Study of Non-Linear Impulsive Neutral Fuzzy Delay Differential Equations with Non-Local Conditions

Abstract

This manuscript aims to investigate the existence and uniqueness of fuzzy mild solutions for non-local impulsive neutral functional differential equations of both first and second order, incorporating finite delay. Furthermore, the study explores the properties of fuzzy set-valued mappings of a real variable, where these mappings exhibit characteristics such as normality, convexity, upper semi-continuity, and compact support. The application of the Banach fixed-point theorem is employed to derive the results. The research extensively employs fundamental concepts from fuzzy set theory, functional analysis, and the Hausdorff metric. Additionally, an illustrative example is provided to exemplify the practical implementation of the proposed concept.

1. Introduction

Differential equations are a fundamental part of pure and practical mathematics that are employed in a wide range of applications. When analyzing a contemporary occurrence, it is frequently necessary to deal with ambiguous notions [1,2,3]. In this case, fuzzy set theory may be one of the better approaches that is not statistical or probabilistic, leading us to examine fuzzy differential equations theory [4,5]. The importance of the branch of fuzzy differential equations in fuzzy analysis is justified by the rich literature in the field. See [6,7,8] for more information. In recent years, the theory of impulsive differential equations has become a hot topic of research. Further, introducing a delay in the fuzzy model allows us to consider more general situations [9,10,11]. For more details, we refer the reader to [12,13,14]. Fuzzy concepts are used to study systems that are subject to uncertainty. In 1965, Zadeh [15] addressed the fuzzy sets by classifying them based on mappings from a set to the unit interval on the real line [16,17]. The term “fuzzy differential equations” was introduced in the year of 1978 by Kandel and Byatt [18] and also M. Mizumoto and K. Tanaka [19]. Puri and Ralescu [20] made this broad generalization and studied by Kaleva [21]. Neutral differential equations are utilized in several areas of mathematical practice due to their wide range of applications. Electricity, heat conduction in material with memory, growth problems, drug administration in the human body, balancing a pencil on a finger tip, and simulation of a jumping ball are some domains where we apply fuzzy differential equations [22,23,24,25]. They have obtained a lot of attention in recent years. These equations considerably influence medicinal and technological processes; for more information, see [10,26]. Bellman and Cooke [27] pioneered the theory of ordinary neutral functional differential equations. Then, Cruz and Hale [7], Hale [9,28], Hale, Meyer, and Henry [29] worked on it. They worked on the fundamental theory of existence and uniqueness, as well as attributes of the solution operator and also worked for stability. Many researchers have contributed to the field of neutral functional differential equations [24,30,31,32,33,34,35]. Benchohra [36] investigated neutral functional differential equations and integro differential equations augmentation in Banach spaces for the non-local Cauchy problems. In addition, Balachandran and Sakthivel [12] investigated the existence of neutral functional integro neutral functional differential equations solutions in Banach spaces. Many works have been published that deal with the existence results in mild solutions of partial neutral differential systems for the first order and second order that are related to (1)–(3) and (4)–(7). Similarly, Dauer and Balachandran proposed the existence solutions in Banach spaces to non-linear neutral integro neutral functional differential equations. Anthoni and Balachandra investigated the existence of solutions to the second-order neutral functional differential equations. See, for example, for the first and second-order cases. Balasubramaniam and Muralisankar [37] have extensively investigated the existence and uniqueness of a fuzzy solution in non-linear fuzzy neutral functional differential equations by working on the Banach Fixed Point theorem. The Banach contraction principle [38], also known as the Banach fixed point theorem, is one of the main pillars of the theory of metric fixed points. According to this principle, if T is a contraction on the non-empty set X, then T has a unique fixed point in X. Utilizing a granular derivative and contraction principle, Acharya F. et al. demonstrated the controlability of fuzzy solutions for neutral impulsive functional neutral functional differential equations with non-local conditions [32]. Using the Banach Fixed Point theorem, they developed certain criteria that ensured the existence and uniqueness of a solution for a non-linear fuzzy fractional neutral functional differential equation [10]. Some fundamental work on fuzzy neutral functional differential equations are available in [26,34,36,39,40,41,42,43]. The study of impulsive functional neutral functional differential equations is associated with the ability to simulate operations and occurrences subject to instantaneous perturbations during their evolution [44,45,46]. The disruptions are carried out in discrete steps, and their duration is not significant in perspective to the duration of the period for the processes and anomalies [16]. They refer to Simeonov’s and Bainov’s monographs, Samoilenko, Benchohra, Lakshmikantham, and Perestyuk et al., who investigate plenty of properties of their solutions and detailed bibliographies. The existence of a fuzzy solution for numerous non-specific initial value problems of first-order and second-order impulsive neutral functional differential equations is examined in this paper. Furthermore, the author knows of just a few articles dealing with fuzzy impulsive differential equations of the second order.

In this paper, we use a few key definitions and background information to examine the first order of the non-local initial value problem. In addition, the second-order non-local initial value problem concerned with a suitable examples is provided in the relevant area to illustrate the idea with a proper conclusion.

2. Preliminaries

Definition 1.

{fuzzy set} Let X be a non-empty set. A fuzzy set A in X is characterized by its membership function $A:X\to [0,1]$ and $A\left(x\right)$ is interpreted as the degree of membership of element x in fuzzy set for $x\in X$. The value zero is used to represent complete non-membership, the value one is used to represent complete membership, and values in between are used to represent intermediate degrees of membership. The mapping A is called the membership function of fuzzy set A.

Example 1.

The membership function of the fuzzy set of $\mathbb{R}$ “near to one” can be classified as $B\left(ð\right)=exp(-\alpha {(ð-1)}^2)$, where α is a ${\mathbb{R}}_{+}$.

Example 2.

Let the membership function for the fuzzy set of $\mathbb{R}$ “near to 0” classified as

$$A\left(x\right)=\frac{1}{1-y^3}$$

Using this function, we can determine the membership grade of each real number in this fuzzy set, which signifies the degree to which that number is close to zero. For instance, the number 3 is assigned a grade of 0.035, the number 1 a grade of 0.5, and the number 0 a grade of 1.

Let X and Y be two non-empty bounded subsets of ${\mathbb{R}}^n$. The distance between X and Y, denoted as $H_d(X,Y)$, is defined using the Hausdorff metric. It is given by

$$H_d(X,Y)=max\left\{\underset{m\in X}{sup}\underset{n\in Y}{inf}\parallel m-n\parallel ,\underset{n\in Y}{sup}\underset{m\in X}{inf}\parallel m-n\parallel \right\}$$

The Euclidean norm, denoted as $\parallel .\parallel$, is used to measure the distance in Rn. With respect to the Hausdorff metric, $(CC\left({\mathbb{R}}^n\right),H_d)$ forms a complete and separable metric space.

Moreover, the supremum metric $d_{\infty }$ on $M^n$ is defined as

$$d_{\infty }(c,d)=\underset{0<\alpha \le 1}{sup}{H}_d({\left[c\right]}^{\alpha },{\left[d\right]}^{\alpha })$$

∀$c,d\in M^n$. This metric establishes $(M^n,d_{\infty })$ as a complete metric space.

Additionally, the supremum metric $H_1$ on $C([0,1],M^n)$ is defined as

$$H_1(u,v)=\underset{ð\in J}{sup}{d}_{\infty }(u\left(ð\right),v\left(ð\right)).$$

where J represents the interval [0, 1]. The metric space $(C([0,1],M^n),H_1)$ is classified as complete.

Definition 2

([30,47]). A mapping ζ: $J\to M^n$ is considered strongly measurable if, the set valued map ${\zeta }_{\alpha }$: $J\to CC\left({\mathbb{R}}^n\right)$ defined by ${\zeta }_{\alpha }\left(ð\right)={\left[\zeta \left(ð\right)\right]}^{\alpha }$ is Lebesgue measurable when $CC\left({\mathbb{R}}^n\right)$ has the topology induced by the Hausdorff metric.

Definition 3

([30,47]). A mapping ζ: $[0,1]\to M^n$ is said to be levelwise continuous at ${ð}_0\in [0,1]$ if the multi-valued map ${\zeta }_{\alpha }\left(ð\right)={\left[\zeta \left(ð\right)\right]}^{\alpha }$ is continuous at $ð={ð}_0$ with respect to the Hausdorff metric d∀$\alpha \in [0,1].$

A mapping ζ: $[0,1]\to M^n$ is said to be integrably bounded, if there is an integrable function $l\left(ð\right)$ ∋ $\parallel y\left(ð\right)\parallel \le l\left(ð\right)$ for every $y\left(ð\right)\in {\zeta }_0\left(ð\right)$.

Definition 4.

Let ζ: $[0,1]\to M^n$. The integral of ζ over $[0,1]$, signified ${\displaystyle {\int }_0^1\zeta \left(ð\right)dð}$ is defined by the equation

$$\begin{array}{cc}\hfill {\left[{\int }_0^1\zeta \left(ð\right)dð\right]}^{\alpha }& ={\int }_0^1{\zeta }_{\alpha }\left(ð\right)dð\hfill \\ & =\left\{{\int }_0^{1}u\left(ð\right)dð|u:[0,1]\to {\mathbb{R}}^{n}is a measurable selection for{\zeta }_{\alpha }\right\}\hfill \end{array}$$

∀$\alpha \in (0,1$].

Definition 5.

The mapping ζ: $[0,1]\times M^n\to M^n$ is levelwise continuous at point $({ð}_0,u_0)\in [0,1]\times M^n$ provided, for any fixed $\alpha \in [0,1]$ and arbitrary $ϵ>0$, there exists a $\delta (ϵ,\alpha )>0$∋

$$H_d({\left[\zeta (ð,u)\right]}^{\alpha },{\left[\zeta (ð,u_0)\right]}^{\alpha })<ϵ$$

whenever $|ð-{ð}_0|<\delta (ϵ,\alpha )$ and $H_d({\left[u\right]}^{\alpha },{\left[u_0\right]}^{\alpha })<\delta (ϵ,\alpha )$∀$ð\in [0,1],u\in M^n.$

Definition 6

([48]). Let $T:M^n\to M^n$ and $(M^n,H_d)$ a Hausdorff metric space then T is a contraction if there exists a fixed point $r<1$ such that

$$H_d(T\left(a\right),T\left(b\right))\le r{H}_d(a,b),\forall a,b\in M^n.$$

Theorem 1

([48]). Each contraction map $T:M^n\to M^n$ on a complete Hausdorff metric space $(M^n,H_d)$ has a unique fixed point.

3. Existence Results for First-Order System

3.1. Impulsive Functional Differential Equations

In this section, we consider the first order non-local initial value problem

$$\begin{array}{cc}\hfill {\displaystyle u^{\prime }\left(ð\right)}& =\mathcal{A}u\left(ð\right)+\zeta (ð,u_{ð}),a.e.ð\in J=[0,\mathrm{\Gamma }],ð\ne {ð}_k,k=1,\cdots ,m,\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \Delta u\left({ð}_k\right)& =I_k\left(u\left({ð}_k^{-}\right)\right),k=1,\cdots ,m\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill u\left(ð\right)& +(\aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_p}))\left(ð\right)=\phi \left(ð\right),ð\in [-r,0],\hfill \end{array}$$

(3)

where the set $M^n$ contains all the upper semi-continuous, normal, convex fuzzy numbers with bounded $\alpha$-level and $\mathcal{A}:J\to M^n$ is the fuzzy coefficient $\zeta$: ${\displaystyle J\times C([-r,0],M^n)\to M^{n}and\aleph :{\left(C([-r,0],M^n)\right)}^p\to M^n,\phi :[-r,0]\to M^n,0<{ð}_1<{ð}_2<\cdots <{ð}_m\le \mathrm{\Gamma },I_k\in C(M^n,M^n),\Delta u\left({ð}_k\right)=u\left({ð}_k^{+}\right)-u\left({ð}_k^{-}\right),u\left({ð}_k^{+}\right)=\underset{h\to 0^{+}}{lim}u({ð}_k+h)}$ and ${\displaystyle u\left({ð}_k^{-}\right)=\underset{h\to 0^{+}}{lim}u({ð}_k-h)}$ represents the left limit and right limit of $u\left(ð\right)$ at $ð={ð}_k$, respectively, $k=1,\cdots ,m.$

For any function y derived on $[-r,\mathrm{\Gamma }]$ and any $ð\in J$ we denote by $y_{\zeta }$ the element of $C([-r,0],M^n)$ defined by $v_{ð}\left(\theta \right)=v(ð+\theta ),\theta \in [-r,0]$. Here, $v_{ð}$ represents the history of the state from time $ð-r$, up to the present time $ð.$

Here, we will consider the space $\Omega$ in order to define the solution for Equations (1)–(3) where,

$\Omega =${ $u:u$ is absolutely continuous from J to $M^n$} and there exists $u\left({ð}_k^{-}\right)$ and $u\left({ð}_k^{+}\right)$ where $k=1,\cdots ,m$, with $u\left({ð}_k^{-}\right)=u\left({ð}_k\right)$.

Definition 7.

A function $u\in \Omega$ is said to be a mild solution of Equations (1)–(3), if u satisfies the Equations (1)–(3). Moreover, If u is an integral solution of Equations (1)–(3), then u is given by

$$\begin{array}{cc}\hfill u\left(ð\right)& =\mathrm{\Gamma }\left(ð\right)\left[\phi \left(0\right)-\aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_p})\left(0\right)\right]+{\int }_0^{ð}\mathrm{\Gamma }(ð-s)\zeta (s,u_s)ds+\sum _{0<{ð}_k<ð}\mathrm{\Gamma }(ð-{ð}_k)I_k\left(u\left({ð}_k^{-}\right)\right),\hfill \\ \hfill \mathit{if}ð\in J.\end{array}$$

Now we will prove the existence result for the problem Equations (1)–(3). To study this problem we will formulate the following hypotheses.

(H1)

There exists a constant $d_1$∋

$$H_d({\left[\zeta (ð,u)\right]}^{\alpha },{\left[\zeta (ð,v)\right]}^{\alpha })\le d_1{H}_d({\left[u\left(\theta \right)\right]}^{\alpha },{\left[v\left(\theta \right)\right]}^{\alpha }),$$

∀$ð\in J$ and all $u,v\in C([-r,0],M^n)$, and $\theta \in [-r,0].$

(H2)

If ℵ is continuous and there exists constants $G_k,k=1,2,\cdots ,p$, ∋

$$H_d({[\aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_p})\left(s\right)]}^{\alpha },{[\aleph (v_{{\tau }_1},\cdots ,v_{{\tau }_p})\left(s\right)]}^{\alpha }\le \sum _{k=1}^p{G}_k{H}_d({\left[u_{{ð}_k}\left(s\right)\right]}^{\alpha },{\left[v_{{ð}_k}\left(s\right)\right]}^{\alpha }),$$

∀$s\in [-r,0]$ and all $u_{{\tau }_k},v_{{\tau }_k}\in C([-r,0],M^n),k=1,\cdots ,p.$

(H3)

∃ a non-negative constant $d_k$∋

$$H_d({\left[I_k\left(u\left({ð}_k^{-}\right)\right)\right]}^{\alpha },{\left[I_k\left(v\left({ð}_k^{-}\right)\right)\right]}^{\alpha })\le d_k{H}_d({\left[u\left(ð\right)\right]}^{\alpha },{\left[v\left(ð\right)\right]}^{\alpha }),$$

$k=1,\cdots ,m$, for each $u,v\in M^n$ and $\parallel \mathrm{\Gamma }\left(ð\right)\parallel \le E,ð\in J$

Theorem 2.

Assume that the hypotheses $\left(H1\right)-\left(H3\right)$ are satisfied. If

$$\left(E\sum _{k=1}^m{G}_k+E{d}_1\mathrm{\Gamma }+E\sum _{k=1}^m{d}_k\right)<1$$

then the initial value problem Equations (1)–(3) has a unique fuzzy solution on $[-r,\mathrm{\Gamma }].$

Proof. 

We transform the problem Equations (1)–(3) into a fixed point problem. The solution of the problem Equations (1)–(3) is a fixed point of the operator $\Phi :C([-r,\mathrm{\Gamma }],M^n)\to C([-r,\mathrm{\Gamma }],M^n)$, and is defined by

$\Phi \left(u\right)\left(ð\right)=\left\{\begin{array}{cc}\phi \left(ð\right)-\aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_m})\left(ð\right),\hfill & ifð\in [-r,0],\hfill \\ \mathrm{\Gamma }\left(ð\right)\left[\phi \left(0\right)-\aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_p})\left(0\right)\right]+{\int }_0^{ð}\mathrm{\Gamma }(ð-s)\zeta (s,u_s)ds\hfill \\ +{\sum }_{0<{ð}_k<ð}\mathrm{\Gamma }(ð-{ð}_k)I_k\left(u\left({ð}_k^{-}\right)\right),\hfill & \mathit{if}ð\in J.\hfill \end{array}\right.$

Now, we shall prove that $\Phi$ is a contraction operator. Consider $u,v\in C([-r,\mathrm{\Gamma }],M^n)$ and $\alpha \in (0,1]$, then

$$\begin{array}{cc}\hfill H_d& \left({[\Phi u\left(ð\right)]}^{\alpha },{[\Phi v\left(ð\right)]}^{\alpha }\right)\hfill \\ & \le H_d\left(\right[\mathrm{\Gamma }\left(ð\right)\left[\phi \left(0\right)-\aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_m})\left(0\right)\right]+{\int }_0^{ð}\mathrm{\Gamma }(ð-s)\zeta (s,u_s)ds\hfill \\ & +\sum _{0<{ð}_k<ð}\mathrm{\Gamma }(ð-{ð}_k)I_k\left(u\left({ð}_k^{-}\right)\right){]}^{\alpha },[\mathrm{\Gamma }\left(ð\right)\left[\phi \left(0\right)-\aleph (v_{{\tau }_1},\cdots ,v_{{\tau }_m})\left(0\right)\right]\hfill \\ & +{\int }_0^{ð}\mathrm{\Gamma }(ð-s)\zeta (s,v_s)ds+\sum _{0<{ð}_k<ð}\mathrm{\Gamma }(ð-{ð}_k)I_k\left(v\left({ð}_k^{-}\right)\right){]}^{\alpha })\hfill \\ & \le H_d\left({\left[\mathrm{\Gamma }\left(ð\right)\aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_m})\left(0\right)\right]}^{\alpha },{\left[\mathrm{\Gamma }\left(ð\right)\aleph (v_{{\tau }_1},\cdots ,v_{{\tau }_m})\left(0\right)\right]}^{\alpha }\right)\hfill \\ & +H_d\left({\left[{\int }_0^{ð}\mathrm{\Gamma }(ð-s)\zeta (s,u_s)ds\right]}^{\alpha },{\left[{\int }_0^{ð}\mathrm{\Gamma }(ð-s)\zeta (s,v_s)ds\right]}^{\alpha }\right)\hfill \\ & +H_d{\left({\left[\sum _{0<{ð}_k<ð}\mathrm{\Gamma }(ð-{ð}_k)I_k\left(u\left({ð}_k^{-}\right)\right)\right]}^{\alpha },\sum _{0<{ð}_k<ð}\mathrm{\Gamma }(ð-{ð}_k)I_k\left(v\left({ð}_k^{-}\right)\right)\right]}^{\alpha })\hfill \\ & \le M\sum _{k=1}^m{G}_k{d}_{\infty }\left(u\left({\tau }_k\right),v\left({\tau }_k\right)\right)+E{d}_1\mathrm{\Gamma }{d}_{\infty }\left(u(ð+\omega ),v(ð+\omega )\right)+E\sum _{k=1}^m{d}_k{d}_{\infty }\left(u\left(ð\right),v\left(ð\right)\right)\hfill \\ & \le E\sum _{k=1}^m{G}_k{H}_1(u,v)+E{d}_1\mathrm{\Gamma }{H}_1(u,v)+E\sum _{k=1}^m{d}_k{H}_1(u,v)\hfill \\ & \le \left(E\sum _{k=1}^m{G}_k+E{d}_1\mathrm{\Gamma }+E\sum _{k=1}^m{d}_k\right)H_1(u,v)\hfill \end{array}$$

As a result, $\Phi$ is a mapping contraction. According to the Banach Fixed Point theorem, $\Phi$ has a unique fixed point that is a solution to Equations (1)–(3). □

3.2. Impulsive Neutral Functional Differential Equations

In this section, we consider the first-order non-local initial value problem

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dð}[u\left(ð\right)-h(ð,u_{ð})]}& =\mathcal{A}u\left(ð\right)+\zeta (ð,u_{ð}),a.e.ð\in J=[0,\mathrm{\Gamma }],ð\ne {ð}_k,k=1,\cdots ,m,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \Delta u\left({ð}_k\right)& =I_k\left(u\left({ð}_k^{-}\right)\right),k=1,\cdots ,m\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill u\left(ð\right)+(\aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_p}))\left(ð\right)& =\phi \left(ð\right),ð\in [-r,0],\hfill \end{array}$$

(6)

where $M^n,\mathcal{A},\zeta ,\aleph ,I$ are as in problem Equations (1)–(3) and the function $h:J\times C([-r,0],M^n)$.

Definition 8.

A function $u\in \Omega$ is said to be a mild solution of Equations (4)–(6), ifu satisfies the Equations (4)–(6). Moreover, if u is an integral solution of Equations (4)–(6), then u is given by

$$\begin{array}{cc}\hfill u\left(ð\right)& =\mathrm{\Gamma }\left(ð\right)\left[\phi \left(0\right)-\aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_p})\left(0\right)-h(0,\phi )\right]+h(ð,u_{ð})+{\int }_0^{ð}\mathcal{A}\mathrm{\Gamma }(ð-s)h(s,u_s)ds\hfill \\ & +{\int }_0^{ð}\mathrm{\Gamma }(ð-s)\zeta (s,u_s)ds+\sum _{0<{ð}_k<ð}\mathrm{\Gamma }(ð-{ð}_k)I_k\left(u\left({ð}_k^{-}\right)\right),\mathit{if}ð\in J.\hfill \end{array}$$

Now we will prove the existence result for the problem Equations (4)–(6). To study this problem we will formulate the following hypotheses.

(H4)

There exists a constant $d_2$∋

$$H_d({\left[h(ð,u)\right]}^{\alpha },{\left[h(ð,v)\right]}^{\alpha })\le d_2{H}_d({\left[u\left(\theta \right)\right]}^{\alpha },{\left[v\left(\theta \right)\right]}^{\alpha }),$$

∀$ð\in J$ and all $u,v\in C([-r,0],M^n),\theta \in [-r,0].$

And $\parallel \mathcal{A}\mathrm{\Gamma }\left(ð\right)\parallel \le E_1,ð\in J$

Theorem 3.

Assume that the hypotheses (H1)–(H4) are satisfied. If

$$\left(E\sum _{k=1}^m{G}_k+d_2+E_1{d}_2\mathrm{\Gamma }+E{d}_1\mathrm{\Gamma }+E\sum _{k=1}^m{d}_k\right)<1$$

then the initial value problem Equations (1)–(3) has a unique fuzzy solution on $[-r,\mathrm{\Gamma }].$

Proof. 

We transform the problem Equations (1)–(3) into a fixed point problem. The solution of the problem Equations (1)–(3) is a fixed point of the operator $\Phi :C([-r,\mathrm{\Gamma }],M^n)\to C([-r,\mathrm{\Gamma }],M^n)$, and is defined by

$\Phi \left(u\right)\left(ð\right)=\left\{\begin{array}{c}\phi \left(ð\right)-\aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_p})\left(ð\right),ifð\in [-r,0],\hfill \\ T\left(ð\right)\left[\phi \left(0\right)-\aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_p})\left(0\right)-h(0,\phi )\right]+h(ð,u_{ð})\hfill \\ +{\int }_0^{ð}\mathcal{A}\mathrm{\Gamma }(ð-s)h(s,u_s)ds+{\int }_0^{ð}\mathrm{\Gamma }(ð-s)\zeta (s,u_s)ds\hfill \\ +{\sum }_{0<{ð}_k<ð}\mathrm{\Gamma }(ð-{ð}_k)I_k\left(u\left({ð}_k^{-}\right)\right),\mathit{if}ð\in J.\hfill \end{array}\right.$

Now, we shall prove that $\Phi$ is a contraction operator. Consider $u,v\in C([-r,\mathrm{\Gamma }],M^n)$ and $\alpha \in (0,1]$, then

$$\begin{array}{cc}\hfill H_d& \left({[\Phi u\left(ð\right)]}^{\alpha },{[\Phi v\left(ð\right)]}^{\alpha }\right)\hfill \\ & \le H_d\left(\right[\mathrm{\Gamma }\left(ð\right)\left[\phi \left(0\right)-\aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_m})\left(0\right)-h(0,\phi )\right]+h(ð,u_{ð})+{\int }_0^{ð}\mathcal{A}\mathrm{\Gamma }(ð-s)h(s,u_s)ds\hfill \\ & +{\int }_0^{ð}\mathrm{\Gamma }(ð-s)\zeta (s,u_s)ds+\sum _{0<{ð}_k<ð}\mathrm{\Gamma }(ð-{ð}_k)I_k\left(u\left({ð}_k^{-}\right)\right){]}^{\alpha },\hfill \\ & [\mathrm{\Gamma }\left(ð\right)\left[\phi \left(0\right)-\aleph (v_{{\tau }_1},\cdots ,v_{{\tau }_m})\left(0\right)-h(0,\phi )\right]+h(ð,v_{ð})\hfill \\ & +{\int }_0^{ð}\mathcal{A}\mathrm{\Gamma }(ð-s)h(s,v_s)ds+{\int }_0^{ð}\mathrm{\Gamma }(ð-s)\zeta (s,v_s)ds+\sum _{0<{ð}_k<ð}\mathrm{\Gamma }(ð-{ð}_k)I_k\left(v\left({ð}_k^{-}\right)\right){]}^{\alpha })\hfill \\ & \le H_d\left({\left[\mathrm{\Gamma }\left(ð\right)\aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_m})\left(0\right)\right]}^{\alpha },{\left[\mathrm{\Gamma }\left(ð\right)\aleph (v_{{\tau }_1},\cdots ,v_{{\tau }_m})\left(0\right)\right]}^{\alpha }\right)+H_d\left({\left[h(ð,u_{ð})\right]}^{\alpha },{\left[h(ð,v_{ð})\right]}^{\alpha }\right)\hfill \\ & +H_d\left({\left[{\int }_0^{ð}\mathcal{A}\mathrm{\Gamma }(ð-s)h(s,u_s)ds\right]}^{\alpha },{\left[{\int }_0^{ð}\mathcal{A}\mathrm{\Gamma }(ð-s)h(s,v_s)ds\right]}^{\alpha }\right)\hfill \\ & +H_d\left({\left[{\int }_0^{ð}\mathrm{\Gamma }(ð-s)\zeta (s,u_s)ds\right]}^{\alpha },{\left[{\int }_0^{ð}\mathrm{\Gamma }(ð-s)\zeta (s,v_s)ds\right]}^{\alpha }\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +H_d{\left({\left[\sum _{0<{ð}_k<ð}\mathrm{\Gamma }(ð-{ð}_k)I_k\left(u\left({ð}_k^{-}\right)\right)\right]}^{\alpha },\sum _{0<{ð}_k<ð}\mathrm{\Gamma }(ð-{ð}_k)I_k\left(v\left({ð}_k^{-}\right)\right)\right]}^{\alpha })\hfill \\ & \le E\sum _{k=1}^m{G}_k{d}_{\infty }\left(u\left({\tau }_k\right),v\left({\tau }_k\right)\right)+d_2{d}_{\infty }\left(u(ð+\omega ),v(ð+\omega )\right)+E_1{d}_2\mathrm{\Gamma }{d}_{\infty }\left(u(ð+\omega ),v(ð+\omega )\right)\hfill \\ & +E{d}_1\mathrm{\Gamma }{d}_{\infty }\left(u(ð+\omega ),v(ð+\omega )\right)+E\sum _{k=1}^m{d}_k{d}_{\infty }\left(u\left(ð\right),v\left(ð\right)\right)\hfill \\ & \le E\sum _{k=1}^m{G}_k{H}_1(u,v)+d_2{H}_1(u,v)+E_1{d}_2\mathrm{\Gamma }{H}_1(u,v)+E{d}_1\mathrm{\Gamma }{H}_1(u,v)+E\sum _{k=1}^m{d}_k{H}_1(u,v)\hfill \\ & \le \left(E\sum _{k=1}^m{G}_k+d_2+E_1{d}_2\mathrm{\Gamma }+E{d}_1\mathrm{\Gamma }+E\sum _{k=1}^m{d}_k\right)H_1(u,v)\hfill \end{array}$$

As a result, $\Phi$ is a mapping contraction. According to the Banach Fixed Point theorem, $\Phi$ has a unique fixed point that is a solution to Equations (4)–(6). □

4. Existence Results for Second-Order System

4.1. Impulsive Functional Differential Equations

In this section we consider the second order non-local initial value problem

$$\begin{array}{cc}\hfill {\displaystyle u^{″}\left(ð\right)}& =\mathcal{A}u\left(ð\right)+\zeta (ð,u_{ð}),a.e.ð\in J=[0,\mathrm{\Gamma }],ð\ne {ð}_k,k=1,\cdots ,m,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill \Delta u\left({ð}_k\right)& =I_k\left(u\left({ð}_k^{-}\right)\right),k=1,\cdots ,m\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill \Delta u^{{}^{\prime }}\left({ð}_k\right)& =J_k\left(u\left({ð}_k^{-}\right)\right),k=1,\cdots ,m\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill u\left(ð\right)& +(\aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_m}))\left(ð\right)=\phi \left(ð\right),ð\in [-r,0],u^{{}^{\prime }}\left(o\right)=\eta \hfill \end{array}$$

(10)

where $J_k\in C(M^n,M^n),v_0\in M^n$ and $\mathcal{A},\zeta ,\aleph ,I_k$ and $\phi$ are as in Section 3.

Now we will prove the existence and uniqueness result for the initial value problem Equations (7)–(10). Here, we will consider the space ${\Omega }^{\prime }$ in order to define the solution for Equations (7)–(10) where ${\Omega }^{\prime }=${ $u:u$ and $u^{\prime }$ are absolutely continuous from J to $M^n$}.

Definition 9.

A function $u\in {\Omega }^{\prime }$ is said to be a mild solution of Equations (7)–(10), ifx satisfies the equation $\left(7\right)-\left(10\right)$. Moreover, If u is an integral solution of Equations (7)–(10), then u is given by

$$\begin{array}{cc}\hfill u\left(ð\right)& =C\left(ð\right)\left[\phi \left(0\right)-\aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_m})\left(0\right)\right]+S\left(ð\right)y_0+{\int }_0^{ð}S(ð-s)\zeta (s,u_s)ds\hfill \\ & +\sum _{0<{ð}_k<ð}C(ð-{ð}_k)I_k\left(u\left({ð}_k^{-}\right)\right)+\sum _{0<{ð}_k<ð}S(ð-{ð}_k)J_k\left(u\left({ð}_k^{-}\right)\right),\mathit{if}ð\in J.\hfill \end{array}$$

Assume that

(H5)

There exists a non-negative constant $d_k^{\prime }$∋

$$\begin{array}{c}\hfill H_d({\left[J_k\left(u\left({ð}_k^{-}\right)\right)\right]}^{\alpha },{\left[J_k\left(v\left({ð}_k^{-}\right)\right)\right]}^{\alpha })\le d_k^{\prime }{H}_d({\left[u\left(ð\right)\right]}^{\alpha },{\left[v\left(ð\right)\right]}^{\alpha }),k=1,\cdots ,m,\mathrm{for}\mathrm{each}u,v\in M^n\\ \hfill \parallel C\left(ð\right)\parallel \le E_2\mathrm{and}\parallel S\left(ð\right)\parallel \le E_3\end{array}$$

Theorem 4.

Assume (H1)–(H5) are satisfied. If

$$\left(E_2\sum _{k=1}^m{G}_k+E_3{d}_1\mathrm{\Gamma }+E_2\sum _{0<{ð}_k<ð}{d}_k+E_3\sum _{0<{ð}_k<ð}{d}_k^{\prime }\right)<1,$$

then the initial value problem Equations (4)–(7) has a unique fuzzy solution on $[-r,\mathrm{\Gamma }].$

Proof. 

We transform problem Equations (7)–(10) into a fixed point problem. The solutions of the problem Equations (7)–(10) are fixed points of the operator ${\Phi }_1$: $C([-r,\mathrm{\Gamma }],M^n)\to C([-r,\mathrm{\Gamma }],M^n)$, and is defined by

${\Phi }_1\left(u\right)\left(ð\right)=\left\{\begin{array}{c}\phi \left(ð\right)-\aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_p})\left(ð\right),ifð\in [-r,\mathrm{O}],\hfill \\ +C\left(ð\right)\left[\phi \left(0\right)-\aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_m})\left(0\right)\right]+S\left(ð\right)v_0+{\int }_0^{ð}S(ð-s)\zeta (s,u_s)ds\hfill \\ +{\sum }_{0<{ð}_k<ð}C(ð-{ð}_k)I_k\left(u\left({ð}_k^{-}\right)\right)+{\sum }_{0<{ð}_k<ð}S(ð-{ð}_k)J_k\left(u\left({ð}_k^{-}\right)\right),ð\in J.\hfill \end{array}\right.$

Now, we shall prove that ${\Phi }_1$ is a contraction operator. Consider $u,v\in C([-r,\mathrm{\Gamma }],M^n)$ and $\alpha \in (0,1$]; then

$$\begin{array}{cc}\hfill H_d& \left({\left[{\Phi }_{1}u\left(ð\right)\right]}^{\alpha },{\left[{\Phi }_{1}v\left(ð\right)\right]}^{\alpha }\right)\hfill \\ & =H_d\left(\right[C\left(ð\right)\left[\phi \left(0\right)-\aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_m})\left(0\right)\right]+S\left(ð\right)v_0+{\int }_0^{ð}S(ð-s)\zeta (s,u_s)ds\hfill \\ & +\sum _{0<{ð}_k<ð}C(ð-{ð}_k)I_k\left(u\left({ð}_k^{-}\right)\right)+\sum _{0<{ð}_k<ð}S(ð-{ð}_k)J_k\left(u\left({ð}_k^{-}\right)\right){]}^{\alpha },\hfill \\ & [C\left(ð\right)\left[\phi \left(0\right)-\aleph (v_{{\tau }_1},\cdots ,v_{{\tau }_m})\left(0\right)\right]+S\left(ð\right)v_0+{\int }_0^{ð}S(ð-s)\zeta (s,v_s)ds\hfill \\ & +\sum _{0<{ð}_k<ð}C(ð-{ð}_k)I_k\left(v\left({ð}_k^{-}\right)\right)+\sum _{0<{ð}_k<ð}S(ð-{ð}_k)J_k\left(v\left({ð}_k^{-}\right)\right){]}^{\alpha })\hfill \\ & \le H_d\left({\left[C\left(ð\right)\aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_m})\left(0\right)\right]}^{\alpha },{\left[C\left(ð\right)\aleph (v_{{\tau }_1},\cdots ,v_{{\tau }_m})\left(0\right)\right]}^{\alpha }\right)\hfill \\ & +H_d\left({\left[{\int }_0^{ð}S(ð-s)\zeta (s,u_s)ds\right]}^{\alpha },{\left[{\int }_0^{ð}S(ð-s)\zeta (s,v_s)ds\right]}^{\alpha }\right)\hfill \\ & +H_d\left({\left[\sum _{0<{ð}_k<ð}C(ð-{ð}_k)I_k\left(u\left({ð}_k^{-}\right)\right)\right]}^{\alpha },{\left[\sum _{0<{ð}_k<ð}C(ð-{ð}_k)I_k\left(v\left({ð}_k^{-}\right)\right)\right]}^{\alpha }\right)\hfill \\ & +H_d\left({\left[\sum _{0<{ð}_k<ð}S(ð-{ð}_k)J_k\left(u\left({ð}_k^{-}\right)\right)\right]}^{\alpha },{\left[\sum _{0<{ð}_k<ð}S(ð-{ð}_k)J_k\left(v\left({ð}_k^{-}\right)\right)\right]}^{\alpha }\right)\hfill \\ & \le E_2\sum _{k=1}^m{G}_k{H}_d\left({\left[u_{{\tau }_k}\left(0\right)\right]}^{\alpha },{\left[v_{{\tau }_k}\left(0\right)\right]}^{\alpha }\right)+E_3{d}_1{\int }_0^{ð}{H}_d\left({\left[u_s\left(\omega \right)\right]}^{\alpha },v_s{\left(\omega \right)]}^{\alpha }\right)ds\hfill \\ & +E_2\sum _{0<{ð}_k<ð}{H}_d\left({\left[I_k\left(u\left({ð}_k^{-}\right)\right)\right]}^{\alpha },{\left[I_k\left(v\left({ð}_k^{-}\right)\right)\right]}^{\alpha }\right)+E_3\sum _{0<{ð}_k<ð}{H}_d\left({\left[J_k\left(u\left({ð}_k^{-}\right)\right)\right]}^{\alpha },{\left[J_k\left(v\left({ð}_k^{-}\right)\right)\right]}^{\alpha }\right)\hfill \\ & \le E_2\sum _{k=1}^m{G}_k{d}_{\infty }\left(u\left({\tau }_k\right),v({\tau }_k\right)+E_3{d}_1{\int }_0^{ð}{d}_{\infty }\left(u(s+\omega ),v(s+\omega )\right)ds\hfill \\ & +E_2\sum _{0<{ð}_k<ð}{d}_k{d}_{\infty }\left(u\left(ð\right),v\left(ð\right)\right)+E_3\sum _{0<{ð}_k<ð}{d}_k^{\prime }{d}_{\infty }\left(u\left(ð\right),v\left(ð\right)\right))\hfill \\ & \le E_2\sum _{k=1}^m{G}_k{H}_1(u,v)+E_3{d}_1\mathrm{\Gamma }{H}_1(u,v)+E_2\sum _{0<{ð}_k<ð}{d}_k{H}_1(u,v)+E_3\sum _{0<{ð}_k<ð}{d}_k^{\prime }{H}_1(u,v)\hfill \\ & \le \left(E_2\sum _{k=1}^m{G}_k+E_3{d}_1\mathrm{\Gamma }+E_2\sum _{0<{ð}_k<ð}{d}_k+E_3\sum _{0<{ð}_k<ð}{d}_k^{\prime }\right)H_1(u,v)\hfill \end{array}$$

As a result, $\Phi$ is a mapping contraction. According to the Banach Fixed Point theorem, $\Phi$ has a unique fixed point that is a solution to Equations (7)–(10). □

4.2. Impulsive Neutral Functional Differential Equations

In this section, we consider the second order non-local initial value problem

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dð}[u^{\prime }\left(ð\right)-h(ð,u_{ð})]}& =Au\left(ð\right)+\zeta (ð,u_{ð}),a.e.ð\in J=[0,\mathrm{\Gamma }],ð\ne {ð}_k,k=1,\cdots ,m,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill \Delta u\left({ð}_k\right)& =I_k\left(u\left({ð}_k^{-}\right)\right),k=1,\cdots ,m\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill \Delta u^{\prime }\left({ð}_k\right)& =J_k\left(u\left({ð}_k^{-}\right)\right),k=1,\cdots ,m\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill u\left(ð\right)+(\aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_m}))\left(ð\right)& =\phi \left(ð\right),ð\in [-r,0],u^{\prime }\left(o\right)=v_0\hfill \end{array}$$

(14)

where $J_k\in C(M^n,M^n),v_0\in M^n$ and $\mathcal{A},\zeta ,\aleph ,h,I_k$ and $\phi$ are as in Section 3.

Now, we will prove the existence and uniqueness result for the initial value problem Equations (11)–(14). Here, we will consider the space ${\Omega }^{\prime }$ in order to define the solution for Equations (11)–(14) where ${\Omega }^{\prime }=${ $u:u$ and $u^{\prime }$ are absolutely continuous from J to $M^n$}.

Definition 10.

A function $u\in {\Omega }^{\prime }$ is said to be a mild solution of Equations (11)–(14), ifu satisfies the Equations (11)–(14). Moreover, if u is an integral solution of Equations (11)–(14), then u is given by

$$\begin{array}{cc}\hfill u\left(ð\right)& =C\left(ð\right)\left[\phi \left(0\right)-\aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_m})\left(0\right)-h(0,\phi )\right]+S\left(ð\right)[v_0-h(0,\phi )]+{\int }_0^{ð}C(ð-s)h(s,v_s)ds\hfill \\ & +{\int }_0^{ð}S(ð-s)\zeta (s,u_s)ds+\sum _{0<{ð}_k<ð}C(ð-{ð}_k)I_k\left(u\left({ð}_k^{-}\right)\right)+\sum _{0<{ð}_k<ð}S(ð-{ð}_k)J_k\left(u\left({ð}_k^{-}\right)\right),\mathit{if}ð\in J.\hfill \end{array}$$

Hence, Φ is a contraction mapping. By the Banach Fixed Point theorem, Φ has a unique fixed point which is a solution to Equations (11)–(14).

Theorem 5.

Assume (H1)–(H5) are satisfied. If

$$\left(E_2\sum _{k=1}^m{G}_k+E_2{d}_2\mathrm{\Gamma }+E_3{d}_1\mathrm{\Gamma }+E_2\sum _{0<{ð}_k<ð}{d}_k+E_3\sum _{0<{ð}_k<ð}{d}_k^{\prime }\right)<1,$$

then the initial value problem Equations (11)–(14) has a unique fuzzy solution on $[-r,\mathrm{\Gamma }].$

Proof. 

We transform problem Equations (11)–(14) into a fixed point problem. The solutions of the problem Equations (11)–(14) are fixed points of the operator ${\Phi }_1$: $C([-r,\mathrm{\Gamma }],M^n)\to C([-r,\mathrm{\Gamma }],M^n)$, and is defined by

${\Phi }_1\left(u\right)\left(ð\right)=\left\{\begin{array}{c}\phi \left(ð\right)-\aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_p})\left(ð\right),ifð\in [-r,\mathrm{O}],\hfill \\ +C\left(ð\right)\left[\phi \left(0\right)-\aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_m})\left(0\right)-h(0,\phi )\right]+S\left(ð\right)[v_0-h(0,\phi )]\hfill \\ +{\int }_0^{ð}C(ð-s)h(s,u_s)ds+{\int }_0^{ð}S(ð-s)\zeta (s,u_s)ds\hfill \\ +{\sum }_{0<{ð}_k<ð}C(ð-{ð}_k)I_k\left(u\left({ð}_k^{-}\right)\right)+{\sum }_{0<{ð}_k<ð}S(ð-{ð}_k)J_k\left(u\left({ð}_k^{-}\right)\right),ð\in J.\hfill \end{array}\right.$

Now, we shall prove that ${\Phi }_1$ is a contraction operator. Consider $u,v\in C([-r,\mathrm{\Gamma }],M^n)$ and $\alpha \in (0,1$]; then

$$\begin{array}{cc}\hfill H_d& \left({\left[{\Phi }_{1}u\left(ð\right)\right]}^{\alpha },{\left[{\Phi }_{1}v\left(ð\right)\right]}^{\alpha }\right)\hfill \\ & =H_d\left(\right[C\left(ð\right)\left[\phi \left(0\right)-\aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_m})\left(0\right)-h(0,\phi )\right]+S\left(ð\right)[v_0-h(0,\phi )]+{\int }_0^{ð}C(ð-s)h(s,u_s)ds\hfill \\ & +{\int }_0^{ð}S(ð-s)\zeta (s,u_s)ds+\sum _{0<{ð}_k<ð}C(ð-{ð}_k)I_k\left(u\left({ð}_k^{-}\right)\right)+\sum _{0<{ð}_k<ð}S(ð-{ð}_k)J_k\left(u\left({ð}_k^{-}\right)\right){]}^{\alpha },\hfill \\ & [C\left(ð\right)\left[\phi \left(0\right)-\aleph (v_{{\tau }_1},\cdots ,v_{{\tau }_m})\left(0\right)-h(0,\phi )\right]+S\left(ð\right)[v_0-h(0,\phi )]+{\int }_0^{ð}C(ð-s)h(s,v_s)ds\hfill \\ & +{\int }_0^{ð}S(ð-s)\zeta (s,v_s)ds+\sum _{0<{ð}_k<ð}C(ð-{ð}_k)I_k\left(v\left({ð}_k^{-}\right)\right)+\sum _{0<{ð}_k<ð}S(ð-{ð}_k)J_k\left(v\left({ð}_k^{-}\right)\right){]}^{\alpha })\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le H_d\left({\left[C\left(ð\right)\aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_m})\left(0\right)\right]}^{\alpha },{\left[C\left(ð\right)\aleph (v_{{\tau }_1},\cdots ,v_{{\tau }_m})\left(0\right)\right]}^{\alpha }\right)+H_d({\left[{\int }_0^{ð}C(ð-s)h(s,u_s)ds\right]}^{\alpha },\hfill \\ & {\left[{\int }_0^{ð}C(ð-s)h(s,v_s)ds\right]}^{\alpha })+H_d\left({\left[{\int }_0^{ð}S(ð-s)\zeta (s,u_s)ds\right]}^{\alpha },{\left[{\int }_0^{ð}S(ð-s)\zeta (s,v_s)ds\right]}^{\alpha }\right)\hfill \\ & +H_d\left({\left[\sum _{0<{ð}_k<ð}C(ð-{ð}_k)I_k\left(u\left({ð}_k^{-}\right)\right)\right]}^{\alpha },{\left[\sum _{0<{ð}_k<ð}C(ð-{ð}_k)I_k\left(v\left({ð}_k^{-}\right)\right)\right]}^{\alpha }\right)\hfill \\ & +H_d\left({\left[\sum _{0<{ð}_k<ð}S(ð-{ð}_k)J_k\left(u\left({ð}_k^{-}\right)\right)\right]}^{\alpha },{\left[\sum _{0<{ð}_k<ð}S(ð-{ð}_k)J_k\left(v\left({ð}_k^{-}\right)\right)\right]}^{\alpha }\right)\hfill \\ & \le E_2\sum _{k=1}^m{G}_k{H}_d\left({\left[u_{{\tau }_k}\left(0\right)\right]}^{\alpha },{\left[v_{{\tau }_k}\left(0\right)\right]}^{\alpha }\right)+E_2{d}_2{\int }_0^{ð}{H}_d\left({\left[u_s\left(\omega \right)\right]}^{\alpha },v_s{\left(\omega \right)]}^{\alpha }\right)ds\hfill \\ & +E_3{d}_1{\int }_0^{ð}{H}_d\left({\left[u_s\left(\omega \right)\right]}^{\alpha },v_s{\left(\omega \right)]}^{\alpha }\right)ds+E_2\sum _{0<{ð}_k<ð}{H}_d\left({\left[I_k\left(u\left({ð}_k^{-}\right)\right)\right]}^{\alpha },{\left[I_k\left(v\left({ð}_k^{-}\right)\right)\right]}^{\alpha }\right)\hfill \\ & +E_3\sum _{0<{ð}_k<ð}{H}_d\left({\left[J_k\left(u\left({ð}_k^{-}\right)\right)\right]}^{\alpha },{\left[J_k\left(v\left({ð}_k^{-}\right)\right)\right]}^{\alpha }\right)\hfill \\ & \le E_2\sum _{k=1}^m{G}_k{d}_{\infty }\left(u\left({\tau }_k\right),v({\tau }_k\right)+E_2{d}_2{\int }_0^{ð}{d}_{\infty }\left(u(s+\omega ),v(s+\omega )\right)ds\hfill \\ & +E_3{d}_1{\int }_0^{ð}{d}_{\infty }\left(u(s+\omega ),v(s+\omega )\right)ds+E_2\sum _{0<{ð}_k<ð}{d}_k{d}_{\infty }\left(u\left(ð\right),v\left(ð\right)\right)+E_3\sum _{0<{ð}_k<ð}{d}_k^{\prime }{d}_{\infty }\left(u\left(ð\right),v\left(ð\right)\right))\hfill \\ & \le E_2\sum _{k=1}^m{G}_k{H}_1(u,v)+E_2{d}_2\mathrm{\Gamma }{H}_1(u,v)+E_3{d}_1\mathrm{\Gamma }{H}_1(u,v)+E_2\sum _{0<{ð}_k<ð}{d}_k{H}_1(u,v)+E_3\sum _{0<{ð}_k<ð}{d}_k^{\prime }{H}_1(u,v)\hfill \\ & \le \left(E_2\sum _{k=1}^m{G}_k+E_2{d}_2\mathrm{\Gamma }+E_3{d}_1\mathrm{\Gamma }+E_2\sum _{0<{ð}_k<ð}{d}_k+E_3\sum _{0<{ð}_k<ð}{d}_k^{\prime }\right)H_1(u,v)\hfill \end{array}$$

As a result, $\Phi$ is a mapping contraction. According to the Banach Fixed Point theorem, $\Phi$ has a unique fixed point that is a solution to Equations (11)–(14). □

5. Examples

Example 3.

In the study, the fuzzy solution of non-linear fuzzy neutral impulsive functional neutral functional differential equations with non-local condition of the form

$$\begin{array}{cc}\hfill u^{\prime }\left(ð\right)& =2\left[u\left(ð\right)\right]+3ðu{(ð+h)}^3,\hfill \\ \hfill u\left(0\right)& +\sum _{k=1}^p{c}_{k}u\left({ð}_k\right)=0\in M^n\hfill \\ \hfill I_k\left(u\left({ð}_k^{-}\right)\right)& =\frac{1}{1+u\left({ð}_k\right),}\hfill \end{array}$$

where ${\displaystyle \zeta (ð,u_{ð})=3{ð}^{2}u{(ð+h)}^2,\aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_p})=\sum _{k=1}^p{c}_{k}u\left({ð}_k\right)}$. Using the properties of fuzzy numbers, refer [31], we define the α-level set. The α-level set of fuzzy number 2 is, [2 ${]}^{\alpha }=[\alpha +1,3-\alpha ]$, ∀$\alpha \in [0,1]$ The α-level set of fuzzy number 3 is, [3 ${]}^{\alpha }=[\alpha +2,4-\alpha ]$, ∀$\alpha \in [0,1]$

The α-level set of ${\left[\zeta (ð,u_{ð})\right]}^{\alpha }$ is

$${\left[3ðu{(ð+h)}^2\right]}^{\alpha }=ð{\left[3\right]}^{\alpha }{\left[u{(ð+h)}^2\right]}^{\alpha }$$

$$=ð[\alpha +2,4-\alpha ][{\left(u_l^{\alpha }(ð+h)\right)}^2,{\left(u_r^{\alpha }(ð+h)\right)}^2]$$

$$=ð[(\alpha +2)\left(u_l^{\alpha }{(ð+h)}^2\right),(4-\alpha ){\left(u_r^{\alpha }(ð+h)\right)}^2]$$

Then,

$$H_d{({\left[\zeta (ð,u_{ð})\right]}^{\alpha },\left[\zeta (ð,v_{ð})\right)]}^{\alpha })$$

$$=H_d(ð[(\alpha +2){\left(u_l^{\alpha }(ð+h)\right)}^2,(4-\alpha ){\left(u_r^{\alpha }(ð+h)\right)}^2],ð[(\alpha +2){\left(v_l^{\alpha }(ð+h)\right)}^2,$$

$$(4-\alpha ){\left(v_r^{\alpha }(ð+h)\right)}^2\left]\right)$$

$$=tmax\left\{(\alpha +2)\right|{\left(u_l^{\alpha }(ð+h)\right)}^2-{\left(v_l^{\alpha }(ð+h)\right)}^2|,(4-\alpha )|{\left(u_r^{\alpha }(ð+h)\right)}^2-{\left(v_r^{\alpha }(ð+h)\right)}^2\left|\right\}$$

$$\le 4b|u_r^{\alpha }(ð+h)+v_r^{\alpha }(ð+h)|max\{\left|\right(u_l^{\alpha }(ð+h)-(v_l^{\alpha }(ð+h|(u_r^{\alpha }(ð+h)-(v_r^{\alpha }(ð+h\}$$

$$=d_1{H}_d({\left[u(ð+h)\right]}^{\alpha },{\left[v(ð+h)\right]}^{\alpha })$$

where ${\left[u(ð+h)\right]}^{\alpha }=[u_l^{\alpha }(ð+h),u_r^{\alpha }(ð+h)]$

The α-level set of ${\displaystyle \aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_p})=\sum _{k=1}^p{c}_{k}u\left({ð}_k\right)}$. Then

$$H_d({[\aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_p})]}^{\alpha },{[\aleph (v_{{\tau }_1},\cdots ,v_{{\tau }_p})]}^{\alpha }$$

$$=H_d({\left[\sum _{k=1}^p{c}_k\left(u\left({ð}_k\right)\right)\right]}^{\alpha },{\left[\sum _{k=1}^p{c}_k\left(v\left({ð}_k\right)\right)\right]}^{\alpha })$$

$$=H_d([\sum _{k=1}^p{c}_k{u}_l^{\alpha }\left({ð}_k\right),\sum _{k=1}^p{c}_k{u}_r^{\alpha }\left({ð}_k\right)],[\sum _{k=1}^p{c}_k{v}_l^{\alpha }\left({ð}_k\right),\sum _{k=1}^p{c}_k{v}_r^{\alpha }\left({ð}_k\right)])$$

$$\le |\sum _{k=1}^p{c}_k|\underset{k}{max}\left\{\right|u_l^{\alpha }\left({ð}_k\right)-v_l^{\alpha }\left({ð}_k\right)|,|u_r^{\alpha }\left({ð}_k\right)-v_r^{\alpha }\left({ð}_k\right)\left|\right\}$$

$$\le \sum _{k=1}^p{G}_k{H}_d({\left[u_{{ð}_k}\left(s\right)\right]}^{\alpha },{\left[v_{{ð}_k}\left(s\right)\right]}^{\alpha })$$

The α-level set of ${\left[I_k\left(u\left({ð}_k\right)\right)\right]}^{\alpha }$ is

$${\left[I_k\left(u\left({ð}_k\right)\right)\right]}^{\alpha }={\left[\frac{1}{1+u\left({ð}_k\right)}\right]}^{\alpha }=[\frac{1}{1+u_l^{\alpha }\left({ð}_k\right)},\frac{1}{1+u_r^{\alpha }\left({ð}_k\right)}]$$

Thus,

$$H_d{\left(\left[I_{k}u\left({ð}_k\right)\right)\right]}^{\alpha },{\left[I_k\left(v\left({ð}_k\right)\right)\right]}^{\alpha })$$

$$=H_d({\left[\frac{1}{1+u\left({ð}_k\right)}\right]}^{\alpha },{\left[\frac{1}{1+v\left({ð}_k\right)}\right]}^{\alpha })$$

$$=H_d([\frac{1}{1+u_l^{\alpha }\left({ð}_k\right)},\frac{1}{1+u_r^{\alpha }\left({ð}_k\right)}],[\frac{1}{1+v_l^{\alpha }\left({ð}_k\right)},\frac{1}{1+v_r^{\alpha }\left({ð}_k\right)}])$$

$$\le \underset{k}{max}\left\{\right|\frac{1}{1+u_l^{\alpha }\left(ð\right)}-\frac{1}{1+v_l^{\alpha }\left(ð\right)}|,|\frac{1}{1+u_r^{\alpha }\left(ð\right)}-\frac{1}{1+v_r^{\alpha }\left(ð\right)}\left|\right\}$$

$$\le \underset{k}{max}\left\{\right|\frac{u_l^{\alpha }\left(ð\right)-v_l^{\alpha }\left(ð\right)}{(1+v_l^{\alpha }\left({ð}_k\right))(1+u_l^{\alpha }\left({ð}_k\right))}|,|\frac{u_r^{\alpha }\left(ð\right)-v_r^{\alpha }\left(ð\right)}{(1+u_r^{\alpha }\left({ð}_k\right))(1+v_r^{\alpha }\left({ð}_k\right))}\left|\right\}$$

$$\le \underset{k}{max}\{\frac{|u_l^{\alpha }\left({ð}_k\right)-v_l^{\alpha }\left({ð}_k\right)|}{(1+|v_l^{\alpha }\left({ð}_k\right)\left|\right)(1+|u_l^{\alpha }\left({ð}_k\right)\left|\right)},(1+|u_r^{\alpha }\left({ð}_k\right)\left|\right)(1+|v_r^{\alpha }\left({ð}_k\right)\left|\right)|u_r^{\alpha }\left({ð}_k\right)-v_r^{\alpha }\left({ð}_k\right)\left|\right\}$$

$$\le d_k\underset{k}{max}\left\{\right|u_l^{\alpha }-v_l^{\alpha }|,|u_r^{\alpha }-v_r^{\alpha }\left|\right\}$$

$$=d_k{H}_d({\left[u\right]}^{\alpha },{\left[v\right]}^{\alpha })$$

where $d_k=1/(1+|u_r^{\alpha }\left({ð}_k\right)\left|\right)(1+|v_r^{\alpha }\left({ð}_k\right)$ Similiarly $d_k^{\prime }$ also holds.

The constant terms $d_1,G_k,d_k$ and $d_k^{\prime }$ satisfy the Lipschitz condition. Then, from Theorems 1 and 5 the fuzzy neutral functional differential equations have a unique fuzzy solution.

Example 4.

In the study, the fuzzy solution of non-linear fuzzy neutral impulsive functional neutral functional differential equations with non-local condition of the form

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dð}[u\left(ð\right)}& -2ðu{(ð+h)}^2{]=2\left[u\left(ð\right)\right]+3ðu(ð+h)}^3,\hfill \\ \hfill u\left(0\right)& +\sum _{k=1}^p{c}_{k}u\left({ð}_k\right)=0\in M^n\hfill \\ \hfill I_k\left(u\left({ð}_k^{-}\right)\right)& =\frac{1}{1+u\left({ð}_k\right),}\hfill \end{array}$$

where ${\displaystyle h(ð,u_{ð})=2ðu{(ð+h)}^2,\zeta (ð,u_{ð})=3{ð}^{2}u{(ð+h)}^2,\aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_p})=\sum _{k=1}^p{c}_{k}u\left({ð}_k\right)}$.

Using the properties of fuzzy nos., refer [34], we define the α-level set.

The α-level set of fuzzy number 2 is,

[2 ${]}^{\alpha }=[\alpha +1,3-\alpha ]$, ∀$\alpha \in [0,1]$ The α-level set of fuzzy nos. 3 is,

[3 ${]}^{\alpha }=[\alpha +2,4-\alpha ]$, ∀$\alpha \in [0,1]$

The α-level set of ${\left[h(ð,u_{ð})\right]}^{\alpha }$ is

$${\left[2ðu{(ð+h)}^2\right]}^{\alpha }=ð{\left[2\right]}^{\alpha }{\left[u{(ð+h)}^2\right]}^{\alpha }$$

$$=ð[\alpha +1,3-\alpha ][{\left(u_l^{\alpha }(ð+h)\right)}^2,{\left(u_r^{\alpha }(ð+h)\right)}^2]$$

$$=ð[(\alpha +1){\left(u_l^{\alpha }(ð+h)\right)}^2,(3-\alpha ){\left(u_r^{\alpha }(ð+h)\right)}^2]$$

Then,

$$H_d{({\left[h(ð,u_{ð})\right]}^{\alpha },h(ð,v_{\zeta })]}^{\alpha })$$

$$=H_d(ð[(\alpha +1){\left(u_l^{\alpha }(ð+h)\right)}^2,(3-\alpha ){\left(u_r^{\alpha }(ð+h)\right)}^2],ð[(\alpha +1){\left(v_l^{\alpha }(ð+h)\right)}^2,$$

$$(3-\alpha ){\left(v_r^{\alpha }(ð+h)\right)}^2\left]\right)$$

$$=max\left\{ð(\alpha +1)\right|{\left(u_l^{\alpha }(ð+h)\right)}^2-{\left(v_l^{\alpha }(ð+h)\right)}^2|,t(3-\alpha )|{\left(u_r^{\alpha }(ð+h)\right)}^2-{\left(v_r^{\alpha }(ð+h)\right)}^2\left|\right\}$$

$$\le 3b|u_r^{\alpha }(ð+h)+v_r^{\alpha }(ð+h)|max\{|\left(u_l^{\alpha }(ð+h)\right)-(v_l^{\alpha }(ð+h|\left(u_r^{\alpha }(ð+h)\right)-(v_r^{\alpha }(ð+h\}$$

$$=d_2{H}_d({\left[u(ð+h)\right]}^{\alpha },{\left[v(ð+h)\right]}^{\alpha })$$

where ${\left[u(ð+h)\right]}^{\alpha }=[u_l^{\alpha }(ð+h),u_r^{\alpha }(ð+h)]$ The α-level set of ${\left[\zeta (ð,u_{ð})\right]}^{\alpha }$ is

$${\left[3ðu{(ð+h)}^2\right]}^{\alpha }=ð{\left[3\right]}^{\alpha }{\left[u{(ð+h)}^2\right]}^{\alpha }$$

$$=ð[\alpha +2,4-\alpha ][{\left(u_l^{\alpha }(ð+h)\right)}^2,{\left(u_r^{\alpha }(ð+h)\right)}^2]$$

$$=ð[(\alpha +2)\left(u_l^{\alpha }{(ð+h)}^2\right),(4-\alpha ){\left(u_r^{\alpha }(ð+h)\right)}^2]$$

Then,

$$H_d{({\left[\zeta (ð,u_{ð})\right]}^{\alpha },\left[\zeta (ð,v_{ð})\right)]}^{\alpha })$$

$$=H_d(ð[(\alpha +2){\left(u_l^{\alpha }(ð+h)\right)}^2,(4-\alpha ){\left(u_r^{\alpha }(ð+h)\right)}^2],ð[(\alpha +2){\left(v_l^{\alpha }(ð+h)\right)}^2,$$

$$(4-\alpha ){\left(v_r^{\alpha }(ð+h)\right)}^2\left]\right)$$

$$=tmax\left\{(\alpha +2)\right|{\left(u_l^{\alpha }(ð+h)\right)}^2-{\left(v_l^{\alpha }(ð+h)\right)}^2|,(4-\alpha )|{\left(u_r^{\alpha }(ð+h)\right)}^2-{\left(v_r^{\alpha }(ð+h)\right)}^2\left|\right\}$$

$$\le 4b|u_r^{\alpha }(ð+h)+v_r^{\alpha }(ð+h)|max\{\left|\right(u_l^{\alpha }(ð+h)-(v_l^{\alpha }(ð+h|(u_r^{\alpha }(ð+h)-(v_r^{\alpha }(ð+h\}$$

$$=d_1{H}_d({\left[u(ð+h)\right]}^{\alpha },{\left[v(ð+h)\right]}^{\alpha })$$

where ${\left[u(ð+h)\right]}^{\alpha }=[u_l^{\alpha }(ð+h),u_r^{\alpha }(ð+h$

The α-level set of ${\displaystyle \aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_p})=\sum _{k=1}^p{c}_{k}u\left({ð}_k\right)}$. Then

$$H_d({[\aleph (u_{{\tau }_1},\cdots ,u_{{\tau }_p})]}^{\alpha },{[\aleph (v_{{\tau }_1},\cdots ,v_{{\tau }_p})]}^{\alpha }$$

$$=H_d({\left[\sum _{k=1}^p{c}_k\left(u\left({ð}_k\right)\right)\right]}^{\alpha },{\left[\sum _{k=1}^p{c}_k\left(v\left({ð}_k\right)\right)\right]}^{\alpha })$$

$$=H_d([\sum _{k=1}^p{c}_k{u}_l^{\alpha }\left({ð}_k\right),\sum _{k=1}^p{c}_k{u}_r^{\alpha }\left({ð}_k\right)],[\sum _{k=1}^p{c}_k{v}_l^{\alpha }\left({ð}_k\right),\sum _{k=1}^p{c}_k{v}_r^{\alpha }\left({ð}_k\right)])$$

$$\le |\sum _{k=1}^p{c}_k|\underset{k}{max}\left\{\right|u_l^{\alpha }\left({ð}_k\right)-v_l^{\alpha }\left({ð}_k\right)|,|u_r^{\alpha }\left({ð}_k\right)-v_r^{\alpha }\left({ð}_k\right)\left|\right\}$$

$$\le \sum _{k=1}^p{G}_k{H}_d({\left[u_{{ð}_k}\left(s\right)\right]}^{\alpha },{\left[v_{{ð}_k}\left(s\right)\right]}^{\alpha })$$

The α-level set of ${\left[I_k\left(u\left({ð}_k\right)\right)\right]}^{\alpha }$ is

$${\left[I_k\left(u\left({ð}_k\right)\right)\right]}^{\alpha }={\left[\frac{1}{1+u\left({ð}_k\right)}\right]}^{\alpha }=[\frac{1}{1+u_l^{\alpha }\left({ð}_k\right)},\frac{1}{1+u_r^{\alpha }\left({ð}_k\right)}]$$

Thus,

$$H_d{\left(\left[I_{k}u\left({ð}_k\right)\right)\right]}^{\alpha },{\left[I_k\left(v\left({ð}_k\right)\right)\right]}^{\alpha })$$

$$=H_d({\left[\frac{1}{1+u\left({ð}_k\right)}\right]}^{\alpha },{\left[\frac{1}{1+v\left({ð}_k\right)}\right]}^{\alpha })$$

$$=H_d([\frac{1}{1+u_l^{\alpha }\left({ð}_k\right)},\frac{1}{1+u_r^{\alpha }\left({ð}_k\right)}],[\frac{1}{1+v_l^{\alpha }\left({ð}_k\right)},\frac{1}{1+v_r^{\alpha }\left({ð}_k\right)}])$$

$$\le \underset{k}{max}\left\{\right|\frac{1}{1+u_l^{\alpha }\left(ð\right)}-\frac{1}{1+v_l^{\alpha }\left(ð\right)}|,|\frac{1}{1+u_r^{\alpha }\left(ð\right)}-\frac{1}{1+v_r^{\alpha }\left(ð\right)}\left|\right\}$$

$$\le \underset{k}{max}\left\{\right|\frac{u_l^{\alpha }\left(ð\right)-v_l^{\alpha }\left(ð\right)}{(1+v_l^{\alpha }\left({ð}_k\right))(1+u_l^{\alpha }\left({ð}_k\right))}|,|\frac{u_r^{\alpha }\left(ð\right)-v_r^{\alpha }\left(ð\right)}{(1+u_r^{\alpha }\left({ð}_k\right))(1+v_r^{\alpha }\left({ð}_k\right))}\left|\right\}$$

$$\le \underset{k}{max}\{\frac{|u_l^{\alpha }\left({ð}_k\right)-v_l^{\alpha }\left({ð}_k\right)|}{(1+|v_l^{\alpha }\left({ð}_k\right)\left|\right)(1+|u_l^{\alpha }\left({ð}_k\right)\left|\right)},(1+|u_r^{\alpha }\left({ð}_k\right)\left|\right)(1+|v_r^{\alpha }\left({ð}_k\right)\left|\right)|u_r^{\alpha }\left({ð}_k\right)-v_r^{\alpha }\left({ð}_k\right)\left|\right\}$$

$$\le d_k\underset{k}{max}\left\{\right|u_l^{\alpha }-v_l^{\alpha }|,|u_r^{\alpha }-v_r^{\alpha }\left|\right\}$$

$$=d_k{H}_d({\left[u\right]}^{\alpha },{\left[v\right]}^{\alpha })$$

where $d_k=1/(1+|u_r^{\alpha }\left({ð}_k\right)\left|\right)(1+|v_r^{\alpha }\left({ð}_k\right)$ Similiarly $d_k^{\prime }$ also holds.

The constant terms $d_1,d_2,G_k,d_k$ and $d_k^{\prime }$ satisfy the Lipschitz condition. Then from Theorems 2 and 4 the fuzzy neutral functional differential equations has a unique fuzzy solution.

6. Conclusions

This article employs the Banach fixed point theorem as a fundamental tool to establish the existence of fuzzy solutions for non-local impulsive neutral functional differential equations in both first and second-order systems. The primary focus of this research is to explore the existence of such solutions, and a comprehensive example is provided to illustrate the obtained results. Furthermore, the applicability of these findings can be extended to the study of system controllability, including both systems or inclusions, as well as neutral integer powers. To further enrich the discussion on existence and controllability, the investigation of fuzzy neutral functional differential equations encompasses the incorporation of fractional powers $\alpha$. Moreover, these findings can be applied to real-world phenomena by examining a pendulum problem and formulating a fuzzy and impulsive controller in ${\mathbb{R}}^n$. Subsequently, MATLAB v 2022 can be employed to simulate the proposed adaptive fuzzy and impulsive controllers, aiming to regulate the inverted pendulum. To ensure the uniform constraint of all involved signals, stability analysis can be employed. At this juncture, all design parameters possess numerical values that yield graphs of the adaptive control input signal, the state, and the target value. This expansion broadens the scope of analysis and facilitates deeper insights into the subject matter.