The Cauchy problem for intuitionistic fuzzy differential equations

: In this paper we discuss the existence and uniqueness theorem of a solution of the cauchy problem of intuitionistic fuzzy differential equation.


Introduction
One of the generalizations of fuzzy sets theory [15] can be considered the proposed intuitionistic fuzzy sets(IFS). Later on Atanassov generalized the concept of fuzzy set and introduced the idea of intuitionistic fuzzy set [1][2][3]. They are very necessary and powerful tool in modeling imprecision, valuable applications of IFSs have been flourished in many different field [4,6,8,9].
For intuitionistic fuzzy concepts, recently the authors [5,[11][12][13] established, the theory of metric space of intuitionistic fuzzy sets, intuitionistic fuzzy differential equations, intuitionistic fuzzy fractional equation and the Cauchy problem for complex intuitionistic fuzzy differential equations. They proved the existence and uniqueness of the intuitionistic fuzzy solution for these intuitionistic fuzzy differential equations using different concepts. This paper is to investigate the existence and uniqueness theorem of intuitionistic fuzzy solutions for the follwing intuitionistic fuzzy differential equations: ( u, v (t) = f (t, ( u, v (t)), u, v (t 0 ) = u 0 , v 0 (1.1) when u 0 , v 0 is an intuitionistic fuzzy quantity and f satisfies the generalized Lipschitz condition. The paper is organized as follows. In Section 2, we collect the fundamental notions and facts which will be used in the rest of the article and we list several comparison propositions on classical ordinary differential equations in [7]. In Section 3 we show the relation between a solution and its approximate solution to the Cauchy problem of the intuitionistic fuzzy differential equation, and furthermore, in Section 4, we prove the existence and uniqueness theorem for a solution to the Cauchy problem of the intuitionistic fuzzy differential equation.

Preliminaries
Throughout this paper, (R n , B(R n ), µ) denotes a complete finite measure space. Let us P k (R n ) the set of all non empty compact convex subsets of R n . we denote by An element u, v of IF n is said an intuitionistic fuzzy number if it satisfies the following conditions (i) u, v is normal i.e there exists x 0 , x 1 ∈ R n such that u(x 0 ) = 1 and v(x 1 ) = 1.
(ii) u is fuzzy convex and v is fuzzy concave.
(iii) u is upper semi-continuous and v is lower semi-continuous so we denote the collection of all intuitionistic fuzzy numbers by IF n On the space IF n we will consider the following metric, where . denotes the usual Euclidean norm in R n . The norm of an intuitionistic fuzzy number u, v ∈ IF n is defined by 1. An intuitionistic fuzzy set u, v is called convex intuitionistic fuzzy set if and only if u is convex fuzzy set and v is concave fuzzy set.
The question that arises, is what IF n with addition and multiplication by a scalar is a vector Theorem 2.3 ( [13]). There exists a normed space X and a function j : IF n −→ X with properties: In the following we list several comparison propositions on classical ordinary differential equations following [7] Proposition 2.1. Let G ⊂ R 2 be an open set and g ∈ C[G, R], (t 0 , x 0 ) ∈ G. Suppose r(t) is the maximum solution to the initial value problem , then there exists an ε 0 > 0 such that the maximum solution r(t, ε) to the initial value problem where D is one of the four Dini derivatives (see [7]), G at most is a countable set on t. Then we must have The relation between a solution and its approximate solution to intuitionistic fuzzy differential equations In the following we give the relation between a solution and its approximate solutions. We For fixed t 1 ∈ [t 0 , t 0 + r] and any t ∈ [t 0 , t 0 + r], t > t 1 , denote It is well known that whenever t 1 < t < t 1 + δ 1 and d n ∞ z, w (t), u, v (t 1 ) < δ 1 with z, w ∈ B u, v 0 , q ]. Take natural number N > 0 such hat Take δ > 0 such that δ < δ 1 and By the definition of F (t, n) and (3.2), we have We choose ϕ ∈ X * such that ϕ = 1 and where t 1 ≤ t ≤ t. In view of (3.9), we have From (3.7) and (3.8) we know that Hence by (3.6) and (3.10) we have , f t 1 , u, v n (t 1 ) + 2ε n < ε 4 + ε 4 + 2ε n < ε whenever n > N and t 1 < t < t 1 + δ. Now let n −→ ∞, and applying Eq. (3.5), we have On the other hand, from the assumption of Theorem 3.1, there exists an δ(t 1 ) ∈ (0, δ) such that the H-differences u, v n (t) u, v n (t 1 ) exist for all t ∈ [t 1 , t 1 + δ(t 1 )] and n = 1, 2, ... Let z, w n (t) = u, v n (t) u, v n (t 1 ). We verify that the intuitionistic fuzzy number-valued sequence { z, w n (t)} uniformly converges on [t 1 , t 1 + δ(t 1 )] In fact, from the assumption d n Since (IF n , d n ∞ ) is complete, there exists an intuitionistic fuzzy number-valued mapping such that { z, w n (t)} uniformly converges to z, w (t) on [t 1 , t 1 + δ(t 1 )] as n −→ ∞ In addition, we have Hence the H-differences u, v (t) u, v (t 1 ) exist for all t 1 ∈ [t 1 , t 1 + δ(t 1 )] Thus from (3.11) we have From t 1 ∈ [t 0 , t 0 + r] is arbitrary, we know that Eq. (3.4) holds true and Thus, we conclude the proof.
and retain other assumptions, then the conclusions also hold true.
Proof 2. This is completely similar to the proof of Theorem 3.1.

Existence and uniqueness theorem for a solution
x 2 )), the initial value problem has only the solution Thus by the inductive method we know Assume m ≥ n, and in view of (4.9) and (4.7) we get we deduce that Since g t, x n−1 (t) uniformly converges to 0, then for arbitrary ε > 0 there exists a natural number N such that Here D + is the Dini derivative (see [7]). From the fact that d n ∞ u, v n (t 0 ), u, v m (t 0 ) = 0 < ε and by proposition 2.1, we have where w(t, ε) is the maximum solution to the initial value problem By proposition 2.2 we know that w(t, ε) uniformly converges to the maximum solution x(t) ≡ 0 of problem (4.1) on t 0 ≤ t ≤ t 0 + r as ε → 0. Thus, according to (4.10) and that (IF n , d n ∞ ) is complete, we know that there exists an intuitionistic fuzzy set-valued mapping u, v : uniformly converges to 0 as n −→ ∞. Applying (4.4) and Corollary (3.1) we have u, v ∈ C 1 [[t 0 , t 0 + r], B( u, v 0 , q)] and u, v (t) is the solution of the initial value problem (3.4). Finally, we prove the uniqueness. Suppose z, w (t) is another solution of initial value problem (3.4). Let Then m(t 0 ) ≡ 0 , f t, z, w (t) ) ≤ g(t, m(t)).
Hence from proposition 2.2 we know d n