Existence of local and global solutions to fractional order fuzzy delay differential equation with non-instantaneous impulses

1.   Introduction

The dynamics of many evolutionary processes are characterized by the fact that at a specific moment of time they experience a sudden change in their state, such as harvesting, natural disasters and shocks, etc. These processes are subject to short-term perturbations, whose time period is minimal in analogy with the whole evolution. In dynamical systems associated with such sudden changes, we assume these changes in the form of impulses. Therefore, impulsive differential equations have been developed to model these types of situations. In literature, there are two types of impulses, one is instantaneous and another one is non-instantaneous impulses. For more detail, one can see [1-4].

The theory of fractional calculus deals with the integral and derivative of any arbitrary (real or complex) order. It was first proposed in the works by mathematicians Leibniz, Abel, L'Hopital, Riemann, Liouville [5,6]. Because of the nonlocality and inherent properties of numerous complex systems, fractional calculus is important to model many physical applications in dissimilar branches of science and engineering. Memory and hereditary are additionally significant properties of various materials and processes in biomechanics, electrical circuits, electrochemistry, biology, electromagnetic processes, control and porous media, which are widely recognized to be well predicted by using fractional differential operators [7-9]. In the existing literature, there are numerous definitions for fractional operators, for example, Riemann-Liouville, Grunwald-Letnikov, Hadamard, Caputo, Riesz-Caputo and so on.

More recently, in $2015$, Caputo and Fabrizio introduced a new fractional derivative known as Caputo-Fabrizio fractional derivative which is given by

where $\varsigma \in \mathbb{R}$ is the order of the derivative[10,11]. A year after, Atangana and Baleanu proposed another definition of nonlocal derivatives with non-singular kernel relied on the Mittag-Leffler function

where ${\mathcal{P}} \in C^F(I) \cap L^F(I)$ is the fuzzy function and $B(\varsigma) = (1 - \varsigma) + \frac{\varsigma}{\Gamma(\varsigma)}$ is known as a normalization function which satisfies $B(0) = B(1) = 1$ and $0 < \varsigma < 1.$ This definition upheld the Caputo-Fabrizio's one relied on the exponential function. It expand the profundity of the connection between the Mittag-Leffler function and fractional calculus which leads to the significant applications such as thermal physics, population dynamics, control problems and so on [12-15].

On the other hand, in many real applications there is some uncertainty that occurs in the system due to that the behaviour of the system is affected. Therefore, to overcome this type of issue, Zadeh [16] in 1965, presented the theory of fuzzy set by using the membership function. The theory of fuzzy set is a well-built tool for modelling the uncertainty, ambiguity and vague information such as particle system, medicine, quantum optics, civil engineering, computational biology, bioinformatics and hydraulic process [17-20] etc. Moreover, the nonlocal effects, as well as uncertainty behaviours, represent interesting phenomena and hence nowadays many researchers working on the fuzzy fractional operator which combine fractional calculus with the fuzzy set theory. In the last few decays, many authors established some results on the existence, uniqueness and stability of the solution for fuzzy differential equations of an integer as well as fractional order [21-28]. In particular, S. Seikkala [21], established the existence and uniqueness of the solution for a fuzzy initial value problem. In [25], authors investigated the existence and uniqueness of solution for the fractional fuzzy differential equation. In [26], authors examined the uniqueness of the solution for a nonlinear impulsive fuzzy integro-differential equation. In [27] authors examined the existence and uniqueness of a mild solution to a nonlinear fuzzy differential equation with time delay. Moreover, there are only a few papers that established the existence and uniqueness results for ABC fuzzy fractional differential equation [29,30]. For instance, in [30] authors considered the ABC fractional fuzzy differential equation and establish the existence and uniqueness of the solution. As per the author's knowledge, there is not a single paper that established the existence of local and global solutions for the ABC fractional fuzzy delay differential system with impulsive effects.

Therefore, motivated by the above facts, in this paper, we will study the existence of local and global solution for fractional order fuzzy delay differential equation with non-instantaneous impulsive condition of the form

where ${\mathcal{P}}$ is a state fuzzy function and $^{ABC}_{\ \ \ \ \ 0} D_{\xi}^{\varsigma}$ is the ABC derivative of order $\varsigma \in (0, 1)$. The points $k_j$ and $\xi_j$ satisfies the sequence $0 = k_0 = \xi_0 < \xi_1 < k_1 < \xi_2 < \cdots < \xi_n < k_n < \xi_{n+1} = T < \infty$. ${\mathcal{H}}$ is the nonlinear function which is defined from $(k_j, \xi_{j+1}] \times\mathbb{K}^n \times \mathbb{K}^n$ into $\mathbb{K}^n$, where $\mathbb{K}^n$ denotes the set of all upper semi continuous convex normal fuzzy number with bounded $\iota$-level intervals and $j = 0, 1, \cdots, n$ (which will be specified in Definition 2.9). The functions $h_j(\xi, {\mathcal{P}}(\xi_j^-))$ represent non-instantaneous impulses during the interval $(\xi_j, k_j], \ j = 1, 2, \cdots, n$. ${\mathcal{P}}(\xi_j^-), \ {\mathcal{P}}(\xi_j^{+})$ represent the left and right limit of the state fuzzy function ${\mathcal{P}}$ at $\xi_j$. For $\delta > 0, \;$ $\Psi : [-\delta, 0] \rightarrow \mathbb{K}^n$ is a continuous function.

The structure of the manuscript is as follows: In Section 2, we have given the fundamental definitions and some important lemmas. In Section 3, we establish the local existence and uniqueness results for the solution. Section 4 is devoted to establishing the global solution for the considered system and in the last Section 5, an example is given to validate the obtained analytical results.

2.   Preliminaries

In this section, we briefly describe some notations, fundamental definitions and important lemmas which are useful to prove the main results. $C^{F}(I = [0, T])$ denote the set of all continuous fuzzy valued functions on $I$ and $L^{F}(I)$ denote the set of all Lebesgue integrable fuzzy valued functions on $I$. Also we define $PC(J; \mathbb{K}^n)$ as: $PC(J; \mathbb{K}^n) = \{{\mathcal{P}} : J \rightarrow \mathbb{K}^n : {\mathcal{P}} \in C^{F}([-\delta, 0);\ \mathbb{K}^n) \ \cup \ C^{F}((\xi_j, \xi_{j+1}]; \ \mathbb{K}^n), \ j = 0, 1, \cdots, n$ and there exists ${\mathcal{P}}(\xi_j^-)$ and ${\mathcal{P}}(\xi_j^+), j = 1, 2, \cdots, n,$ with ${\mathcal{P}}(\xi_j^-) = {\mathcal{P}}(\xi_j) \}$ for the space of piecewise continuous functions, where $J = [-\delta, 0)\cup [0, T].$

Next, we define the definition of ABC derivative and integrals.

Definition 2.1. [31] The Atangana-Baleanu fractional integral of order $\varsigma \in (0, 1),$ is given by

where ${\mathcal{P}} \in C^F(I) \cap L^F(I)$ is the fuzzy function and $B(\varsigma) = (1 - \varsigma) + \frac{\varsigma}{\Gamma(\varsigma)}$ is known as the normalization function which satisfies $B(0) = B(1) = 1$.

Definition 2.2. [31] The Atangana-Baleanu fractional fuzzy derivative in Caputo sense is defined by

where ${\mathcal{P}} \in C^F(I) \cap L^F(I)$ is the fuzzy function and $E_{\varsigma}$ is the Mittag-Leffler function.

Definition 2.3. For ABC derivative, we have the following important properties of Laplace transformation

1) $L(^{ABC} D_{a+}^{\varsigma} {\mathcal{P}}(\xi)) = \frac{B(\varsigma)}{1- \varsigma} \frac{s^{\varsigma -1}}{s^\varsigma + \frac{\varsigma}{1 - \varsigma}} (s{\mathcal{P}}(s) - {\mathcal{P}}(0)),$

2) $L(\xi^{\varsigma}) = \frac{\varsigma}{s^{\varsigma+1}},$

3) $L({\mathcal{H}}^{n}(\xi)) = s^{n} L({\mathcal{H}}(\xi)) - s^{n-1}{\mathcal{H}}(0)\cdots - {\mathcal{H}}^{(n-1)}(0),$

4) $L(u(\xi) * {\mathcal{P}}(\xi)) = L(u(\xi)) L({\mathcal{P}}(\xi)).$

Now, we define some important definitions and lemmas which are often used.

Definition 2.4. [32] Let $\varkappa$ and $\alpha$ be two nonempty bounded subsets of $\mathbb{R}^n$ then we define

where $||. ||$ denotes the usual Euclidean norm in $\mathbb{R}^n$. Clearly, $d_{\cal H}(\varkappa, \alpha) = d_{\cal H}(\alpha, \varkappa)$, i.e., it is symmetric on $\varkappa$ and $\alpha$.

Consequently, for any nonemptey subsets $\varkappa, \alpha$ and $\Theta$ of $\mathbb{R}^ n$, we have

$(1)$ $d_{\cal H}(\varkappa, \alpha) \geq 0$ with $d_{\cal H}(\varkappa, \alpha) = 0$ iff $\varkappa = \alpha$,

$(2)$ $d_{\cal H}(\varkappa, \alpha) = d_{\cal H}(\alpha, \varkappa),$

$(3)$ $d_{\cal H}(\varkappa, \alpha) \leq d_{\cal H}(\varkappa, \Theta) + d_{\cal H}(\Theta, \alpha).$

We denote $\mathbb{K}^{n}(\mathbb{R}^n)$ for the family of all nonempty subset of $\mathbb{R}^n$ which are convex and compact. The scalar multiplication and addition in $\mathbb{K}^{n}(\mathbb{R}^n)$ are defined as

for all $\varrho \geq 0$ and $\varkappa, \alpha \in \mathbb{K}^{n}(\mathbb{R}^n)$.

Definition 2.5. [32] We define

Clearly, $(\mathbb{K}^{n}(\mathbb{R}^n), d_{\cal H})$ forms a complete metric space.

Definition 2.6. [32] The supremium metric $d_\infty$ on $\mathbb{K}^n$ is defined by

Clearly, we can see that $d_\infty$ is a metric in $\mathbb{K}^n$ and $(\mathbb{K}^n, d_\infty)$ forms a complete metric space.

Suppose that $J = [-\delta, T] \subset \mathbb{R}$ be a compact interval and $PC(J; \mathbb{K}^n)$ denotes tha space of all fuzzy functions which are piece-wise continuous from $J$ to $\mathbb{K}^n$. We define the metric ${H}_1$ on $PC(J; \mathbb{K}^n)$ by

Clearly, $(PC(J; \mathbb{K}^n), {H}_1)$ is a complete metric space.

Definition 2.7. [33] A membership function $\Omega_\mathcal{D} : \Xi \rightarrow [0, 1]$ of fuzzy set $\mathcal{D}$ satisfy the following:

1) If $\Omega_\mathcal{D}(\rho) = 1$, then $\rho$ is completely belongs to $\mathcal{D}$,

2) If $0 < \Omega_\mathcal{D}(\rho) < 1$, then $\rho$ is partially belongs to $\mathcal{D}$,

3) If $\Omega_\mathcal{D}(\rho) = 0$, then $\rho \notin \mathcal{D}$.

Definition 2.8. [33] A fuzzy set $\mathcal{D}$ is said to be fuzzy number if it satisfy the following properties:

1) $\mathcal{D}$ is normal, i.e., $\exists \ \rho_0 \in \mathbb{R}$ with $\Omega_\mathcal{D}(\rho_0) = 1.$

2) $\mathcal{D}$ is fuzzy convex, i.e., $\Omega_\mathcal{D}(\xi\rho + (1-\xi)\hat{\rho}) \geq min\{\Omega_\mathcal{D}(\rho), \Omega_\mathcal{D}(\hat{\rho})\}, \ \forall \ \xi \in [0, 1], \rho, \hat{\rho} \in \mathbb{R}^n.$

3) $\mathcal{D}$ is upper semi continuous on $\mathbb{R}^n$, i.e., $\forall \ \epsilon > 0, \ \exists \ \tau > 0$ such that $\Omega_\mathcal{D}(\rho) - \Omega_\mathcal{D}(\rho_0) < \epsilon, |\rho - \rho_0| < \tau.$

4) $\mathcal{D}$ is compactly supported, i.e., $cl\{\rho \in \mathbb{R}^n; \Omega_\mathcal{D}(\rho) > 0\}$ is compact.

Definition 2.9. [34] The $\iota$ - level set of fuzzy set $\mathcal{D}$ is defined by

and for $\iota = 0$, we have

Definition 2.10. [34] A fuzzy number $g\in \mathbb{R}$ is called positive if for two arbitrary fuzzy number $g_1, g_2$, it holds $0 < g_1 < g_2$ for the support $\psi_g = [g_1, g_2]$ of g, i.e., $\psi_g$ is in the positive real line. Similarly, g is called negative if $g_1 \leq g_2 < 0$ and zero if $g_1 \leq 0 \leq g_2.$

Lemma 2.1. [35] If $g, h\in \mathbb{K}^n$, then for $\iota \in (0, 1]$,

Definition 2.11. [35] We define the fuzzy integral as follows

provided that the right side Lebesgue integrals in the above equation are exists. Also, the fuzzy integral is a fuzzy number.

Lemma 2.2. [30] If $y(\xi) \in C^{F}(I)\cap L^{F}(I), \ 0 < \varsigma < 1,$ then the unique solution of following problem

is given by

Lemma 2.3. A function ${\mathcal{P}}\in PC(J; \mathbb{K}^{n})$ is the solution of the considered system (1.1) if ${\mathcal{H}}(0, {\mathcal{P}}(0), {\mathcal{P}}(-\delta)) = 0$ holds and solution is given by

Proof. From Lemma 2.2, for any $\xi \in [0, \xi_1]$, we have

Now, for any $\xi \in (\xi_1, k_1]$,

Also, for any $\xi \in (k_1, \xi_2]$,

Now, for any $\xi \in (\xi_2, k_2]$,

Also, for any $\xi \in (k_2, \xi_3],$

By using the similar process, we will get for $\xi \in (k_j, \xi_{j+1}],$

which has the form (2.1). Hence, the result follows. For more detail on solution, please see [36].

3.   Existence of local solution

In this section, we state and prove the existence and uniqueness of local solution to the system (1.1). For this purpose, the following assumptions are required:

(B1) The function ${\mathcal{H}} : (k_j, \xi_{j +1}] \times \mathbb{K}^n \times \mathbb{K}^n \rightarrow \mathbb{K}^n, \ j = 0, 1, \cdots, n,$ is continuous and there exists positive constant $M_{{\mathcal{H}}_1}$ and $M_{{\mathcal{H}}_2}$ such that

(B2) The functions $h_j : (\xi_j, k_j] \times \mathbb{K}^n \rightarrow \mathbb{K}^n, \ j = 1, 2, \cdots, n,$ are continuous and there exists a positive constants $M_{h_j} < 1$ such that

For the convenience, we use the following notations throughout the manuscript $L = \max_{1\leq j \leq n} \{L_1, L_{2_j}\},$ where $L_1 = \left(\frac{1 - \varsigma}{B(\varsigma)} (M_{{\mathcal{H}}_1} + M_{{\mathcal{H}}_2}) + \; \frac{\varsigma}{B(\varsigma) \Gamma(\varsigma) }(M_{{\mathcal{H}}_1} + M_{{\mathcal{H}}_2})T^{\varsigma}\right)$ and $L_{2_j} = \left(M_{h_{j}}+ \frac{1 - \varsigma}{B(\varsigma)}(M_{{\mathcal{H}}_1} + M_{{\mathcal{H}}_2}) + \; \frac{\varsigma}{B(\varsigma) \Gamma(\varsigma) }(M_{{\mathcal{H}}_1} + M_{{\mathcal{H}}_2})T^{\varsigma}\right),$ $j = 1, 2, \cdots, n.$

Theorem 3.1. If the assumptions $(B1)$ and $(B2)$ are satisfied then the problem (1.1) has a unique local solution on $J$.

Proof. For each $\eta\in PC(J; \mathbb{K}^{n})$, we define an operator $\Lambda : PC(J; \mathbb{K}^{n}) \rightarrow PC(J; \mathbb{K}^{n})$ such that

Here, we need to show that the operator $\Lambda$ has a fixed point, which is the solution of our considered system (1.1). The proof of this theorem is divided into the following cases:

Case 1: For $\xi \in [-\delta, 0), \ \eta, \gamma \in PC(J; \mathbb{K}^{n})$,

Hence,

Case 2: For $\xi \in [0, \xi_1], \ \eta, \gamma \in PC(J; \mathbb{K}^{n})$,

Therefore,

Therefore,

Thus,

Case 3: For $\xi\in (\xi_j, k_j], \ j = 1, 2, \cdots, n$ and $\eta, \gamma \in PC(J; \mathbb{K}^{n})$,

Therefore,

Thus,

Hence,

Case 4: For $\xi\in (k_j, \xi_{j+1}], \ j = 1, 2, \cdots, n$ and $\eta, \gamma \in PC(J; \mathbb{K}^{n})$,

Therefore,

Therefore,

Hence,

From the above four cases, we conclude that

Thus, for sufficiently small $T$, $\Lambda$ is a strict contraction mapping and hence by Banach fixed point theorem $\Lambda$ has a unique fixed point which is the solution of system (1.1). The Theorem 3.1 is existence of local solution because our mapping $\Lambda : PC(J:\mathbb{K}^n) \rightarrow PC(J:\mathbb{K}^n)$ is not strict contraction for all values of $T$. In Eq (3.1), we can see that a constant $L < 1$ if $T$ is sufficiently small. Thus, we can say that our solution exists locally.

4.   Existence of global solution

To show the existence of global solution, we need the Gronwall's inequality:

Lemma 4.1. [37] (Gronwall's inequality) Let $\mathfrak{F} (\xi, k) \geq 0$ be a continuous function on $0 \leq k < \xi \leq T.$ If, there are positive constant $a, b, \varsigma$ such that

then there is a constant $C$ such that $\mathfrak{F}(\xi, k) \leq C \ \mathit{\text{for}} \ 0 \leq k < \xi \leq T.$

For the convenience, we set the following notations

$C_1 = \max_{1 \leq j \leq n} \{a_1, c_{1_j}, a_{2_j}\}$, $C_2 = \max_{1 \leq j \leq n} \{ b_1, b_{2_j}\}$,

$C_3 = \max_{-\delta \leq \xi \leq T}\bigg\{ C_1 \exp\bigg(C_2 \int_{-\delta}^{T} |(\xi - \vartheta)^{\varsigma -1}|d\vartheta\bigg)\bigg\}$,

$a_1 = \left(\frac{1}{L_3} A_1 + \frac{1}{L_3} \frac{1 - \varsigma}{B(\varsigma)} K(T) + \frac{1}{L_3} \frac{\varsigma}{B(\varsigma) \Gamma(\varsigma) } K(T) T^{\varsigma}\right)$, $L_3 = \left(1 - \frac{1 - \varsigma}{B(\varsigma)}2K(T)\right)$, $A_1 = d_{\infty}(\Psi(0), 0)$,

$c_{1_j} = \frac{1}{1-M_{h_j}}$, $a_{2_j} = \left(\frac{1}{L_{4_j}} \frac{1 - \varsigma}{ B(\varsigma)} K(T) + \frac{1}{L_{4_j}} \frac{\varsigma}{B(\varsigma) \Gamma(\varsigma) }K(T) (T)^{\varsigma}\right)$, $b_1 = \left(\frac{1}{L_3} \frac{\varsigma}{B(\varsigma) \Gamma(\varsigma) } 2 K(T) \right),$

$L_{4_j} = \left(1 - M_{h_j} - \frac{1 - \varsigma}{B(\varsigma)} 2K(T)\right),$ $b_{2_j} = \left(\frac{1}{L_{4_j}} \frac{\varsigma}{B(\varsigma) \Gamma(\varsigma) }2 K(T)\right), \ j = 1, 2, \cdots, n.$

Theorem 4.1. Let the function ${\mathcal{H}} : (k_j, \xi_{j+1}] \times \mathbb{K}^{n} \times \mathbb{K}^{n} \rightarrow \mathbb{K}^{n}$ satisfies the assumptions $(B1)$ and $(B2)$ and there exists a real valued function $K(\xi)$ which is continuous and non decreasing such that

Then, the Eq (1.1) has a unique solution ${\mathcal{P}}$ which exists for all $\xi \in [-\delta, T]$.

Proof. By Theorem 3.1, we can continue the solution of system (1.1) as long as $\|{\mathcal{P}}\|$ stays bounded. Therefore, we need to show that if ${\mathcal{P}}$ exists on $[-\delta, T),$ then it is bounded as $\xi \uparrow T$. Also, the solution of system $(1.1)$ is given by

The proof of this theorem is divided into following four cases:

Case 1: For $\xi \in [-\delta, 0)$, we have ${\mathcal{P}}(\xi) = \Psi (\xi)$.

In this case, we get

Case 2: For $\xi \in [0, \xi_1],$

Now, we have

Therefore,

Thus,

Case 3: Similarly, for $\xi\in (\xi_j, k_j], \ j = 1, 2, \cdots, n$ and ${\mathcal{P}}(\xi) = h_j(\xi, {\mathcal{P}}(\xi_j^-))$,

Therefore,

Thus,

Case 4: For $\xi\in (k_j, \xi_{j+1}], \ j = 1, 2, \cdots, n$, we have

Now, we have

Therefore,

Thus,

From the above four cases, we conclude that, for $\xi \in [-\delta, T]$

So that,

Thus, $H_1({\mathcal{P}}, 0) = \|{\mathcal{P}}\| \leq C_3.$ Hence, from Lemma 4.1, ${\mathcal{P}}$ is bounded. Therefore, we can extend our solution to the whole interval $[-\delta, T].$ Thus, our solution is global. For more details, please see[37]).

Remark 5. By using the above argument, we can prove that the system (1.1) has atleast one solution under the following weak assumptions

(A1) Function ${\mathcal{H}} :(k_j, \xi_{j+1}] \times\mathbb{K}^n \times \mathbb{K}^n \rightarrow \mathbb{K}^n, \ j = 0, 1, \cdots, n,$ is continuous and there exists a positive constant $M_H > 0$ such that

(A2) Functions $h_j : (\xi_j, k_j] \times \mathbb{K}^n \rightarrow \mathbb{K}^n, \ j = 1, 2, \cdots, n$ are continuous and there exists a positive constants $M_{j} > 0$ such that

6.   Example

We consider the following retarded fractional differential system with non-instantaneous impulsive condition

Here, we have $\varsigma = \frac{1}{2}, \ \xi \in [-\frac{1}{2}, 2], \ 0 = k_{0} < \xi_1 = 1 < k_1 = 1.5 < \xi_2 = 2 = T, \ {\mathcal{H}}(\xi, {\mathcal{P}}(\xi), {\mathcal{P}}(\xi - \delta)) = \bar{2}\xi{\mathcal{P}}^{2}(\xi) + \bar{2}\xi^2{\mathcal{P}}^{2}(\xi - \frac{1}{2})$ and impulsive function $h_j(\xi, {\mathcal{P}}(\xi_j^-)) = \frac{\sin(j \xi)}{e^{j \xi}} {\mathcal{P}}(\xi_{j}^{-}), \ j = 1$.

The $\iota$-level of fuzzy number $\bar{2}$ is $[2]^{\iota} = [ \iota + 1, 3 - \iota], \ \forall \ \iota \in [0, 1].$ Then, the $\iota$-level set of ${\mathcal{H}}(\xi, {\mathcal{P}}(\xi), {\mathcal{P}}(\xi - \delta))$ is

Now, we have

where $c_1 = (3 - \iota) T \max_{\frac{-1}{2} \leq \xi \leq 2}\{| \eta_{q}^{\iota}(\xi) + \gamma_{q}^{\iota}(\xi) |\}$, $c_2 = (3 - \iota) T^2 \max_{\frac{-1}{2} \leq \xi \leq 2}\{|\eta_{r}^{\iota}(\xi - \frac{1}{2}) + \gamma_{r}^{\iota}(\xi - \frac{1}{2})|\}$ satisfies the condition $(B1)$.

Now, the $\iota$-level set of fuzzy number $\bar{1}$ is $[\bar{1}]^{\iota} = [\iota, 2 - \iota], \ \forall \ \iota \in [0, 1]$ and $\iota$-level set of impulsive function $h_j(\xi, {\mathcal{P}}(\xi_j^-))$ is

Therefore,

where $c_3 = (2 -\iota) \frac{\sin(jT)}{e^{jT}}, \ j = 1,$ satisfies the condition $(B2)$.

Thus, all the conditions of Theorem 3.1 are fulfilled. Hence, system (5.1) has a unique fuzzy solution.

7.   Conclusions

In this work, we have considered the fractional order fuzzy delay differential system with non-instantaneous impulses. The main aim of this work is to establish the existence of local and global solutions to the considered system. In Section 3, we have studied the existence of a local solution and in Section 4, we have extended the local solution of Section 3 to a global solution. Fuzzy set theory, Banach fixed point theorem and non-linear function analysis are the major tools to establish these results. In Section 5, an example is given to validate obtained outcomes.

Acknowledgments

The researchers would like to thank the Deanship of Scientific Research, Qassim University for funding the publication of this project. We are also thankful to the associate editor and anonymous reviewers for their constructive comments and suggestions which help us to improve the manuscript.

Conflict of interest

The authors declare no conflict of interest.