The existence and uniqueness theorems of fuzzy delay differential equations

Delay differential equations (DDEs) arise many different phenomena including in physics, biology and chemistry. In many cases of the modeling of real world problems, information about the behaviour of a dynamical system is uncertain. In order to obtain a more realistic model, we have to take into account these uncertainties. Therefore, in this paper, we propose the existence and uniqueness theorems for fuzzy timedelay dynamical systems. We finally present some conclusions and new directions for further research in this area.


INTRODUCTION
Delay differential equations (DDEs) are differential equations in which the derivative of the unknown function at certain time is given in term of the value of the function at previous times.These types of equations are a large and important class of dynamical systems.They often arise in either natural or technological control problems.The delay may appear because of physical properties of equipment used in the system, signal transmission or measurement of system variables.For example, actuators, sensors and field networks which are involved in feedback loops may exhibit delays.Time-delay systems are also used to model several different mechanisms in the dynamics of epidemics [1].
In many cases of the modelling of real world phenomena, information about the behaviour of a dynamical system is uncertain.In order to obtain a more realistic model, we have to take into account these uncertainties.Fuzzy differential equations are a natural way to model dynamical systems under uncertainty.This type of system will provide a better representation of the real world problems.Therefore the study of this topic has been rapidly growing in recent years.It was first started by Chang and Zadeh [2], who introduced the concept of fuzzy derivative.It was followed up by Dubois and Prade [3] in 1982.They used the extension principle in their approach.In 1980, Kandel and Byatt [4] applied the concept of fuzzy differential equation to the analysis of fuzzy dynamical problems.The researches have been continuing unlimited to the theory of fuzzy differential equations but also the application in the real problems as were reported in many literatures [5,6,7,8,9].The existence and uniqueness theorem is one of the most important and fundamental theorems in the theory of classical differential equation.The theorems that deal with fuzzy-set function were also discussed in many literatures.Some of them can be found in [10,11,12].Xiaowei and Zhongfeng [10] proved a new existence and uniqueness theorem for fuzzy differential equation.This theorem was different from previous works since, they use Liu process [10].Balachandran and Prakash [11] proved the solutions of fuzzy delay differential equations with nonlocal condition.In [12], Lupulescu and Abbas proved a local existence and uniqueness result for fuzzy delay differential equations driven by Liu process.
Therefore, in this paper we will derive the existence and uniqueness theorems of a specific fuzzy delay differential equation (FDDE).This paper will focus on the existence and uniqueness theorems of fuzzy time-delay dynamical systems.The organization of this paper is as follows.In Section 2, some notations, concepts and the basic definitions of delay differential equations are briefly presented.In Section 3, introduces fuzzy delay differential equations and the existence and uniqueness theorems for fuzzy time-delay dynamical systems.Finally, Section 4 presents concluding remarks.

PRELIMINARIES
Definition 4: [13] The function is a solution of the initial value problem (4) on , if is a solution of (2) on , and .
Lemma 1: [13] If is continuous on , , then is a continuous function of for ∈ , .

FUZZY DELAY DIFFERENTIAL EQUATION
Consider the first-order fuzzy time-delay initial value differential equation given by where and ∈ : are dimensional fuzzy functions of , every element of matrices , ∈ and , ∈ are supposed to be fuzzy numbers where represents the fuzzy sets defined on .The function ′ is the fuzzy derivative of at ∈ and is a fuzzy number and the time-delay is a known positive rational number.
Most of these steps follow from [13]. a fuzzy delay differential equation on .
Definition 7: Let ∶ → .A function is said to be a solution of ( 8 is a subset of and is Lipschitz on .(The Lipschitz constant for and depends, in general, on the particular set ).

Existence and Uniqueness Theorems for Fuzzy Time-Delay Dynamical Systems
Theorem 1: (Existence) Let : , → be continuous and locally Lipschitz.Then for each ∈ , there exists such that the fuzzy time-delay initial value problem ( 6) has a unique solution on , Δ for some Δ 0.  Since the right hand side tend to 0, it follows that the left hand side must be 0. Hence satisfies (11) which means that is a solution of (6).

Proof
Theorem 2: (Uniqueness) Let : , → be continuous and locally Lipschitz on its domain.Then given any ∈ , there is at most one solution of the initial value problem (6) on , for any ∈ , .
Proof Suppose (for contradiction) that for some From this and the Reid's Lemma it follows that 0, and hence on , , contradicting the definition of .Since , therefore there cannot be two different solutions on its domain.

CONCLUSION
In this paper we introduced specific fuzzy delay differential equation which is fuzzy time-delay dynamical systems.By Banach Space, we proved an existence and uniqueness theorem for fuzzy time-delay dynamical systems under Lipschitz condition.For further research, numerial solution and stability analysis on this system will be considered.

Definition 5 :
If is a function defined at least on , → then we define a new function ∶ .Let ∈ .We will take the norm on this space to be | | sup || ||, where ||.|| is the usual Euclidean norm on .Further, for ⊆ let , 0 → be the set of continuous fuzzy functions , 0 into .In the following, unless otherwise stated, we will take ⊆ and ⊆ to be open sets.Definition 6: If ∶ → is a given fuzzy functional and ′ represent the right hand derivative, we call the relation ′ , 8 Now, the series ∑ 27 converges, the convergence of the sequence ℓ follows by application of the comparison test of each component of as ℓ → ∞ of this inequality then gives In the following, unless otherwise stated, we will take ⊆ and ⊆ to be open sets.
. Let ∈ .We will take the norm on this space to be || || sup || ||, where ||.|| is the usual Euclidean norm on .With this norm, is a Banach space.Further, for ⊆ let , 0 → be the set of continuous fuzzy functions , 0 into .Definition 3: [13] Let ∶ → .A function is said to be a solution of (2) on , if there are ∈ and such that i. ∈ , , , ii. , ⊂ , iii.satisfies 2 for ∈ , .For a given ∈ and ∈ , the initial value problem associated with the FDDE (2) is If satisfies Condition (C) then a continuous function : , → is a solution of Equation 4 if and only if