System of intuitionistic fuzzy differential equations with intuitionistic fuzzy initial values

In this paper, we have studied the system of differential equations with intuitionistic fuzzy initial values under the interpretation of (i,ii)-GH differentiability concepts and Zadeh’s extension principle interpretation. And we have given some numerical examples.


Introduction
Fuzzy set theory was firstly introduced by L. A. Zadeh in 1965 [46]. He defined fuzzy set concept by introducing every element with a function µ : X → [0, 1], called membership function. Later, some extensions of fuzzy set theory were proposed [8,30,37]. One of these extensions is Atanassov's intuitionistic fuzzy set (IFS) theory [8].
In 1983, Atanassov [7] introduced the concept of intuitionistic fuzzy sets and carried out rigorous researches to develop the theory [8][9][10][11][12][13][14][15][16]. In this set concept, Apart from the membership function, he introduced a new degree ν : X → [0, 1], called non-membership function, such that the sum µ + ν is less than or equal to 1. Hence the difference 1 − (µ + ν) is regarded as degree of hesitation. Since intuitionistic fuzzy set theory contains membership function, non-membership function and the degree of hesitation, it can be regarded as a tool which is more flexible and closer to human reasoning in handling uncertainty due to imprecise knowledge or data.
In literature, there are different approaches for solving fuzzy differential equations. Each method has advantages and disadvantages in the applications. One of the commonly used method is based on Zadeh's extension principle. In this method, the fuzzy solution is obtained from the crisp solution by using the well-known Zadeh's extension principle [22]. However there is no definition of fuzzy derivative in this approach . Hence some other methods based on fuzzy derivative concept were also proposed and used. One of the earliest method is Hukuhara differentiability concept [41]. However, this approach has also a weak point which is that the solution becomes fuzzier as time passes by [25]. Hence the length of the support of the fuzzy solution increases. To overcome this disadvantage some methods such as differential inclusions [32] and strongly generalized differentiability concept [19] were coined. The method based on strongly generalized differentiability concept allows us to obtain the solutions with decreasing length of support [18][19][20][21]. Hence the drawback of Hukuhara differentiability can be overcome with strongly generalized Hukuhara differentiability concept. Besides, this approach shows to be more favorable in applications [21].
This paper is prepared as follows. In Section 2, some fundamental definitions and theorems in fuzzy sets and intuitionistic fuzzy sets are given. In Section 3, some definitions and theorems related to (i)-GH and (ii)-GH are extended from fuzzy case to intuitionistic fuzzy case by using the definitions and theorems in Section 2. In Section 4, we study system of differential equation on intuitionistic fuzzy environment under (i)-GH and (ii)-GH differentiability and the intuitionistic Zadeh's extension principle interpretation. Besides we give some numerical examples in this section. Finally we conclude the paper by giving summary and results in Section 6.

Preliminaries
is called an intuitionistic fuzzy set in R n . Here µ A and ν A are called membership and nonmembership functions, respectively.
We will denote set of all intuitionistic fuzzy sets in R n by IF (R n ).
Definition 2.8. [39] A triangular intuitionistic fuzzy number (TIFN)Ā i ∈ IF N (R) is defined with the following membership and non-membership functions: Here a * 1 ≤ a 1 ≤ a 2 ≤ a 3 ≤ a * 3 and it is denoted byĀ i = (a 1 , a 2 , a 3 ; a * 1 , a 2 , a * 3 ). Note that its α and β cuts can be obtained as Definition 2.9. [26] Let A and B be two nonempty subsets of R n and c ∈ R. The Minkowski addition and scalar multiplication of sets are defined as follows: The family of all compact and convex subsets of R n is closed under Minkowski addition and scalar multiplication.
Addition and scalar multiplication of fuzzy numbers in IF N (R n ) is defined as follows: . Basic end-point arithmetic operations of these closed and bounded intervals are as follows: 4. Division: Assume the interval B does not contain zero. Then Let us define the following distance functions Here d H is Hausdorff metric [49]. The function 14. Let f : (a, b) → IF N (R) be an intuitionistic fuzzy number valued function and x 0 , x 0 + h ∈ (a, b). f is called Hukuhara differentiable at x 0 if there exists an element f H (x 0 ) ∈ IF N (R) such that for all h > 0 the following is satisfied be an intuitionistic fuzzy number valued function and such that for all h > 0 at least one of the followings is satisfied: exist and the following limits exist such that exist and the following limits exist such that exist and the following limits exist such that exist and the following limits exist such that is called the Heaviside step function.
Definition 2.17. [16] Let X and Y be two sets and f : X → Y be a function. LetĀ i be an intuitionistic fuzzy set over X.

(i)-GH and (ii)-GH differentiability in an intuitionistic fuzzy environment
and its α and β cuts be given by such that the end-points of its α and β cuts are differentiable at t 0 ∈ (a, b).

For every
is a non-empty compact and convex set in R n .
be an intuitionistic fuzzy number valued function and t 0 ∈ (a, b).

If for every
then there exists an intuitionistic fuzzy numberĀ i such that lim t→t 0 f (t) =Ā i . And the α and β cuts Proof. Assume the conditions (1) and (2) then by the definition of the metric D we can write that lim and its α and β-cuts be given by 1. If f is strongly generalized differentiable at t 0 ∈ (a, b) as in case (1) 2. If f is strongly generalized differentiable at t 0 ∈ (a, b) as in case (2) . So by the interval operations we can write that So by the equality of intervals we obtain that and Similar results can be obtained for f (t) H f (t − h). Hence we obtain So by Lemma 3.2 we can write that 2. The proof can be done in a similar way.
). If f is strongly generalized differentiable at t 0 as in case (3) or (4) of Definition 2.17. Then f (t) ∈ R for all t ∈ (a, b).
Theorem 3.5. Let f and g be intuitionistic fuzzy number valued functions. If f and g are both (i)-GH differentiable or both (ii)-GH differentiable, then β)] be α and β cuts of f ; and g(t, α) = [g 1 (t, α), g 2 (t, α)] and g * (t, β) = [g * 1 (t, β), g * 2 (t, β)] be α and β cuts of g. Assume f and g be (i)-GH differentiable then we can write that and 2. The proof can be done in a similar way. 4 Application to a system of intuitionistic fuzzy differential equations In this section we will study the following system of first order differential equations in intuitionistic fuzzy environment under (i,ii)-GH differentiability and the intuitionistic Zadeh's extension principle interpretation.

Solving a system of intuitionistic fuzzy differential equations under Zadeh's extension principle interpretation
Let A and B be positive real numbers. Now let us consider the following system of intuitionistic fuzzy differential equation with the following triangular intuitionistic fuzzy numbers x(t 0 ) = (a 1 , a 2 , a 3 ; a * 1 , a 2 , a * 3 ) and under intuitionistic Zadeh's Extension Principle. Firstly we need to find the crisp solution of with x(t 0 ) = a 2 and y(t 0 ) = b 2 . The crisp solution of this system is We know that when we replace crisp initial values with intuitionistic fuzzy ones we obtain the intuitionistic fuzzy solution by the intuitionistic Zadeh's extension principle. Hence we can write the following α and β cuts: and So by end-point interval arithmetics and Heaviside function we the following obtain the points of x(t, α) and x * (t, β) as follows: ABt )(a 2 + (a * 3 − a 2 )β − (a 2 + (a * 1 − a 2 )β) +a 2 + (a * 1 − a 2 )β))) + ( Example 4.2.1. Let us find the intuitionistic fuzzy solution of the following system of differential equations with the following triangular intuitionistic fuzzy initial values x(0) = (1, 1, 3; −2, 1, 6) and y(0) = (−2, 3, 0; −4, 3, 2). We can obtain the end points of α and β cuts of the solution as follows:

Summary and conclusions
The main goal of this paper is to give solutions to system of differential equations with intuitionistic fuzzy initial values under the interpretation of (i,ii)-GH differentiability and intuitionistic Zadeh's extension principle concept. To do this we have firstly extended some theorems and definitions about (i,ii)-GH differentiability in fuzzy set theory in Section 3. Later, we have given a procedure to find the solutions to system of ordinary differential equations with triangular intuitionistic fuzzy initial values in Section 4. And we have given some numerical results in Section 4. Under (i,ii)-GH differentiability concept or Zadeh's extension principle interpretation, we have observed that the endpoints of α or β of solutions may switch on subintervals where crisp solution exists. That is why, this fact makes the solution to be exists locally on subintervals. To cope with this Heaviside function can be used to write the endpoints of α or β of the solutions.