Fixed point approach for solving a system of Volterra integral equations and Lebesgue integral concept in F$_{\text{CM}}$-spaces

1.   Introduction

Let $X$ be a nonempty set. In 2011, Berinde and Borcut ^[1] initiated the concept of a tripled fixed point of a mapping $H: X\times X\times X\rightarrow X$. Some tripled fixed (coincidence) point theorems are obtained via the mixed monotone ($g$-monotone) property, see ^[2]. Their obtained results are generalizations and extensions of the work due to Bhaskar and Lakshmikantham ^[3]. As an application, they studied the existence of solutions of a periodic boundary value problem whose coupled fixed point technique cannot solve such a problem.

The concept of cone metric spaces was reintroduced in 2007 by Huang and Zhang ^[4] by replacing the set of real numbers with an ordered Banach space. This concept was first initiated in literature by showing its importance via a numerical approach by Kantorovich ^[5]. Note that cone (normed) metric spaces have interesting applications in fixed point theory and the numerical analysis. For instance, see ^[6,7,8,9,10].

The notion of a fuzzy set was appeared in 1965 by Zadeh ^[11]. This notion has hardly been studied and extended. Its application on variant fields became fruitful and needful. Data analysis, computational intelligence and artificial intelligence are intensively developed. This theory is also generalized and extended in many directions by means of the theories of aggregation operators and triangular norms and co-norms, see ^{[12,13,14,15]}. Kramosil and Michalek ^[16] initiated the notion of a fuzzy metric space. Defining a fuzzy metric is one of the essential problems in fuzzy mathematics, which was frequently used in pattern recognition and fuzzy optimization. In 1994, George and Veeramani ^[17] presented some fixed point results in fuzzy metric spaces.

On the other hand, by combining the concepts of a cone metric space setting and a fuzzy set, Oner et al. ^[18] presented in 2015 the notion of a fuzzy cone metric space (as an abbreviation, (F$_{\text{CM}}$-spaces) and the fuzzy cone Banach contraction result was established. Further results in this direction have been investigated, see ^{[19,20,21,22,23]}. Very recently, Waheed et al. ^[24] established some coupled fixed point results in F$_{\text{CM}}$-spaces. Going in the same direction, the aim of this work is to present some tripled fixed point results in this setting for new contractive mappings via a triangular property. We also give nontrivial examples and two illustrated applications making effective the presented results. To our knowledge, this work is the first time to deal with tripled fixed point notion in F$_{\text{CM}}$-spaces.

In mathematics, the Volterra integral equations are a special type of integral equations. They are divided into two groups referred to as the first and the second kind. It is known that linear and nonlinear Volterra integral equations arise in many scientific fields such as the population dynamics, spread of epidemics, and semi-conductor devices. Finding solutions of linear or nonlinear Volterra integral equations is highly interesting for researchers and scientists in this field and there are available studies to find analytical or numerical solutions for Volterra integral equations (see ^{[25,26,27,28]}). In fact, Volterra integral equations are usually solved analytically or numerically by finding approximate solutions to the problems using numerical or analytical approximation methods. For instance, well-posedness and regularity of backward (doubly) stochastic Volterra integral equations have been studied in ^[29,30]. Among the methods used to solve Volterra integral equations are Sinc-collocation method ^[31], Barycentric Lagrange interpolation and the equidistance Chebyshev interpolation nodes ^[32], relaxed Monte Carlo method ^[33], etc. We will be concerned in this work to use a fixed point technique via a tripled fixed point approach to solve a system of Volterra integral equations. We will also present an application on Lebesgue integral type mappings using a tripled fixed point result.

This paper is organized as follows: Section 2 giving the essential definitions and known results in the literature that help us in the rest of the paper. In Section 3, we prove our main tripled fixed point results and we provide some concrete examples. The aim of Section 4 is to apply our obtained results by ensuring the existence of a tripled fixed point for a Lebesgue integral mapping and a unique solution for a system of Volterra integral equations.

2.   Necessary facts

Definition 2.1. ^[34] An operation $\star :\left[ 0, 1\right] \times \left[ 0, 1 \right] \rightarrow \left[ 0, 1\right]$ is described as a continuous $\nu -$ norm if it fulfills the following:

● $\star$ is associative and commutative,

● $\star$ is continuous,

● for each $\lambda \in \lbrack 0, 1],$ $1\star \lambda = \lambda,$

● for each $\lambda _{1}, \lambda _{2}, \beta _{1}, \beta _{2}\in \lbrack 0, 1],$ if $\lambda _{1}\leq \lambda _{2}$ and $\beta _{1}\leq \beta _{2}$, then $\lambda _{1}\star \beta _{1}\leq \lambda _{2}\star \beta _{2}.$

Here $\mathbb{N},$ $\beth$, $\vartheta$ and $\nu -$norm, represent the set of natural numbers, the real Banach space, a zero element in $\beth$ and a continuous $\nu -$norm, respectively.

Definition 2.2. ^[4] A subset $\Upsilon \in \beth$ is called a cone if the following hold:

$(1)$ $\Upsilon \neq \emptyset$ is closed and $\Upsilon \neq \left\{ \vartheta \right\};$

$(2)$ If $\lambda _{1}, \beta _{1}\in \left(0, \infty \right)$ and $\theta, \rho \in \Upsilon$, then $\lambda _{1}\theta +\beta _{1}\rho \in \Upsilon;$

$(3)$ If both $\theta \in \Upsilon$ and $-\theta \in \Upsilon,$ then $\theta = \vartheta.$

A partial ordering on a cone $\Upsilon$ is described as $\theta \preceq \rho \Longleftrightarrow \rho -\theta \in \Upsilon$. $\theta \prec \rho$ stands for $\theta \preceq \rho$ and $\theta \neq \rho,$ while $\theta \ll \rho$ stands for $\rho -\theta \in int(\Upsilon).$ Here each cone has non-empty interior.

Definition 2.3. ^[18] A $3-$tuple $(\Omega, \Theta _{\varpi }, \star)$ is called an $F_{CM}-$space if $\Upsilon$ is a cone in $\beth,$ $\Omega$ is an arbitrary set, $\star$ is a $\nu -$norm, and $\Theta _{\varpi }$ is a fuzzy set on $\Omega ^{2}\times int(\Upsilon)$ such that the following are satisfied, for all $\theta, \rho, \delta \in \Omega$ and $\nu, \mu \in int(\Upsilon),$

$(\heartsuit _{1})$ $\Theta _{\varpi }(\theta, \rho, \nu) > \vartheta$ and $\Theta _{\varpi }(\theta, \rho, \nu) = 1\Leftrightarrow \theta = \rho;$

$(\heartsuit _{2})$ $\Theta _{\varpi }(\theta, \rho, \nu) = \Theta _{\varpi }(\rho, \theta, \nu);$

$(\heartsuit _{3})$ $\Theta _{\varpi }(\theta, \rho, \nu)\star \Theta _{\varpi }(\rho, \delta, \mu)\leq \Theta _{\varpi }(\theta, \delta, \nu +\mu);$

$(\heartsuit _{4})$ $\Theta _{\varpi }(\theta, \rho, \cdot):int(\Upsilon)\rightarrow \lbrack 0, 1]$ is continuous.

Definition 2.4. ^[18] Let $(\Omega, \Theta _{\varpi }, \star)$ be an $F_{CM}-$ space, $\theta \in \Omega$ and $\left(\theta _{i}\right)$ be a sequence in $\Omega.$

● $\left(\theta _{i}\right)$ is called convergent to some $\theta$ if for $\nu \gg \vartheta$ and $0 < u < 1,$ there exists $i_{1}\in \mathbb{N}$ such that $\Theta _{\varpi }(\theta _{i}, \theta, \nu) > 1-u,$ $\forall i > i_{1},$ and we can write $\lim_{i\rightarrow \infty }\theta _{i} = \theta$.

● $\left(\theta _{i}\right)$ is called a Cauchy sequence if for $\nu \gg \vartheta$ and $0 < u < 1,$ there exists $i_{1}\in \mathbb{N}$ such that

● If every Cauchy sequence is convergent in $\Omega,$ then the triple $(\Omega, \Theta _{\varpi }, \star)$ is called complete.

● $\left(\theta _{i}\right)$ is called a fuzzy cone contraction (F$_{ \text{cc}}$) if there exists $\beta \in \left(0, 1\right)$ such that

Definition 2.5. ^[35] Assume that $(\Omega, \Theta _{\varpi }, \star)$ is an $F_{CM}-$space. Then the fuzzy cone metric $\Theta _{\varpi }$ is called triangular if

Lemma 2.6. ^[18] Suppose that $(\Omega, \Theta _{\varpi }, \star)$ is an $F_{CM}-$space, $\theta \in \Omega$ and $\left(\theta _{i}\right)$ is asequence in $\Omega,$ then

Definition 2.7. ^[18] Let $(\Omega, \Theta _{\varpi }, \star)$ be an $F_{CM}-$ space and $\Xi :\Omega \rightarrow \Omega.$ Then $\Xi$ is called an F$_{ \text{cc}}$ if there exists $g\in \left(0, 1\right)$ such that

Definition 2.8. ^[3] A pair $(\theta, \rho)$ is called a coupled FP of the mapping $\Xi :\Omega \times \Omega \rightarrow \Omega$ if

Definition 2.9. ^[1] Let $\Omega \neq \emptyset$. Then a triple $(\theta, \rho, \delta)\in \Omega ^{3}$ is called a TFP of the mapping $\Xi :\Omega ^{3}\rightarrow \Omega$ if $\theta = \Xi (\theta, \rho, \delta),$ $\rho = \Xi (\rho, \delta, \theta)$ and $\delta = \Xi (\delta, \theta, \rho).$

Example 2.10. Let $\Omega = [0, \infty)$ and $\Xi :\Omega ^{3}\rightarrow \Omega$ be a mapping given by

Then $\Xi$ has a TFP when $\theta = \rho = \delta.$

3.   Results and examples

This part is concerned with presenting the main theoretical results of our paper. In addition, some supporting examples are provided.

Theorem 3.1. Let $\Xi :\Omega ^{3}\rightarrow \Omega$ be a mapping defined ona complete $F_{CM}-$space$\ (\Omega, \Theta _{\varpi }, \star)$ in which $\Theta _{\varpi }$ is triangular and fulfills

where

for all $\varkappa, \widehat{\varkappa }, \widetilde{\varkappa }, \varrho, \widehat{\varrho }, \widetilde{\varrho }\in \Omega,$ $\nu \gg \vartheta$and $u_{1}\in \lbrack 0, 1)$ and $u_{2}\geq 0$ with $u_{1}+4u_{2} < 1.$ Then $\Xi$ has a unique TFP in $\Omega.$

Proof. Consider $\varkappa _{0}, \widehat{\varkappa }_{0}, \widetilde{\varkappa } _{0}\in \Omega.$ Describe sequences $\{\varkappa _{\alpha }\},$ $\{\widehat{ \varkappa }_{\alpha }\}$ and $\{\widetilde{\varkappa }_{\alpha }\}$ in $\Omega$ such that

By (3.1), for $\nu \gg \vartheta,$ we have

where

By (3.4) and (3.3), for $\nu \gg \vartheta,$ we have

By simple calculations, we obtain that

where $\nabla = \frac{u_{1}+2u_{2}}{1-2u_{2}} < 1.$ Analogously, one can write

By induction and from (3.5) and (3.6), for $\nu \gg \vartheta,$ we conclude that

This implies that $\{\varkappa _{\alpha }\}$ is an F$_{\text{cc}}$ and hence

Now, for $\ell > \alpha$ and $\nu \gg \vartheta,$ we get

This proves that the sequence $\{\varkappa _{\alpha }\}$ is Cauchy. Again, regarding to the sequence $\widehat{\varkappa }_{\alpha },$ by (3.1), for $\nu \gg \vartheta,$ we get

where

It follows from (3.7) and (3.8) that, for $\nu \gg \vartheta,$

Again, by simple calculations, we obtain that

where $\nabla = \frac{u_{1}+2u_{2}}{1-2u_{2}} < 1.$ Similarly, we have

From (3.9), (3.10) and by induction for $\nu \gg \vartheta,$ we get

This yields that the sequence $\{\widehat{\varkappa }_{\alpha }\}$ is an F$_{ \text{cc}}$ and hence

Now, for $\ell > \alpha$ and $\nu \gg \vartheta,$ we get

Hence, $\{\widehat{\varkappa }_{\alpha }\}$ is a Cauchy sequence. With the same approach, one can prove that the sequence $\{\widetilde{\varkappa } _{\alpha }\}$ is also Cauchy in $\Omega$. Since $\Omega$ is complete, there are $\varkappa,$ $\widehat{\varkappa }$ and $\widetilde{ \varkappa }$ in $\Omega$ such that $\varkappa _{\alpha }\rightarrow \varkappa,$ $\widehat{\varkappa }_{\alpha }\rightarrow \widehat{\varkappa }$ and $\widetilde{\varkappa }_{\alpha }\rightarrow \widetilde{\varkappa }$ as $\alpha \rightarrow \infty.$ So, we can write

Hence

Therefore, a mapping $\Xi$ has a TFP $\left(\varkappa, \widehat{\varkappa }, \widetilde{\varkappa }\right) \ $in $\Omega ^{3}.$

For uniqueness, assume that $\left(\varkappa _{1}, \widehat{\varkappa }_{1}, \widetilde{\varkappa } _{1}\right)$ is another TFP of $\Xi$ such that $\left(\varkappa _{1}, \widehat{\varkappa }_{1}, \widetilde{\varkappa }_{1}\right) \neq \left(\varkappa, \widehat{\varkappa }, \widetilde{\varkappa }\right).$ From (3.1), for $\nu \gg \vartheta,$ we can write

where

From (3.12) in (3.11), we obtain that

where $u_{1}+2u_{2} < 1.$ This implies that $\Theta _{\varpi }\left(\varkappa, \varkappa _{1}, \nu \right) = 1,$ for $\nu \gg \vartheta$. Thus, $\varkappa = \varkappa _{1}.$ By the same manner, one can obtain $\widehat{ \varkappa } = \widehat{\varkappa }_{1}$ and $\widetilde{\varkappa } = \widetilde{ \varkappa }_{1}.$ This completes the proof.

Corollary 3.2. Theorem 3.1 is also true if we replace the condition (3.1) with one of the following:

$(\clubsuit _{1})$ For $\varkappa, \widehat{\varkappa }, \widetilde{\varkappa }, \varrho, \widehat{\varrho }, \widetilde{\varrho }\in \Omega,$ $\nu \gg \vartheta,$

for any $u_{1}\in \lbrack 0, 1)$ and $u_{2}\geq 0$ with $\left(u_{1}+2u_{2}\right) < 1.$

$(\clubsuit _{2})$ For $\varkappa, \widehat{\varkappa }, \widetilde{\varkappa }, \varrho, \widehat{\varrho }, \widetilde{\varrho }\in \Omega,$ $\nu \gg \vartheta,$

for any $u_{1}\in \lbrack 0, 1)$ and $u_{2}\geq 0$ with $u_{1}+2u_{2} < 1.$

Example 3.3. Let $\Omega = (0, \infty),$ $\star$ be a $\nu -$norm and $\Theta _{\varpi }:\Omega ^{2}\times (0, \infty)\rightarrow \lbrack 0, 1]$ be described as

for all $\varkappa, \varrho \in \Omega,$ for $\nu > 0.$ Clearly, $(\Omega, \Theta _{\varpi }, \star)$ is a complete $F_{CM}-$space. Define the mapping $\Xi :\Omega ^{3}\rightarrow \Omega$ by

Then, for $\nu \gg \vartheta,$

Hence, all the conditions of Theorem 3.1 are fulfilled with $u_{1} = u_{2} = \frac{1}{21}.$ Therefore, $\Xi$ possesses a point $(3, 3, 3)$ as unique TFP, that is,

Theorem 3.4. Let $(\Omega, \Theta _{\varpi }, \star)$ be a complete $F_{CM}-$space, $\Xi :\Omega ^{3}\rightarrow \Omega$ be a given mapping and $\Theta_{\varpi }$ be triangular satisfying

for all $\varkappa, \widehat{\varkappa }, \widetilde{\varkappa }, \varrho, \widehat{\varrho }, \widetilde{\varrho }\in \Omega,$ $\nu \gg \vartheta$and $u_{1}\in \lbrack 0, 1)$ and $u_{2}, u_{3}\geq 0$ with $u_{1}+2u_{2}+u_{3} < 1.$ Then $\Xi$ has a unique TFP in $\Omega.$

Proof. Assume that the sequences given in (3.2) are valid. Then from (3.13), for $\nu \gg \vartheta,$ we get

After a routine calculation, we have

where $\gimel = \frac{u_{1}+u_{2}}{1-u_{2}-u_{3}} < 1.$ Similarly,

It follows from (3.14), (3.15) and induction that, for $\nu \gg \vartheta,$

This implies that $\{\varkappa _{\alpha }\}$ is an F$_{\text{cc}}$ and so we get

Now, for $\ell > \alpha$ and $\nu \gg \vartheta,$ we obtain

This proves that the sequence $\{\varkappa _{\alpha }\}$ is Cauchy. In the same scenario, it can be shown that the sequences $\{\widehat{\varkappa } _{\alpha }\}$ and $\{\widetilde{\varkappa }_{\alpha }\}$ are Cauchy. Since $\Omega$ is complete, there are $\varkappa,$ $\widehat{\varkappa }$ and $\widetilde{\varkappa }$ in $\Omega$ such that $\varkappa _{\alpha }\rightarrow \varkappa,$ $\widehat{\varkappa }_{\alpha }\rightarrow \widehat{\varkappa }$ and $\widetilde{\varkappa }_{\alpha }\rightarrow \widetilde{\varkappa }$ as $\alpha \rightarrow \infty.$ Hence, one can write

Thus,

Therefore, a mapping $\Xi$ possesses a TFP $\left(\varkappa, \widehat{ \varkappa }, \widetilde{\varkappa }\right) \ $in $\Omega ^{3}.$

Now, let $\left(\varkappa _{1}, \widehat{\varkappa }_{1}, \widetilde{ \varkappa }_{1}\right)$ be a TFP of $\Xi$ such that $\left(\varkappa _{1}, \widehat{\varkappa }_{1}, \widetilde{\varkappa }_{1}\right) \neq \left(\varkappa, \widehat{\varkappa }, \widetilde{\varkappa }\right).$ From (3.1), for $\nu \gg \vartheta,$ we have

Therefore, $\Theta _{\varpi }\left(\varkappa, \varkappa _{1}, \nu \right) = 1$. Thus it follows that $\varkappa = \varkappa _{1}.$ Analogously, one can obtain $\widehat{\varkappa } = \widehat{\varkappa }_{1}$ and $\widetilde{ \varkappa } = \widetilde{\varkappa }_{1}.$ This finishes the proof.

Corollary 3.5. Theorem 3.4 is valid if we replace the condition (3.13) with the following condition: For $\varkappa, \widehat{\varkappa }, \widetilde{\varkappa }, \varrho, \widehat{\varrho }, \widetilde{\varrho }\in\Omega,$ $\nu \gg \vartheta,$

for any $u_{1}\in \lbrack 0, 1)$ and $u_{3}\geq 0$ with $u_{1}+2u_{3} < 1.$

In order to support Theorem 3.4, the example below is considered.

Example 3.6. Assume that all the requirements of Example 3.3 hold. Then, from (3.13), for $\nu \gg \vartheta,$ we get

It is easy to prove that all the requirements of Theorem 3.4 are satisfied with $u_{1} = u_{2} = \frac{1}{21}$ and $u_{3} = 0.$ Hence, $\Xi$ has a unique strong TFP in $\Omega$, which is $(3, 3, 3).$

4.   Applications

4.1.   Lebesgue integral type mappings

This part is devoted to discuss an application on Lebesgue integral type mappings to strengthen our theoretical results.

In 2002, Branciari ^[36] presented the following theorem:

Theorem 4.1. Let $\Xi$ be a mapping defined in a complete metric space $(\Omega, d)$ satisfying

where $\rho \in \lbrack 0, 1)$ and $\omega : \mathbb{R} ^{+}\rightarrow \mathbb{R} ^{+}$ is a Lebesgue integrable function which is summable, positive and suchthat $\int_{0}^{\epsilon }\omega (\eta)d\eta > 0$ for each $\epsilon > 0.$Then $\Xi$ has a unique fixed point $q\in \Omega.$ Moreover, for all $\varkappa \in \Omega,$ $\lim\limits_{\alpha \rightarrow \infty }\Xi^{\alpha }\varkappa = q.$

According to the above idea, we obtain a unique TFP result in $F_{CM}-$ space.

Theorem 4.2. Let $\Xi :\Omega ^{3}\rightarrow \Omega$ be a mapping defined ona complete $F_{CM}-$space$\ (\Omega, \Theta _{\varpi }, \star)$ in which $\Theta _{\varpi }$ is triangular and fulfills

where $\left(\frac{1}{Z\left(\Xi, \left(\varkappa, \widehat{\varkappa }, \widetilde{\varkappa }\right), \left(\varrho, \widehat{\varrho }, \widetilde{\varrho }\right), \nu \right) }-1\right)$ is defined in Theorem 3.1, for all $\varkappa, \widehat{\varkappa }, \widetilde{\varkappa }, \varrho, \widehat{\varrho }, \widetilde{\varrho }\in \Omega,$ $\nu \gg \vartheta$and $u_{1}\in \lbrack 0, 1)$, $u_{2}\geq 0$ with $u_{1}+4u_{2} < 1$ and $\omega : \mathbb{R} ^{+}\rightarrow \mathbb{R} ^{+}$ is a Lebesgue integrable function which is summable, positive and suchthat $e^{\epsilon } > 0$ for each $\epsilon > 0.$ Then $\Xi$ possesses a uniqueTFP in $\Omega.$

Proof. Define a sequences $\{\varkappa _{\alpha }\},$ $\{\widehat{\varkappa } _{\alpha }\}$ and $\{\widetilde{\varkappa }_{\alpha }\}$ as (3.2). Then by (4.1) and from some statements of the proof of Theorem 3.1, for $\nu \gg \vartheta,$ we get

where $\nabla = \frac{u_{1}+2u_{2}}{1-2u_{2}} < 1.$ Similarly, by using the same arguments, we obtain that

From (4.2), (4.3) and by induction for $\nu \gg \vartheta,$ we can write

This proves that the sequence $\{\varkappa _{\alpha }\}$ is an F$_{\text{cc}}$ and hence

Thus

Now, for $\ell > \alpha$ and $\nu \gg \vartheta,$ we get

It follows that

This proves that $\{\varkappa _{\alpha }\}$ is a Cauchy sequence in $\Omega$. By the same approach, we can show that $\{\widehat{\varkappa }_{\alpha }\}$ and $\{\widetilde{\varkappa }_{\alpha }\}$ are Cauchy sequences in $\Omega$. Since $\Omega$ is complete, there are $\varkappa,$ $\widehat{ \varkappa }$ and $\widetilde{\varkappa }$ in $\Omega$ such that $\varkappa _{\alpha }\rightarrow \varkappa,$ $\widehat{\varkappa }_{\alpha }\rightarrow \widehat{\varkappa }$ and $\widetilde{\varkappa }_{\alpha }\rightarrow \widetilde{\varkappa }$ as $\alpha \rightarrow \infty.$ Therefore,

Hence

For uniqueness, suppose that $\left(\varkappa _{1}, \widehat{\varkappa }_{1}, \widetilde{\varkappa }_{1}\right)$ is another TFP of $\Xi$ such that $\left(\varkappa _{1}, \widehat{\varkappa }_{1}, \widetilde{\varkappa }_{1}\right) \neq \left(\varkappa, \widehat{\varkappa }, \widetilde{\varkappa }\right).$ By a similar method to the proof of Theorem 3.1 and using (4.1), for $\nu \gg \vartheta,$ we have

Hence, we obtain $\Theta _{\varpi }\left(\varkappa, \varkappa _{1}, \nu \right) = 1,$ for $\nu \gg \vartheta$. This implies that $\varkappa = \varkappa _{1}.$ Similarly, $\widehat{\varkappa } = \widehat{\varkappa }_{1}$ and $\widetilde{\varkappa } = \widetilde{\varkappa }_{1}.$ This finishes the required.

4.2.   Solving a system of Volterra integral equations

In this part, we apply Theorem 3.1 to ensure the existence of a solution to the system of Volterra integral equations (see also ^[37,38]). Solving this system is equivalent to find a TFP of the mapping $\Xi$.

Let $B = (C[0, 1], \mathbb{R})$ be the space of all real continuous functions and describe a supremum norm on $B$ by

Define a distance $d:B\times B\rightarrow \mathbb{R}$ by

Since $\star$ is a $\nu -$norm, we get $\star (r, s) = rs,$ $\forall r, s\in \lbrack 0, 1].$ Define a fuzzy metric $\Theta _{\varpi }:B\times B\times (0, \infty)\rightarrow \lbrack 0, 1]$ by

for $\varkappa, \varrho \in B$ and $\nu \gg \vartheta.$ Obviously, $\Theta _{\varpi }$ is triangular and $\left(B, \Theta _{\varpi }, \star \right)$ is a complete $F_{CM}-$space. Consider the following system:

where $\eta \in \mathbb{R},$ and $\xi _{1}, \xi _{2}, \xi _{3}\in B.$

To study the existence of the solution to system (4.5), we consider the following:

(h$_{1}$) The functions $\xi _{i}:[0, 1]\rightarrow \mathbb{R}$ and $\mho _{i}:[0, 1]\times \lbrack 0, 1]\times \mathbb{R} \rightarrow \mathbb{R},$ $(i = 1, 2, 3)$ are continuous;

(h$_{2})$ For $\varrho, \widehat{\varphi }, \widetilde{\kappa }\in A,$ $\widehat{\varrho }, \widetilde{\varphi }, \kappa \in G$ and $\widetilde{ \varrho }, \varphi, \widehat{\kappa }\in H,$ where $A, G, H\subset B$ such that

(h$_{3})$ There exists $\gamma \in \lbrack 0, 1]$ such that

where

Analogously, $\mho \left(G_{(\varphi, \widehat{\varphi }, \widetilde{\varphi })}, H_{(\kappa, \widehat{\kappa }, \widetilde{\kappa })}\right)$ and $\mho \left(A_{(\varkappa, \widehat{\varkappa }, \widetilde{\varkappa })}, H_{(\kappa, \widehat{\kappa }, \widetilde{\kappa })}\right),$ where

$A_{\left(\varrho, \widehat{\varrho }, \widetilde{\varrho }\right) }, G_{(\varphi, \widehat{\varphi }, \widetilde{\varphi })}, H_{(\kappa, \widehat{\kappa }, \widetilde{\kappa })}, A_{\xi _{1}}, G_{\xi _{2}}, H_{\xi _{3}}, A_{\varrho }, G_{\varphi }, H_{\kappa }\in B.$

Theorem 4.3. Problem (4.5) has a unique solution, provided that theconditions $(h_{1})-(h_{3})\ $ hold.

Proof. Define an operator $\Xi :B^{3}\rightarrow B$ by

Now, we need to assume the the following cases:

($\heartsuit _{1}$) If $\mho \left(A_{(\varrho, \widehat{\varrho }, \widetilde{\varrho })}, G_{(\varphi, \widehat{\varphi }, \widetilde{\varphi })}\right) = \left\Vert A_{\varkappa }-G_{\ell }\right\Vert,$ then by (4.4) and (4.5), we get

for $\nu \gg \vartheta,$ and for $\varrho, \widehat{\varphi }, \in A,$ $\widehat{\varrho }, \widetilde{\varphi }\in G$ and $\widetilde{\varrho }, \varphi \in H.$ Therefore $\Xi$ fulfills all the conditions of Theorem 3.1 with $u_{1} = \gamma$ and $u_{2} = 0$. Thus, the problem (4.5) has a unique solution in $B$.

($\heartsuit _{2}$) If

then from (4.4) and (4.5), we have

for $\nu \gg \vartheta,$ and for $\varrho, \widehat{\varphi }, \in A,$ $\widehat{\varrho }, \widetilde{\varphi }\in G$ and $\widetilde{\varrho }, \varphi \in H.$ Therefore, $\Xi$ satisfies all the conditions of Theorem 3.1 with $u_{1} = 0$ and $u_{2} = \gamma$. Thus the system (4.5) has a unique solution in $B$.

Similarly, we can finish the proof if we consider $\mho \left(G_{(\varphi, \widehat{\varphi }, \widetilde{\varphi })}, H_{(\kappa, \widehat{ \kappa }, \widetilde{\kappa })}\right)$ and $\mho \left(A_{(\varkappa, \widehat{\varkappa }, \widetilde{\varkappa })}, H_{(\kappa, \widehat{\kappa }, \widetilde{\kappa })}\right)$ under the same cases $(\heartsuit _{1})$ and $(\heartsuit _{2})$.

Acknowledgments

The authors express many thanks to the Editor-in-Chief, handling editor, and the reviewers for their outstanding comments that improve our paper.

Conflict of interest

The authors declare that they have no competing interests concerning the publication of this article.