Existence of extremal solutions to interval-valued delay fractional differential equations via monotone iterative technique

Abstract

In this paper the interval-valued delay fractional differential equations (IDFDEs) under the Caputo generalized Hukuhara differentiability are introduced. By establishing some necessary comparison results and using the monotone iterative technique combined with the method of upper and lower solutions, we investigate the existence of extremal solutions for interval-valued delay fractional differential equations. Several examples are presented to illustrate the concepts and results.

Keywords

Interval-valued functions interval-valued fractional integral and derivatives interval-valued delay fractional differential equation

1 Introduction

Fractional calculus is a generalization of the ordirany differentiation and integration to arbitrary non-integer order. In recent years, the fractional differential equations have become an important tool in mathematical modelling the anomalous dynamics of various processes related to complex systems in most areas of engineering and science. Many applications of fractional differential equations in signal processing, complex dynamics in biological tissues and viscoelastic materials can be found in [24 , 32]. The most recent papers and books which deal with the fractional calculus, results of various types of fractional differential equations and their applications are Samko et al. [37], Podlubny [34] and Kilbas et al. [16].

In recent years, interval and fuzzy analysis were proposed to handle interval uncertainty due to incomplete information that appears in many mathematical or computer models, and the interval-valued differential equations have become a suitable tool to model dynamic systems in which uncertainties or vagueness pervade. The main theoretical and practical results in the fields of the interval analysis, and a wide number of applications in many different real problems can be found in several works [5 , 38]. First works devoted to the subject of fractional differential equations with uncertainty were introduced by Agarwal et al. [1, 2]. They defined the Riemann-Liouville differentiability concept under the Hukuhara differentiability to solve fuzzy fractional differential equations. Some basic results on the existence and uniqueness of solutions to initial value problem of fuzzy fractional differential equations under Hukuhara fractional-order Riemann-Liouville differentiability are studied by Arshad and Lupulescu in [8]. Later on, the new concepts about generalized Hukuhara fractional Riemann-Liouville and Caputo-type differentiability of fuzzy-valued functions were proposed and studied by Allahviranloo et al. [3] and Mazandarani et al. [30]. In [23], Lupulescu gave some new concepts of the Riemann-Liouville and Caputo fractional derivatives for interval-valued functions, and the basic theories of the fractional calculus for interval-valued functions were proposed and developed by using generalization of the Hukuhara difference for closed interval on the real line. Various approaches and methods, based on Hukuhara differentiability or generalized Hukuhara differentiability were then considered to investigate interval or fuzzy fractional differential equations in a number of papers in literature (see for example [6 , 35] and [19 –21]).

The monotone iterative technique with the method of upper and lower solutions is an effective tool that offers the existence result of solution in a closed set generated by the lower and upper solutions. The idea of this method shows that if we can find a lower solution X^L and an upper solution X^U of IDFDE, and if, furthermore X^L ⪯ X^U, then there exists a solution satisfying X^L ⪯ X ⪯ X^U . There has been extensive works done using this technique and a variety of nonlinear problems have been tackled. To get a comprehensive view on this technique refers [17, 18]. There has been continuous development in this area and some of the papers recently published in this area involving various problems are given in the references [28 , 40].

The main purpose of this paper is to prove the existence of extremal solutions of interval-valued delay fractional differential equations by using several tools from fractional interval-valued calculus to approximate its extremal solutions in a given interval functional interval by the method of upper and lower solutions and monotone iterative technique. Our paper has the following structure: in Section 2, we recall some well-known definitions of the fractional integral and derivative for interval-valued functions, and some necessary properties of interval fractional derivatives are also provided. By establishing some necessary comparison results and using the monotone iterative technique, the existence of extremal solutions of the interval-valued delay fractional integral and differential equation are proved in Section 3.

2 The fundamental theorem of calculus

Let K_c denote the set of all nonempty compact intervals of the real line $ℝ$ . If A = [a^-, a⁺], B = [b^-, b⁺] ∈ K_c, then the usual interval operations, i.e. Minkowski addition and scalar multiplication, are defined by $\begin{matrix} A + B & = & [a^{-}, a^{+}] + [b^{-}, b^{+}] \\ = & [a^{-} + b^{-}, a^{+} + b^{+}], \\ λ A & = & λ [a^{-}, a^{+}] = {\begin{matrix} [λ a^{-}, λ a^{+}] & if λ > 0 \\ 0 & if λ = 0 \\ [λ a^{+}, λ a^{-}] & if λ < 0, \end{matrix} \end{matrix}$ respectively. With respect to the above operations, K_c is a quasilinear space (Markov [29]).

Generalized Hukuhara difference. Denote the width of A ∈ K_c by w (A) = a⁺ - a^- . The generalized Hukuhara difference (or gH-difference for short) of two intervals A, B ∈ K_c is defined as follows [29, 38] $\begin{matrix} A \underset{g}{⊖} B \\ = C \Leftrightarrow {\begin{matrix} (i) A = B + C if w (A) \geq w (B), \\ (ii) B = A + (- 1) C if w (A) \leq w (B) . \end{matrix} \end{matrix}$

The Hausdorff-Pompeiu metric H in K_c is defined as follows: $H (A, B) = max {| a^{-} - b^{-} |, | a^{+} - b^{+} |} .$ (2.1) The following properties are well known: $H (A, B) = 0 \Leftrightarrow A = B, H (λ A, λ B) = | λ | H (A, B), H (A, B) = H (A \underset{g}{⊖} B, 0) .$ We notice that (K_c, H) is a complete, separable and locally compact metric space. We define the magnitude of A ∈ K_c by ||A|| : = H (A, 0) = max {|a^-|, |a⁺|}. Then || · || is a norm on K_c and $H (A, B) = | | A \underset{g}{⊖} B | |$ . Since (K_c, + , · , H) is a quasilinear metric space, then the concepts of continuity and limit for interval-valued functions are understood in the sense of the metric H.

Lemma 1. [14] If $(X^{m})_{m \in ℕ} \subset K_{c}$ and A ∈ K_c are given, then X^m → X as m→ ∞ if and only if $X^{m} \underset{g}{⊖} A \to X \underset{g}{⊖} A$ as m → ∞ .

For X ∈ K_c, we consider the following partial ordering in K_c .

Definition 1. Let X, Y ∈ K_c . We say that X ⪯ Y (X ≽ Y) if and only if x^- ≤ y^- and x⁺ ≤ y⁺ (x^- ≥ y^- and x⁺ ≥ y⁺).

Some interesting properties on the partial ordering ⪯ of the space of interval-valued functions were presented in [36]. Let C ([a, b] , K_c) denote the set of continuous interval-valued functions from [a, b] into K_c. For γ ∈ [0, 1), we introduce the space C_γ ([a, b] , K_c) of interval-valued functions F : (a, b] → K_c such that the function (· - a) ^γF (·) ∈ C ([a, b] , K_c). The space C_γ ([a, b] , K_c) is a complete metric space with respect to the metric $| | F | |_{C_{γ}} : = sup_{a \leq t \leq b} t^{γ} H (F (t), 0) .$

A function F : [a, b] → K_c is said to be absolutely continuous if, for each ɛ > 0, there exists δ > 0 such that, for each family {(s_k, t_k) ; k = 1, 2, . . . , n} of disjoint open intervals in [a, b] with $\sum_{k = 1}^{n} (t_{k} - s_{k}) < δ,$ we have $\sum_{k = 1}^{n} H (F (t_{k}), F (s_{k})) < ɛ .$ Let AC ([a, b] , K_c) denote the set of all absolutely continuous interval-valued functions from [a, b] into K_c.

We say that $(F_{k})_{k \in ℕ} \subseteq K_{c}$ is a nondecreasing sequence if F_k ⪯ F_k+1 for all $k \in ℕ .$ Analogously, we say that $(G_{k})_{k \in ℕ} \subseteq K_{c}$ is a nonincreasing sequence if G_k+1 ⪯ G_k for all $k \in ℕ .$ Consider interval functions F, G : [a, b] → K_c . The partial ordering ⪯ can be extended to the space of interval functions, as follows: for all t ∈ [a, b] $\begin{matrix} F & ⪯ & G if and only if f^{-} (t) \leq g^{-} (t) and f^{+} (t) \\ \leq & g^{+} (t) . \end{matrix}$

Theorem 1. The following properties hold (see [36]):

If $(F_{k})_{k \in ℕ} \subseteq C ([a, b], K_{c})$ is a nondecreasing (respectively, nonincreasing) sequence such that F_k → F in C ([a, b] , K_c), then $F_{k} ⪯ F (respectively, F ⪯ F_{k}), \forall k \in ℕ .$

Let F_k : [a, b] → K_c, G : [a, b] → K_c . If $F_{k} ⪯ G, \forall k \in ℕ$ , and F_k (t) converges to F (t) in K_c, for all t ∈ [a, b], then F ⪯ G .

Let F_k, G_k : [a, b] → K_c, F, G : [a, b] → K_c . If $F_{k} ⪯ G_{k}, \forall k \in ℕ,$ and F_k (t) converges to F and G_k (t) converges to G (t) in K_c, for all t ∈ [a, b] , then F ⪯ G .

If $(F_{k})_{k \in ℕ} \subseteq C ([a, b], K_{c})$ is a nondecreasing (respectively, nonincreasing) sequence such that there exists a subsequence (F_{k
_l}) → X in C ([a, b] , K_c), then (F_k) → F in C ([a, b] , K_c) .

Generalized Hukuhara derivative. The generalized Hukuhara derivative (gH-derivative for short) of F : [a, b] → K_c at t₀ ∈ [a, b] is defined by (provided it exists) $F^{'} (t_{0}) = lim_{h \to 0} \frac{F (t_{0} + h) ⊖_{g} F (t_{0})}{h} .$ (2.2)

Denote by C¹ ([a, b] , K_c) the space of interval-valued functions which are continuous gH-differentiable on [a, b] . We recall that an interval-valued function F : [a, b] → K_c is w-increasing (w-decreasing) on [a, b] if the real function t ↦ w (F (t)) increasing (decreasing) on [a, b]. If F is w-increasing or w-decreasing on [a, b], then we say that F is w-monotone on [a, b]. It is well known that a real monotone function on [a, b] is differentiable almost everywhere (a.e for short) on [a, b]. Hence, if t ↦ w (F (t)) is monotone on [a, b], then t ↦ w (F (t)) is differentiable a.e. on [a, b]. But the differentiability a.e. of t ↦ w (F (t)) on [a, b] does not imply that F is gH-differentiabe a.e. on [a, b] as is shown in the next example.

Example 1. Let F : [a, b] → K_c be an interval function given by F (t) = [f^- (t) , f⁺ (t)], where $\begin{matrix} f^{-} (t) = {\begin{matrix} t & if t \in [0, 1] \cap ℚ, \\ - 2 t & if t \notin [0, 1] \cap ℚ, \end{matrix} \\ f^{+} (t) = {\begin{matrix} 2 t & if t \in [0, 1] \cap ℚ, \\ - t & if t \notin [0, 1] \cap ℚ . \end{matrix} \end{matrix}$

Since w (F (t)) = t, t ∈ [a, b], then w is differentiable on [a, b] and F is w-increasing on [a, b], but f^- and f⁺ are not differentiable on [a, b]. Therefore, it follows that F cannot be gH-differentiabe a.e. on [a, b]. Also, notice that F is w-increasing on [a, b], but f^- and f⁺ are not monotone on [a, b].

For 1≤ p ≤ ∞, let L^p ([a, b] , K_c) the set of all interval-valued functions F : [a, b] → K_c such that the real function t ↦ ||F (t) || belongs to L^p [a, b]. Then L^p ([a, b] , K_c) is a complete metric space with respect to the metric $H_{p}$ defined by $H_{p} (F, G) : = | | F \underset{g}{⊖} G | |_{p}$ (see [23]), where $| | F | |_{p} : = {\begin{matrix} {(\int_{a}^{b} | | F (t) | |^{p} dt)}^{1 / p} & if 1 \leq p < \infty, \\ \underset{t \in [a, b]}{ess sup} | | F (t) | | & if p = \infty . \end{matrix}$

Interval-valued Riemann-Liouville fractional integral. The Riemann–Liouville derivative of order α ∈ (0, 1] for a real function f ∈ L¹ [a, b] is defined for a.e. t ∈ [a, b] by (see [16]) $\begin{matrix} (D_{a^{+}}^{α} f) (t) & = & \frac{d}{dt} (I_{a^{+}}^{1 - α} f) (t) \\ = & \frac{d}{dt} \int_{a}^{t} \frac{(t - s)^{- α}}{Γ (1 - α)} f (s) ds . \end{matrix}$

If f ∈ L¹ [a, b] is a real function such that $D_{a^{+}}^{α} f$ exists a.e. on [a,b], then the Caputo fractional derivative $C D_{a^{+}}^{α} f$ of order α ∈ (0, 1) is defined for a.e. t ∈ [a, b] by (see [16]) $(^{C} D_{a^{+}}^{α} f) (t) : = (D_{a^{+}}^{α} [f (\cdot) - f (a)]) (t) .$

Let F ∈ L^p ([a, b] , K_c), 1≤ p ≤ ∞. The interval-valued Riemann-Liouville integral of interval-valued function F is defined for a.e. t ∈ [a, b] by $(ℑ_{a^{+}}^{α} F) (t) = \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} F (s) ds .$ (2.3)

If F (t) = [f^- (t) , f⁺ (t)] ∈ L^p ([a, b] , K_c) and α > 0, then we can indicate the interval-valued Riemann-Liouville integral of interval-valued function F based on lower and upper functions; that is, $(ℑ_{a^{+}}^{α} F) (t) = [(I_{a^{+}}^{α} f^{-}) (t), (I_{a^{+}}^{α} f^{+}) (t)] .$ (2.4)Remark 1. (a) If F ∈ L^p ([a, b] , K_c), 1≤ p ≤ ∞, then $ℑ_{a^{+}}^{α} F \in L^{p} ([a, b], K_{c})$ and $| | ℑ_{a^{+}}^{α} F | |_{p} \leq \frac{(b - a)^{α}}{Γ (1 + α)} | | F | |_{p} .$

Consequently, if $(F_{m})_{m \in ℕ}$ is a sequence in L^p ([a, b] , K_c), 1≤ p ≤ ∞, such that F_m converges to F in L^p ([a, b] , K_c), then $ℑ_{a^{+}}^{α} F_{m}$ converges to $ℑ_{a^{+}}^{α} F \in L^{p} ([a, b], K_{c})$ . In particular, if F ∈ L^∞ ([a, b] , K_c), then $ℑ_{a^{+}}^{α} F \in C ([a, b], K_{c})$ and $(ℑ_{a^{+}}^{α} F) (a) = 0$ .

(b) If F ∈ C_γ ([a, b] , K_c), 0 ≤ γ < 1, then we have $\begin{matrix} | | (ℑ_{a^{+}}^{α} F) (t) | | \leq \frac{Γ (1 - γ) (t - a)^{α - γ}}{Γ (1 + α - γ)} | | F | |_{γ} . \end{matrix}$ We distinguish the following cases.

If α < γ < 1, then $ℑ_{a^{+}}^{α}$ is an operator from C_γ ([a, b] , K_c) into C_γ-α ([a, b] , K_c) and $| | ℑ_{a^{+}}^{α} F | |_{C_{γ - α}} \leq \frac{Γ (1 - γ)}{Γ (1 + α - γ)} | | F | |_{C_{γ}} .$ Also, in this case we may expect $ℑ_{a^{+}}^{α} F$ to be unbounded at t = a.

If 0 < γ ≤ α, then $ℑ_{a^{+}}^{α}$ is an operator from C_γ ([a, b] , K_c) into C ([a, b] , K_c) and $| | ℑ_{a^{+}}^{α} F | | \leq \frac{Γ (1 - γ) (b - a)^{α - γ}}{Γ (1 + α - γ)} | | F | |_{C_{γ}} .$

In this case we have $(ℑ_{a^{+}}^{α} F) (a) = 0$ .

Interval-valued Riemann-Liouville fractional derivative. For F = [f^- (t) , f⁺ (t)] ∈ L¹ ([a, b] , K_c) and α ∈ (0, 1], we define F_1-α : [a, b] → K_c by $F_{1 - α} (t) = (ℑ_{a^{+}}^{1 - α} F) (t) : = \int_{a}^{t} \frac{(t - s)^{- α}}{Γ (1 - α)} F (s) ds .$

If the gH-derivative $F_{1 - α}^{'} (t)$ exists for a.e. t ∈ [a, b], then $F_{1 - α}^{'} (t)$ is called the interval-valued Riemann-Liouville fractional derivative of order α ∈ (0, 1]. The Riemann-Liouville gH-fractional derivative of F will be denoted by $D_{a^{+}}^{α} F$ . Therefore, $(D_{a^{+}}^{α} F) (t) : = {(ℑ_{a^{+}}^{1 - α} F)}^{'} (t) .$ Remark 2. (Lupulescu [23]) Let F = [f^-, f⁺] ∈ AC ([a, b] , K_c). Then F_1-α ∈ AC ([a, b] , K_c) and

if either F is w-increasing on [a, b] or F is w-decreasing and F_1-α is w-increasing on [a, b], then $(D_{a^{+}}^{α} F) (t) = [(D_{a^{+}}^{α} f^{-}) (t), (D_{a^{+}}^{α} f^{+}) (t)];$ (2.5)

if F_1-α is w-decreasing on [a, b], then $(D_{a^{+}}^{α} F) (t) = [(D_{a^{+}}^{α} f^{+}) (t), (D_{a^{+}}^{α} f^{-}) (t)] .$ (2.6)

Lemma 2. If F ∈ L¹ ([a, b] , K_c) is a w-monotone interval-valued function such that $(D_{a^{+}}^{α} F) (t)$ exists for a.e. t ∈ [a, b] and $G (t) : = F (t) \underset{g}{⊖} F (a)$ . Then

$ℑ_{a^{+}}^{1 - α} G \in AC ([a, b], K_{c})$ and $\frac{d}{dt} w ((ℑ_{a^{+}}^{1 - α} G) (t)) \geq 0$ for a.e. t ∈ [a, b].

$w ((D_{a^{+}}^{α} F) (t)) \geq \frac{(t - a)^{- α}}{Γ (1 - α)} w (F (a))$ for a.e. t ∈ [a, b] if F is w-increasing on [a, b].

$w ((D_{a^{+}}^{α} F) (t)) \leq \frac{(t - a)^{- α}}{Γ (1 - α)} w (F (a))$ for a.e. t ∈ [a, b] if F is w-decreasing and F_1-α is w-increasing on [a, b].

Proof. Since F is w-monotone on [a, b], then by Lemma 1 from [9] it follows that G is w-increasing on [a, b] and thus, by Lemma 1 in [23], $ℑ_{a^{+}}^{1 - α} G$ is w-increasing on [a, b]. Since $(D_{a^{+}}^{α} F) (t)$ exists for a.e. t ∈ [a, b], then $ℑ_{a^{+}}^{1 - α} F \in AC ([a, b], K_{c})$ and so $ℑ_{a^{+}}^{1 - α} G \in AC ([a, b], K_{c})$ . Therefore, we obtain that $\frac{d}{dt} w ((ℑ_{a^{+}}^{1 - α} G) (t)) \geq 0$ for a.e. t ∈ [a, b]. Next, if F is w-increasing on [a, b], then we have $\begin{matrix} w ((D_{a^{+}}^{α} F) (t)) - \frac{(t - s)^{- α}}{Γ (1 - α)} w (F (a)) \\ = \frac{d}{dt} I_{a^{+}}^{1 - α} w (F (t)) - \frac{d}{dt} I_{a^{+}}^{1 - α} w (F (a)) \\ \frac{d}{dt} I_{a^{+}}^{1 - α} [w (F (t)) - w (F (a))] \\ = \frac{d}{dt} w (ℑ_{a^{+}}^{1 - α} G (t)) \geq 0 \end{matrix}$ for a.e. t ∈ [a, b] and thus (b) holds. The proof of (c) is similar to the case of (b). □

Proposition 1. [23] If F ∈ L^p ([a, b] , K_c) (1≤ p ≤ ∞), then $(D_{a^{+}}^{α} ℑ_{a^{+}}^{α} F) (t) = F (t) for a . e . t \in [a, b] .$

Proposition 2. [23] Let F ∈ L¹ ([a, b] , K_c) be such that F_1-α ∈ AC ([a, b] , K_c). If either $\frac{d}{dt} w (F_{1 - α} (t)) \geq 0$ for a.e. t ∈ [a, b] or $\frac{d}{dt} w (F_{1 - α} (t)) \leq 0$ for a.e. t ∈ [a, b], then the gH-difference $F (t) \underset{g}{⊖} \frac{(t - a)^{α - 1}}{Γ (α)} F_{1 - α} (a)$ exists for a.e. t ∈ [a, b], and $(ℑ_{a^{+}}^{α} D_{a^{+}}^{α} F) (t) = F (t) \underset{g}{⊖} \frac{(t - a)^{α - 1}}{Γ (α)} F_{1 - α} (a) .$

Interval-valued Caputo fractional derivative. Let F ∈ L¹ ([a, b] , K_c) such that the Riemann-Liouville gH-fractional derivative $D_{a^{+}}^{α} F$ , α ∈ (0, 1], exists a.e. on [a, b]. In this case, the interval-valued Caputo fractional derivative $(C D_{a^{+}}^{α} F) (t)$ of order α ∈ (0, 1] of F is defined by $\begin{matrix} (C D_{a^{+}}^{α} F) (t) & : = & (D_{a^{+}}^{α} [F (\cdot) \underset{g}{⊖} F (a)]) (t) \\ for a . e . t \in [a, b] . \end{matrix}$

Theorem 2. Let F : [a, b] → K_c be such that $D_{a^{+}}^{α} F$ , α ∈ (0, 1], exists a.e. on [a, b].

If either F is w-increasing on [a, b] or F is w-decreasing and F_1-α is w-increasing on [a, b], then $(C D_{a^{+}}^{α} F) (t) = (D_{a^{+}}^{α} F) (t) ⊖ \frac{(t - a)^{- α}}{Γ (1 - α)} F (a) .$ (2.7)

If F_1-α are w-decreasing on [a, b], then

$\begin{matrix} (C D_{a^{+}}^{α} F) (t) \\ = (D_{a^{+}}^{α} F) (t) + \frac{(t - a)^{- α}}{Γ (1 - α)} (- F (a)) . \end{matrix}$ (2.8)

Proof. Let us consider the constant interval-valued function G : [a, b] → K_c, given by G (t) : = F (a). Then $G_{1 - α} (t) = (ℑ_{a^{+}}^{1 - α} F (a)) (t) = \frac{(t - a)^{1 - α}}{Γ (2 - α)} F (a)$ , and so G_1-α is w-increasing on [a, b]. Now, if F is w-increasing on [a, b] or F is w-decreasing and F_1-α is w-increasing on [a, b], then by Theorem 4 (Lupulescu [23]) it follows that $\begin{matrix} (C D_{a^{+}}^{α} F) (t) & = & (D_{a^{+}}^{α} [F (\cdot) \underset{g}{⊖} F (a)]) (t) \\ = & (D_{a^{+}}^{α} F) (t) \underset{g}{⊖} (D_{a^{+}}^{α} G) (t) \\ = & (D_{a^{+}}^{α} F) (t) \underset{g}{⊖} \frac{(t - a)^{- α}}{Γ (1 - α)} F (a), \end{matrix}$ that is, (2.7). Also, if F_1-α is w-increasing on [a, b], then by Theorem 4 (Lupulescu [23]) it follows that $\begin{matrix} (C D_{a^{+}}^{α} F) (t) & = & (D_{a^{+}}^{α} F) (t) + (- (D_{a^{+}}^{α} G) (t)) \\ = & (D_{a^{+}}^{α} F) (t) + \frac{(t - a)^{- α}}{Γ (1 - α)} (- F (a)), \end{matrix}$ that is, (2.8). □

Remark 3. If F = [f^- (t) , f⁺ (t)] ∈ AC ([a, b] , K_c), F is w-monotone and α ∈ (0, 1], then it is obviously that

$(C D_{a^{+}}^{α} F) (t) = [(^{C} D_{a^{+}}^{α} f^{-}) (t), (C D_{a^{+}}^{α} f^{+}) (t)]$ for a.e. t ∈ [a, b], if F is w-increasing;

$(C D_{a^{+}}^{α} F) (t) = [(C D_{a^{+}}^{α} f^{+}) (t), (^{C} D_{a^{+}}^{α} f^{-}) (t)]$ for a.e. t ∈ [a, b], if F is w-decreasing.

Proposition 3. [23] If F ∈ AC ([a, b] , K_c) is a w-monotone interval-function and α ∈ (0, 1], then $(ℑ_{a^{+}}^{α} C D_{a^{+}}^{α} F) (t) = F (t) \underset{g}{⊖} F (a) .$ (2.9)

Proposition 4. [23] Let F ∈ L^∞ ([a, b] , K_c) be such that either F is w-increasing on [a, b], or F is w-decreasing on [a, b] and $t \mapsto F_{α} (t) : = ℑ_{a^{+}}^{α} F (t)$ is w-increasing on [a, b], then $(C D_{a^{+}}^{α} ℑ_{a^{+}}^{α} F) (t) = F (t) for a . e . t \in [a, b] .$

Theorem 3. Let H be an interval-valued function such that t ↦ H (t) belongs to C ([a, b] , K_c) , and $t \mapsto (ℑ_{a^{+}}^{α} H) (t)$ be w-increasing on [a, b]. Then there is a w-monotone unique solution X ∈ C ([a, b] , K_c) of the initial value problem $(^{C} D_{a^{+}}^{α} X) (t) = H (t), X (a) = X_{0} \in K_{c}$ (2.10) given by the integral equation $X (t) \underset{g}{⊖} X_{0} = \int_{a}^{t} \frac{(t - s)^{α - 1}}{Γ (α)} H (s) ds, t \in [a, b] .$ (2.11)

Proof. The proof of this theoren is similar to the proof in [14]. □

Let us suppose that X ∈ C ([a, b] , K_c) is w-monotone on [a, b] and satisfies (2.11). Since X is w-monotone on [a, b], then by Lemma 1 from [9] it follows that $X (t) \underset{g}{⊖} X_{0}$ is w-increasing on [a, b]. Hence, from (2.11) it follows that the interval-valued function $t \mapsto (ℑ_{a^{+}}^{α} H) (t)$ must be w-increasing on [a, b]. Therefore, the w-monotone interval-valued functions satisfying (2.11) must be sought in the set of all w-monotone interval-valued functions X ∈ C ([a, b] , K_c) for which the interval-valued function $t \mapsto (ℑ_{a^{+}}^{α} H) (t)$ is w-increasing on [a, b].

If X ∈ C ([a, T] , K_c) satisfies (2.11), then $X_{a} (t) : = X (t) \underset{g}{⊖} X (a)$ , t ∈ [a, b] is called a condensed solution of (2.11). Obviously, if X ∈ C ([a, T] , K_c) is a solution of (2.10), then $X_{a}$ is a condensed solution of (2.11), but conversely is not true (see [14]) if $t \mapsto (ℑ_{a^{+}}^{α} H) (t)$ is not w-increasing on [a, b]. Moreover, if X ∈ C ([a, T] , K_c) is a w-monotone solution of (2.11), then the condensed solution $X_{a}$ is w-increasing on [a, b]. Also, it is clear that a condensed solution $X_{a} \in C ([a, T], K_{c})$ may produce two solutions of (2.11): a w-increasing solution $X^{↑} (t) = X (a) + X_{a} (t)$ , t ∈ [a, b], if X is w-increasing on [a, b], and a w-decreasing solution $X^{↓} (t) = X (a) ⊖ (- X_{a} (t))$ , t ∈ [a, b], if X is w-decreasing on [a, b].

Example 2. Let us consider the following initial value problem: ${\begin{matrix} (^{C} D_{0^{+}}^{1 / 2} X) (t) = [t, 1], t \in [0, 1] \\ X (0) = [0, 1], \end{matrix}$ (2.12) and its associated integral equation $X (t) \underset{g}{⊖} X (0) = (ℑ_{0^{+}}^{1 / 2} F) (t), t \in [0, 1],$ (2.13) where F (t) : = [t, 1], t ∈ [0, 1]. We have that $(ℑ_{0^{+}}^{1 / 2} F) (t) = [\frac{4}{3 \sqrt{π}} t^{3 / 2}, \frac{2}{\sqrt{π}} t^{1 / 2}],$ where t ∈ [t₀, 1]. If we put $Y_{0} (t) : = (ℑ_{0^{+}}^{1 / 2} F) (t)$ , t ∈ [t₀, 1], then $\frac{d}{dt} w (Y_{0} (t)) = \frac{t^{- 1 / 2}}{\sqrt{π}} (1 - 2 t)$ , t ∈ (t₀, 1]. It follows that Y₀ is w-increasing on [t₀, t₁] and w-decreasing on [t₁, 1], where $t_{1} = \frac{1}{2}$ . Then the condensed solution $X_{0}$ of integral equation (2.13), namely: $X_{0} (t) : = X_{0} (t) \underset{g}{⊖} X_{0} (t_{0}) = [\frac{4}{3 \sqrt{π}} t^{3 / 2}, \frac{2}{\sqrt{π}} t^{1 / 2}],$ produces a w-monotone solution X₀ of (2.12) only on the interval [t₀, t₁]. We obtain the w-increasing solution $\begin{matrix} X_{0}^{↑} (t) & = & [0, 1] + [\frac{4}{3 \sqrt{π}} t^{3 / 2}, \frac{2}{\sqrt{π}} t^{1 / 2}] \\ = & [\frac{4}{3 \sqrt{π}} t^{3 / 2}, 1 + \frac{2}{\sqrt{π}} t^{1 / 2}], t \in [t_{0}, t_{1}], \end{matrix}$ and the w-decreasing solution $\begin{matrix} X_{0}^{↓} (t) & = & [0, 1] ⊖ [- \frac{2}{\sqrt{π}} t^{1 / 2}, - \frac{4}{3 \sqrt{π}} t^{3 / 2}] \\ = & [\frac{2}{\sqrt{π}} t^{1 / 2}, 1 + \frac{4}{3 \sqrt{π}} t^{3 / 2}], t \in [t_{0}, t_{1}] . ∥ \end{matrix}$ The solution $X_{0}$ can be extended to the right of the point t₁ up to a point t₂ ∈ (t₁, 1] such that $(ℑ_{t_{1}^{+}}^{1 / 2} F) (t)$ is w-increasing on [t₁, t₂]. The extension of $X_{0}$ up to t₂ is an interval-valued function $X_{1} : [t_{0}, t_{2}] \to K_{c}$ such that $X_{1} (t) = X_{0} (t)$ for t ∈ [t₀, t₁] and X₁ is the solution of the following initial value problem: ${\begin{matrix} (^{C} D_{t_{n}^{+}}^{1 / 2} X) (t) = [t, 1], t \in [t_{n}, 1] \\ X (t_{n}) = X_{n - 1} (t_{n}), \end{matrix}$ (2.14) where $X_{0} (t_{1}) = X_{0} (t_{1}) \underset{g}{⊖} X_{0} (t_{0}) = (ℑ_{t_{0}^{+}}^{1 / 2} F) (t_{1})$ ; that is, $X (t_{1}) = X_{0}^{↑} (t_{1}) = X_{0} (t_{0}) + (ℑ_{t_{0}^{+}}^{1 / 2} F) (t_{1}),$ if X₀ is w-increasing and $X (t_{1}) = X_{0}^{↓} (t_{1}) = X_{0} (t_{0}) ⊖ (- (ℑ_{t_{0}^{+}}^{1 / 2} F) (t_{1}))$ if X₀ is w-decreasing. Here X₀ (t₀) = [0, 1], $X (t_{1}) = X_{0}^{↑} (t_{1}) = [\frac{1}{3} \sqrt{\frac{2}{π}}, 1 + \sqrt{\frac{2}{π}}]$ if X is w-increasing, and $X (t_{1}) = X_{0}^{↓} (t_{1}) = [\sqrt{\frac{2}{π}}, 1 + \frac{1}{3} \sqrt{\frac{2}{π}}]$ if X is w-decreasing. The integral equation associated to (2.14) is $X (t) \underset{g}{⊖} X (t_{1}) = (ℑ_{t_{1}^{+}}^{1 / 2} F) (t), t \in [t_{1}, 1] .$ (2.15)

Next, we have that $\begin{matrix} (ℑ_{t_{1}^{+}}^{1 / 2} F) (t) & = & [\frac{4 t + 2 t_{1}}{3 \sqrt{π}} \sqrt{t - t_{1}}, \frac{2}{\sqrt{π}} \sqrt{t - t_{1}}], \\ t \in [t_{1}, 1] . \end{matrix}$ If we put $Y_{1} (t) : = (ℑ_{t_{1}^{+}}^{1 / 2} F) (t)$ , t ∈ [t₁, 1], then $\frac{d}{dt} w (Y_{1} (t)) = \frac{1}{2 \sqrt{π (t - t_{1})}} (1 + t_{1} - 2 t)$ , t ∈ (t₁, 1]. It follows that Y₁ is w-increasing on [t₁, t₂] and w-decreasing on [t₂, 1], where $t_{2} = \frac{1}{2} (1 + t_{1}) = \frac{3}{4}$ . Then the condensed solution $X_{1}$ of integral Equation (2.13) is $\begin{matrix} X_{1} (t) & : = & X_{1} (t) \underset{g}{⊖} X_{1} (t_{1}) \\ = & [\frac{4 t + 2 t_{1}}{3 \sqrt{π}} \sqrt{t - t_{1}}, \frac{2}{\sqrt{π}} \sqrt{t - t_{1}}], \end{matrix}$ t ∈ [t₁, t₂], and it produces four w-monotone solutions for initial value problem (2.14), namely: two w-increasing solutions $\begin{matrix} X_{1}^{↑ ↑} (t) & = & X_{0}^{↑} (t_{1}) + (ℑ_{t_{1}^{+}}^{1 / 2} F) (t) \\ = & [\frac{1}{3} \sqrt{\frac{2}{π}} + \frac{4 t + 2 t_{1}}{3 \sqrt{π}} \sqrt{t - t_{1}}, 1 \\ + \sqrt{\frac{2}{π}} + \frac{2}{\sqrt{π}} \sqrt{t - t_{1}}], \end{matrix}$ t ∈ [t₁, t₂] , and $\begin{matrix} X_{1}^{↓ ↑} (t) & = & X_{1}^{↓} (t) + (ℑ_{t_{1}^{+}}^{1 / 2} F) (t) \\ = & [\sqrt{\frac{2}{π}} + \frac{4 t + 2 t_{1}}{3 \sqrt{π}} \sqrt{t - t_{1}}, 1 \\ + \frac{1}{3} \sqrt{\frac{2}{π}} + \frac{2}{\sqrt{π}} \sqrt{t - t_{1}}], \end{matrix}$ t ∈ [t₁, t₂] , and two w-decreasing solutions $\begin{matrix} X_{1}^{↑ ↓} (t) & = & X_{1}^{↑} (t) ⊖ (- (ℑ_{t_{1}^{+}}^{1 / 2} F) (t)) \\ = & [\frac{1}{3} \sqrt{\frac{2}{π}}, 1 + \sqrt{\frac{2}{π}}] \\ ⊖ [- \frac{2}{\sqrt{π}} \sqrt{t - t_{1}}, - \frac{4 t + 2 t_{1}}{3 \sqrt{π}} \sqrt{t - t_{1}}] \\ = & [\frac{1}{3} \sqrt{\frac{2}{π}} + \frac{2}{\sqrt{π}} \sqrt{t - t_{1}}, 1 \\ + \sqrt{\frac{2}{π}} + \frac{4 t + 2 t_{1}}{3 \sqrt{π}} \sqrt{t - t_{1}}], \end{matrix}$ t ∈ [t₁, t₂] , and $\begin{matrix} X_{1}^{↓ ↓} (t) & = & X_{1}^{↓} (t) ⊖ (- (ℑ_{t_{1}^{+}}^{1 / 2} F) (t)) \\ = & [\sqrt{\frac{2}{π}}, 1 + \frac{1}{3} \sqrt{\frac{2}{π}}] \\ ⊖ [- \frac{2}{\sqrt{π}} \sqrt{t - t_{1}}, - \frac{4 t + 2 t_{1}}{3 \sqrt{π}} \sqrt{t - t_{1}}] \\ = & [\sqrt{\frac{2}{π}} + \frac{2}{\sqrt{π}} \sqrt{t - t_{1}}, 1 \\ + \frac{1}{3} \sqrt{\frac{2}{π}} + \frac{4 t + 2 t_{1}}{3 \sqrt{π}} \sqrt{t - t_{1}}], \end{matrix}$ for t ∈ [t₁, t₂] . In fact, $X_{1}^{↑ ↑}$ is the w-increasing solution and $X_{1}^{↑ ↓}$ is the w-decreasing solution of (2.14) if we use the initial condition $X (t_{1}) = X_{0}^{↑} (t_{1})$ . Similarly, $X_{1}^{↓ ↑}$ is the w-increasing solution and $X_{1}^{↓ ↓}$ is the w-decreasing solution of (2.14) if we use the initial condition $X (t_{1}) = X_{0}^{↓} (t_{1})$ . Now, it is easy to check that the interval-valued function X^↑ : [0, 1] → K_c given by $X^{↑} (t) = {\begin{matrix} X_{0}^{↑} (t) & if t \in [t_{0}, t_{1}], \\ X_{1}^{↑ ↑} (t) & if t \in [t_{1}, t_{2}], \end{matrix}$ is a w-increasing solution of (2.12) on [t₀, t₂]. Similar, the interval-valued function X^↓ : [0, 1] → K_c given by $X^{↓} (t) = {\begin{matrix} X_{0}^{↓} (t) & if t \in [t_{0}, t_{1}], \\ X_{1}^{↓ ↓} (t) & if t \in [t_{1}, t_{2}], \end{matrix}$ is a w-decreasing solution of (2.12) on [t₀, t₂]. By mathematical induction we can show that for any n ≥ 1 the solution $X_{0}$ can be extended to the right of the point t_n up to a point t_n+1 ∈ (t_n, 1] such that $(ℑ_{t_{n}^{+}}^{1 / 2} F) (t)$ is w -increasing on [t_n, t_n+1], where $t_{n + 1} = \frac{1}{2} (1 + t_{n})$ , n ≥ 0; that is, t_n = 1 - (1/2) ⁿ, n ≥ 0. Indeed, suppose that $X_{0}$ was extended up to the point t_n such that by $\begin{matrix} X_{n - 1} (t) & : = & X_{n - 1} (t) \underset{g}{⊖} X_{n - 1} (t_{n - 1}) \\ = & [\frac{4 t + 2 t_{n - 1}}{3 \sqrt{π}} \sqrt{t - t_{n - 1}}, \frac{2}{\sqrt{π}} \sqrt{t - t_{n - 1}}], \end{matrix}$ t ∈ [t_n-1, t_n] . The extension of $X_{0}$ up to t_n+1 is an interval-valued function $X_{n} : [t_{0}, t_{n + 1}] \to K_{c}$ such that $X_{n} (t) = X_{n - 1} (t)$ for t ∈ [t₀, t_n] and X_n is the solution of the following initial value problem: ${\begin{matrix} (^{C} D_{t_{n}^{+}}^{1 / 2} X) (t) = [t, 1], t \in [t_{n}, 1] \\ X (t_{n}) = X_{n - 1} (t_{n}), \end{matrix}$ (2.16) where $X_{n - 1} (t_{n}) = X_{n - 1} (t_{n}) \underset{g}{⊖} X_{n - 1} (t_{n - 1}) = (ℑ_{t_{n - 1}^{+}}^{1 / 2} F) (t_{n})$ . Next, we have that $\begin{matrix} (ℑ_{t_{n - 1}^{+}}^{1 / 2} F) (t) = [\frac{4 t + 2 t_{n}}{3 \sqrt{π}} \sqrt{t - t_{n}}, \frac{2}{\sqrt{π}} \sqrt{t - t_{n}}], \end{matrix}$ t ∈ [t_n, 1] . If we put $Y_{n} (t) : = (ℑ_{t_{n}^{+}}^{1 / 2} F) (t)$ , t ∈ [t_n, 1], then $\frac{d}{dt} w (Y_{n} (t)) = \frac{1}{2 \sqrt{π (t - t_{n})}} (1 + t_{n} - 2 t)$ , t ∈ (t_n, 1]. It follows that Y_n is w-increasing on [t_n, t_n+1] and w-decreasing on [t_n+1, 1], where $t_{n + 1} = \frac{1}{2} (1 + t_{n})$ . Then the condensed solution $X_{n}$ of integral equation associated to (2.16) is given by $\begin{matrix} X_{n} (t) & : = & X_{n} (t) \underset{g}{⊖} X_{n} (t_{n}) \\ = & [\frac{4 t + 2 t_{n}}{3 \sqrt{π}} \sqrt{t - t_{n}}, \frac{2}{\sqrt{π}} \sqrt{t - t_{n}}], \\ t \in [t_{n}, t_{n + 1}], \end{matrix}$ and it produces 2ⁿ w-increasing solutions and 2ⁿ w-decreasing solutions for initial value problem (2.16). A reasoning, not so difficult, leads us to establish the extended monotone solutions of (2.16) on [0, 1]. We obtain the w-increasing solution X^↑ : [0, 1] → K_c given by $X^{↑} (t) = {\begin{matrix} X_{0}^{↑} (t) & if t \in [t_{0}, t_{1}], \\ X_{n}^{↑^{n}} (t) & if t \in [t_{n}, t_{n + 1}], n \geq 1, \end{matrix}$ where $\begin{matrix} X_{n}^{↑^{n}} (t) \\ = [a_{n} + \frac{4 t + 2 t_{n}}{3 \sqrt{π}} \sqrt{t - t_{n}}, b_{n} + \frac{2}{\sqrt{π}} \sqrt{t - t_{n}}], \\ t \in [t_{n}, t_{n + 1}], n \geq 1 \\ a_{n} = \frac{2}{3 \sqrt{π}} \sum_{k = 1}^{n} (3 - \frac{1}{2^{k - 2}}) \sqrt{\frac{1}{2^{k}}}, \\ b_{n} = 1 + \frac{2}{\sqrt{π}} \sum_{k = 1}^{n} \sqrt{\frac{1}{2^{k}}}, n \geq 1, \end{matrix}$ and ↑ⁿ means $\underset{(n + 1) - times}{\underset{︸}{↑ ↑ . . . ↑}}$ .

Also, the w-decreasing solution X^↓ : [0, 1] → K_c is given by $X^{↓} (t) = {\begin{matrix} X_{0}^{↓} (t) & if t \in [t_{0}, t_{1}], \\ X_{n}^{↓^{n}} (t) & if t \in [t_{n}, t_{n + 1}], n \geq 1, \end{matrix}$ where $\begin{matrix} X_{n}^{↓^{n}} (t) \\ = [c_{n} + \frac{2}{\sqrt{π}} \sqrt{t - t_{n}}, d_{n} + \frac{4 t + 2 t_{n}}{3 \sqrt{π}} \sqrt{t - t_{n}}], \\ t \in [t_{n}, t_{n + 1}], n \geq 1, \\ c_{n} = \frac{2}{\sqrt{π}} \sum_{k = 1}^{n} \sqrt{\frac{1}{2^{k}}}, \\ d_{n} = 1 + \frac{2}{3 \sqrt{π}} \sum_{k = 1}^{n} (3 - \frac{1}{2^{k - 2}}) \sqrt{\frac{1}{2^{k}}}, n \geq 1, \end{matrix}$ and ↓ⁿ means $\underset{(n + 1) - times}{\underset{︸}{↓ ↓ . . . ↓}}$ .

3 Existence of extremal solutions

For a positive number σ, we denote by C_σ the space C ([- σ, 0] , K_c) equipped with the metric defined by $H_{σ} (X, Y) = sup_{t \in [- σ, 0]} H (X (t), Y (t)) .$

Define I = [a, b] , J = [a - σ, a] ∪ I = [a - σ, b] . Then, for each t ∈ I we denote by X_t the element of C_σ defined by X_t (s) = X (t + s), s ∈ [- σ, 0] .

Consider the following interval delay fractional differential equation of order α ∈ (0, 1) with the initial condition: ${\begin{matrix} (^{C} D_{a^{+}}^{α} X) (t) = F (t, X (t), X_{t}), \\ X (t) = φ (t - a), t \in [a - σ, a], \end{matrix}$ (3.17) where F : [a, b] × K_c × C_σ → K_c, φ ∈ C_σ. Let X : [a, b] → K_c be a w-increasing (w-decreasing) interval-valued function which is Caputo gH-fractional differentiability on [a, b]. If X and its derivative satisfy problem (3.17), we say that X is a (i)-solution ((ii)-solution) of problem (3.17). Denote by C^1,F ([a, b] , K_c) the space of interval-valued functions which are continuous Caputo gH-fractional differentiable on [a, b] . In the following, for given interval-valued functions F : [a, b] × K_c × C_σ → K_c we will denote F (t, X (t) , X_t) by $F (t)$ .

Lemma 3. Let F be interval-valued functions such that $t \mapsto F (t)$ belongs to C ([a, b] , K_c), for any X ∈ K_c . A w-monotone interval-valued function X ∈ C ([a, b] , K_c) is a solution of initial value problem (3.17), if and only if X satisfies the following interval fractional integral equation ${\begin{matrix} X (t) = φ (t - a), t \in [a - σ, a] \\ X (t) \underset{g}{⊖} φ (0) = \int_{a}^{t} \frac{(t - s)^{α - 1}}{Γ (α)} F (s, X (s), X_{s}) ds, \end{matrix}$ (3.18) and the interval-valued function $t \mapsto ℑ_{a^{+}}^{α} F (t)$ is w-increasing on [a, b] .

Proof. Assume that X ∈ C ([a, b] , K_c) be a w-monotone solution of (3.17) and let $X (t) : = X (t) \underset{g}{⊖} X (a) = X (t) \underset{g}{⊖} φ (0)$ , t ∈ [a, b]. As X is w-monotone on [a, b], it follows that $t \mapsto X (t)$ is w-increasing on [a, b]. From (3.17) and $(C D_{a^{+}}^{α} X) (t) : = (D_{a^{+}}^{α} X) (t) = {(ℑ_{a^{+}}^{1 - α} X)}^{'} (t)$ , $X_{1 - α} (a) : = (ℑ_{a^{+}}^{1 - α} X) (a) = 0$ , t ∈ [a, b], by Lemma 2, we get $\frac{d}{dt} w (ℑ_{a^{+}}^{1 - α} X (t)) \geq 0$ for all t ∈ [a, b]. Hence, by Proposition 2, we have that $\begin{matrix} (ℑ_{a^{+}}^{α} C D_{a^{+}}^{α} X) (t) \\ = (ℑ_{a^{+}}^{α} D_{a^{+}}^{α} [X (\cdot) \underset{g}{⊖} X (a)]) (t) \\ = X (t) \underset{g}{⊖} \frac{(t - a)^{α - 1}}{Γ (α)} X_{1 - α} (a) \\ = X (t), t \in [a, b] . \end{matrix}$

Since $F \in C ([a, b], K_{c})$ and from $(C D_{a^{+}}^{α} X) (t) = F (t)$ , t ∈ [a, b], it shows that $(ℑ_{a^{+}}^{α} C D_{a^{+}}^{α} X) (t) = (ℑ_{a^{+}}^{α} F) (t)$ , t ∈ [a, b], and thus $X (t) = (ℑ_{a^{+}}^{α} F) (t)$ , t ∈ [a, b]. Therefore, we obtain X satisfying (3.18). Moreover, as $t \mapsto X (t) \underset{g}{⊖} φ (0)$ is w-increasing on [a, b], it infers that $t \mapsto (ℑ_{a^{+}}^{α} F) (t)$ is also w-increasing on [a, b].

Conversely, suppose that X ∈ C ([a, b] , K_c) is a w-monotone interval-valued function satisfying (3.18) and such that $t \mapsto (ℑ_{a^{+}}^{α} F) (t)$ is w-increasing on [a, b]. Since $F \in C ([a, b], K_{c})$ , the interval-valued function $t \mapsto (ℑ_{a^{+}}^{α} F) (t)$ is continuous on [a, b] and $(ℑ_{a^{+}}^{α} F) (a) = 0$ . Then $X (a) = 0$ , and thus X (a) = φ (0). Furthermore, since $t \mapsto (ℑ_{a^{+}}^{α} F) (t)$ is w-increasing on [a, b], then applying $D_{a^{+}}^{α}$ in (3.18) we have that $(D_{a^{+}}^{α} X) (t) = (D_{a^{+}}^{α} ℑ_{a^{+}}^{α} F) (t)$ , t ∈ [a, b]. By Proposition 1 it follows that $(D_{a^{+}}^{α} X) (t) = F (t)$ , t ∈ [a, b], and thus $(^{C} D_{a^{+}}^{α} X) (t) = F (t, X (t), X_{t}), t \in [a, b],$ that is, (3.17) holds.

Remark 4. From the result of Example 2, we notice that if a w-monotone interval-valued function X is a solution of (3.17) such that the function $t \mapsto ℑ_{a^{+}}^{α} F (t))$ is w-increasing on [a, b], then X is a w-monotone solution of (3.17). If a w-monotone interval-valued function X is a w-monotone solution of (3.17) on [a, b], then X is a solution of (3.18) on [a, b] , but conversely is not true if the function $t \mapsto ℑ_{a^{+}}^{α} F (t))$ is not w-increasing on [a, b] . Moreover, if X ∈ C ([a, b] , K_c) is a w-increasing on [a, b], then (3.18) can be written as ${\begin{matrix} X (t) = φ (t - a), t \in [a - σ, a] \\ X (t) = φ (0) + \int_{a}^{t} \frac{(t - s)^{α - 1}}{Γ (α)} F (s, X (s), X_{s}) ds, \end{matrix}$ (3.19) if X ∈ C ([a, b] , K_c) is a w-decreasing on [a, b], then (3.18) can be written as ${\begin{matrix} X (t) = φ (t - a), t \in [a - σ, a] \\ X (t) = φ (0) ⊖ \frac{(- 1)}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} F (s, X (s), X_{s}) ds . \end{matrix}$ (3.20)

Definition 2. A function X^L ∈ C ([a - σ, b] , K_c) ∩ C^1,F ([a, b] , K_c) is a lower (i)-solution for (3.17) if ${\begin{matrix} (^{C} D_{a^{+}}^{α} X^{L}) (t) \leq F (t, X^{L} (t), X_{t}^{L}), t \in [a, b] \\ X^{L} (t) = ξ (t - a) \leq φ (t - a), t \in [a - σ, a], \end{matrix}$ (3.21) where X^L is w-increasing on [a, b] and ξ (t - a) ∈ C_σ. A function X^U ∈ C ([a - σ, b] , K_c) ∩ C^1,F ([a, b] , K_c) is an upper (i)-solution for (3.17) if it satisfies the reverse inequalities of (3.21).

Analogously, definitions can be given for lower (ii)-solution and upper (ii)-solution for (3.17).

Definition 3. A function Y^U ∈ C ([a - σ, b] , K_c) ∩ C^1,F ([a, b] , K_c) is an upper (ii)-solution for (3.17) if ${\begin{matrix} (^{C} D_{a^{+}}^{α} Y^{U}) (t) ≽ F (t, Y^{U} (t), Y_{t}^{U}), t \in [a, b] \\ Y^{U} (t) = ψ (t - a) ≽ φ (t - a), t \in [a - σ, a], \end{matrix}$ (3.22) where Y^U is w-decreasing and ψ (t - a) ∈ C_σ. A function Y^L ∈ C ([a - σ, b] , K_c) ∩ C^1,F ([a, b] , K_c) is a lower (ii)-solution for (3.17) if it satisfies the reverse inequalities of (3.22).

Definition 4. Let X^L, X^U be lower and upper (i)-solutions for the Equation (3.17) such that [X^L, X^U] = {X ∈ C ([a - σ, b] , K_c) : X^L ⪯ X ⪯ X^U}. We say that the solution X^min ∈ [X^L, X^U] is a minimal (i)-solution of (3.17), if X^min is w-increasing and $X^{min} (t) ⪯ X (t), t \in [a - σ, b],$ for any (i)-solution X ∈ [X^L, X^U]. We define a maximal (i)-solution X^max ∈ [X^L, X^U] as a function satisfying the reverse inequalities.

Definition 5. Let Y^L, Y^U be lower and upper (ii)-solutions for the Equation (3.17) such that [Y^L, Y^U] = {Y ∈ C ([a - σ, b] , K_c) : Y^L ⪯ Y ⪯ Y^U}. We say that the solution Y^min ∈ [Y^L, Y^U] is a minimal (ii)-solution of (3.17), if Y^min is w-decreasing and $Y^{min} (t) ⪯ Y (t), t \in [a - σ, b],$ for any (ii)-solution Y ∈ [Y^L, Y^U]. We define a maximal (ii)-solution Y^max ∈ [Y^L, Y^U] as a function satisfying the reverse inequalities.

The following result is a comparison theorem which will need for our main result.

Theorem 4. Let X, Y ∈ AC ([a, b] , K_c) be equally w-monotonic on [a, b]. If there exists t₁ ∈ (a, b] such that X (t₁) = Y (t₁) and X (t) ⪯ Y (t) on [a, b], then it follows that $(C D_{a^{+}}^{α} X) (t_{1}) ≽ (C D_{a^{+}}^{α} Y) (t_{1})$ .

Proof. Using Remark 3 and Theorem 2, we have that for t₁ ∈ (a, b]

$\begin{matrix} (C D_{a^{+}}^{α} [X \underset{g}{⊖} Y]) (t_{1}) = (D_{a^{+}}^{α} [X \underset{g}{⊖} Y]) (t_{1}) \\ \underset{g}{⊖} \frac{(t_{1} - a)^{- α}}{Γ (1 - α)} [X \underset{g}{⊖} Y] (a), ≽ (D_{a^{+}}^{α} [X \underset{g}{⊖} Y]) (t_{1}) . \end{matrix}$ (3.23)

In addition, we know that $\begin{matrix} (D_{a^{+}}^{α} [X \underset{g}{⊖} Y]) (t) \\ = (\int_{a}^{t} \frac{(t - s)^{- α}}{Γ (1 - α)} [X \underset{g}{⊖} Y] (s) ds)^{'} . \end{matrix}$

Let $M (t) = \int_{a}^{t} (t - s)^{- α} [X \underset{g}{⊖} Y] (s) ds$ and for h > 0,

$\begin{matrix} M (t_{1} + h) \underset{g}{⊖} M (t_{1}) \\ = \int_{a}^{t_{1} + h} (t_{1} + h - s)^{- α} [X \underset{g}{⊖} Y] (s) ds \\ \underset{g}{⊖} \int_{a}^{t_{1}} (t_{1} - s)^{- α} [X \underset{g}{⊖} Y] (s) ds \\ = \int_{a}^{t_{1}} [(t_{1} + h - s)^{- α} - (t_{1} - s)^{- α}] [X \underset{g}{⊖} Y] (s) ds \\ + \int_{t_{1}}^{t_{1} + h} (t_{1} + h - s)^{- α} [X \underset{g}{⊖} Y] (s) ds . \end{matrix}$ (3.24)

As [(t₁ + h - s) ^-α - (t₁ - s) ^-α] <0 for s ∈ [a, t₁) and X (s) ⪯ Y (s), we obtain $\int_{a}^{t_{1}} [(t_{1} + h - s)^{- α} - (t_{1} - s)^{- α}] [X \underset{g}{⊖} Y] (s) ds ≽ 0 .$ (3.25)

Because of the continuous property of $(X \underset{g}{⊖} Y) (t)$ and X (t₁) = Y (t₁), there exists a positive interval C₁ (t₁) and a negative interval C₂ (t₁) , such that, for s ∈ [t₁, t₁ + h] , $C_{2} (t_{1}) ⪯ (X \underset{g}{⊖} Y) (s) ⪯ C_{1} (t_{1}) .$

Then, we get

$\begin{matrix} \int_{t_{1}}^{t_{1} + h} (t_{1} + h - s)^{- α} [X \underset{g}{⊖} Y] (s) ds \\ ≽ C_{2} (t_{1}) \int_{t_{1}}^{t_{1} + h} (t_{1} + h - s)^{- α} ds = \frac{C_{2} (t_{1})}{1 - α} h^{1 - α} . \end{matrix}$ (3.26)

Therefore, from (3.24), (3.25) and (3.26), letting h → 0, we obtain $M^{'} (t_{1}) ≽ 0$ . This infers $(D_{a^{+}}^{α} [X \underset{g}{⊖} Y]) (t_{1}) = \frac{1}{Γ (1 - α)} M^{'} (t_{1}) ≽ 0$ and from (3.23) we get $(C D_{a^{+}}^{α} [X \underset{g}{⊖} Y]) (t_{1}) ≽ 0 .$ Since X, Y are equally w-monotonic, we infer that $(C D_{a^{+}}^{α} X) (t_{1}) ≽ (C D_{a^{+}}^{α} Y) (t_{1})$ by using the result of Theorem 9 ([23]). The proof is complete.

Lemma 4. Let F ∈ C ([a, b] × K_c × C_σ, K_c) and F (t, A₁, A₂) be nondecreasing in A₁ and A₂ for each t ∈ [a, b] . Assume that X, Y ∈ C ([a - σ, b] , K_c) ∩ C^1,F ([a, b] , K_c) (in the same case of w-monotonic on [a, b]) and X (t) = ξ (t - a) ≺ Y (t) = ψ (t - a) , t ∈ [a - σ, a] . Assume further that

$\begin{array}{l} ^{C} D_{a^{+}}^{α} X (t) \leq F (t, X (t), X_{t}), \\ ^{C} D_{a^{+}}^{α} Y (t) ≻ F (t, Y (t), Y_{t}), \end{array}$ (3.27) for t ∈ [a, b] . Then, $X (t) ≺ Y (t), t \in [a, b] .$ (3.28)

Proof. If the assertion (3.28) is false, then the set $ⅅ = {t \in [a, b] : X (t) ≽ Y (t)}$ is nonempty. Because of ξ (t - a) ≺ ψ (t - a) and the continuity of the functions involved, there exists a t₁ such that a < t₁ ≤ b and $X (t_{1}) = Y (t_{1}), X (t) ≺ Y (t), t \in [a, t_{1}) .$ (3.29)

Then by Theorem 4, we obtain $^{C} D_{a^{+}}^{α} X (t_{1}) ≽^{C} D_{a^{+}}^{α} Y (t_{1}) .$ (3.30)

From X (t - a) ≺ Y (t - a) and (3.29), we deduce that X (t₁ - a) ⪯ Y (t₁ - a) , which, in view of the nondecreasing property of the functions F (t, X (t) , X_t), yields $F (t_{1}, X (t_{1}), X_{t_{1}}) ⪯ F (t_{1}, Y (t_{1}), Y_{t_{1}}) .$ (3.31)

On the other hand, the relations (3.27) and (3.30) lead to the inequality $F (t_{1}, X (X_{t 1}), X_{t 1}) \underline{≻} C D_{α +}^{α} X (t_{1}) \underline{≻} C D_{α +}^{α} Y (t_{1}) ≻ F (t_{1}, Y (t_{1}), Y_{t 1}),$ which is incompatible with (3.31) because of (3.29). Consequently, the set $ⅅ$ is empty, and (3.28) is true. The proof is complete.

Remark 5. The conclusion (3.28) remains valid even when the inequalities (3.27) are replaced by $^{C} D_{a^{+}}^{α} X (t) ≺ F (t, X (t), X_{t}),^{C} D_{a^{+}}^{α} Y (t) ≽ F (t, Y (t), Y_{t}) .$

Theorem 5. Assume that

Z, W ∈ C ([a - σ, b] , K_c) ∩ C^1,F ([a, b] , K_c) (in the same case of w-monotonic on [a, b]), F (t, A, B) ∈ C ([a, b] × K_c × C_σ, K_c) is nondecreasing in A and B for each t and $\begin{array}{l} ^{C} D_{a^{+}}^{α} Z (t) \leq F (t, Z (t), Z_{t}), \\ ^{C} D_{a^{+}}^{α} W (t) ≽ F (t, W (t), W_{t}), t \in [a, b]; \end{array}$

there exists a positive real-valued number L so that $\begin{matrix} F (t, A_{1}, A_{2}) ⪯ F (t, B_{1}, B_{2}) \\ + \frac{L}{1 + t} [A_{1} ⊖ B_{1} + sup_{θ \in [- σ, 0]} (A_{2} (θ) ⊖ B_{2} (θ))], \end{matrix}$ where t ∈ [a, b] and A₁ ≽ B₁, A₂ ≽ B₂ .

Then Z (t) = ξ (t - a) ⪯ W (t) = ψ (t - a) , t ∈ [a - σ, a] implies Z (t) ⪯ W (t) , t ∈ [a, b] , provided L (t - a) ^α-1 < 1/Γ (α) .

Proof. Choose a positive real number ɛ and consider W_ɛ (t) = W (t) + ɛ (1 + t). Then, we have W_ɛ,t = W_t + ɛ (1 + t + s) and W_ɛ,t ≻ W_t, t ∈ [a, b] , s ∈ [- σ, 0] . Therefore, we obtain $\begin{array}{l} ^{C} D_{a^{+}}^{α} W_{ε} (t) =^{C} D_{a^{+}}^{α} W (t) + ε^{C} D_{a^{+}}^{α} (1 + t) \\ ≽ F (t, W_{ε} (t), W_{ε, t}) + \frac{ε {(t - a)}^{1 - α}}{Γ (α)} \\ ⊖ \frac{L}{1 + t} [(W_{ε} (t) ⊖ W (t)) \\ + \sup_{θ \in [- σ, 0]} (W_{ε, t} (θ) ⊖ W_{t} (θ))] \\ = F (t, W_{ε} (t), W_{ε, t}) + ε [\frac{{(t - a)}^{1 - α}}{Γ (α)} - L] \\ ≽ F (t, W_{ε} (t), W_{ε, t}) . \end{array}$

Here we have utilized the assumptions (H1)-(H2) and the condition L (t - a) ^α-1 < 1/Γ (α). We can now apply Lemma 4 to Z and W_ɛ to yield Z (t) ≺ W_ɛ (t) , t ∈ [a, b] . As ɛ > 0 is arbitrary, we conclude that Z (t) ⪯ W (t) . The proof is complete.

Corollary 1. Let Z, W ∈ C ([a - σ, b] , K_c) ∩ C^1,F ([a, b] , K_c) (in the same case of w-monotonic on [a, b]), $M (t) \in C ([a, b], K_{c})$ . Suppose that $C D_{a +}^{a} Z (t) \underline{≺} M (t)$ and $C D_{a +}^{a} W (t) \underline{≺} M (t)$ for t ∈ [a, b] . Then Z (t) = ξ (t - a) ⪯ W (t) = ψ (t - a) , t ∈ [a - σ, a] implies Z (t) ⪯ W (t) , t ∈ [a, b] .

Theorem 6. Let F ∈ C ([a, b] × K_c × C_σ, K_c) and F (t, A, B) be nondecreasing in A and B for each t ∈ [a, b] . Suppose that F maps bounded sets in [a, b] × K_c × C_σ to bounded sets in K_c. Moreover, assume one of the following conditions is verified:

there exists a lower (i)-solution X^L ∈ C ([a - σ, b] , K_c) ∩ C^1,F ([a, b] , K_c) and an upper (i)-solution X^U ∈ C ([a - σ, b] , K_c) ∩ C^1,F ([a, b] , K_c) of the Equation (3.17) satisfying X^L (t)⪯ X^U (t) , t ∈ [a - σ, b] ;

there exists a lower (ii)-solution Y^L ∈ C ([a - σ, b] , K_c) ∩ C^1,F ([a, b] , K_c) and an upper (ii)-solution Y^U ∈ C ([a - σ, b] , K_c) ∩ C^1,F ([a, b] , K_c) of the Equation (3.17) satisfying Y^L (t)⪯ Y^U (t) , t ∈ [a - σ, b] ;

Then, there exist the coupled minimal and maximal (i)-solutions X^min, X^max ∈ [X^L, X^U] in the case of the condition (H3) for problem (3.17) and the coupled minimal and maximal (ii)-solutions Y^min, Y^max ∈ [Y^L, Y^U] in the case of the condition (H4) for problem (3.17). Moreover, if F ∈ C ([a, b] × K_c × C_σ, K_c) satisfies the condition (H2) of Theorem 5, then problem (3.17) has a unique (i)-solution in the case of the condition (H3) and (ii)-solution in the case of the condition (H4).

Proof. Step 1. We show that there exists at least a solution X ∈ [X^L, X^U] in the case of the condition (H3) for problem (3.17) and Y ∈ [Y^L, Y^U] in the case of the condition (H4) for problem (3.17) in some intervals $[a - σ, T],$ with $T \leq b$ .

Let ρ > 0 be a given real number and let B (ρ, X₀) : = {X ∈ K_c | H (X, X₀) ≤ ρ}. Since the mapping F maps bounded sets in [a, b] × K_c × C_σ to bounded sets in K_c, there exists a positive constant M such that H (F (t, X, X_t) , 0) ≤ M. Choose t^* > a such that $t^{*} - a \leq (\frac{ρ Γ (1 + α)}{M})^{1 / α}$ , and put $T : = min {t^{*}, b} .$ Let $B$ be a set of continuous interval functions X such that X (t) = φ (t - a) for t ∈ [a - σ, a] and X ∈ B (ρ, X₀) . Next, we consider the sequence of continuous interval functions ${X^{n}}_{n = 0}^{\infty}$ given by: X⁰ (t) : = X₀, where ${\begin{matrix} X^{0} (t) = φ (t - a), t \in [a - σ, a] \\ X^{0} (t) = φ (0), t \in [a, T] \end{matrix}$ and for n = 1, 2, . . . ${\begin{matrix} X^{n} (t) = φ (t - a), t \in [a - σ, a] \\ X^{n} (t) \underset{g}{⊖} φ (0) \\ = \int_{a}^{t} \frac{(t - s)^{α - 1}}{Γ (α)} F (s, X^{n - 1} (s), X_{s}^{n - 1}) ds, \end{matrix}$ (3.32) for $t \in [a, T],$ where the interval-valued function $t \mapsto ℑ_{a^{+}}^{α} F^{n - 1} (t) : = ℑ_{a^{+}}^{α} F (t, X^{n - 1} (t), X_{t}^{n - 1})$ is w-increasing on $[a, T] .$ For all n ≥ 0, it follows that Xⁿ (t) ∈ B (ρ, X₀) if and only if $X^{n} (t) \underset{g}{⊖} φ (0) \in B (ρ, 0) .$ If we suppose that $X^{n - 1} (t) \in B$ and for a given n ≥ 2, then from $H (X^{n} (t) \underset{g}{⊖} φ (0), 0) \leq \frac{M (t - a)^{α}}{Γ (1 + α)} \leq ρ$ it follows that $X^{n} (t) \in B .$ Hence, by mathematical induction, we have that $X^{n} (t) \in B$ for all n ≥ 1 . Next, for any $t_{1}, t_{2} \in [a, T]$ with t₁ < t₂, we have $\begin{matrix} H (X^{n} (t_{1}) \underset{g}{⊖} φ (0), X^{n} (t_{2}) \underset{g}{⊖} φ (0)) \\ \leq \frac{M}{α} [2 (t_{2} - t_{1})^{α} + (t_{1} - a)^{α} - (t_{2} - a)^{α}] \\ \leq \frac{2 M}{α} (t_{2} - t_{1})^{α} \end{matrix}$ or $H (X^{n} (t_{1}), X^{n} (t_{2})) \leq \frac{2 M}{α} (t_{2} - t_{1})^{α} .$ Therefore, for any ɛ > 0 and any n ≥ 1, we have that H (Xⁿ (t₁) , Xⁿ (t₂)) < ɛ, provided that $| t_{2} - t_{1} | < δ_{0} : = (\frac{ɛ Γ (1 + α)}{2 M})^{1 / α} .$ It then follows that the family ${X^{n} (\cdot), n \in ℕ}$ is equicontinuous and uniformly bounded. Then by Arzela-Ascoli theorem for multi-valued functions [33], the family ${X^{n} (\cdot), n \in ℕ}$ is relative compact in $C ([a, T], K_{c}) .$ Therefore, there exists a subsequence of ${X^{n}}_{n \in ℕ}$ which uniformly converges to a continuous interval-valued function $X : [a - σ, T] \to K_{c} .$ Next, for any $t \in [a, T]$ we obtain $\begin{matrix} H (X^{n_{k}} (t) \underset{g}{⊖} φ (0), \int_{a}^{t} \frac{(t - s)^{α - 1}}{Γ (α)} F (s, X, X_{s}) ds) \\ \leq \int_{a}^{t} \frac{(t - s)^{α - 1}}{Γ (α)} H (F (s, X^{n_{k}}, X_{s}^{n_{k}}), F (s, X, X_{s})) ds . \end{matrix}$

Since F is continuous function, then the summands in the last expression converge to zero as n_k → ∞ . Therefore, for $t \in [a, T]$ and from Lemma 1 we get $\begin{matrix} X (t) \underset{g}{⊖} φ (0) & = & lim_{n_{k} \to \infty} [X^{n_{k}} (t) \underset{g}{⊖} φ (0)] \\ = & \int_{a}^{t} \frac{(t - s)^{α - 1}}{Γ (α)} F (s, X (s), X_{s}) ds \end{matrix}$ and the interval-valued function $t \mapsto ℑ_{a^{+}}^{α} F (t)$ is w-increasing on $[a, T] .$

Next, we show that X ∈ [X^L, X^U]. For any ɛ > 0, we consider $X_{ɛ}^{U} (t) = X^{U} (t) + ɛ (1 + t)$ and $X_{ɛ}^{L} (t) + ɛ (1 + t) = X^{L} (t)$ . Then, we have $X_{ɛ, t}^{U} = X_{t}^{U} + ɛ (1 + t + s)$ and $X_{ɛ, t}^{U} ≻ X_{t}^{U}, t \in [a, b], s \in [- σ, 0] .$ Similarly, we get $X_{ɛ, t}^{L} + ɛ (1 + t + s) = X_{t}^{L}$ and $X_{ɛ, t}^{L} ≺ X_{t}^{L}, s \in [- σ, 0), t \in [a, b] .$ Thus we obtain $X_{ɛ}^{L} (t) ⪯ X^{L} (t) ⪯ X^{U} (t) ⪯ X_{ɛ}^{U} (t), t \in [a - σ, a]$ , $X_{ɛ}^{L} (t) ⪯ X^{L} (t) ⪯ X^{U} (t) ⪯ X_{ɛ}^{U} (t), t \in [a, b]$ and $X_{ɛ}^{L} (a) ⪯ X^{L} (a) ⪯ X^{U} (a) ⪯ X_{ɛ}^{U} (a) .$ As X^L, X^U are lower and upper solutions of the Equation (3.17), we have that $X_{ɛ}^{L} (t) ⪯ X^{L} (t) ⪯ X (t) ⪯ X^{U} (t) ⪯ X_{ɛ}^{U} (t), t \in [a - σ, a]$ and $X_{ɛ}^{L} (a) ⪯ X (a) ⪯ X_{ɛ}^{U} (a),$ where X (t) is a solution of the Equation (3.17). Now we have to show that $X_{ɛ}^{L} (t) ≺ X (t) ≺ X_{ɛ}^{U} (t), t \in [a, b] .$ If the above assertion is not true, then there exists a t₁ ∈ (a, b) such that $X (t_{1}) = X_{ɛ}^{U} (t_{1}), X_{ɛ}^{L} (t) ≺ X (t) ≺ X_{ɛ}^{U} (t),$ (3.33) for t ∈ [a, b] ∖ {t₁} . Hence by Theorem 4 we get $(C D_{a^{+}}^{α} X) (t_{1}) ≽ (C D_{a^{+}}^{α} X_{ɛ}^{U}) (t_{1})$ which yields

$\begin{matrix} F (t_{1}, X (t_{1}), X_{t_{1}}) \\ = (C D_{a^{+}}^{α} X) (t_{1}) ≽ (C D_{a^{+}}^{α} X_{ɛ}^{U}) (t_{1}) \\ ≽ F (t_{1}, X_{ɛ}^{U} (t_{1}), X_{ɛ, t_{1}}^{U}) . \end{matrix}$ (3.34)

Moreover, from $X (t) ⪯ X_{ɛ}^{U} (t), t \in [a - σ, a]$ and (3.29), we deduce $X (t_{1} + s) ≺ X_{ɛ}^{U} (t_{1} + s),$ where s ∈ [- σ, 0) . In view of the nondecreasing property of the function F (t, A, B) in A, B, yields $F (t, X_{ɛ}^{U} (t_{1}), X_{ɛ, t_{1}}^{U}) ≻ F (t, X (t_{1}), X_{t_{1}}) .$ (3.35)

This is a contradiction and thus we have that $X (t) ≺ X_{ɛ}^{U} (t) .$ Similarly, we can show that $X_{ɛ}^{L} ≺ X (t), t \in [a - σ, b]$ and hence the relation $X_{ɛ}^{L} (t) ≺ X (t) ≺ X_{ɛ}^{U} (t)$ holds for all t ∈ [a - σ, b]. Now as ɛ → 0, we conclude that X^L (t) ⪯ X (t) ⪯ X^U (t) .

Step 2. In this part, we prove that there exist maximal and minimal (ii)-solutions in the case of the condition (H4) for Equation (3.17). The proof of the case (H3) is absolutely similar.

We first suppose the hypotheses (H4) is satisfied. For each n ≥ 0 consider interval fractional delay fractional differential equation under the forms $^{C} D_{a^{+}}^{α} V^{n + 1} (t) = F (t, V^{n} (t), V_{t}^{n}), t \in [a, b],$ (3.36) $^{C} D_{a^{+}}^{α} W^{n + 1} (t) = F (t, W^{n} (t), W_{t}^{n}), t \in [a, b],$ (3.37) where Vⁿ⁺¹ (t) = Wⁿ⁺¹ (t) = φ (t - a) for t ∈ [a - σ, a] , and Vⁿ⁺¹ and Wⁿ⁺¹ are w-decreasing. We can begin the iterations by setting W⁰ = Y^U and V⁰ = Y^L. The solutions of (3.36) and (3.37) are denoted by Vⁿ⁺¹, Wⁿ⁺¹ respectively. We claim that

$\begin{matrix} Y^{L} & = & V^{0} ⪯ V^{1} ⪯ V^{2} ⪯ \dots V^{n} ⪯ \dots ⪯ W^{n} \\ ⪯ & \dots W^{2} ⪯ W^{1} ⪯ W^{0} = Y^{U} . \end{matrix}$ (3.38)

To confirm (3.38), first note from (3.36) and (3.37) for n = 0 that $\begin{array}{l} ^{C} D_{a^{+}}^{α} V^{1} (t) = F (t, V^{0} (t), V_{t}^{0}), t \in [a, b], \\ ^{C} D_{a^{+}}^{α} W^{1} (t) = F (t, W^{0} (t), W_{t}^{0}), t \in [a, b] . \end{array}$ where V¹ (t) = W¹ (t) = φ (t - a) , t ∈ [a - σ, a]. We now claim that V⁰ ⪯ V¹ ⪯ W¹ ⪯ W⁰ on [a - σ, b]. Indeed, we can write $\begin{array}{l} ^{C} D_{a^{+}}^{α} V^{1} (t) ≽ F (t, V^{0} (t), V_{t}^{0}), t \in [a, b], \\ V^{1} (t) = φ (t - a), t \in [a - σ, a], \end{array}$ and since V⁰ = Y^L, we obtain $\begin{array}{l} ^{C} D_{a^{+}}^{α} V^{0} (t) \leq F (t, V^{0} (t), V_{t}^{0}), t \in [a, b], \\ V^{0} (t) = ξ (t - a), t \in [a - σ, a], \end{array}$ with V⁰ (t) = ξ (t - a) ⪯ V¹ (t) = φ (t - a) , t ∈ [a - σ, a] . Utilizing Corollary 1, we get V⁰ ⪯ V¹ . We now show that V¹ ⪯ W¹ by using the nondecreasing property of the function F, $\begin{array}{l} ^{C} D_{a^{+}}^{α} V^{1} (t) = F (t, V^{0} (t), V_{t}^{0}) \leq F (t, W^{0} (t), W_{t}^{0}), \\ ^{C} D_{a^{+}}^{α} W^{1} (t) ≽ F (t, W^{0} (t), W_{t}^{0}), \end{array}$ with V¹ (t) = W¹ (t) = φ (t - a) , t ∈ [a - σ, a] . Applying Corollary 1, we conclude that V¹ ⪯ W⁰ . Similarly, it can be proved that W¹ ⪯ W⁰ . Hence we have showed that V⁰ ⪯ V¹ ⪯ W¹ ⪯ W⁰, t ∈ [a - σ, b] . We assume inductively for t ∈ [a - σ, b] $\begin{matrix} Y^{L} & = & V^{0} ⪯ V^{n - 1} ⪯ V^{n} ⪯ W^{n} \\ ⪯ & W^{n - 1} ⪯ W^{0} = Y^{U}, \end{matrix}$ and since F is nondecreasing, we conclude that $\begin{array}{l} ^{C} D_{a^{+}}^{α} V^{n} (t) = F (t, V^{n - 1} (t), V_{t}^{n - 1}) \\ \leq F (t, V^{n} (t), V_{t}^{n}), \end{array}$ Vⁿ (t) = φ (t - a) = Vⁿ⁺¹ (t) , t ∈ [a - σ, a] . Using Corollary 1 and from (3.36), we obtain Vⁿ (t) ⪯ Vⁿ⁺¹ (t) , t ∈ [a - σ, b] . Similarly, it can be proved that Wⁿ⁺¹ (t) ⪯ Wⁿ (t) , t ∈ [a - σ, b] . Next we demonstrate $V^{n} ⪯ W^{n}, \forall n \in ℕ .$ Indeed, we have $\begin{array}{l} ^{C} D_{a^{+}}^{α} W^{n} (t) = F (t, W^{n - 1}, W_{t}^{n - 1}) \\ ≽ F (t, V^{n - 1}, V_{t}^{n - 1}), \\ ^{C} D_{a^{+}}^{α} V^{n} (t) = F (t, V^{n - 1}, V_{t}^{n - 1}) \\ \leq F (t, V^{n - 1}, V_{t}^{n - 1}), \end{array}$ Wⁿ (t) = φ (t - a) = Vⁿ (t) , t ∈ [a - σ, a] , ∀ n > 1 . Hence we apply Corollary 1 and conclude that $V^{n} ⪯ W^{n}, \forall n \in ℕ .$ This completes the proof of the assertion (3.38).

Since the sequences of functions Vⁿ and Wⁿ satisfy the relation (3.38), they are uniformly bounded. We obtain the uniform boundedness and the equicontinuity of the sequences Vⁿ and Wⁿ similarly as in Step 1. Therefore, Vⁿ and Wⁿ are relatively compact in C ([a - σ, b] , K_c) . Then applying Arzela-Ascoli Theorem, Theorem 1-(v) and monotonicity of Vⁿ and Wⁿ, there exist continuous functions Y^min and Y^max such that the subsequences V^{n
_k} and W^{n
_k} converge uniformly to Y^min and Y^max on [a - σ, b], respectively. By using the convergence properties in the relations (3.36) and (3.37) we deduce that $\begin{array}{l} ^{C} D_{a^{+}}^{α} Y^{\min} (t) = F (t, Y^{\min} (t), Y^{\min_{t}}), t \in [a, b], \\ Y^{\min} (t) = φ (t - a), t \in [a - σ, a], \end{array}$ (3.39) $\begin{array}{l} ^{C} D_{a^{+}}^{α} Y^{\max} (t) = F (t, Y^{\max} (t), Y^{\max_{t}}), t \in [a, b], \\ Y^{\max} (t) = φ (t - a), t \in [a - σ, a], \end{array}$ (3.40) and that Y^L = V⁰ ⪯ Y^min ⪯ Y^max ⪯ W⁰ = Y^U on [a - σ, b] .

Next we show that Y^min and Y^max are coupled minimal and maximal (ii)-solutions of the problem (3.17). We assume that Y (t) is any (ii)-solution of (3.17) such that Y^L = V⁰ ⪯ Y ⪯ W⁰ on [a - σ, b]. We have to prove that V⁰ ⪯ Y^min ⪯ Y ⪯ Y^max ⪯ W⁰ on [a - σ, b]. Indeed, since Y^L = V⁰ ⪯ Y ⪯ W⁰ = Y^U, we obtain Vⁿ ⪯ Y ⪯ Wⁿ on [a - σ, b] for some $n \in ℕ$ . Employing the nondecreasing of F and applying Corollary 1, we get Vⁿ⁺¹ ⪯ Y . In the similar way, we obtain Y ⪯ Wⁿ⁺¹ . Therefore, for all n, we have Vⁿ⁺¹ ⪯ Y ⪯ Wⁿ⁺¹ on [a - σ, b]. Taking limits as n→ ∞, we get Y^L = V⁰ ⪯ Y^min ⪯ Y ⪯ Y^max ⪯ W⁰ = Y^U on [a - σ, b]. Moreover, applying Lemma 3 for (3.39) and (3.40), we get the following coupled minimal and maximal (ii)-solutions of the problem (3.17) ${\begin{matrix} Y^{min} (t) = φ (t - a), for t \in [a - σ, a] \\ Y^{min} (t) = φ (0) ⊖ \int_{a}^{t} \frac{(- 1) (t - s)^{α - 1}}{Γ (α)} \\ F (s, Y^{min}, Y_{s}^{min}) ds, \end{matrix}$ (3.41) ${\begin{matrix} Y^{max} (t) = φ (t - a), for t \in [a - σ, a] \\ Y^{max} (t) = φ (0) ⊖ \int_{a}^{t} \frac{(- 1) (t - s)^{α - 1}}{Γ (α)} \\ F (s, Y^{max}, Y_{s}^{max}) ds . \end{matrix}$ (3.42)

Step 3: In sequel, we prove the uniqueness of (ii)-solution in the case of the condition (H4) for Equation (3.17). Let Y^min, Y^max be coupled minimal and maximal (ii)-solutions of the problem (3.17). As Y^min ⪯ Y^max on [a - σ, b], there exist two real-valued functions $P^{-} (t), P^{+} (t) \in C ([a - σ, b], ℝ^{+})$ and P^- (t) ≤ P⁺ (t) such that Y^max = Y^min + [P^- (t) , P⁺ (t)] , ∀ t ∈ [a - σ, b] . Now, we have to show that Y^max = Y^min . Indeed, since Y^max, Y^min are the coupled minimal and maximal (ii)-solutions of (3.17) respectively and thus they satisfy the integral Equations (3.41) and (3.42) respectively. From the assumption (H2), we have $\begin{matrix} Y^{min} + [P^{-} (t), P^{+} (t)] = Y^{max} = φ (0) \\ ⊖ \frac{(- 1)}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} F (s, Y^{max} (s), Y_{s}^{max}) ds \\ ⪯ Y^{min} (t) ⊖ \frac{(- 1) L}{Γ (α)} \int_{a}^{t} \frac{(t - s)^{α - 1}}{1 + s} \\ sup_{r \in [s - σ, s]} (Y^{max} (r) ⊖ Y^{min} (r)) ds . \end{matrix}$

Then, we can obtain $\begin{matrix} P^{-} (t) & \leq & \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} \frac{L}{1 + s} sup_{r \in [s - σ, s]} P^{+} (r) dr, \\ P^{+} (t) & \leq & \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} \frac{L}{1 + s} sup_{r \in [s - σ, s]} P^{-} (r) dr \end{matrix}$

If we let $a (s) = sup_{r \in [s - σ, τ]} P^{-} (r), b (s) = sup_{r \in [s - σ, τ]} P^{+} (r),$ and L₁ (s) = L/(1 + s), s ∈ [a, t] ⊂ [a, b] , then we have $\begin{matrix} (\begin{matrix} a (t) \\ b (t) \end{matrix}) \\ \leq \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} (\begin{matrix} 0 & L_{1} (s) \\ L_{1} (s) & 0 \end{matrix}) (\begin{matrix} a (s) \\ b (s) \end{matrix}) ds . \end{matrix}$

We apply Gronwal inequality to get a (t) ≤0 and b (t) ≤0 for all t ∈ [a, b] . This prove the uniqueness of the (ii)-solution for (3.17). The proof is complete.

Example 3. Let us consider the interval-valued delay fractional differential equation ${\begin{matrix} ^{C} D_{0^{+}}^{α} X (t) = (1 + \sin (t)) X (t - \frac{π}{16}), t \in [0, \frac{π}{16}] \\ X (t) = φ (t), t \in [- \frac{π}{16}, 0], \end{matrix}$ (3.43) where φ (t) = [1, 2 - t]. In this example we shall solve (3.43) on [0, π/16] .

We notice that $F (t, X (t), X_{t}) = (1 + sin (t)) X (t - \frac{π}{16})$ satisfies the conditions of Theorem 6.

Case 1: We show that (3.43) exits extremal (i)-solutions on $[0, \frac{π}{16}]$ . First, assume that $\begin{matrix} X^{L} (t) = [(1 - ɛ) + \frac{(1 - ɛ) t^{α}}{Γ (α + 1)}, (2 - ɛ) + \frac{1}{Γ (α + 1)} (A_{1} t^{α} + \frac{t^{α + 1}}{(α + 1)})], \\ X^{U} (t) = [(1 + ɛ) + \frac{B (1 + ɛ) t^{α}}{Γ (α + 1)}, (2 + ɛ) + \frac{B}{Γ (α + 1)} (A_{2} t^{α} + \frac{t^{α + 1}}{(α + 1)})] \end{matrix}$ are lower and upper (i)-solutions on $[0, \frac{π}{16}]$ of (3.43) respectively, where A₁ = (2 + π/16 - ɛ) , A₂ = (2 + π/16 + ɛ) , B = 1 + sin(π/16) and ɛ > 0. Then, we have $\begin{array}{l} {\begin{matrix} ^{C} D_{0^{+}}^{α} X^{L} (t) \leq B X^{L} (t - \frac{π}{16}), t \in [0, π / 16] \\ X^{L} (t) = φ_{1} (t) = [1 - ε, 2 - t - ε] \leq φ (t), \end{matrix} \\ {\begin{matrix} ^{C} D_{0^{+}}^{α} X^{U} (t) ≽ X^{U} (t - \frac{π}{16}), t \in [0, π / 16] \\ X^{U} (t) = φ_{2} (t) = [1 + ε, 2 - t + ε] ≽ φ (t) . \end{matrix} \end{array}$

Next, let us construct the sequences by $\begin{array}{l} {\begin{matrix} ^{C} D_{0^{+}}^{α} V^{n + 1} (t) = V^{n} (t - \frac{π}{16}) - \frac{\hat{ε}}{n + 1}, \\ t \in [0, π / 16] \\ V^{n + 1} (t) = φ (t) - \frac{\hat{ε}}{n + 1}, t \in [- \frac{π}{16}, 0], \end{matrix} \\ {\begin{matrix} ^{C} D_{0^{+}}^{α} W^{n + 1} (t) = B W^{n} (t - \frac{π}{16}) \\ + \frac{\hat{ε}}{n + 1}, t \in [0, π / 16], \\ W^{n + 1} (t) = φ (t) + \frac{\hat{ε}}{n + 1}, t \in [- \frac{π}{16}, 0], \end{matrix} \end{array}$ for all n = 0, 1, . . . and $\hat{ɛ} > 0$ . We verify that monotone sequences of the above contructions satisfy

$(W^{n} (t)) \overset{n \to \infty}{\to} X^{\max} (t)$ and X^max is a maximal (i)-solution of (3.43).

$(V^{n} (t)) \overset{n \to \infty}{\to} X^{\max} (t)$ and X^min is a minimal (ii)-solution of (3.43).

First, we prove (a). Indeed, let $0 < {\hat{ɛ}}_{2} < {\hat{ɛ}}_{1} ⩽ \hat{ɛ}$ , then for each positive integer n we obtain $W^{n + 1} (t, {\hat{ɛ}}_{2}) ⪯ W^{n + 1} (t, {\hat{ɛ}}_{1}), t \in [- \frac{π}{16}, 0]$ and $\begin{array}{l} ^{C} D_{0^{+}}^{α} W^{n + 1} (t, {\hat{ε}}_{2}) \leq B W^{n} (t - \frac{π}{16}) + \frac{{\hat{ε}}_{2}}{n + 1}, \\ ^{C} D_{0^{+}}^{α} W^{n + 1} (t, {\hat{ε}}_{1}) ≽ B W^{n} (t - \frac{π}{16}) + \frac{{\hat{ε}}_{2}}{n + 1} . \end{array}$

By using Corollary 1, we obtain $W^{n + 1} (t, {\hat{ɛ}}_{2}) ⪯ W^{n + 1} (t, {\hat{ɛ}}_{1})$

for all $t \in [- \frac{π}{16}, \frac{π}{16}]$ and ${\hat{ɛ}}_{2} < {\hat{ɛ}}_{1} .$ On the other hand Wⁿ⁺¹ (t) ⪯ Wⁿ (t) with $\hat{ɛ}$ fixed. Since the family of functions $(W^{n + 1} (t, \hat{ɛ}))$ is equicontinuous and uniformly bounded on $[0, \frac{π}{16}]$ , there exists a decreasing sequence $(\frac{\hat{ɛ}}{{(n + 1)}_{k}}) \to 0$ and uniform limit $X^{max} (t) = lim_{k \to \infty} W^{{(n + 1)}_{k}} (t, \hat{ɛ})$ exits on $[0, \frac{π}{16}]$ . Since ${\begin{matrix} W^{{(n + 1)}_{k}} (t, \hat{ɛ}) = φ (0) + \frac{\hat{ɛ}}{(n + 1)_{k}} \\ + \frac{B}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} W^{{(n + 1)}_{k}} (s - \frac{π}{16}) ds, \\ t \in [0, \frac{π}{16}] \\ W^{(n + 1)_{k}} (t, \hat{ɛ}) = φ (t) + \frac{\hat{ɛ}}{(n + 1)_{k}}, t \in [- \frac{π}{16}, 0], \end{matrix}$ we obtain the maximal (i)-solution of (3.43) on [0, π/16] as below $\begin{matrix} X^{max} (t) & = & [1 + \frac{B t^{α}}{Γ (α + 1)}, 2 + \frac{B}{Γ (α + 1)} \\ ((2 + π / 16) t^{α} + \frac{t^{α + 1}}{α + 1})] . \end{matrix}$

Next we show that X^max (t) is a required maximal (i)-solution of (3.43) on $[0, \frac{π}{16}]$ . For this purpose, we observe that $X (t) = φ (t) ≺ φ (t) + \frac{\hat{ɛ}}{(n + 1)} = W^{n + 1} (t, \hat{ɛ}), t \in [- π / 16, 0]$ and F is nondecreasing, hence we get $\begin{array}{l} ^{C} D {}_{+}^{α}X (t) ≺ (1 + \sin (t)) X (t - \frac{π}{16}) + \frac{\hat{ε}}{(n + 1)} \\ \leq B W^{n} (t - \frac{π}{16}, \hat{ε}) + \frac{\hat{ε}}{n + 1} \\ \leq^{C} D_{0^{+}}^{α} W^{n + 1} (t, \hat{ε}) . \end{array}$

By using Corollary 1, then we get $X (t) ≺ W^{n + 1} (t, \hat{ɛ})$ on $[- \frac{π}{16}, \frac{π}{16}]$ . The uniqueness of maximal (i)-solution X^max (t) shows that $W^{n + 1} (t, \hat{ɛ})$ tends uniformly to X^max (t) and it is the maximal (i)-solution of (3.43) with $X^{max} (t) = lim_{k \to \infty} W^{{(n + 1)}_{k}} (t, \hat{ɛ}) .$

Next, we prove (b). Proceeding similarly as above, let $0 < {\hat{ɛ}}_{2} < {\hat{ɛ}}_{1} ⩽ \hat{ɛ}$ , then for each positive integer n we obtain $V^{n + 1} (t, {\hat{ɛ}}_{1}) ⪯ V^{n + 1} (t, {\hat{ɛ}}_{2}), t \in [- \frac{π}{16}, 0]$ and $\begin{array}{l} ^{C} D_{0^{+}}^{α} V^{n + 1} (t, {\hat{ε}}_{1}) \leq V^{n} (t - \frac{π}{16}) - \frac{{\hat{ε}}_{1}}{n + 1}, \\ ^{C} D_{0^{+}}^{α} V^{n + 1} (t, {\hat{ε}}_{2}) ≽ V^{n} (t - \frac{π}{16}) - \frac{{\hat{ε}}_{1}}{n + 1} . \end{array}$

Hence we apply Corollary 1 and get that $V^{n + 1} (t, {\hat{ɛ}}_{1}) ⪯ V^{n + 1} (t, {\hat{ɛ}}_{2})$ for all $t \in [- \frac{π}{16}, \frac{π}{16}]$ and ${\hat{ɛ}}_{2} < {\hat{ɛ}}_{1} .$ On the other hand Vⁿ (t) ⪯ Vⁿ⁺¹ (t) with $\hat{ɛ}$ fixed. Since the family of functions $(V^{n + 1} (t, \hat{ɛ}))$ is equicontinuous and uniformly bounded on $[0, \frac{π}{16}]$ , there exists a decreasing sequence $(\frac{\hat{ɛ}}{{(n + 1)}_{k}}) \to 0$ and uniform limit $X^{min} (t) = lim_{k \to \infty} V^{{(n + 1)}_{k}} (t, \hat{ɛ})$ exits on $[0, \frac{π}{16}]$ . Since ${\begin{matrix} V^{{(n + 1)}_{k}} (t, \hat{ɛ}) = φ (0) - \frac{\hat{ɛ}}{(n + 1)_{k}} \\ + \frac{1}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} V^{{(n + 1)}_{k}} (s - \frac{π}{16}) ds, \\ t \in [0, \frac{π}{16}] \\ V^{(n + 1)_{k}} (t, \hat{ɛ}) = φ (t) - \frac{\hat{ɛ}}{(n + 1)_{k}}, t \in [- \frac{π}{16}, 0], \end{matrix}$ we obtain the minimal (i)-solution of (3.43) on [0, π/16] as below $\begin{matrix} X^{min} (t) & = & [1 + \frac{t^{α}}{Γ (α + 1)}, 2 \\ + \frac{1}{Γ (α + 1)} ((2 + π / 16) t^{α} + \frac{t^{α + 1}}{α + 1})] . \end{matrix}$

Next we show that X^min (t) is a required minimal (i)-solution of (3.43) on $[0, \frac{π}{16}]$ . For this purpose, we observe that $X (t) = φ (t) ≻ φ (t) - \frac{\hat{ɛ}}{(n + 1)} = V^{n + 1} (t, \hat{ɛ}), t \in [- π / 16, 0]$ and F is nondecreasing, hence we obtain $\begin{array}{l} ^{C} D_{0^{+}}^{α} X (t) ≻ (1 + \sin (t)) X (t - \frac{π}{16}) - \frac{\hat{ε}}{(n + 1)} \\ ≽ V^{n} (t - \frac{π}{16}, \hat{ε}) - \frac{\hat{ε}}{n + 1} ≽^{C} D_{0^{+}}^{α} V^{n + 1} (t, \hat{ε}) . \end{array}$

By using Corollary 1, then we get $V^{n + 1} (t, \hat{ɛ}) ≺ X (t)$ on $[- \frac{π}{16}, \frac{π}{16}]$ . The uniqueness of minimal (i)-solution X^min (t) shows that $V^{n + 1} (t, \hat{ɛ})$ tends uniformly to X^min (t) and it is the minimal (i)-solution of (3.43) with $X^{min} (t) = lim_{k \to \infty} V^{{(n + 1)}_{k}} (t, \hat{ɛ}) .$

Case 2: In this case, we show that (3.43) exits the extremal (ii)-solutions on $[0, \frac{π}{16}] .$ Let $\begin{matrix} Y^{L} (t) & = & [(1 - ɛ) + \frac{1}{Γ (α + 1)} [A_{1} t^{α} + \frac{t^{α + 1}}{(α + 1)}, (2 - ɛ) + \frac{1}{Γ (α + 1)} t^{α}]], \\ Y^{U} (t) & = & [(1 + ɛ) + \frac{B}{Γ (α + 1)} [A_{2} t^{α} + \frac{t^{α + 1}}{(α + 1)}, (2 + ɛ) + \frac{B (1 + ɛ)}{Γ (α + 1)} t^{α}]] \end{matrix}$ are the lower and upper (ii)-solutions on $[0, \frac{π}{16}]$ of (3.43) respectively. Proceeding similarly as Case 1, we obtain $Y^{min} (t) = lim_{k \to \infty} V^{{(n + 1)}_{k}} (t, \hat{ɛ}), Y^{max} (t) = lim_{k \to \infty} W^{{(n + 1)}_{k}} (t, \hat{ɛ}),$ where $\begin{matrix} {\begin{matrix} V^{{(n + 1)}_{k}} (t, \hat{ɛ}) = φ (0) - \frac{\hat{ɛ}}{(n + 1)_{k}} \\ ⊖ \int_{0}^{t} \frac{(- 1) t - s)^{α - 1}}{Γ (α)} (V^{{(n + 1)}_{k}} (s - \frac{π}{16}) ds, t \in [0, \frac{π}{16}] \\ V^{(n + 1)_{k}} (t, \hat{ɛ}) = φ (t) - \frac{\hat{ɛ}}{(n + 1)_{k}}, t \in [- \frac{π}{16}, 0], \end{matrix} \\ {\begin{matrix} W^{{(n + 1)}_{k}} (t, \hat{ɛ}) = φ (0) + \frac{\hat{ɛ}}{(n + 1)_{k}} \\ ⊖ \int_{0}^{t} \frac{(- 1) B (t - s)^{α - 1}}{Γ (α)} W^{{(n + 1)}_{k}} (s - \frac{π}{16}) ds, t \in [0, \frac{π}{16}] \\ W^{(n + 1)_{k}} (t, \hat{ɛ}) = φ (t) + \frac{\hat{ɛ}}{(n + 1)_{k}}, t \in [- \frac{π}{16}, 0] . \end{matrix} \end{matrix}$

Therefore, we obtain the maximal and minimal (ii)-solutions of (3.43) on [0, π/16] as below $\begin{matrix} X^{min} (t) & = & [1 + \frac{1}{Γ (α + 1)} [(2 + π / 16) t^{α} + \frac{t^{α + 1}}{(α + 1)}, 2 + \frac{1}{Γ (α + 1)} t^{α}]] \\ X^{max} (t) & = & [1 + \frac{B}{Γ (α + 1)} [(2 + π / 16) t^{α} + \frac{t^{α + 1}}{(α + 1)}, 2 + \frac{B}{Γ (α + 1)} t^{α}]] . \end{matrix}$

4 Conclusions

A purpose of the paper is to present some necessary comparison results for interval-valued delay fractional differential equations. This result is used to study the existence of extremal solutions of IDFDEs.

This approach is motivated since this topic has not yet been addressed, and by the fact that this result is a powerful for the study of the fuzzy delay fractional differential equations. From this perspective, we think that the presented results will be useful for the development of the theory of the fuzzy differential equations.

Acknowledgments

The authors would like to express deep gratitude to the Editor-in-Chief Professor Reza Langari for providing his support.

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