e_n}=S^n({\overline{u}}_{e_1})$ is a Cauchy sequence.

Proof. 

Consider $G(\overline{u})=\mathit{\psi }({\overline{u}}_{e_1},S({\overline{u}}_{e_1}),{\displaystyle \frac{1}{\overline{u}}})$ and $H(\overline{u})=\mathit{\omega }({\overline{u}}_{e_1},S({\overline{u}}_{e_1}),{\displaystyle \frac{1}{\overline{u}}})$, then maps G and H are increasing and decreasing, respectively, from $(0,\infty )$ to $[0,1]$. Consider $h<\mathit{\rho }<1$, then by Lemmas 9 and 10, we have

$$l{t}_{n\to \infty }\prod _{i=n}^{\infty }{M}_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_1},S{\overline{u}}_{e_1},{\displaystyle \frac{1}{{(h/\mathit{\rho })}^i}})=1_{L^{*}}.$$

(12)

As $\mathit{\rho }<1$, ${\sum }_{n=1}^{\infty }{\mathit{\rho }}^n<\infty$, so for every ${\mathit{ϵ}}_o>0$ there exist $n_o$ so that ${\sum }_{n=n_o}^{\infty }{\mathit{\rho }}^n<{\mathit{ϵ}}_o$. Now, if $\mathit{\rho }>n>n_o$ and $t>{\mathit{ϵ}}_o$, we have

$$\begin{array}{cc}\hfill M_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_n},{\overline{u}}_{e_p},t)& {\ge }_{L^{*}}{M}_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_n},{\overline{u}}_{e_p},{\mathit{ϵ}}_o){\ge }_{L^{*}}\prod _{i=n}^{p-1}{M}_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_i},{\overline{u}}_{e_{i-1}},{\mathit{\rho }}^i)\hfill \\ \hfill & {\ge }_{L^{*}}\prod _{i=n}^{p-1}{M}_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_1},S{\overline{u}}_{e_1},{\displaystyle \frac{{\mathit{\rho }}^i}{h^i}})=\prod _{i=n}^{p-1}{M}_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_1},S{\overline{u}}_{e_1},{\displaystyle \frac{1}{{(h/\mathit{\rho })}^i}}).\hfill \end{array}$$

(13)

Thus, from Equation (12), we have $l{t}_{n\to \infty }{M}_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_n},S{\overline{u}}_{e_p},t)=1_{L^{*}}$ where $p>n$. Thus, $\{{\overline{u}}_{e_n}\}$ is Cauchy. □

Theorem 12. 

Consider $(\overline{\mathit{\chi }},M_{\mathit{\omega },\mathit{\psi }},\tau )$ to be complete MIFSMS, so that for some ${\mathit{\rho }}_o\in (0,1)$ and ${\overline{u}}_{e_o}\in SP(\overline{\mathit{\chi }})$, (14) holds

$$\begin{array}{c}\hfill l{t}_{n\to \infty }\prod _{i=n}^{\infty }{M}_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_o},S{\overline{u}}_{e_o},{\displaystyle \frac{1}{{\mathit{\rho }}_o^i}})=1_{L^{*}}.\end{array}$$

(14)

Consider $h\in (0,1)$ and $S:SP(\overline{\mathit{\chi }})\to SP(\overline{\mathit{\chi }})$ that satisfies

$$\begin{array}{c}\hfill M_{\mathit{\omega },\mathit{\psi }}(S{\overline{u}}_{e_1},S{\overline{v}}_{e_2},ht){\ge }_{L^{*}}{M}_{\mathit{\omega },\mathit{\psi }}({\overline{u}}_{e_1},{\overline{v}}_{e_2},t)\end{array}$$

for each ${\overline{u}}_{e_1},{\overline{v}}_{e_2}\in SP(\overline{\mathit{\chi }})$. Then, S possesses a fixed point which is unique as well.

Proof. 

We will be proving Theorem 12 by the above lemmas.

As $(\overline{\mathit{\chi }},M_{\mathit{\omega },\mathit{\psi }},\tau )$ is complete MIFSMS, there exists $\overline{v}\in SP(\overline{\mathit{\chi }})$ so that $l{t}_{n\to \infty }{\overline{u}}_{e_n}=\overline{v}$. Now, it can be easily proven that S possesses $\overline{v}$ as its fixed point which is unique as well by the similar argument as used in Theorem 8. This completes the proof of Theorem 12. □

Example 4. 

Take $\overline{\mathit{\chi }}=[0,1]\cap Q$ and define $\Theta (\overline{\mathcal{ȷ}},\overline{\mathcal{\ell }})=(max\{{\overline{\mathcal{ȷ}}}_1+{\overline{\mathcal{\ell }}}_1-1,0\},min\{{\overline{\mathcal{ȷ}}}_2+{\overline{\mathcal{\ell }}}_2,1\})$. Take $\mathit{\omega },\mathit{\psi }$ soft fuzzy sets on $SP(\overline{\mathit{\chi }})\times SP(\overline{\mathit{\chi }})\times (0,\infty )$ given as $M_{\mathit{\omega },\mathit{\psi }}(\overline{v},\overline{w},\mathit{\varrho })=(1-\frac{max\{\overline{v},\overline{w}\}}{1+\mathit{\varrho }},\frac{max\{\overline{v},\overline{w}\}}{1+\mathit{\varrho }})$, for all $\overline{v},\overline{w}\in SP(\overline{\mathit{\chi }})$, $\mathit{\varrho }>0$. Then, $(\overline{\mathit{\chi }},M_{\mathit{\omega },\mathit{\psi }},\Theta )$ is a complete MIFSMS. Define a self map $T:SP(\overline{\mathit{\chi }})\to SP(\overline{\mathit{\chi }})$ so that

$$\begin{array}{c}\hfill T(\overline{\mathit{\iota }})=\left\{\begin{array}{c}\frac{\overline{\mathit{\iota }}}{4}\mathit{if}\overline{\mathit{\iota }}\in [0,\frac{1}{2}]\cap Q\hfill \\ \frac{\overline{\mathit{\iota }}}{2}\mathit{if}\overline{\mathit{\iota }}\in (\frac{1}{2},1]\cap Q.\hfill \end{array}\right.\end{array}$$

Then, T satisfies Theorem 12 and possesses 0 as its unique fixed point.

6. Application

Now, we are giving an application of Theorem 8 in solving integral equation.

Consider the following integral equation:

$$\begin{array}{c}\hfill {\overline{\mathit{\iota }}}_e(r)={\int }_0^{r}K(r,s,{\overline{\mathit{\iota }}}_e(s))ds\end{array}$$

(15)

for all $r\in [0,I]$, where $I>0$ and $K\in C([0,I]\times [0,I]\times R,R)$. Consider $\overline{\mathit{\chi }}=C([0,I],R)$ be the space consisting of continuous functions on $[0,I]$ with the norm $\Vert {\overline{\mathit{\iota }}}_e\Vert =su{p}_{r\in [0,I]} | {\overline{\mathit{\iota }}}_e(r) |$, where ${\overline{\mathit{\iota }}}_e\in \overline{\mathit{\chi }}$ and the induced soft metric is defined as $\mu ({\overline{\mathit{\iota }}}_e,{\overline{\mathit{\kappa }}}_e)=su{p}_{r\in [0,I]} | {\overline{\mathit{\iota }}}_e(r)-{\overline{\mathit{\kappa }}}_e(r) |$, for all ${\overline{\mathit{\iota }}}_e,{\overline{\mathit{\kappa }}}_e\in \overline{\mathit{\chi }}$. Let the MIFSM $M_{\mathit{\omega },\mathit{\psi }}$ be defined as

$$\begin{array}{c}\hfill M_{\mathit{\omega },\mathit{\psi }}({\overline{\mathit{\iota }}}_e,{\overline{\mathit{\kappa }}}_e,\mathit{\varrho })=(\frac{\mathit{\varrho }}{\mathit{\varrho }+su{p}_{r\in [0,I]} | {\overline{\mathit{\iota }}}_e(r)-{\overline{\mathit{\kappa }}}_e(r) |},\frac{su{p}_{r\in [0,I]} | {\overline{\mathit{\iota }}}_e(r)-{\overline{\mathit{\kappa }}}_e(r) |}{\mathit{\varrho }+su{p}_{r\in [0,I]} | {\overline{\mathit{\iota }}}_e(r)-{\overline{\mathit{\kappa }}}_e(r) |}),\end{array}$$

where $\mathit{\varrho },m,n\in R^{+}$ and $\tau (\overline{u},\overline{v})=({\overline{u}}_1{\overline{v}}_1,min({\overline{u}}_2+{\overline{v}}_2,1))$. Then, $(\overline{X},M_{\mathit{\omega },\mathit{\psi }},\tau )$ is a complete MIFSMS.

Now, claim the existence of a solution of (15).

Let function K satisfy the following conditions:

(i)

$K(r,s,{\overline{\mathit{\iota }}}_e(s))\ge 0$, for all $r,s\in [0,I]$ and ${\overline{\mathit{\iota }}}_e\in \overline{\mathit{\chi }}$;

(ii)

There exist $\mathit{\lambda }>0$ so that

$$\begin{array}{c}\hfill | K(r,s,{\overline{\mathit{\iota }}}_e(s))-K(r,s,{\overline{\mathit{\kappa }}}_e(s))\mid \le \mathit{\lambda }su{p}_{r\in [0,I]}\mid {\overline{\mathit{\iota }}}_e(r)-{\overline{\mathit{\kappa }}}_e(r)\mid ,\end{array}$$

for all $r,s\in [0,I]$;

(iii)

There exist $h\in (0,1)$ so that $\mathit{\lambda }r<h$.

Consider $\{t_s\}$ to be any s-increasing sequence, so that $t_{s+1}-t_s\to \infty$ as $s\to \infty$, thus $l{t}_{n\to \infty }{\prod }_{i=1}^n{M}_{\mathit{\omega },\mathit{\psi }}({\overline{\mathit{\iota }}}_e,{\overline{\mathit{\kappa }}}_e,t_i)=1_{L^{*}}$.

Define a self map $S:\overline{\mathit{\chi }}\to \overline{\mathit{\chi }}$ as

$$\begin{array}{c}\hfill S({\overline{\mathit{\iota }}}_e)(\mathcal{ȷ})={\int }_0^{\mathcal{ȷ}}K(\mathcal{ȷ},\mathcal{\ell },{\overline{\mathit{\iota }}}_e(\mathcal{\ell }))d\mathcal{\ell },\end{array}$$

then, we have

$$\begin{array}{c}\hfill S({\overline{\mathit{\iota }}}_e)(\mathcal{ȷ})-S({\overline{\mathit{\kappa }}}_e)(\mathcal{ȷ})={\int }_0^{\mathcal{ȷ}}K(\mathcal{ȷ},\mathcal{\ell },{\overline{\mathit{\iota }}}_e(\mathcal{\ell }))-K(\mathcal{ȷ},\mathcal{\ell },{\overline{\mathit{\kappa }}}_e(\mathcal{\ell }))d\mathcal{\ell },\end{array}$$

thus

$$\begin{array}{cc}\hfill | S({\overline{\mathit{\iota }}}_e)(\mathcal{ȷ})-S({\overline{\mathit{\kappa }}}_e)(\mathcal{ȷ})\mid =& | {\int }_0^{\mathcal{ȷ}}K(\mathcal{ȷ},\mathcal{\ell },{\overline{\mathit{\iota }}}_e(\mathcal{\ell }))-K(\mathcal{ȷ},\mathcal{\ell },{\overline{\mathit{\kappa }}}_e(\mathcal{\ell }))d\mathcal{\ell } |\hfill \\ \hfill & \le {\int }_0^{\mathcal{ȷ}}\mid K(\mathcal{ȷ},\mathcal{\ell },{\overline{\mathit{\iota }}}_e(\mathcal{\ell }))-K(\mathcal{ȷ},\mathcal{\ell },{\overline{\mathit{\kappa }}}_e(\mathcal{\ell }))\mid d\mathcal{\ell }\hfill \\ \hfill & \le \mathit{\lambda }\mathcal{ȷ}su{p}_{\mathcal{ȷ}\in [0,I]}\mid {\overline{\mathit{\iota }}}_e(\mathcal{ȷ})-{\overline{\mathit{\kappa }}}_e(\mathcal{ȷ}) |\hfill \\ \hfill & <hsu{p}_{\mathcal{ȷ}\in [0,I]}\mid {\overline{\mathit{\iota }}}_e(\mathcal{ȷ})-{\overline{\mathit{\kappa }}}_e(\mathcal{ȷ})\mid .\hfill \end{array}$$

Now, we have

$$\begin{array}{cc}\hfill M_{\mathit{\omega },\mathit{\psi }}(S{\overline{\mathit{\iota }}}_e(\mathcal{ȷ}),S{\overline{\mathit{\kappa }}}_e(\mathcal{ȷ}),h\mathit{\varrho })=& (\frac{h\mathit{\varrho }}{h\mathit{\varrho }+su{p}_{\mathcal{ȷ}\in [0,I]}\mid S{\overline{\mathit{\iota }}}_e(\mathcal{ȷ})-S{\overline{\mathit{\kappa }}}_e(\mathcal{ȷ}) |},\frac{su{p}_{\mathcal{ȷ}\in [0,I]}\mid S{\overline{\mathit{\iota }}}_e(\mathcal{ȷ})-S{\overline{\mathit{\kappa }}}_e(\mathcal{ȷ}) |}{h\mathit{\varrho }+su{p}_{\mathcal{ȷ}\in [0,I]}\mid S{\overline{\mathit{\iota }}}_e(\mathcal{ȷ})-S{\overline{\mathit{\kappa }}}_e(\mathcal{ȷ}) |})\hfill \\ \hfill & =(\frac{\mathit{\varrho }}{\mathit{\varrho }+\frac{su{p}_{\mathcal{ȷ}\in [0,I]}\mid S{\overline{\mathit{\iota }}}_e(\mathcal{ȷ})-S{\overline{\mathit{\kappa }}}_e(\mathcal{ȷ}) |}{h}},\frac{\frac{su{p}_{\mathcal{ȷ}\in [0,I]}\mid S{\overline{\mathit{\iota }}}_e(\mathcal{ȷ})-S{\overline{\mathit{\kappa }}}_e(\mathcal{ȷ}) |}{h}}{\mathit{\varrho }+\frac{su{p}_{\mathcal{ȷ}\in [0,I]}\mid S{\overline{\mathit{\iota }}}_e(\mathcal{ȷ})-S{\overline{\mathit{\kappa }}}_e(\mathcal{ȷ}) |}{h}})\hfill \\ \hfill & {\ge }_{L^{*}}(\frac{\mathit{\varrho }}{\mathit{\varrho }+su{p}_{\mathcal{ȷ}\in [0,I]}\mid {\overline{\mathit{\iota }}}_e(\mathcal{ȷ})-{\overline{\mathit{\kappa }}}_e(\mathcal{ȷ}) |},\frac{su{p}_{\mathcal{ȷ}\in [0,I]}\mid {\overline{\mathit{\iota }}}_e(\mathcal{ȷ})-{\overline{\mathit{\kappa }}}_e(\mathcal{ȷ}) |}{\mathit{\varrho }+su{p}_{\mathcal{ȷ}\in [0,I]}\mid {\overline{\mathit{\iota }}}_e(\mathcal{ȷ})-{\overline{\mathit{\kappa }}}_e(\mathcal{ȷ}) |})\hfill \\ \hfill & =M_{\mathit{\omega },\mathit{\psi }}({\overline{\mathit{\iota }}}_e(\mathcal{ȷ}),{\overline{\mathit{\kappa }}}_e(\mathcal{ȷ}),\mathit{\varrho }).\hfill \end{array}$$

Thus, $M_{\mathit{\omega },\mathit{\psi }}(S{\overline{\mathit{\iota }}}_e(\mathcal{ȷ}),S{\overline{\mathit{\kappa }}}_e(\mathcal{ȷ}),h\mathit{\varrho }){\ge }_{L^{*}}{M}_{\mathit{\omega },\mathit{\psi }}({\overline{\mathit{\iota }}}_e(\mathcal{ȷ}),{\overline{\mathit{\kappa }}}_e(\mathcal{ȷ}),\mathit{\varrho })$. Therefore, every assertion of Theorem 8 holds. Hence, S possesses a fixed point $\overline{\mathit{\zeta }}\in \overline{\mathit{\chi }}$ which is unique as well so that $S\overline{\mathit{\zeta }}=\overline{\mathit{\zeta }}$, thus $\overline{\mathit{\zeta }}\in C([0,I],R)$ satisfies integral Equation (15).

7. Conclusions

We have defined basic notions of MIFSMS in this paper. Some Theorems of MIFMS have been broadened in MIFSMS. FPT’s are also proven in our new space along with an application to the integral equation.

8. Discussion

The new results and examples formulated in this work lay the foundation of new results in the future. Moreover, to prove the validity of new results, an application is given in solving the integral equation.