Application of Contractive-like Mapping Principles to Fuzzy Functional Differential Equation

In this paper, we prove the existence and uniqueness of solution for the fuzzy functional differential equation under generalized Hukuhara derivative via contractive-like mapping principles.


Introduction
Fuzzy differential equations (FDEs) forms a suitable setting for the mathematical modeling of real-world problems in which uncertainty or vagueness pervades.A rich collection of results from the theory of FDEs is contained in the monograph of Lakshmikantham and Mohapatra [1] and references therein.
The method of fuzzy mapping was initially introduced by Chang and Zadeh [2].Later, Dubois and Prade [3,4] presented a form of elementary fuzzy calculus based on the extension principle [5].Puri and Ralescu [6] suggested two definitions for the fuzzy derivative of fuzzy functions.The first method was based on H-difference notation and was further investigated by Kaleva [7].Several approaches were later proposed for FDEs and the existence of their solutions (e.g.[8,9]).There are several approaches to the study of fuzzy differential equations.One popular approach is based on H-differentiability.The approach based on Hderivative has the disadvantage that it leads to solutions which have an increasing length of their support.
Bede and Gal [10] solved the above mentioned approach under strongly generalized differentiability of fuzzy-number-valued functions.In this case the derivative exists and the solution of FDEs may have decreased the length of the support, but the uniqueness is lost.Other researchers have proposed several approaches to the solutions of FDEs (e.g.[11][12][13][14], [15]).Almost authors applied the fixed point theorems like the Darbo's theorem and the classical Banach fixed point theorem [16], Schauder's fixed point theorem [17] and the method of successive approximations [8],. . .as a tool to prove the existence and uniqueness of the solution of FDEs.In [13], Lupulescu proved the local existence and uniqueness via the method of successive approximations and for global existence and uniqueness via the Banach fixed-point theorem.Allahviranloo et al. [17] also investigated existence and uniqueness of the solution of nonlinear fuzzy Volterra integral equations.Using Arzela-Ascoli's theorem and Schauder's fixed point theorem, authors proved thr existence and uniqueness of the solution for this kind

Preliminaries
In this section, we shall review some results the existence and uniqueness of fixed points for mappings defined in partially ordered sets.Also, we recall some fundamental results of fuzzy sets, ordering relations over fuzzy sets and fuzzy distance.
Definition 2.1 (see [18]) An altering distance function is a function Λ : R + → R + which satisfies (i) Λ is continuous and non-decreasing.
Theorem 2.1 (see [18]) Let (X, ≤) be a be a partially ordered set and suppose that there exists a metric d in X such that (X, d) is a complete metric space.Let F : X → X be a monotone non-decreasing mapping such that for some altering distance functions Λ and Υ. Suppose that either F is continuous or X is such that if a non-decreasing sequence (x n ) n∈N → 0 in X, then x n ≤ x for all n ∈ N.
If there exists x 0 ∈ X with x 0 ≤ F(x 0 ), then f has a fixed point.
Theorem 2.2 (see [18]) Let (X, ≤) be a be a partially ordered set and suppose that there exists a metric d in X such that (X, d) is a complete metric space.Let F : X → X be a monotone non-decreasing mapping such that for some altering distance functions Λ and Υ. Suppose that either f is continuous or X is such that if a non-increasing sequence (x n ) n∈N → 0 in X, then x n ≥ x for all n ∈ N.
If there exists x 0 ∈ X with x 0 ≥ F(x 0 ), then F has a fixed point.
The addition and scalar multiplication in K c (R d ), we define as usual, i.e.A, B ∈ K c (R d ) and λ ∈ R, then we have Then E d is called the space of fuzzy sets.
The supremum on E d is defined by It is easy to see that D is a metric in E d .In fact, (E d , D) is a complete metric space.
For every x, y, z ∈ E d and λ ∈ R + , we have We say that x is differentiable at t, if there exists x (t) ∈ E d such that (i)for all h > 0 sufficiently small, there exist x (t + h) x(t), x(t) x(t − h) and the limits lim or (ii) for all h > 0 sufficiently small, there exist x (t) x(t + h), x(t − h) x(t) and the limits lim ITM Web of Conferences 20, 02008 (2018) https://doi.org/10.1051/itmconf/20182002008ICM 2018 In the sequel, we define the following partial orderings ≤ and in the space E d and E 1 : and and for fuzzy functions,
(ii) f is weakly contractive for comparable elements, i.e., for some altering distance functions Λ and Υ, it holds Then, the problem (1) has a unique (i)-solution on [−σ, a].
Proof.We consider the space C([−σ, a], E d ) equipped with the complete metric It is easy to see that this metric is equivalent to the metric D(u, v), because In order to prove the theorem, we shall show the conditions of Theorem (2.1) are satisfied.First, we prove that the operator T is a non-decreasing and continuous.Indeed, the continuity is trivial and take u v on intervals [0, a] and for every t ∈ [−σ, 0], we have

[T u](t) [T v](t) ϕ(t).
For every t ∈ [0, a], we obtain From the assumption (ii), we have Next, we will prove that the inequality (6) holds, for all u v. Suppose that and Λ is an altering distance function.Then we have ITM Web of Conferences 20, 02008 (2018) https://doi.org/10.1051/itmconf/20182002008ICM 2018 By the inequality ( 6) and ( 7), we derive that Combine assumption (ii) and ( 8), it follows that 0 ≤ −Υ D σ (u, v) , for all u v, and therefore, Υ D σ (u, v) 0. This implies D σ (u, v) 0. Thus, we have 0. This contradict the assumption above, that is, for all u v we have Since Λ is an altering distance function and by the inequality (10), we infer that Finally, using the existence of the lower (i)-solution and Theorem 2.1, we shall prove that µ is such that µ T µ.Indeed, if t ∈ [−σ, 0], µ 0 ϕ then µ(t) ϕ(t) [T µ](t) and for t ∈ [0, a],

Theorem 3.3 Replacing the existence of a lower (i)-solution of the problem (1) by the existence of a upper (i)-solution of the problem (1), the conclusion of Theorem 3.2 is still valid.
Proof.The proof of Theorem 3.3 is similarly the proof Theorem 3.2.
Proof.By the condition (i), we imply that the existence of Hukuhara differences ϕ(0) + (−1) E d ) is a solution of the problem (1) and conversely.
Similarly to Theorem 3.2, we shall show the conditions of Theorem (2.1) are satisfied.For all u s v s on [−σ, a], we have Therefore, A has the operator non-decreasing.
For all u By similar calculations as (10), we have.
Therefore, if Λ is some increasing altering distance function, it holds Then, from 4 and (8), we derive that where Finally, using the existence of the lower (ii)-solution and Theorem 3.1, we shall prove that µ is such that µ

Thus µ(t)
[A µ](t).We see that the operator A verifies all conditions of Theorem 2.1, that is, A has a fixed point in C([−σ, E d ]).Given that C([−σ, E d ]) verifies that every pair of elements of C([−σ, E d ]) has an upper bound, the operator A has a unique fixed point.Theorem 3.5 Replacing the existence of a lower (ii)-solution of the problem (11) by the existence of a upper (ii)-solution of the problem (11), the conclusion of Theorem 3.5 where u(t) is the population at time t, u 0 (−1, 0, 1) and λ > 0.
But this example, we want to prove that the conditions of Theorem 3.2 (or Theorem 3.2) are satisfied.Now, we check the existence of a lower (i)-solution for the problem (13).
Since Λ is an altering distance function and from (15), we infer that where Υ(t) (1 − λ)t.By using Theorem 3.2, the existence of a lower (i)-solution for the problem provides the existence of a unique solution on [−1, ∞).
It is easy to see that the conditions of Theorem (3.3) are satisfied, that is, the problem (13) has a unique (ii)-solution on [−1, ∞).

( 5 )
If there exists u ∈ C([−σ, a], E d ) is a fixed point of T , then u ∈ C 1 ([0, a],E d) is a solution of the problem (1) and conversely.
Since every pair of functions in C([−σ, a], E d ) has an upper bound, the operator T has a unique fixed point u ∈ C([−σ, a], E d ) and u is the unique solution of the problem (1) on [−σ, a].