Intuitionistic fuzzy semigroup

: In this work, we generalize the deﬁnition of a intuitionistic fuzzy strongly continuous semi-group and its generator. We establish some of their properties and some results about the existence and uniqueness of solutions for intuitionistic fuzzy equation.


Introduction
The initial value crisp problem    x(t) = Ax(t) + f (t, x(t)) t ∈ [0, T ] has a unique mild solution under assumption some conditions, if A is the generator of a C 0semigroup, (S(t)) t≥0 on a Banach space X, the system (1.1) has a unique mild solution x ∈ C ([0, T ]). In [7], C. G. Gal and S. G. Gal studied, with more details, fuzzy linear and semilinear (additive and positive homogeneous) operators theory, introduced semigroups of operators of fuzzy-number-valued functions, and gave various applications to fuzzy differential equation.
In this work, we study the existence and uniqueness of mild solution for fractional differential equation with intuitionistic fuzzy data of the following form:

Preliminaries
In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. Let X = R and we denote by P k (R) the set of all nonempty compact convex subsets of R.
Definition 2.1. We denote 2. u is fuzzy convex and v is fuzzy concave.
3. u is upper semicontinuous and v is lower semicontinuous Definition 2.2. The intuitionistic fuzzy zero is intuitionistic fuzzy set defined by 1 and λ ∈ R, we define the addition by : According to Zadeh's extension principle, we have addition and scalar multiplication in intuitionistic fuzzy number space IF 1 as follows : We denote iii) for any nondecreasing sequence The space IF 1 is metrizable by the distance of the following form :   1]. Then λ α , λ α , µ α , µ α are measurable.
Definition 2.5. Suppose A = [a, b], F : [a, b] → IF 1 is integrably bounded and strongly measurable for each α ∈ (0, 1], If F : [a, b] → IF 1 is integrable then in view of Lemma (2.1) F is obtained by integrating the α-level curves, that is 3 The embedding theorem IF 1 with addition and multiplication by scalar laws is not a vector space. In order to extend the Radstrom embedding theorem to IF 1 , we need to define a linear structure is defined in IF 1 by There exists a normed space X and a function j : Proof. Define an equivalence relation in . It is easy to check that j : is an isometry, and properties (2), (3) follow the definition.

Intuitionistic fuzzy strongly continuous semi-group
In this section we give the approach of the concept of intuitionistic fuzzy Semi-group.
Definition 4.1. We called an intuitionistic fuzzy C 0 -Semi-group (one parameter, strongly continuous, nonlinear) the whole family {T (t), t ≥ 0} of operators from IF 1 into itself satisfying the following conditions • T (0) = i, the identity mapping on IF 1 .
• The function g : • There exist two constants M > 0 and ω ∈ R such that In particular, if M = 1 and ω = 0, we say that {T (t); t ≥ 0} is a contraction intuitionistic fuzzy semigroup.
Remark 4.1. The continuity of g at 0, implies the continuity of g at t 0 ≥ 0.
whenever this limit exists in the metric space IF 1 , d ∞ . Then the operator A : is called the infinitesimal generator of the intuitionistic fuzzy semigroup {T (t), t ≥ 0}. Proof. We assume that A is the generator of an intuitionistic fuzzy semigroup {T (t); t ≥ 0} on IF 1 , then we have for all (u, v) ∈ j −1 (D(A)) lim h→0 + Remark 4.2. Since the infinitesimal generator A 1 of {T 1 (t); t ≥ 0} is unique, we deduce that the infinitesimal generator A of an intuitionistic fuzzy semigroup {T (t); t ≥ 0} is also unique. Lemma 4.1. Let A be the generator of an intuitionistic fuzzy semigroup {T (t); t ≥ 0} on IF 1 , then for all (u, v) ∈ IF 1 such that T (t)(u, v) ∈ D(A) for all t ≥ 0, the mapping t → g(t) = T (t)(u, v) is differentiable and g (t) = AT (t)(u, v) Proof. Let (u, v) ∈ IF 1 , for t, h ≥ 0 we have , for h > 0. Using the continuity of g and the definition of A, we have Hence, g is differentiable and g (t) = AT (t)(u, v), for all t ≥ 0.