Existence of Fuzzy Solutions For Nonlocal Impulsive Neutral Functional Differential Equations

In this paper, the existence of fuzzy solutions for first and second order nonlocal impulsive neutral functional differential equations are studied using Banach fixed point theorem. We used the techniques of fuzzy set theory, functional analysis, and Hausdorff metric. Example is provide to illustrate the theory.


Introduction
Fuzzy differential and integrodifferential equations are a field of interest, due to their applicability to the analysis of phenomena with memory where imprecision is inherent.However, the concrete example is the radiocardiogram, where the two compartments correspond to the left and right ventricles of the pulmonary and systematic circulation.Pipes coming out from and returning into the same compartment may represent shunts, and the equation representing this model is a nonlinear neutral Volterra integrodifferential equation.These classes of equations also arise, for example, in the study of problems such as heat conduction in materials with memory or population dynamics for spatially distributed populations; Balasubramaniam and Muralisankar [1] proved the existence and uniqueness of fuzzy solutions for the semilinear fuzzy integrodifferential equation with nonlocal initial condition.They considered the semilinear one-dimensional heat equation on a connected domain (0, 1) for material with memory.Generally, several systems are mostly related to uncertainty and inexactness.The problem of inexactness is considered in general exact science, and that of uncertainty is considered as vague or fuzzy and accident.Ding and Kandel [2] analyzed a way to combine differential equations with fuzzy sets to form a fuzzy logic system called a fuzzy dynamical system, which can be regarded to form a fuzzy neutral functional differential equation.For fuzzy concepts, the theory of metric space of fuzzy sets and fuzzy Volterra integral equations are developed.For the system which are related to uncertainty is studied using fuzzy concepts.Zadeh [3] in 1965, introduced the concept of fuzzy sets by defining them in terms of mappings from a set into the unit interval on the real line.Agarwal et al [4] carried out the work on the fuzzy solutions for multipoint boundary value problems.Balasubramaniam and Muralisankar [1] explicated the existence and uniqueness of fuzzy solution for the nonlinear fuzzy integrodifferential equations.Recently, Ramesh and Vengataasalam [5] examined the solutions of fuzzy impulsive delay integrodifferential equations with nonlocal condition.Vengataasalam and Ramesh [6] studied the fuzzy solutions for impulsive semilinear differential equations.For more on fuzzy differential equations, refer to [7,8].Neutral differential equations are used in many areas of applied mathematics and due to its vast applications these equations are given much attention in recent years.These equations have major impact in the field of biological and engineering processes; for details, see [9,10].The theory of ordinary Neutral Functional Differential Equations (NFDE) was initially developed by Bellman and Cooke [11].Then it was developed by Cruz and Hale [12], Hale [13,14], Hale and Meyer [15], and Henry [16].They developed the basic theory of existence and uniqueness, and also the properties of the solution operator and stability.Many authors contributed to the field of NFDE [17,18,19,20].Benchohra [18] studied neutral functional differential and integrodifferential inclusions for nonlocal Cauchy problems in Banach spaces.Balachandran and Sakthivel [19] examined the existence of solutions of neutral functional integrodifferential equation in Banach spaces.Many works dealing the existence results of mild solutions for first and second order abstract partial neutral differential systems which are similar to (3.1) − (3.3) and (3.4) − (3.7), were published.Balachandran and Dauer [20] investigated the existence of solutions of nonlinear neutral integrodifferential equations in Banach spaces.Balachandran and Anthoni [21] analyzed the existence of solutions of second order neutral functional differential equations.See, for example, [18,22,23] for the first order case and [24,22,25] for the second order.The study of impulsive functional differential equation is linked to their utility in simulating processes and phenomena subject to short-time perturbations during their evolution.The perturbations are performed discretely and their duration is negligible in comparison with the total duration of the processes and phenomena.We refer to the monographs of Bainov and Simeonov [26], Benchohra et al. [27], Lakshmikantham et al. [28], and Samoilenko and Perestyuk [29] where numerous properties of their solutions are studied, and the detailed bibliographies are given.This paper is concerned with the existence of fuzzy solutions for more general initial value problems of first and second order impulsive neutral functional differential equations.Moreover, to the author's knowledge, there are few papers dealing with fuzzy impulsive differential equation of second order.This paper has four sections.In Section 2, we will briefly recall some basic definitions and preliminary facts which will be used in the later sections.In section 3, we consider the first order nonlocal initial value problem.Section 4 deals with the second order nonlocal initial value problem.We conclude the article with section 5.

Preliminaries
In this section, we introduce notations, definitions and preliminary facts which are used throughout this paper.Definition 2.1.{fuzzy set} Let X be a nonempty set.A fuzzy set A in X is characterized by its membership function A : X → [0, 1] and A(x) is interpreted as the degree of membership of element x in fuzzy set A for each x ∈ 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 2.1.The membership function of the fuzzy set of real numbers "close to one" can be defined as , where β is a positive real number.
Example 2.2.Let the membership function for the fuzzy set of real numbers "close to zero" defined as follows 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 CC(R n ) denotes the set of all nonempty compact, convex subsets of R n .Denote by, E n = {u : R n → [0, 1] such that they satisfy (i) − (iv) mentioned below}, (i) u is normal i.e., there exists an We call u ∈ E n an n-dimension fuzzy number.
For any Let A, B be two nonempty bounded subsets of R n .The distance between A and B is defined by the Hausdorff metric where ∥.∥ denotes the usual Euclidean norm in R n .Then (CC(R n ), H d ) is a complete and seperable metric space [30].
We define the supremum metric A strongly measurable and integrably bounded map f : is strongly measurable and integrably bounded, then f is integrable.
exist and are equal to f ′ (t 0 ).Here the limit is taken in the metric space (E n , H d ).At the end points of [0, 1], we consider only the one-sided derivatives.The subtraction is one of Hukuhara and does not obey definition of g(u, v).
, then we say that f ′ (t 0 ) is the fuzzy derivative of f (t) at the point t 0 or the Hukuhara derivative of f (t) at t 0 , usually denoted by D H f (t 0 ).For the concepts of fuzzy measurability and fuzzy continuity, we refer to [8].
and arbitrary ε > 0, there exists a δ (ε, α) > 0 such that 3 First Order Fuzzy Solutions For Impulsive NFDEs In this section, we consider the first order nonlocal initial value problem where E n be the set of all upper semi continuous, convex, normal fuzzy numbers with bounded α-level and f : ) where Moreover, If x is an integral solution of (3.1) − (3.3), then x is given by Now we will prove the existence result for the problem (3.1) − (3.3).To study this problem we will formulate the following hypotheses.
(H4) There exists a non-negative constant d k such that and is defined by Now, we shall prove that Φ is a contraction operator.Consider x, y ∈ C([−r, T ], E n ) and α ∈ (0, 1], then Thus, for each t ∈ J

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Hence, Φ is a contraction mapping.By the Banach fixed point theorem, Φ has a unique fixed point which is a solution to (3.1) − (3.3).

Second Order Fuzzy Solutions for NFDEs
In this section, we will prove the existence and uniqueness result for the initial value problem (3.4) − (3.7).Here, we will consider the space Ω ′ in order to define the solution for (3.4) − (3.7)where

Assume that
(H5) There exists a non-negative constant and is defined by International Scientific Publications and Consulting Services Thus, for each t ∈ J, ) Hence, Φ 1 is a contraction mapping.By Banach fixed point theorem, Φ 1 has a unique fixed point which is a solution of (3.4) − (3.7).
The α− level set of fuzzy number 2 is, The α− level set of fuzzy number 3 is, Then, Then, ] α , ] , [

Conclusion
In this paper, we have studied the existence of fuzzy solutions for nonlocal impulsive neutral functional differential equations for the first and second order systems using Banach fixed point theorem.One can extend the same results and study the controllability of both of the systems or inclusions.Also the integer powers of NFDE are replaced by some fractional power α for the further discussion of existence and controllability.

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International Scientific Publications and Consulting Servicesfor all α ∈ (0, 1]. and f , g, h, I k and φ are as in problem (3.1) − (3.3).In this section, we shall deal with the existence of the fuzzy solutions for problem (3.1) − (3.3) and in section 4, we shall deal with the existence of the fuzzy solutions for problem (3.4) − (3.7).Here we will consider the space Ω in order to define the solution for (3.1) − (3.3) where, Ω = {x : x is absolutely continuous from J to E n } and there exists x(t − k ) and x(t + k ) where k = 1, • • • , m, with x(t − k ) = x(t k ).
n, we call the ordered one-dimension fuzzy number class u 1 , u 2 , ..., u n (i.e. the Cartesian product of one-dimension fuzzy number u 1 , u 2 , ..., un) an n-dimension fuzzy vector, denote it as (u 1 , u 2 , ..., u n ), and call the collection of all n-dimension fuzzy vectors (i.e. the Cartesian product E × E × • • • × E) n-dimensional fuzzy vector space, and denote it as Definition 3.1.A function x ∈ Ω is said to be a solution of (3.1) − (3.3), if x satisfies the equation d dt [x(t) − h(t, x t )] = f (t, x t ) a.e. on J, ∆x(t k )