Boundary Value Problems for a Class of First-Order Fuzzy Delay Differential Equations

: In this paper, we study a class of fuzzy differential equations with variable boundary value conditions. Applying the upper and lower solutions method and the monotone iterative technique, we provide some sufﬁcient conditions for the existence of solutions, which can be applied to discuss some dynamical models in biology and economics.


Introduction
Delay differential equations are frequently used to study system models in biology, economics, physics, engineering and so on [1]. However, deterministic differential equations cannot be applied to systems with uncertainty. For example, if the dynamical system depends upon a subjective decision, state variables or parameters will be inaccuracies [2]. To consider these issues in mathematical models, we might need to utilize the aspect of fuzziness.
Fuzzy differential equations have been studied frequently during the last few years. We can find many papers concerned with the existence of solutions for fuzzy differential equations. With the strongly generalized differentiability concept introduced in [3,4], B. Bede et al. presented a variation of constants formulas for first-order, linear, fuzzy differential equations in [5]. Based on these results, some new theorems about the existence of fuzzy differential equations were obtained. We refer to [6][7][8][9] and the references therein.
The upper and lower solutions method is considered an important way to study fuzzy differential equations. For example, R. Rodríguez-López et al. discussed initial value problems of fuzzy differential equations with the upper and lower solutions method in [10,11], R. Alikhani and F. Bahrami [9] discussed a first-order, nonlinear, fuzzy integro-differential equation by using the upper and lower solutions method.
We can also find some new results about fuzzy delay differential equations. A. Khastan et al. studied the existence of solutions to an initial value problem of the fuzzy delay differential equation in [12]. Some other results can be found in [13][14][15]. The methods involved in these papers are fixed point theories, the variation of constants method and the upper and lower solutions method.
In [7,16,17], J. J. Nieto and R. Rodríguez-López studied boundary value problems of fuzzy differential equations. For example, the authors considered periodic boundary value conditions in [7] where {t 1 , t 2 , · · · , t m } ∈ [0, T], a, σ are piecewise continuous on [0, T]. The authors changed the fuzzy differential Equation (1) into two crisp ordinary differential equations, which were deduced from the level set equations of Equation (1). A similar method also had been used to study periodic boundary value problems in [6,18,19].
In general, there are few papers discussing fuzzy delay differential equations, especially the boundary value problems of fuzzy delay differential equations. In our previous work [20], we studied boundary value problems . From then on, we can find few papers on the topics of fuzzy differential equations with variable parameters in the boundary value conditions. Motivated by [12][13][14][15], we will extend the results in our previous work [20] to the fuzzy delay differential equation.
In the present paper, we consider where Applying the upper and lower solutions method and the monotone iterative technique, we provide some sufficient conditions for the existence of maximal and minimal solutions to (2). The results can be applied to discuss some dynamical models in biology. At the end of this paper, we provide an example to verify our results.

Preliminaries
Let R F be the class of fuzzy subsets of the real axis, u : R → [0, 1], which satisfies: (iii) u is upper semi-continuous on R; (iv) The closure of {t ∈ R|u(t) > 0} is compact.
on R F and (R F , D) is a complete metric space.

Definition 1 ([21]
). For every u, v ∈ R F , if there exists z ∈ R F such that u = v + z, then z is said to be the H-difference of u and v. We denote z = u v.

Definition 2 ([3,4])
. A function f : [a, b] → R F is a strongly generalized differentiable at t ∈ (a, b); if there exists f (t) ∈ R F such that ∀h > 0 sufficiently small, H-difference and limits in the following formulas exist with metric D: Proof. (i) is from Theorem 2.5.2 in [8]. The proof of (ii) is similar; here we omit it.

Now, we define an ordering relation in
If v ≤ u, then cv ≤ cu for c ∈ (0, +∞) and cu ≤ cv for c ∈ (−∞, 0); x is strongly generalized differentiable as in the Definition 2(i) or (ii).

Existence of Solutions to
We consider sufficient conditions for the existence of solutions to (2) in this section.
Then there exist maximal and minimal (i)-differentiable solutions to (2) in [v, u].
Here we claim that A is nondecreasing in the interval [v, u]. In fact, let x, y ∈ [v, u] and x ≤ y, nondecreasing property of f provides that f (t, x(λ(t))) ≤ f (t, y(λ(t))), ∀t ∈ [0, T]. We can conclude from Lemma 6(v) that Together with Lemma 4(i), we obtain that Ax(t) ≤ Ay(t), t ∈ [0, T]. Consequently, A is nondecreasing in [v, u]. Now we demonstrate that v ≤ Av ≤ Au ≤ u. Av ≤ Au can be deduced directly by nondecreasing property of A, we just need to prove v ≤ Av and Au ≤ u.
∀t ∈ [0, T] and m > n > 1, Lemma 1, Lemma 2(iii) and (7) imply that That is, {A n v} is a Cauchy sequence. According to the completeness of C([0, T], R F ), {A n v} is convergent; that is, there exists v * ∈ C([0, T], R F ) such that {A n v} converges uniformly to v * . Easily we can check that v * satisfies Av * = v * . As a result, v * is a (i)-differentiable solution to (2). An analogous result can be obtained for {A n u}.
Suppose that A n u → u * and x ∈ [v, u] is also a (i)-differentiable solution to (2); that is, Ax = x. Applying nondecreasing property of A, we have A n v ≤ x ≤ A n u for n ≥ 1. Lemma 4(v) implies v * ≤ x ≤ u * . Consequently, v * , u * are minimal and maximal (i)-differentiable solutions for (2) in the interval [v, u]. where Then there exist maximal and minimal (ii)-differentiable solutions for (2) Proof. Referring to (4), we denote For every x ∈ C([0, T], R F ), Bx is (ii)-differentiable and Bx(0) = αBx(T). Lemma 5(ii) implies that any x ∈ C([0, T], R F ) satisfying Bx = x is a (ii)-differentiable solution to (2). Here we prove that B is nondecreasing in the interval [v, u].
Applying the same method in the proof of Theorem 1, we can prove that there exist maximal and minimal (ii)-differentiable solutions to (2) in the interval [v, u].
If α ∈ (−∞, 0), the upper and lower solutions method can not be applied directly. In fact, referring to (3) and (4), the corresponding integral operators for α ∈ (−1, 0) and α ∈ (−∞, −1) can be written as The nondecreasing or nonincreasing properties of f are not enough to guarantee the monotonicity of F and H; we need more hypotheses to discuss the existence of solution to (2).

Condition (ii) implies that F is contraction mapping on
By Banach contraction mapping principle, there exists at least one (i)-differentiable solution for (2) in the interval [v, u].  Then there exists at least one (ii)-differentiable solution to (2) in [v, u].

Proof.
For every x, y ∈ C([0, T], R F ) and t ∈ [0, T], we can check that On the other hand, for every x, y ∈ [v, u] and t ∈ [0, T], Condition (ii) implies that H has at least one fixed point in [v, u].

Conclusions
First-order delay differential equations are frequently applied to study models in economics, biology and so on. For example, the exponential growth model can be extended to x (t) = rx(λ(t)). Some researchers try to consider the models in fuzzy cases; there are also some literatures discussing numerical algorithms of boundary value problems of fuzzy differential equations. However, we believe that it is necessary to discuss the existence of solutions before calculating the numerical solutions. In this paper, we provide some sufficient conditions for the existence of solutions to fuzzy delay differential equations. The results can be applied to estimate the existence and position of the solutions to fuzzy delay differential equation models. Now, we introduce an example to verify our theorems.
In addition, Thus, Theorem 1(i) and (ii) are all satisfied; there exist maximal and minimal (i)-differentiable solutions to (22). We can get the approximate maximal and minimal (i)-differentiable solutions by calculating A n u and A n v.

Conflicts of Interest:
The author declares no conflict of interest. The funders had no role in the decision to publish the results.