Stability of intuitionistic fuzzy nonlinear fractional differential equations

: In this paper, we study the existence and uniqueness results of solution for the intuitionistic fuzzy nonlinear fractional differential equations involving the Caputo concepts of fractional derivative. In addition, we establish essentially the Mittag-Lefﬂer stability result for the intuitionistic fuzzy nonlinear fractional differential equations by giving some sufﬁcient criteria to guarantee the stability of the zero solution. Finally, some examples are presented to illustrate the proposed stability theorem.

2. u is fuzzy convex and v is fuzzy concave. 3. u is upper semi-continuous and v is lower semi-continuous. 4. sup u, v = {x ∈ R : v(x) < 1} is bounded.
Definition 2.9. [7,18] The generalized Hukuhara difference of two intuitionistic fuzzy numbers u 1 , v 1 , u 2 , v 2 ∈ IF 1 is as follows: We say that f is generalized Hukuhara differentiable at t 0 if there exists f (t 0 ) ∈ IF 1 such that: Then we have the following result.
is a complet metric space.

Fractional integral and fractional derivatives of an intuitionistic fuzzy function
Definition 2.14.
is the Euler gamma function. Proposition 2.15. [10] Let F, G ∈ L([0, T ], IF 1 ) and a ∈ IF 1 , then we have Example 2.17. Consider the intuitionistic fuzzy function u, v (t) = tC, where C = (a 1 ; a 2 ; a 3 ; a 4 ; a 1 ; a 2 ; a 3 ; a 4 ) is a trapezoidal intuitionistic fuzzy number. In this example we calculate the intuitionistic fuzzy Caputo fractional derivative of the function u, v (t). For this purpose, we start by giving the gH-derivative of u, v (t) as follows: This implies that u, v (t) = C.

Laplace transform of Caputo fractional derivative
In order to establish the Laplace transform of Caputo fractional derivative [12], we write the Caputo derivative under the form By using the formula of Laplace transform of Riemann-Liouville fractional integral, we have [1] L{ Were G(s) is given by Finaly, the Laplace transform of Caputo fractional derivative is,

Laplace transform of Mittag-Leffler function
The Mittag-Leffler function is an important function that finds widespread use in the world of fractional calculus. Just as the exponential naturally arises out of the solution to integer order differential equations, the Mittag-Leffler function plays an analogous role in the solution of noninteger order differential equations. In fact, the exponential function itself is a very special form, one of an infinite set of the seemingly ubiquitous functions.
We recall that the Mittag-Leffler function is given by The general form is given by Then we have Indeed, for s > λ, using the series expansion of the exponential function we have Then similarly for the Mittag-Leffler function we obtain The Laplace transform of Mittag-Leffler function in two parameters is given by

Existence and uniqueness resuts
In this section we study the existence and uniqueness of the solution for the following initial value problem.
where c D q is the Caputo derivative of x(t) at order α ∈]0, 1[. For this purpose, we need some notations and definitions: • C([t 0 , T ], IF 1 ) denotes the space of all continuous functions from [t 0 , T ] to IF 1 .
• B r (x 0 , r) = {y ∈ IF 1 , d ∞ (x 0 , y) ≤ r} denotes the closed ball of IF 1 .   1) x satisfies the integral equation Proof. [11] The proof of this lemma is similar to the proof of Theorem 3 in [11].
Theorem 3.5. [9,24] Let C be a non-empty closed subset of a Banach space X and let (k n ) n≥0 be a sequence such that ∞ n 0 k n converges. Moreover, let the operator T : C → C satisfy the following inequality T n x − T n y ≤ k n x − y ∀n ∈ N and ∀x, y ∈ C.
Then the operator T has a uniquely defined fixed point x * . In addition, the sequence {T n x 0 } n≥0 converges to x * for every x 0 ∈ C.

Fundamental theorem
H 2 ) There exists a continuous function k : Then the problem (3) has a unique solution.
Proof. The proof of this theorem will be given in two steps.
Step 1. (Existence of a solution of the problem (3). To show that the problem (3) has at least one solution defined on [t 0 , T ] we use the Schauder fixed point theorem, [22].
Let T : C([t 0 , T ], IF 1 ) → C([t 0 , T ], IF 1 be the operator defined as follows: Hence, Tx ∈ C([t 0 , T ], IF 1 ), it follows that T transforms the ball Hence, lim which shows that TB ρ is equicontinuous, by using Arzelà-Ascoli theorem [23] we deduce that TB ρ is relatively compact.
Finally, by using the Schauder fixed point theorem, we can conclude that the operator T has a fixed point x(t), which is a solution of the problem (3).
Step 2. (Uniqueness of the solution) To show the uniqueness of the solution, we suppose that there exists another solution y(t) : We take the sup on both sides of the previous inequality we will have, , y(t)) .
By the induction method one can conclude that for every n ∈ N * and for every x, y ∈ C([t 0 , T ], IF 1 ): Now we shall prove that for every n ∈ N * we have • For n = 1 the statement (4) is trivially true.

Stability of the solution
Our aim in this section is to study the stability of the solution for the following nonlinear fractional differential equation. For this purpose, we extend the Lyapunov direct method to introduce the stability in the Mittag-Leffler sense of the trivial solution (x = 0 IF ) for the following nonlinear system: where f : [0, +∞[×IF 1 → IF 1 is an intuitionistic fuzzy continuous function in t locally Lipschitz in x such that f (t, 0 IF ) = 0 IF . The existence and uniqueness results are discussed in Section 3 for the case t ∈ [t 0 , T ]. Since the fact f (t, 0 IF ) = 0 IF means that the intuitionistic fuzzy zero function is a solution of initial value problem (5), then our purpose in this work is to study the stability of the trivial solution of the problem (5). There is no loss of generality in doing so because any equilibrium point can be shifted to the origin via a change of variables. Suppose the intuitionistic fuzzy equilibrium point for (5) is x e = 0 IF and consider the change of variable X = x gH x e . The fractional derivative of X(t) is given by where F (t, 0 IF ) = 0 IF and the system has equilibrium at the origin.
Proof. From equations (6) and (7) we get There exists a positive function k(t) satisfying By using the Laplace transform (1) and (2), we obtain It follows that By using the inverse of Laplace transform we obtain .
is a nonnegative function [12], it follows that

Examples
The following examples are used to prove the stability result.
Example 4.7. Consider the following intuitionistic fuzzy fractional problem: where 0 < β < 1. We consider the candidate Lipschitz function V : It follows that c 1 = c 2 = 1.
On the other hand, we have Finally, we apply c 1 = c 2 = 1 and c 3 = −1 in Theorem 4.6, and we obtain Wich implies the Mittag-Leffler stability of system (8).
Example 4.8. We consider the following intuitionistic fuzzy nonlinear fractional system: where 0 < q < 1, x = 0 IF is the equilibrium point of the problem (9) and f : [0, +∞[×IF 1 → IF 1 is an intuitionistic fuzzy Lipschitz function whith Lipschitz constant k > 0. We suppose that there exists a Lyapunov function V (t, x(t)) satisfying the following conditions: where c 1 , c 2 , c 3 are positive constants and V (t, x) = dV (t, x(t)) dt .

Conclusion and future works
In this paper, we studied the existence and stability results of nonlinear intuitionistic fuzzy fractional-order dynamic systems by using the Schauder fixed point theorem and the notions of Mittag-Leffler stability, and we discussed some sufficient criteria to demonstrate the stability of the trivial solution of the proposed system. Our future works include the Mittag-Leffler stability of intuitionistic fuzzy multi-variables fractional-order nonlinear systems.