Existence of extremal solutions for interval-valued functional integro-differential equations

Abstract

This paper is devoted to studying the maximal and minimal solutions for the interval-valued functional integro-differential equations (IFIDEs) under generalized Hukuhara differentiability by the method of upper and lower solutions and monotone iterative technique. Some examples are given to illustrate the results.

Keywords

Interval-valued differential equations functional integro-differential equations method of upper and lower solutions extremal solutions

1 Introduction

In the mathematical description of many phenomena related to scientific research, functional differential equations and systems arise as an important tool to achieve a more adequate explanation and an accurate adjustment to the behavior of the particular magnitude of interest. This is reflected in fields such as biology, engineering, physics and other sciences. There exists an extensive literature dealing with functional differential equations and their applications, the reader is referred to the monographs [20 , 41], and references therein. The set-valued differential and integral equations are an important part of the theory of set-valued analysis. They have the important value in control theory and its applications; and they were studied in 1969 by De Blasi and Iervolino [16]. Recently, set-valued differential equations have been studied by many authors due to their application in many areas. For many results in the theory of set-valued differential and integral equations, the readers can be referred to the following books and papers [1 , 31] and references therein. Given any σ > 0, let $C = C ([- σ, 0], R)$ denote the space of continuous functions on [- σ, 0] . Suppose that $x \in C ([t_{0} - σ, t_{0} + p], ℝ), t_{0} \geq 0, p > 0 .$ For any t ≥ t₀, we let x_t denote a translation of the restriction of x to the interval [t - σ, t]; more specifically x_t is an element of $C$ defined by x_t (s) = x (t + s) , s ∈ [- σ, 0] . In many real word problems, it is desirable to transforms the behavior of a special phenomena into a deterministic initial value problem of functional integro-differential equations, namely,

${\begin{matrix} x^{'} (t) = f (t, x_{t}) + \int_{t_{0}}^{t} g (t, s, x_{s}) ds, t \in [t_{0}, t_{0} + p], \\ x (t) = λ (t - t_{0}), t \in [t_{0} - σ, t_{0}], \end{matrix}$ (1.1) where $f : [t_{0}, t_{0} + p] \times C \to ℝ$ , $g : [t_{0}, t_{0} + p] \times [t_{0}, t_{0} + p] \times C \to ℝ$ and $φ \in C$ . However, the model is not usually perfect due to the lack of certain information of the initial value λ, the functions f or g, which must be estimated through measurements. The analysis of measurements errors leads to the study of qualitative behavior of the solutions of (1.1). In such situations, interval-valued differential equations (IDEs) are common tools if the underlying structure is not probabilistic. The interval-valued analysis and interval-valued differential equations are the particular cases of the set-valued analysis and set differential equations, respectively. In many case, when modeling real-world phenomena, information about the behavior of a dynamic system is uncertain, and we have to consider these uncertainties to gain more models. The interval-valued differential and integro-differential equations can be used to model dynamic systems subject to uncertainties. Recently, many works have been done by several authors in the theory of interval-valued differential equations (see e.g [4 , 48]). These equations can be studied with a framework of the Hukuhara derivative [28]. However it causes that the solutions have increasing length of their values. Stefanini and Bede [48] proposed to consider so-called strongly generalized derivative of interval-valued functions. This concept of differentiability is based on four types of lateral derivative. The differentiability in the first type (i) coincides with Hukuhara differentiability and then the strongly generalized differentiability is more general than Hukuhara differentiability. Contrary to the first type (i), if we consider the problems with differentiability in the second form (ii), the problems can have solutions with decreasing length of their values. On the other hand, as the differentiability in the third type (iii) and the differentiability in the fourth type (iv) are linked with the so called switching points, we can obtain more than two solutions for the problem. Therefore, the concept of strongly generalized differentiability was the starting point for the topic of interval-valued differential equations (see [38, 42]). Besides that, some corrections of mistakes in [48] were corrected by Allahviranloo et al. [9] and some very important extensions of the interval-valued differential equations are the set differential equations (see [1 , 41]). The connection between the fuzzy analysis and the interval analysis is very well known (Moore [45]). Interval analysis and fuzzy analysis were introduced as an attempt to handle interval uncertainty that appears in many mathematical or computer models of some deterministic real-world phenomena. Based on the results in [11], there are some very important extensions and development related to the subject of the present paper in the field of fuzzy sets, i.e., fuzzy calculus and fuzzy differential equations under the generalized Hukuhara derivative. Recently, several works e.g., [8 , 44], have been done on fuzzy differential equations and fuzzy integro-differential equations, fractional fuzzy differential equations [3 , 37], and some methods for solving fuzzy differential equations [6, 7].For a positive number σ, we denote by C_σ = $C ([- σ, 0], K_{C} (ℝ))$ the space of continuous mappings from [- σ, 0] to $K_{C} (ℝ)$ . Let p > 0. Denote I = [t₀, t₀ + p] , J = [t₀ - σ, t₀] ∪ I = [t₀ - σ, t₀ + p] . For any t ∈ I denote X_t by the element of C_σ defined by X_t (s) = X (t + s) for s ∈ [- σ, 0] . Based on the extensions of (1.1) for the interval cases, in [22], authors studied the interval-valued functional integro-differential equations with strongly generalized derivative under the form

${\begin{matrix} D_{H}^{g} X (t) = F (t, X_{t}) + \int_{t_{0}}^{t} G (t, s, X_{s}) ds, t \in I \\ X (t) = φ (t - t_{0}) = φ_{0}, t \in [t_{0} - σ, t_{0}] \end{matrix}$ (1.2) where $F : I \times C_{σ} \to K_{C} (ℝ), G : I \times I \times C_{σ} \to K_{C} (ℝ), φ \in C_{σ},$ $K_{C} (ℝ)$ is the space of interval-valued functions and the symbol $D_{H}^{g}$ denotes the strongly generalized derivative. In [22], authors obtained the existence and uniqueness of solutions to both kinds of IFIDEs. The different types of solutions to IFIDEs are generated by the usage of two different concepts of interval-valued derivative. The proof is based on the application of the Banach fixed point theorem and the method of successive approximations. The monotone iterative technique with the method of upper and lower solutions is an effective tool that offers existence result in a closed set generated by the lower and upper solutions. The idea of this method show that if we can find a lower solution X^L and an upper solution X^U of IFIDE, 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 refer [32]. 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 [10 , 46]. In this study, we consider an initial value problem for an interval-valued functional integro-differential equation, and use several tools from interval calculus to approximate its extremal solutions in a given interval functional interval by the method of upper and lower solutions and monotone iterative technique. For the development of the monotone method, we establish some properties related to order and convergence in the interval space and the space of continuous interval functions defined in a real compact interval. Moreover, the approach is followed to prove the existence and uniqueness properties of solutions to both kinds of IFIDEs. The paper is organized as follows. In Section 2, we recall some basic concepts and notations about interval analysis and interval-valued differential equations. Moreover, we show some properties relative to the convergence and preservation of ordering under convergence in interval spaces, and we obtain a new compactness criteria in spaces of interval functions. In Section 3, we use the method of upper and lower solutions and monotone iterative technique to prove that there exist maximal and minimal solutions and then under more conditions, we show the uniqueness result for the solution of problem (1.2). Finally, we present some examples.

2 Fundamental theorems of spaces of interval and interval functions

Let us denote by $K_{C} (ℝ)$ the set of any nonempty compact intervals of the real line $ℝ .$ The addition and scalar multiplication in $K_{C} (ℝ)$ , are defined as usual, i.e. for $A, B \in K_{C} (ℝ), A = [\underline{A}, \bar{A}], B = [\underline{B}, \bar{B}]$ , where $\underline{A} \leq \bar{A}, \underline{B} \leq \bar{B}$ , and λ ≥ 0, then we have $\begin{matrix} A + B & = [\underline{A} + \underline{B}, \bar{A} + \bar{B}], \\ λ A & = [λ \underline{A}, λ \bar{A}], (- λ A = [- λ \bar{A}, - λ \underline{A}]) . \end{matrix}$

Furthermore, let $A \in K_{C} (ℝ), λ_{1}, λ_{2}, λ_{3}, λ_{4} \in ℝ$ and λ₃λ₄ ≥ 0, then have λ₁ (λ₂A) = (λ₁λ₂) A and (λ₃ + λ₄) A = λ₃A + λ₄A . Let $A, B \in K_{C} (ℝ)$ as above, the Hausdorff metric H in $K_{C} (ℝ)$ is defined as follows: $H [A, B] = max {| \underline{A} - \underline{B} |, | \bar{A} - \bar{B} |} .$ (2.1)

It is known that $(K_{C} (ℝ), H)$ is a complete, separable and locally compact metric space. We define the magnitude and the length of $A \in K_{C} (ℝ)$ by $\begin{matrix} H [A, 0] = ∥ A ∥ = max {| \underline{A} |, | \bar{A} |}, \\ len (A) = \bar{A} - \underline{A}, \end{matrix}$ respectively, where 0 is the zero element of $K_{C} (ℝ)$ which is regarded as a one point. The Hausdorff metric (2.1) satisfies the following properties: $\begin{matrix} H [A + C, B + C] = H [A, B], H [λ A, λ B] \\ = | λ | H [A, B], \end{matrix}$ for all $A, B, C, D \in K_{C} (ℝ)$ and $λ \in ℝ$ . Let $A, B \in K_{C} (ℝ)$ . If there exists an interval $C \in K_{C} (ℝ)$ such that A = B + C, then we call C the Hukuhara difference of A and B. We denote the interval C by A ⊖ B. Note that A ⊖ B ≠ A + (-) B . It is known that A ⊖ B exists in the case len (A) ≥ len (B) . Besides, we can see the properties of Hukuhara difference in [38]. For $X \in K_{C} (ℝ),$ we consider the following two partial orderings in $K_{C} (ℝ) .$

Definition 2.1. Let $X, Y \in K_{C} (ℝ) .$ We say that X ≤ Y (X ≥ Y) if and only if $\underline{X} \leq \underline{Y}$ and $\bar{X} \leq \bar{Y}$ ( $\underline{X} \geq \underline{Y}$ and $\bar{X} \geq \bar{Y}$ ).

We show some interesting properties on the partial ordering ≤ the space of interval-valued functions.

Lemma 2.1. Suppose $X, Y, Z, W \in K_{C} (ℝ)$ and $c \in ℝ^{+} .$

X = Y if and only if X ≤ Y and X ≥ Y .

If X ≤ Y, then X + Z ≤ Y + Z .

If X ≤ Y and Z ≤ W, then X + Z ≤ Y + W .

If X ≤ Y, then cX ≤ cY .

If X ≤ Y, then (-1) X ≥ (-1) Y .

If X ⊖ Y exists, then X ≤ Y ⇔ X ⊖ Y ≤ 0.

If len (X) = len (Y) , then X ⊖ Y ≥ 0 ⇔ Y ⊖ X ≤ 0 .

If the Hukuhara differences X ⊖ Z and Y ⊖ Z exist, then X ≤ Y ⇔ X ⊖ Z ≤ Y ⊖ Z.

If the Hukuhara differences X ⊖ Y and X ⊖ Z exist, then Y ≥ Z ⇔ X ⊖ Y ≤ X ⊖ Z.

If X ≤ Y ≤ Z, then H [X, Y] ≤ H [X, Z] and H [Y, Z] ≤ H [X, Z] .

Proof. It is easy to check that the properties (i)-(vi) satisfied. (vii) If len (X) = len (Y) , then the Hukuhara differences X ⊖ Y and Y ⊖ X exist. Hence, we get $X ⊖ Y \geq 0 \Leftrightarrow {\begin{matrix} \underline{Y} - \underline{X} \leq 0 \\ \bar{Y} - \bar{X} \leq 0 \end{matrix} \Leftrightarrow Y ⊖ X \leq 0 .$ (viii) We have X ≤ Y $\Leftrightarrow {\begin{matrix} \underline{X} - \underline{Z} \leq \underline{Y} - \underline{Z} \\ \bar{X} - \bar{Z} \leq \bar{Y} - \bar{Z} \end{matrix} \Leftrightarrow X ⊖ Z \leq Y ⊖ Z .$ (ix) Proceeding similarly as above, we get $\begin{matrix} Y \geq Z \Leftrightarrow {\begin{matrix} \underline{Y} \geq \underline{Z} \\ \bar{Y} \geq \bar{Z} \end{matrix} & \Leftrightarrow {\begin{matrix} \underline{X} - \underline{Y} \leq \underline{X} - \underline{Z} \\ \bar{X} - \bar{Y} \leq \bar{X} - \bar{Z} \end{matrix} \\ \Leftrightarrow X ⊖ Y \leq X ⊖ Z . \end{matrix}$ (x) From the assumption X ≤ Y ≤ Z, we get $\begin{matrix} H [Y, Z] \leq max {| \underline{X} - \underline{Z} |, | \bar{X} - \bar{Z} |} \leq H [X, Z], \\ H [X, Y] \leq max {| \underline{X} - \underline{Z} |, | \bar{X} - \bar{Z} |} \leq H [X, Z] . \end{matrix}$

On the other hand, we say that $(X_{k})_{k \in ℕ} \subseteq K_{C} (ℝ)$ is a nondecreasing sequence if X_k ≤ X_k+1 for all $k \in ℕ .$ Analogously, we say that $(Y_{k})_{k \in ℕ} \subseteq K_{C} (ℝ)$ is a nonincreasing sequence if Y_k+1 ≤ Y_k for all $k \in ℕ .$

Lemma 2.2. Lemma2-1 On $K_{C} (ℝ)$ the following properties hold:

If $(X_{k})_{k \in ℕ} \subseteq K_{C} (ℝ)$ is a nondecreasing sequence such that X_k → X in $K_{C} (ℝ)$ , then $X_{k} \leq X, \forall k \in ℕ .$

If $(X_{k})_{k \in ℕ} \subseteq K_{C} (ℝ)$ is a nonincreasing sequence such that X_k → X in $K_{C} (ℝ)$ , then $X \leq X_{k}, \forall k \in ℕ .$

If $(X_{k})_{k \in ℕ} \subseteq K_{C} (ℝ), Y \in K_{C} (ℝ)$ are such that $X_{k} \leq Y, \forall k \in ℕ$ , and X_k → X in $K_{C} (ℝ),$ then X ≤ Y .

If $(X_{k})_{k \in ℕ}, (Y_{k})_{k \in ℕ} \subseteq K_{C} (ℝ)$ , and $X, Y \in K_{C} (ℝ)$ are such that $X_{k} \leq Y_{k}, \forall k \in ℕ,$ and X_k → X, Y_k → Y in $K_{C} (ℝ),$ then X ≤ Y .

If $(X_{k})_{k \in ℕ} \subseteq K_{C} (ℝ)$ is a nonincreasing (respectively, nondecreasing) sequence such that there exists a subsequence (X_{k
_l}) → X in $K_{C} (ℝ)$ , then (X_k) → X .

Proof. (i) If $(X_{k})_{k \in ℕ} \subseteq K_{C} (ℝ)$ is a nondecreasing sequence and X_k → X in $K_{C} (ℝ)$ , $({\underline{X}}_{k})_{k \in ℕ} \subseteq ℝ$ and $({\bar{X}}_{k})_{k \in ℕ} \subseteq ℝ$ are nondecreasing sequences and converge to $\underline{X}$ , $\bar{X}$ respectively. Therefore, $({\underline{X}}_{k}) \leq \underline{X}$ and $({\bar{X}}_{k}) \leq \bar{X}$ , for every $k \in ℕ,$ which implies (X_k) ≤ X, for every $k \in ℕ .$ The assertion (ii) is proved similarly. (iii) If X_k ≤ Y and X_k → X in $K_{C} (ℝ),$ we obtain $({\underline{X}}_{k}) \leq \underline{Y}, ({\bar{X}}_{k}) \leq \bar{Y}$ and $\begin{matrix} lim_{k \to + \infty} H [X_{k}, X] \\ = lim_{k \to + \infty} max {| {\underline{X}}_{k} - \underline{X} |, | {\bar{X}}_{k} - \bar{X} |} = 0 . \end{matrix}$

For the following sequences in $ℝ$ , we deduce that ${\underline{X}}_{k} \to \underline{X}$ , ${\bar{X}}_{k} \to \bar{X}$ , as k → + ∞ , and hence, $\underline{X} \leq \underline{Y}, \bar{X} \leq \bar{Y}$ , that is, X ≤ Y . The assertion (iv) is proved similarly. (v) Since (X_{k
_l}) → X in $K_{C} (ℝ)$ , given ɛ > 0, there exists $l_{0} \in ℕ$ such that H [X_{k
_l}, X] < ɛ for $l \in ℕ,$ l ≥ l₀ . Moreover, from the result (ii), in the nonincreasing case, the monotonicity of (X_{k
_l}) and the fact that (X_{k
_l}) converges to X provide that X_{k
_l} ≥ X, for every $l \in ℕ .$ To prove that, for (X_k) nonincreasing, the terms of the sequence satisfy X_k ≥ X, for every $k \in ℕ,$ we take $k \in ℕ,$ then there exists $\hat{l} \in ℕ$ with $k_{\hat{l}} > k,$ and, thus, $X \leq X_{k_{\hat{l}}} \leq X_{k}$ . Then, for every $n \in ℕ, n \geq k_{l_{0}},$ we have X_{k
_{l
₀}} ≥ X_n ≥ X. Therefore, by using the property (xiii) of Lemma 2.1, we obtain $H [X_{n}, X] \leq H [X_{k_{l_{0}}}, X] < ɛ, n \in ℕ, n \geq k_{l_{0}},$ and the convergence of (X_n) towards X is derived. The proof is complete.

Consider interval functions $X, Y : I = [a, b] \to K_{C} (ℝ) .$ The partial ordering ≤ can be extended to the space of interval functions, as follows: $X \leq Y iff . \underline{X} (t) \leq \underline{Y} (t) and \bar{X} (t) \leq \bar{Y} (t), \forall t \in [a, b] .$

Definition 2.2. We say that the interval-valued mapping $X : [a, b] \to K_{C} (ℝ)$ is continuous at the point t ∈ [a, b] if for every ɛ > 0 there exists a δ = δ (t, ɛ) >0 such that H [X (t) , X (s)] ≤ ɛ, for all s ∈ [a, b] with |t - s| < δ .

The space of continuous interval-valued functions is denoted by $C (I, K_{C} (ℝ)) .$ Moreover, considering the metric H₀ on $C (I, K_{C} (ℝ))$ , defined by $\begin{matrix} H_{0} [X, Y] & = & sup_{t \in I} H [X (t), Y (t)], \\ \forall X, Y \in C (I, K_{C} (ℝ)), \end{matrix}$ it holds that $C (I, K_{C} (ℝ), H_{0})$ is a complete metric space. As a corollary of Lemma 2.2, we obtain the following results.

Corollary 2.1.The following properties hold:

If $(X_{k})_{k \in ℕ} \subseteq C (I, K_{C} (ℝ))$ is a nondecreasing sequence such that X_k → X in $C (I, K_{C} (ℝ))$ , then $X_{k} \leq X, \forall k \in ℕ .$

If $(X_{k})_{k \in ℕ} \subseteq C (I, K_{C} (ℝ))$ is a nonincreasing sequence such that X_k → X in $C (I, K_{C} (ℝ))$ , then $X \leq X_{k}, \forall k \in ℕ .$

Let $X_{k} : I \to K_{C} (ℝ), Y : I \to K_{C} (ℝ) .$ If $X_{k} \leq Y, \forall k \in ℕ$ , and X_k (t) converges to X (t) in $K_{C} (ℝ),$ for all t ∈ I, then X ≤ Y .

Let $X_{k}, Y_{k} : I \to K_{C} (ℝ), X, Y : I \to K_{C}$ $(ℝ) .$ If $X_{k} \leq Y_{k}, \forall k \in ℕ,$ and X_k (t) converges to X and Y_k (t) converges to Y (t) in $K_{C} (ℝ),$ for all t ∈ I, then X ≤ Y .

If $(X_{k})_{k \in ℕ} \subseteq C (I, K_{C} (ℝ))$ is a nondecreasing (respectively, nonincreasing) sequence such that there exists a subsequence (X_{k
_l}) → X in $C (I, K_{C} (ℝ))$ , then (X_k) → X in $C (I, K_{C} (ℝ)) .$

Theorem 2.1.theorem2.1 Let I be a compact interval in $ℝ$ , and $B \subseteq C (I, K_{C} (R))$ such that, for all $X \in B$ and t ∈ I, $X (t) = [\underline{X} (t), \bar{X} (t)]$ is a continuous interval-valued function. Consider $\begin{matrix} B_{1} = {\underline{X} : X (t) \in B} \subseteq C (I, ℝ), where \underline{X} : I \to ℝ, \\ B_{2} = {\bar{X} : X (t) \in B} \subseteq C (I, ℝ), where \bar{X} : I \to ℝ . \end{matrix}$

Then, $B$ is a relatively compact set in $C (I, K_{C} (ℝ))$ if and only if $B_{1}$ and $B_{2}$ are relatively compact sets in $(C (I, ℝ), | \cdot |) .$

Proof. Let (X_k) be a sequence in $B,$ and prove that it has a convergent subsequence. First of all, for $X \in B$ , $\underline{X}$ and $\bar{X}$ are continuous on I. Now, by hypothesis, since $B_{1}$ is relatively compact set in $(C (I, ℝ), | \cdot |),$ $({\underline{X}}_{k})$ has a subsequence $({\underline{X}}_{k_{l}})$ converging in $(C (I, ℝ), | \cdot |)$ to $\underline{Y} \in C (I, ℝ)$ . Furthermore, using that $B_{2}$ is also relatively compact set in $(C (I, ℝ), | \cdot |),$ then $({\bar{X}}_{k})$ has a subsequence $({\bar{X}}_{k_{l}})$ converging in $(C (I, ℝ), | \cdot |)$ to $\bar{Y} \in C (I, ℝ) .$ Now, we prove that functions $\underline{Y}$ and $\bar{Y}$ define a interval-valued function $X \in C (I, K_{C} (ℝ))$ with $\underline{X} = \underline{Y}$ , $\bar{X} = \bar{Y},$ and (X_{k
_l}) → X in $C (I, K_{C} (ℝ)) .$ Indeed, we define $X : I \to K_{C} (ℝ)$ by $X (t) = [\underline{Y} (t), \bar{Y} (t)], t \in I .$ We notice that $[\underline{Y} (t), \bar{Y} (t)]$ are nonempty compact intervals of $ℝ$ , since ${\underline{X}}_{k_{l}} (t) \leq {\bar{X}}_{k_{l}} (t), \forall t \in I,$ and, in consequence, passing to the limit as l→ + ∞, $\underline{Y} (t) \leq \bar{Y} (t) .$ Using that $\underline{Y}$ and $\bar{Y}$ are continuous, we show that $X \in C (I, K_{C} (ℝ)) .$ For t₁, t₂ ∈ I, since $\underline{Y}$ and $\bar{Y}$ are continuous on I, we obtain $H [X (t_{2}), X (t_{1})] = H [[\underline{Y} (t_{2}), \bar{Y} (t_{2})], [\underline{Y} (t_{1}), \bar{Y} (t_{1})]]$ $= max {| \underline{Y} (t_{2}) - \underline{Y} (t_{1}) |, | \bar{Y} (t_{2}) - \bar{Y} (t_{1}) |} \to 0$ as t₂ → t₁.

Finally, we have $\begin{matrix} H_{0} [X_{k_{l}}, X] \\ = sup_{t \in I} max {| {\underline{X}}_{k_{l}} (t) - \underline{X} (t) |, | {\bar{X}}_{k_{l}} (t) - \bar{X} (t) |} \\ = max {sup_{t \in I} | {\underline{X}}_{k_{l}} (t) - \underline{Y} (t) |, sup_{t \in I} | {\bar{X}}_{k_{l}} (t) - \bar{Y} (t) |} \\ \to 0 as l \to + \infty, \end{matrix}$ and the convergent subsequence of X_k towards X in $C (I, K_{C} (ℝ))$ is derived.

In the sequel, we show that the opposite implication is true. Consider ${\underline{X}}_{k_{l}}$ and ${\bar{X}}_{k_{l}}$ two sequences in $B_{1}, B_{2}$ respectively, where $X_{k} \in B$ . Now, by hypothesis, since (X_k) is a sequence in $B$ and $B$ is a relatively compact set in $C (I, K_{C} (ℝ))$ , (X_k) has a subsequence (X_{k
_l}) converging in $C (I, K_{C} (ℝ))$ to (X_k) . So, we get $\begin{matrix} H_{0} [X_{k_{l}}, X] \\ = sup_{t \in I} max {| {\underline{X}}_{k_{l}} (t) - \underline{X} (t) |, | {\bar{X}}_{k_{l}} (t) - \bar{X} (t) |} \\ \to 0 as k \to + \infty . \end{matrix}$

which prove that ${\underline{X}}_{k}$ and ${\bar{X}}_{k}$ have two subsequences converging towards $\underline{X}$ and $\bar{X}$ in $C (I, ℝ)$ respectively. This implies that $B_{1}$ and $B_{2}$ are relatively compact sets in $(C (I, ℝ), | \cdot |) .$ The proof is complete.

Definition 2.3. def2.1 [48] Let $X : (a, b) \to K_{C} (ℝ)$ and t ∈ (a, b). We say that X is strongly generalized differentiable at t if there exists a $D_{H}^{g} X (t) \in K_{C} (ℝ)$ such that

for all h > 0 sufficiently small, ∃X (t + h) ⊖ X (t) , ∃ X (t) ⊖ X (t - h) and $\begin{matrix} lim_{h ↘ 0} H [\frac{X (t + h) ⊖ X (t)}{h}, D_{H}^{g} X (t)] = 0, \\ lim_{h ↘ 0} H [\frac{X (t) ⊖ X (t - h)}{h}, D_{H}^{g} X (t)] = 0 \end{matrix}$ or

for all h > 0 sufficiently small, ∃X (t) ⊖ X (t + h) , ∃X (t - h) ⊖ X (t) and $\begin{matrix} lim_{h ↘ 0} H [\frac{X (t) ⊖ X (t + h)}{- h}, D_{H}^{g} X (t)] = 0, \\ lim_{h ↘ 0} H [\frac{X (t - h) ⊖ X (t)}{- h}, D_{H}^{g} X (t)] = 0 \end{matrix}$ or

for all h > 0 sufficiently small, ∃X (t + h) ⊖ X (t) , ∃X (t - h) ⊖ X (t) and $\begin{matrix} lim_{h ↘ 0} H [\frac{X (t + h) ⊖ X (t)}{h}, D_{H}^{g} X (t)] = 0, \\ lim_{h ↘ 0} H [\frac{X (t - h) ⊖ X (t)}{- h}, D_{H}^{g} X (t)] = 0 \end{matrix}$ or

for all h > 0 sufficiently small, ∃X (t) ⊖ X (t + h) , ∃X (t) ⊖ X (t - h) and the limits $\begin{matrix} lim_{h ↘ 0} H [\frac{X (t) ⊖ X (t + h)}{- h}, D_{H}^{g} X (t)] = 0, \\ lim_{h ↘ 0} H [\frac{X (t) ⊖ X (t - h)}{h}, D_{H}^{g} X (t)] = 0 . \end{matrix}$

We say that X is (i)-differentiable or (ii)-differentiable on [a, b], if it is differentiable in the sense (i) or (ii) of Definition 2.3, respectively.

Lemma 2.3.lemma4.1 (see [22]) Assume that $F : I \times C_{σ} \to K_{C} (ℝ), G : I \times I \times C_{σ} \to K_{C} (ℝ)$ is continuous. An interval-valued mapping $X : J \to K_{C} (ℝ)$ is a solution to the problem (1.2) on J if and only if X is a continuous interval-valued mapping and it satisfies to one of the following interval-valued integral equations: $(S 1) {\begin{matrix} X (t) = φ (t - t_{0}) \in C_{σ}, for t \in [t_{0} - σ, t_{0}] \\ X (t) = φ (0) + \int_{t_{0}}^{t} (F (s, X_{s}) + \int_{t_{0}}^{t} G (t, s, X_{s}) ds) ds \end{matrix}$ (2.2)

If X is (i)-differentiable. $(S 2) {\begin{matrix} X (t) = φ (t - t_{0}) \in C_{σ}, for t \in [t_{0} - σ, t_{0}] \\ X (t) = φ (0) ⊖ (- 1) \int_{t_{0}}^{t} (F (s, X_{s}) + \int_{t_{0}}^{t} G (t, s, X_{s}) ds) ds, \end{matrix}$ (2.3)

If X is (ii)-differentiable. We remark that in (2.3), it is hidden the following statement: there exists Hukuhara difference $φ (0) ⊖ (- 1) \int_{t_{0}}^{t} (F (s, X_{s}) + \int_{t_{0}}^{t} G (t, s, X_{s}) ds) ds .$

Lemma 2.4. lemma-c Let $X, Y \in C (I, K_{C} (ℝ))$ and $A \in K_{C} (ℝ)$ and also X ≤ Y, then

$\int_{a}^{t} X (s) ds \leq \int_{a}^{t} Y (s) ds, \forall t \in I .$

$A ⊖ (- 1) \int_{a}^{t} X (s) ds \leq A ⊖ (- 1) \int_{a}^{t} X (s) ds,$ ∀t ∈ I, provided $A ⊖ (- 1) \int_{a}^{t} X (s) ds$ and $A ⊖ (- 1) \int_{a}^{t} Y (s) ds$ are well-defined.

3 Monotone method for interval functional integro-differential equations

Let us consider again the interval-valued functional integro-differential equations with generalized Hukuhara derivative under the form ${\begin{matrix} D_{H}^{g} X (t) = F (t, X_{t}) + \int_{t_{0}}^{t} G (t, s, X_{s}) ds, t \in I \\ X (t) = φ (t - t_{0}) = φ_{0}, t \in [t_{0} - σ, t_{0}] \end{matrix}$ (3.1) where $F : I \times C_{σ} \to K_{C} (ℝ), G : D \times C_{σ} \to K_{C} (ℝ), φ \in C_{σ}, D = {(t, s) | t_{0} \leq s \leq t \leq t_{0} + p} .$

Definition 3.1. Let $X : J \to K_{C} (ℝ)$ be an interval-valued function which is (i)-differentiable (or (ii)-differentiable). If X and its derivative satisfy problem (3.1), we say that X is a (i)-solution (or (ii)-solution) of problem (3.1).

Definition 3.2. A function $X^{L} \in C^{1} ([t_{0} - σ, t_{0} + p], K_{C} (ℝ))$ is a lower (i)-solution for (3.1) if ${\begin{matrix} D_{H}^{g} X^{L} (t) \leq F (t, X_{t}^{L}) + \int_{t_{0}}^{t} G (t, s, X_{s}^{L}) ds, t \in I \\ X^{L} (t) = ξ (t - t_{0}) \leq φ (t - t_{0}), t \in [t_{0} - σ, t_{0}] \end{matrix}$ (3.2) where X^L is (i)-differentiable and ξ (t - t₀) ∈ C_σ. A function $X^{U} \in C^{1} ([t_{0} - σ, t_{0} + p], K_{C} (ℝ))$ is an upper (i)-solution for (3.1) if it satisfies the reverse inequalities of (3.2).

Analogously, definitions can be given for lower (ii)-solution and upper (ii)-solution for (3.1).

Definition 3.3. A function $Y^{U} \in C^{1} ([t_{0} - σ, t_{0} + p], K_{C} (ℝ))$ is a upper (ii)-solution for (3.1) if ${\begin{matrix} D_{H}^{g} Y^{U} (t) \geq F (t, Y_{t}^{U}) + \int_{t_{0}}^{t} G (t, s, Y_{s}^{L}) ds, t \in I \\ Y^{L} (t) = ψ (t - t_{0}) \geq φ (t - t_{0}), t \in [t_{0} - σ, t_{0}] \end{matrix}$ (3.3) where Y^U is (ii)-differentiable and ψ (t - t₀) ∈ C_σ. A function $Y^{L} \in C^{1} ([t_{0} - σ, t_{0} + p], K_{C} (ℝ))$ is an lower (ii)-solution for (3.1) if it satisfies the reverse inequalities of (3.3).

Theorem 3.1.theorem111 Let $F \in C (I \times C_{σ}, K_{C} (ℝ)), G \in C (D \times C_{σ}, K_{C} (ℝ)) .$ Suppose that F, G map bounded sets in I × C_σ to bounded sets in $K_{C} (ℝ)$ . Moreover, assume one of the following conditions is verified:

there exists a lower (i)-solution X^L ∈ C¹ $([t_{0} - σ, t_{0} + p], K_{C} (ℝ))$ and an upper (i)-solution $X^{U} \in C^{1} ([t_{0} - σ, t_{0} + p], K_{C} (ℝ))$ of the Equation (3.1) satisfying X^L (t)≤ X^U (t) , t ∈ [t₀ - σ, t₀ + p] ;

there exists a lower (ii)-solution Y^L ∈ C¹ $([t_{0} - σ, t_{0} + p], K_{C} (ℝ))$ and an upper (ii)-solution $Y^{U} \in C^{1} ([t_{0} - σ, t_{0} + p], K_{C} (ℝ))$ of the Equation (3.1) satisfying Y^L (t)≤ Y^U (t) , t ∈ [t₀ - σ, t₀ + p] ;

Then there exists at least a solution X ∈ [X^L, X^U] in the case (H1) for Equation (3.1) and Y ∈ [Y^L, Y^U] in the case (H2) for Equation (3.1) in some intervals [t₀, t₀ + α] , with α ≤ p.

Proof.

Step 1: (there exists a (ii)-solution for (3.1)) We first suppose the hypothesis (H2) is fulfilled. We know that problem (3.1) is equivalent to the integral Equation (2.3). Now define, $\begin{matrix} (T Y) (t) \\ = {\begin{matrix} φ (t - t_{0}) if t \in [t_{0} - σ, t_{0}] \\ φ (0) ⊖ (- 1) \int_{t_{0}}^{t} (F (s, Y_{s}) + \int_{t_{0}}^{s} G (s, τ, Y_{τ}) d τ) ds \end{matrix} \end{matrix}$

It is easily seen that $T$ is a continuous map from $C ([t_{0} - σ, t_{0} + p], K_{C} (ℝ))$ to $C ([t_{0} - σ, t_{0} + p], K_{C} (ℝ))$ . Utilizing $F \in C (I \times C_{σ}, K_{C} (ℝ)), G \in C (D \times C_{σ}, K_{C} (ℝ))$ , F and G map bounded sets in I × C_σ to bounded sets in $K_{C} (ℝ)$ and the properties of distance H, to prove equi-continuity of $T$ , we conclude $\begin{matrix} H [(T Y) (t_{1}), (T Y) (t_{2})] \\ \leq M_{1} (t_{2} - t_{1}) + M_{2} \frac{(t_{2} - t_{1})^{2}}{2!}, \end{matrix}$ where since F, G map bounded sets to bounded sets in $K_{C} (ℝ)$ , there are real positive numbers M₁, M₂ such that $\begin{matrix} H [F (s, Y_{s}), 0] \leq M_{1}, H [G (t, s, Y_{s}), 0] \leq M_{2} . \end{matrix}$

To prove the uniformly boundedness of (TY) (t) for all t ∈ [t₀ - σ, t₀ + p] , we have $\begin{matrix} H [(T Y) (t), φ (0)] \leq {pM}_{1} + \frac{p^{2} M_{2}}{2!} . \end{matrix}$

Next, let $S$ be the subset of C ([t₀ - σ, t₀ + p]) consisting the functions which are of the form $(𝕋 Y) (t), Y \in S .$ As $𝕋 Y$ is uniformly bounded and equicontinuity, $S$ is uniformly bounded and equicontinuity. Now, we show that $S$ is closed so that we can infer that $S$ is compact. Indeed, let $(T^{n} Y) \subset S$ such that $T^{n} Y \to Y$ as n → ∞ . As $(T^{n} Y)$ is a sequence of uniformly bounded and equicontinuity functions, we infer that $S$ is closed. An application of Schauder’s fixed point theorem now yields the existence of at least one function $Y \in S$ such that $(𝕋 Y) (t) = φ (t - t_{0}), t \in [t_{0} - σ, t_{0}]$ and $(𝕋 Y) (t) = Y (t), t \in [t_{0}, t_{0} + p] .$

Step 2: (To prove Y ∈ [Y^L, Y^U]) For any ɛ > 0 consider $Y_{ɛ}^{U} (t) = Y^{U} (t) + ɛ (1 + t)$ and $Y_{ɛ}^{L} (t) + ɛ (1 + t) = Y^{L} (t)$ then $Y_{ɛ, t}^{U} = Y_{t}^{U} + ɛ (1 +$ t + s) and $Y_{ɛ, t}^{U} > Y_{t}^{U}, t \in [t_{0}, t_{0} + p] .$ Similarly $Y_{ɛ, t}^{L} + ɛ (1 + t + s) = Y_{t}^{L}$ and $Y_{ɛ, t}^{L} < Y_{t}^{L}, t \in [t_{0}, t_{0}$ +p] . Thus we obtain $Y_{ɛ}^{L} (t) \leq Y^{L} (t) \leq Y^{U} (t) \leq Y_{ɛ}^{U} (t), t \in [t_{0} - σ, t_{0}]$ , $Y_{ɛ}^{L} (t) \leq Y^{L} (t) \leq Y^{U} (t) \leq Y_{ɛ}^{U} (t), t \in [t_{0}, t_{0} + p]$ and $Y_{ɛ}^{L} (t_{0}) \leq Y^{L} (t_{0}) \leq Y^{U} (t_{0}) \leq Y_{ɛ}^{U} (t_{0}) .$ As Y^L, Y^U are lower and upper (ii)-solutions of the Equation (3.1), we have that $Y_{ɛ}^{L} (t) \leq Y^{L} (t) \leq Y (t) \leq Y^{U} (t) \leq Y_{ɛ}^{U} (t), t \in [t_{0} - σ, t_{0}]$ and $Y_{ɛ}^{L} (t_{0}) \leq Y (t_{0}) \leq Y_{ɛ}^{U} (t_{0}),$ where Y (t) is a (ii)-solution of the Equation (3.1). Now we have to show that $Y_{ɛ}^{L} (t) < Y (t) < Y_{ɛ}^{U} (t), t \in [t_{0}, t_{0} + p] .$ If the above assertion is not true, then there exists a t₁ ∈ (t₀, t₀ + p) such that $Y (t_{1}) = Y_{ɛ}^{U} (t_{1})$ and $Y_{ɛ}^{L} (t) < Y (t) < Y_{ɛ}^{U} (t), t \in [t_{0}, t_{0} + p] ∖ {t_{1}} .$ This implies that $Y (t_{1}) = Y_{ɛ}^{U} (t_{1}) > Y^{U} (t_{1})$ and $D_{H}^{g} Y (t_{1}) > D_{H}^{g} Y^{U} (t_{1}) .$ By Definition 2.3(ii), together with $D_{H}^{g} Y (t_{1}) > D_{H}^{g} Y^{U} (t_{1})$ , implies that there exists an h > 0 such that $\begin{matrix} \frac{Y (t_{1}) ⊖ Y^{U} (t_{1})}{- h} < \frac{Y^{U} (t_{1} - h) ⊖ Y^{U} (t_{1} - h)}{- h}, \end{matrix}$ t₁ ∈ (t₀, t₀ + p) . This contradicts that Y (t₁) > Y^U (t₁) , t ∈ [t₀ - σ, t₀ + p] . Therefore, we have that $Y (t) < Y_{ɛ}^{U} (t) .$ Similarly, we can show that $Y_{ɛ}^{L} < Y (t), t \in [t_{0} - σ, t_{0} + p]$ and hence the relation $Y_{ɛ}^{L} (t) < Y (t) < Y_{ɛ}^{U} (t)$ holds for all t ∈ [t₀ - σ, t₀ + p]. Now as ɛ → 0, we conclude that Y^L (t) ≤ Y (t) ≤ Y^U (t) . The proof is complete.

Lemma 3.1. lemma-c-1 Let $F \in C (I \times C_{σ}, K_{C} (ℝ)), G \in C (D \times C_{σ}, K_{C} (ℝ))$ and F (t, Z), G (t, s, Z) be nondecreasing in Z for each $t \in I, (t, s) \in D .$ Assume that $Z, W \in C^{1} ([t_{0} - σ, t_{0} + p], K_{C} (ℝ))$ (in the same case of differentiability) and Z (t) = ξ (t - t₀) < W (t) = ψ (t - t₀) , t ∈ [t₀ - σ, t₀] . Assume further that $D_{H}^{g} Z (t) \leq F (t, Z_{t}) + \int_{t_{0}}^{t} G (t, s, Z_{s}) ds, t \in I,$ (3.4) $D_{H}^{g} W (t) > F (t, W_{t}) + \int_{t_{0}}^{t} G (t, s, W_{s}) ds, t \in I .$ (3.5)

Then,

$Z (t) < W (t), t \in I .$ (3.6)

Proof. If the assertion (3.6) is false, then the set $𝔻 = {t \in I : Z (t) \geq W (t)}$ is nonempty. Because of ξ (t - t₀) < ψ (t - t₀) and the continuity of the functions involved, there exists a t₁ such that t₀ < t₁ ≤ t₀ + p and

$Z (t_{1}) = W (t_{1}), Z (t) < W (t), t \in [t_{0}, t_{1}) .$ (3.7)

Thus, we obtain for small h < 0 $\begin{matrix} \frac{Z (t_{1} + h) ⊖ Z (t_{1})}{h} > \frac{W (t_{1} + h) ⊖ W (t_{1})}{h}, \end{matrix}$ which, in turn, implies that

$D_{H}^{g} Z (t_{1}) \geq D_{H}^{g} W (t_{1}) .$ (3.8)

From Z (t - t₀) < W (t - t₀) and (3.7), we deduce that Z (t₁ - t₀) ≤ W (t₁ - t₀) , which, in view of the nondecreasing property of the functions F (t, Z) , G (t, s, Z) in Z, yields F (t₁, Z_{t
₁}) ≤ F (t₁, W_{t
₁}) and G (t₁, s, Z_{t
₁}) ≤ G (t₁, s, W_{t
₁}) . Utilizing Lemma 2.4, we obtain

$\begin{matrix} F (t, Z_{t_{1}}) + \int_{t_{0}}^{t_{1}} G (t_{1}, s, Z_{s}) ds \leq F (t, W_{t_{1}}) \\ + \int_{t_{0}}^{t_{1}} G (t, s, W_{s}) ds . \end{matrix}$ (3.9)

On the other hand, the relations (3.4), (3.5) and (3.8) lead to the inequality $\begin{matrix} F (t, Z_{t_{1}}) + \int_{t_{0}}^{t_{1}} G (t_{1}, s, Z_{s}) ds > F (t, W_{t_{1}}) \\ + \int_{t_{0}}^{t_{1}} G (t, s, W_{s}) ds, \end{matrix}$ which is incompatible with (3.9) because of (3.7). Consequently, the set $𝔻$ is empty, and (3.6) is true. The proof is complete.

Remark 3.1. The conclusion (3.6) remains valid even when the inequalities (3.4) and (3.5) are replaced by $\begin{matrix} D_{H}^{g} Z (t) & < F (t, Z_{t}) + \int_{t_{0}}^{t} G (t, s, Z_{s}) ds, \\ D_{H}^{g} W (t) & \geq F (t, W_{t}) + \int_{t_{0}}^{t} G (t, s, W_{s}) ds . \end{matrix}$

Theorem 3.2.Assume that (H3) $Z, W \in C^{1} ([t_{0} - σ, t_{0} + p], K_{C} (ℝ))$ (in the same case of differentiability), $F (t, Z) \in C (I \times C_{σ}, K_{C} (ℝ)), G (t, s, Z) \in C (D \times C_{σ}, K_{C} (ℝ))$ are nondecreasing in Z for each t and $\begin{matrix} D_{H}^{g} Z (t) & \leq F (t, Z_{t}) + \int_{t_{0}}^{t} G (t, s, Z_{s}) ds, t \in I, \\ D_{H}^{g} W (t) & \geq F (t, W_{t}) + \int_{t_{0}}^{t} G (t, s, W_{s}) ds, t \in I; \end{matrix}$ (H4) there exist two positive real-valued numbers L₁, L₂ so that $\begin{matrix} F (t, Z_{1}) \leq F (t, Z_{2}) + \frac{L_{1}}{1 + t} sup_{θ \in [- σ, 0]} (Z_{1} (θ) ⊖ Z_{2} (θ)), \\ G (t, s, Z_{1}) \leq G (t, s, Z_{2}) + \frac{L_{2}}{1 + t} sup_{θ \in [- σ, 0]} (Z_{1} (θ) ⊖ Z_{2} (θ)), \end{matrix}$ with $t \in I, (t, s) \in D$ , Z₁, Z₂ ∈ C_σ and Z₁ ≥ Z₂ . Then Z (t) = ξ (t - t₀) ≤ W (t) = ψ (t - t₀) , t ∈ [t₀ - σ, t₀] implies Z (t) ≤ W (t) , t ∈ I, provided max {L₁, L₂} (1 + t) <1

Proof. Chose a positive number M such that M = max {L₁, L₂} and a positive real number ɛ . Consider W_ɛ (t) = W (t) + ɛ (1 + t) then W_ɛ,t = W_t + ɛ (1 + t + s) and W_ɛ,t > W_t, t ∈ [t₀, t₀ + p] . Thus we have $\begin{matrix} D_{H}^{g} W_{ɛ} (t) \geq F (t, W_{t}) + \int_{t_{0}}^{t} G (t, s, W_{s}) ds + ɛ \\ \geq F (t, W_{ɛ, t}) ⊖ \frac{L_{1}}{1 + t} sup_{θ \in [- σ, 0]} (W_{ɛ, t} (θ) ⊖ W_{t} (θ)) + ɛ \\ + \int_{t_{0}}^{t} [G (t, s, W_{ɛ, s}) ⊖ \frac{L_{2}}{1 + s} sup_{θ \in [- σ, 0]} (W_{ɛ, s} (θ) ⊖ W_{s} (θ))] ds \\ = F (t, W_{ɛ, t}) + \int_{t_{0}}^{t} G (t, s, W_{ɛ, s}) ds - L_{1} ɛ - L_{2} ɛ (t - t_{0}) + ɛ \\ > F (t, W_{ɛ, t}) + \int_{t_{0}}^{t} G (t, s, W_{ɛ, s}) ds + ɛ (1 - M (1 + t)) \\ \geq F (t, W_{ɛ, t}) + \int_{t_{0}}^{t} G (t, s, W_{ɛ, s}) ds . \end{matrix}$

Here we have utilized the assumptions (H3)-(H4) and the condition M (1 + t) <1. We can now apply Lemma 3.1 to Z and W_ɛ to yield Z (t) < W_ɛ (t) , t ∈ [t₀, t₀ + p] . As ɛ > 0 is arbitrary, we conclude that Z (t) ≤ W (t) . The proof is complete. □

Corollary 3.1. coro-tc Let Z, W ∈ C¹ ([t₀ - σ, t₀ + p] , $K_{C} (ℝ))$ (in the same case of differentiability), $H (t) \in C (I, K_{C} (ℝ))$ . Suppose that $D_{H}^{g} Z (t) \leq H (t)$ and $D_{H}^{g} W (t) \geq H (t)$ for t ∈ I . Then Z (t) = ξ (t - t₀) ≤ W (t) = ψ (t - t₀) , t ∈ [t₀ - σ, t₀] implies Z (t) ≤ W (t) , t ∈ I .

Definition 3.4. Let X^L, X^U be lower and upper (i)-solutions for the Equation (3.1) such that $[X^{L}, X^{U}] = {X \in C ([t_{0} - σ, t_{0} + p], K_{C} (ℝ)) : X^{L} \leq X \leq X^{U}}$ . We say that the solution X^min ∈ [X_L, X_U] is a minimal (i)-solution of (3.1), if X^min is (i)-differentiable and $X^{min} (t) \leq X (t), t \in [t_{0} - σ, t_{0} + p],$ 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 3.5.Let Y^L, Y^U be lower and upper (ii)-solutions for the Equation (3.1) such that $[Y^{L}, Y^{U}] = {Y \in C ([t_{0} - σ, t_{0} + p], K_{C} (ℝ)) : Y^{L} \leq Y \leq Y^{U}}$ . We say that the solution Y^min ∈ [Y_L, Y_U] is a minimal (ii)-solution of (3.1), if Y^min is (ii)-differentiable $Y^{min} (t) \leq Y (t), t \in [t_{0} - σ, t_{0} + p],$ 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.

Theorem 3.3. theorem-main Let all the conditions mentioned in Theorem 3.1 be satisfied. Additionally, assume that if $Y, Z \in C ([t_{0} - σ, t_{0} + p], K_{C} (ℝ))$ such that Y ≤ Z on [t₀ - σ, t₀ + p], then F (t, Y_t) ≤ F (t, Z_t) and G (t, s, Y_t) ≤ G (t, s, Z_t) on [t₀, t₀ + p], where Y_t = Y (t + τ) , Z_t = Z (t + τ), τ ∈ [- σ, 0]. Then

- in the case (H1), there exist monotone sequences $(X^{L, n} (t)), (X^{U, n} (t)) \in C ([t_{0} - σ, t_{0} + p], K_{C} (ℝ))$ such that (X^L,n (t)) → X^min (t) and (X^U,n (t)) → X^max (t) and X^min, X^max ∈ [X^L, X^U] are the coupled minimal and maximal (i)-solutions of (3.1) respectively, that is, they satisfy ${\begin{matrix} X^{min} (t) = φ (t - t_{0}), for t \in [t_{0} - σ, t_{0}] \\ X^{min} (t) = φ (0) \\ + \int_{t_{0}}^{t} (F (s, X_{s}^{min}) + \int_{t_{0}}^{t} G (t, s, X_{s}^{min}) ds) ds, t \in I, \end{matrix}$ (3.10) ${\begin{matrix} X^{max} (t) = φ (t - t_{0}), for t \in [t_{0} - σ, t_{0}] \\ X^{max} (t) = φ (0) \\ + \int_{t_{0}}^{t} (F (s, X_{s}^{max}) + \int_{t_{0}}^{t} G (t, s, X_{s}^{max}) ds) ds, t \in I . \end{matrix}$ (3.11) - in the case (H2), there exist monotone sequences $(Y^{L, n} (t)), (Y^{U, n} (t)) \in C ([t_{0} - σ, t_{0} + p], K_{C} (ℝ))$ such that (Y^L,n (t)) → Y^min (t) and (Y^U,n (t)) → Y^max (t) and Y^min, Y^max ∈ [Y^L, Y^U] are the coupled minimal and maximal (ii)-solutions of (3.1) respectively, that is, they satisfy ${\begin{matrix} Y^{min} (t) = φ (t - t_{0}), for t \in [t_{0} - σ, t_{0}] \\ Y^{min} (t) = φ (0) \\ ⊖ (- 1) \int_{t_{0}}^{t} (F (s, Y_{s}^{min}) + \int_{t_{0}}^{t} G (t, s, Y_{s}^{min}) ds) ds, t \in I, \end{matrix}$ (3.12) ${\begin{matrix} Y^{max} (t) = φ (t - t_{0}), for t \in [t_{0} - σ, t_{0}] \\ Y^{max} (t) = φ (0) \\ ⊖ (- 1) \int_{t_{0}}^{t} (F (s, Y_{s}^{max}) + \int_{t_{0}}^{t} G (t, s, Y_{s}^{max}) ds) ds, t \in I . \end{matrix}$ (3.13)

Moreover, there exist two positive real-valued numbers L₁, L₂ so that $\begin{matrix} F (t, Z_{1}) \leq F (t, Z_{2}) + \frac{L_{1}}{1 + t} sup_{θ \in [- σ, 0]} (Z_{1} (θ) ⊖ Z_{2} (θ)), \\ G (t, s, Z_{1}) \leq G (t, s, Z_{2}) + \frac{L_{2}}{1 + t} sup_{θ \in [- σ, 0]} (Z_{1} (θ) ⊖ Z_{2} (θ)), \end{matrix}$ with $t \in I, (t, s) \in D$ , Z₁, Z₂ ∈ C_σ and Z₁ ≥ Z₂ . Then problem (3.1) has a unique (i)-solution in the case (H1) and (ii)-solution in the case (H2).

Proof. In this proof, we only prove that there exist maximal and minimal solutions in the case (H2) for Equation (3.1). The proof of the case (H1) is similar. We first suppose the hypothesis (H2) is satisfied. For each n ≥ 0 consider IFIDEs with (ii)-differentiability under the forms ${\begin{matrix} D_{H}^{g} V^{n + 1} (t) = F (t, V_{t}^{n}) + \int_{t_{0}}^{t} G (t, s, V_{s}^{n}) ds, t \in I, \\ V^{n + 1} (t) = φ (t - t_{0}), t \in [t_{0} - σ, t_{0}], \end{matrix}$ (3.14) ${\begin{matrix} D_{H}^{g} W^{n + 1} (t) = F (t, W_{t}^{n}) + \int_{t_{0}}^{t} G (t, s, W_{s}^{n}) ds, t \in I, \\ W^{n + 1} (t) = φ (t - t_{0}), t \in [t_{0} - σ, t_{0}] . \end{matrix}$ (3.15)

We can begin the iterations by setting W⁰ = Y^U and V⁰ = Y^L. The solutions of (3.1) and (3.2) are denoted by Wⁿ⁺¹, Vⁿ⁺¹ respectively. We claim that $Y^{L} = V^{0} \leq \dots V^{n} \leq \dots \leq W^{n} \leq \dots W^{0} = Y^{U} .$ (3.16)

To confirm (3.16), first note from (3.14) and (3.15) for n = 0 that ${\begin{matrix} D_{H}^{g} V^{1} (t) = F (t, V_{t}^{0}) + \int_{t_{0}}^{t} G (t, s, V_{s}^{0}) ds, t \in I, \\ V^{1} (t) = φ (t - t_{0}), t \in [t_{0} - σ, t_{0}], \end{matrix}$ (3.17) ${\begin{matrix} D_{H}^{g} W^{1} (t) = F (t, W_{t}^{0}) + \int_{t_{0}}^{t} G (t, s, W_{s}^{0}) ds, t \in I, \\ W^{1} (t) = φ (t - t_{0}), t \in [t_{0} - σ, t_{0}] . \end{matrix}$ (3.18)

We now claim that V⁰ ≤ V¹ ≤ W¹ ≤ W⁰ on [t₀ - σ, t₀ + p]. Indeed, we can write

${\begin{matrix} D_{H}^{g} V^{1} (t) \geq F (t, V_{t}^{0}) + \int_{t_{0}}^{t} G (t, s, V_{s}^{0}) ds, t \in I, \\ V^{1} (t) = φ (t - t_{0}), t \in [t_{0} - σ, t_{0}], \end{matrix}$ (3.19) and since V⁰ = Y^L, we obtain $\begin{matrix} {\begin{matrix} D_{H}^{g} V^{0} (t) \leq F (t, V_{t}^{0}) + \int_{t_{0}}^{t} G (t, s, V_{s}^{0}) ds, t \in I, \\ V^{0} (t) = ξ (t - t_{0}), t \in [t_{0} - σ, t_{0}], \end{matrix} \end{matrix}$ with V⁰ (t) = ξ (t - t₀) ≤ V¹ (t) = φ (t - t₀) , t ∈ [t₀-σ, t₀] . Utilizing Corollary we get V⁰ ≤ V¹ . We now show that V¹ ≤ W¹ by using the nondecreasing property of the functions F and G, $\begin{matrix} D_{H}^{g} V^{1} (t) \leq F (t, W_{t}^{0}) + \int_{t_{0}}^{t} G (t, s, W_{s}^{0}) ds, \\ D_{H}^{g} W^{1} (t) \geq F (t, W_{t}^{0}) + \int_{t_{0}}^{t} G (t, s, W_{s}^{0}) ds, \end{matrix}$ with V¹ (t) = W¹ (t) = φ (t - t₀) , t ∈ [t₀ - σ, t₀] . Applying Corollary 3.1, we conclude that V¹≤W⁰ . Similarly, it can be proved that W¹ ≤ W⁰ . Hence we have showed that V⁰ ≤ V¹ ≤ W¹ ≤ W⁰, t ∈ [t₀ - σ, t₀ + p] . We assume inductively

$V^{0} \leq V^{n - 1} \leq V^{n} \leq W^{n} \leq W^{n - 1} \leq W^{0} .$ (3.20)

From (3.20) and using the assumptions of F, G we conclude that $\begin{matrix} D_{H}^{g} V^{n} (t) \leq F (t, V_{t}^{n}) + \int_{t_{0}}^{t} G (t, s, V_{s}^{n}) ds, \end{matrix}$ Vⁿ (t) = φ (t - t₀) = Vⁿ⁺¹ (t) , t ∈ [t₀ - σ, t₀] . Using Corollary 3.1 and from (3.1), we obtain Vⁿ (t) ≤ Vⁿ⁺¹ (t) , t ∈ [t₀ - σ, t₀ + p] . Similarly, it can be proved that Wⁿ⁺¹ (t) ≤ Wⁿ (t) , t ∈ [t₀ - σ, t₀ + p] . Next we demonstrate $V^{n} \leq W^{n}, \forall n \in ℕ .$ Indeed, we have $\begin{matrix} D_{H}^{g} W^{n} \geq F (t, V_{t}^{n - 1}) + \int_{t_{0}}^{t} G (t, s, V_{s}^{n - 1}) ds, \\ D_{H}^{g} V^{n} \leq F (t, V_{t}^{n - 1}) + \int_{t_{0}}^{t} G (t, s, V_{s}^{n - 1}) ds, \end{matrix}$ Wⁿ (t) = φ (t - t₀) = Vⁿ (t) , t ∈ [t₀ - σ, t₀] , ∀ n > 1 . Hence we apply Corollary 3.1 and conclude that $V^{n} \leq W^{n}, \forall n \in ℕ .$ This completes the proof of the assertion (3.1).Since the sequences of functions (Vⁿ) and (Wⁿ) satisfy the relation (3.1), they are uniformly bounded. Together with the assumptions in Theorem 3.1, we prove the uniformly boundedness and the equi-continuity of the sequences (Vⁿ) and (Wⁿ) below (we only prove for the sequence Wⁿ and the other sequence is similar). For all t ∈ [t₀, t₀ + p] , we have $\begin{matrix} H [W^{n + 1} (t), φ (0)] \leq {pM}_{1} + \frac{p^{2} M_{2}}{2!} \end{matrix}$ and for t₁ < t₂, t₁, t₂ ∈ [t₀, t₀ + p] , we obtain $\begin{matrix} H [W^{n + 1} (t_{1}), W^{n + 1} (t_{2})] \leq M_{1} (t_{2} - t_{1}) \\ + M_{2} \frac{(t_{2} - t_{1})^{2}}{2!} . \end{matrix}$ Therefore, (Vⁿ) and (Wⁿ) are relatively compact in $C ([t_{0} - σ, t_{0} + p], K_{C} (ℝ)) .$ Then applying Arzela-Ascoli Theorem, Corollary 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 [t₀ - σ, t₀ + p], respectively. By using the convergence properties in the relations (3.14) and (3.15) we deduce that ${\begin{matrix} D_{H}^{g} Y^{min} (t) = F (t, Y_{t}^{min}) + \int_{t_{0}}^{t} G (t, s, Y_{s}^{min}) ds, t \in I, \\ Y^{min} (t) = φ (t - t_{0}), t \in [t_{0} - σ, t_{0}], \end{matrix}$ (3.21) ${\begin{matrix} D_{H}^{g} Y^{max} (t) = F (t, Y_{t}^{max}) + \int_{t_{0}}^{t} G (t, s, Y_{s}^{max}) ds, t \in I, \\ Y^{max} (t) = φ (t - t_{0}), t \in [t_{0} - σ, t_{0}], \end{matrix}$ (3.22) and that Y^L = V⁰ ≤ Y^min ≤ Y^max ≤ W⁰ = Y^U on [t₀ - σ, t₀ + p] . Next we show that Y^min and Y^max are coupled minimal and maximal (ii)-solutions of the problem (3.1). We assume that Y (t) is any (ii)-solution of (3.1) such that Y^L = V⁰ ≤ Y ≤ W⁰ on [t₀ - σ, t₀ + p]. We have to prove that V⁰ ≤ Y^min ≤ Y ≤ Y^max ≤ W⁰ on [t₀ - σ, t₀ + p]. Indeed, since Y^L = V⁰ ≤ Y ≤ W⁰ = Y^U, Vⁿ ≤ Y ≤ Wⁿ on [t₀ - σ, t₀ + p], for some n . Employing the nondecreasing of F, G and applying Corollary 3.1, we get Vⁿ⁺¹ ≤ X . In the similar way, we obtain Y ≤ Wⁿ⁺¹ . Therefore, for all n, we have Vⁿ⁺¹ ≤ Y ≤ Wⁿ⁺¹ on [t₀ - σ, t₀ + p]. Taking limits as n→ ∞, we get Y^L = V⁰ ≤ Y^min ≤ Y ≤ Y^max ≤ W⁰ = Y^U on [t₀ - σ, t₀ + p]. Moreover, applying Lemma 2.3 for (3.21) and (3.21), we get assertions (3.12) and (3.13) respectively. In sequel, we prove the uniqueness of solution for two cases (H1) and (H2). For the case (H1), we let X^min, X^max be coupled minimal and maximal (i)-solutions of the problem (3.1). As X^min ≤ X^max on [t₀ - σ, t₀ + p], there exist two real-valued functions $\underline{R} (t), \bar{R} (t) \in C ([t_{0} - σ, t_{0} + p], ℝ^{+})$ and $\underline{R} (t) \leq \bar{R} (t)$ such that $X^{max} = X^{min} + [\underline{R} (t), \bar{R} (t)], \forall t \in [t_{0} - σ, t_{0} + p] .$ Now, we have to show that X^max = X^min . Indeed, since X^max, X^min are the coupled minimal and maximal (i)-solutions of (3.1) respectively and thus they satisfy the integral equations (3.10) and (3.11) respectively. From the assumptions of theorem, we have $X^{min} + [\underline{R} (t), \bar{R} (t)] = X^{max}$ $\begin{matrix} = φ (0) + \int_{t_{0}}^{t} (F (s, X_{s}^{max}) + \int_{t_{0}}^{t} G (t, s, X_{s}^{max}) ds) ds \\ \leq X^{min} (t) \\ + \int_{t_{0}}^{t} L_{1} (s) sup_{θ \in [- σ, 0]} (X^{max} (s + θ) ⊖ X^{min} (s + θ)) ds \\ + \int_{t_{0}}^{t} \int_{t_{0}}^{t} L_{2} (s) sup_{θ \in [- σ, 0]} (X^{max} (s + θ) ⊖ X^{min} (s + θ)) dsds \\ \leq X^{min} (t) + \int_{t_{0}}^{t} L_{1} (s) sup_{r \in [s - σ, s]} (X^{max} (r) ⊖ X^{min} (r)) ds \\ + \int_{t_{0}}^{t} \int_{t_{0}}^{t} L_{2} (s) sup_{r \in [s - σ, s]} (X^{max} (r) ⊖ X^{min} (r)) dsds, \end{matrix}$

where L₁ (s) = L₁/(1 + s) , L₂ (s) = L₂/(1 + s). Hence we obtain

$\underline{R} (t) \leq \int_{t_{0}}^{t} L_{1} (s) sup_{r \in [s - σ, s]} \underline{R} (r) dr + \int_{t_{0}}^{t} \int_{t_{0}}^{t} L_{2} (s)$ $sup_{r \in [s - σ, s]} \underline{R} (r) drdr$

and

$\bar{R} (t) \leq \int_{t_{0}}^{t} L_{1} (s) sup_{r \in [s - σ, s]} \bar{R} (r) dr + \int_{t_{0}}^{t} \int_{t_{0}}^{t} L_{2} (s)$ $sup_{r \in [s - σ, s]} \bar{R} (r) drdr .$

If we let $a (s) = sup_{r \in [s - σ, s]} \underline{R} (r), b (s) = sup_{r \in [s - σ, s]}$ $\bar{R} (r), s \in [t_{0}, t] \subset [t_{0}, t_{0} + p],$ then we have $\begin{matrix} (\begin{matrix} a (t) \\ b (t) \end{matrix}) & = \int_{t_{0}}^{t} (\begin{matrix} L_{1} (s) & 0 \\ 0 & L_{1} (s) \end{matrix}) (\begin{matrix} a (s) \\ b (s) \end{matrix}) ds \\ + \int_{t_{0}}^{t} \int_{t_{0}}^{t} (\begin{matrix} L_{2} (s) & 0 \\ 0 & L_{2} (s) \end{matrix}) (\begin{matrix} a (s) \\ b (s) \end{matrix}) dsds . \end{matrix}$

We apply Gronwal inequality to get a (t) ≤0 and b (t) ≤0 for all t ∈ [t₀, t₀ + p] . This prove the uniqueness of the (i)-solution for (3.1). For the case (H2), in the similar way, as Y^min ≤ Y^max on [t₀ - σ, t₀ + p], this yields $\begin{matrix} {\underline{Y}}^{max} = {\underline{Y}}^{min} + \underline{P} (t), and {\bar{Y}}^{max} = {\bar{Y}}^{min} + \bar{P} (t) . \end{matrix}$

From the assumptions of theorem and the integral equations (3.12), (3.13), we have $\begin{matrix} Y^{min} + [\underline{P} (t), \bar{P} (t)] = Y^{max} \\ = φ (0) ⊖ (- 1) \int_{t_{0}}^{t} (F (s, Y_{s}^{min}) \\ + \int_{t_{0}}^{t} G (t, s, Y_{s}^{min}) ds) ds \\ \leq X^{min} (t) ⊖ (- 1) \int_{t_{0}}^{t} L_{1} (s) \\ sup_{r \in [s - σ, s]} (X^{max} (r) ⊖ X^{min} (r)) ds \\ ⊖ (- 1) \int_{t_{0}}^{t} \int_{t_{0}}^{t} L_{2} (s) sup_{r \in [s - σ, s]} (X^{max} (r) ⊖ X^{min} (r)) dsds . \end{matrix}$

Then, we can obtain

$\underline{P} (t) \leq \int_{t_{0}}^{t} L_{1} (s) sup_{r \in [s - σ, s]} \bar{P} (r) dr + \int_{t_{0}}^{t} \int_{t_{0}}^{t} L_{2} (s)$ $sup_{r \in [s - σ, s]} \bar{P} (r) drdr$

and

$\bar{P} (t) \leq \int_{t_{0}}^{t} L_{1} (s) sup_{r \in [s - σ, s]} \underline{P} (r) dr + \int_{t_{0}}^{t} \int_{t_{0}}^{t} L_{2} (s)$ $sup_{r \in [s - σ, s]} \underline{P} (r) drdr .$

If we let $c (s) = sup_{r \in [s - σ, s]} \underline{P} (r), d (s) = sup_{r \in [s - σ, s]}$ $\bar{P} (r), s \in [t_{0}, t] \subset [t_{0}, t_{0} + p],$ then we have $\begin{matrix} (\begin{matrix} c (t) \\ d (t) \end{matrix}) & = \int_{t_{0}}^{t} (\begin{matrix} 0 & L_{1} (s) \\ L_{1} (s) & 0 \end{matrix}) (\begin{matrix} c (s) \\ d (s) \end{matrix}) ds \\ + \int_{t_{0}}^{t} \int_{t_{0}}^{t} (\begin{matrix} 0 & L_{2} (s) \\ L_{2} (s) & 0 \end{matrix}) (\begin{matrix} c (s) \\ d (s) \end{matrix}) dsds . \end{matrix}$

We apply Gronwal inequality to get c (t) ≤0 and d (t) ≤0 for all t ∈ [t₀, t₀ + p] . This prove the uniqueness of the (ii)-solution for (3.1). The proof is complete.

Example 3.4.exam2 Let us consider the interval-valued functional integro-differential equation under two kinds of Hukuhara derivatives ${\begin{matrix} D_{H}^{g} X (t) = X (t - \frac{1}{2}) + α \int_{0}^{t} e^{(s - t)} X (s - \frac{1}{2}) ds \\ X (t) = φ (t), t \in [- \frac{1}{2}, 0], \end{matrix}$ (3.23) where $φ (t) = [1, 2 - t], α \in ℝ^{+} ∖ {0}$ . In this example we shall solve (3.23) on [0, 1/2] .

Let $X^{L} (t) = [{\underline{X}}^{L} (t), {\bar{X}}^{L} (t)]$ be the lower (i)-solution of (3.23), where $\begin{matrix} {\underline{X}}^{L} (t) = (1 - ɛ) (1 + t + α t) - α (1 - ɛ) (1 - e^{t}), \\ {\bar{X}}^{L} (t) = (2 - ɛ) + (5 / 2 - ɛ + α + (5 / 2 - ɛ) α) t \\ - \frac{1 + α}{2} t^{2} - α (1 - e^{t}) (7 / 2 - ɛ), \end{matrix}$ for t ∈ [0, 1/2] and $\begin{matrix} X^{L} (t) = φ (t) - ɛ = [1 - ɛ, 2 - t - ɛ] \leq φ (t), \end{matrix}$ for $t \in [- \frac{1}{2}, 0] .$ Let $X^{U} (t) = [{\underline{X}}^{U} (t), {\bar{X}}^{U} (t)]$ be the upper (i)-solution of (3.23), where $\begin{matrix} {\underline{X}}^{U} (t) = (1 + ɛ) (1 + t + α t) - α (1 + ɛ) (1 - e^{t}), \\ {\bar{X}}^{U} (t) = (2 + ɛ) + (5 / 2 + ɛ + α + (5 / 2 + ɛ) α) t \\ - \frac{1 + α}{2} t^{2} - α (1 - e^{t}) (7 / 2 + ɛ), \end{matrix}$ for t ∈ [0, 1/2] and $\begin{matrix} X^{U} (t) = φ (t) + ɛ = [1 + ɛ, 2 - t + ɛ] \geq φ (t), \end{matrix}$ for $t \in [- \frac{1}{2}, 0]$ and ɛ > 0.

Moreover, it is easy to check that the conditions of Theorem 3.3 are satisfied for case (H1). Therefore, there exists a unique (i)-solution for this problem. Following the method of steps, we obtain the (i)-solution to (3.23) defined on [0, 1/2] and it is of the form $\begin{matrix} X (t) = [1 + t + α t - α (1 - e^{t}), \\ 2 + (5 / 2 + (7 / 2) α) t - \frac{(1 + α) t^{2}}{2} - \frac{7 α (1 - e^{t})}{2}], \end{matrix}$ t ∈ [0, 1/2]. Next, we consider α ∈ (0, 1/10] . Let $Y^{L} (t) = [{\underline{Y}}^{L} (t), {\bar{Y}}^{L} (t)]$ be the lower (ii)-solution of (3.23), where $\begin{matrix} {\underline{Y}}^{L} (t) = (1 - ɛ) + (5 / 2 - ɛ + α + (5 / 2 - ɛ) α) t \\ - \frac{1 + α}{2} t^{2} - α (1 - e^{t}) (7 / 2 - ɛ), \\ {\bar{Y}}^{L} (t) = (2 - ɛ) + (1 + α) (1 - ɛ) t \\ - α (1 - ɛ) (1 - e^{t}), \end{matrix}$ for t ∈ [0, 1/2] and $\begin{matrix} Y^{L} (t) = φ (t) - ɛ = [1 - ɛ, 2 - t - ɛ] \leq φ (t), \end{matrix}$ for $t \in [- \frac{1}{2}, 0] .$ Let $Y^{U} (t) = [{\underline{Y}}^{U} (t), {\bar{Y}}^{U} (t)]$ be the upper (ii)-solution of (3.23), where $\begin{matrix} {\underline{Y}}^{U} (t) = (1 + ɛ) + (5 / 2 + ɛ + α + (5 / 2 + ɛ) α) t \\ - \frac{(1 + α) t^{2}}{2} - α (1 - e^{t}) (7 / 2 + ɛ), \\ {\bar{Y}}^{U} (t) = (2 + ɛ) + (1 + α) (1 + ɛ) t - \\ α (1 + ɛ) (1 - e^{t}), \end{matrix}$ for t ∈ [0, 1/2] and $\begin{matrix} Y^{U} (t) = φ (t) + ɛ = [1 + ɛ, 2 - t + ɛ] \geq φ (t), \end{matrix}$ for $t \in [- \frac{1}{2}, 0] .$ In the same way, we can obtain a unique (ii)-solution for this problem. Following the method of steps, we obtain the (ii)-solution to (3.23) defined on [0, 1/2] and it is of the form $\begin{matrix} Y (t) = [1 + (5 / 2 + (7 / 2) α) t \\ - \frac{(1 + α) t^{2}}{2} - \frac{7 α (1 - e^{t})}{2}, \\ 2 + (1 + α) t - α (1 - e^{t})], t \in [0, 1 / 2] . \end{matrix}$

In Figs. 1 and 2, (i)-solution and (ii)-solution curves of (3.23) are given.

Example 3.5.exam1 Let us consider the interval-valued functional integro-differential equation under two kinds of Hukuhara derivatives

${\begin{matrix} D_{H}^{g} X (t) = (1 + sin (t)) X (t - \frac{π}{12}) \\ + \int_{0}^{t} e^{(s - t)} (1 + sin (s)) X (s - \frac{π}{12}) ds \\ X (t) = φ (t), t \in [- \frac{π}{12}, 0], \end{matrix}$ (3.24) where φ (t) = [1, 2 - t]. In this example we shall solve (3.24) on [0, π/12] .

We notice that F (t, X_t) = (1 + sin(t)) X $(t - \frac{π}{12}),$ G (t, s, X_t) = e^(s-t) (1 + sin(s)) X $(s - \frac{π}{12})$ satisfy the conditions of Theorem 3.3 Case 1: we show that (3.24) exits extremal (i)-solutions on $[0, \frac{π}{12}]$ . First, assume that $X^{L} (t) = [{\underline{X}}^{L} (t), {\bar{X}}^{L} (t)]$ is the lower (i)-solution on $[0, \frac{π}{12}]$ , where ${\underline{X}}^{L} (t) = (1 - ɛ) (2 t + e^{t}),$ $\begin{matrix} {\bar{X}}^{L} (t) & = (2 - ɛ) + (4 - 3 ɛ + π / 12) t \\ - t^{2} + (e^{t} - 1) (2 - 2 ɛ + π / 12) \end{matrix}$ and $X^{U} (t) = [{\underline{X}}^{U} (t), {\bar{X}}^{U} (t)]$ is the upper (i)-solution on $[0, \frac{π}{12}]$ , where ${\underline{X}}^{U} (t) = (1 + ɛ) (1 + 2 α t + α (e^{t} - 1)),$ $\begin{matrix} {\bar{X}}^{U} (t) & = (2 + ɛ) + (2 α (2 + ɛ + \frac{π}{12}) + ɛ) t \\ - α t^{2} + (e^{t} - 1) (α (2 + ɛ + \frac{π}{12}) + ɛ), \end{matrix}$ where we put α = (1 + sin(π/12)) and ɛ > 0. Indeed, we check that ${\begin{matrix} D_{H}^{g} X^{L} (t) \leq α X^{L} (t - \frac{π}{12}) \\ + α \int_{0}^{t} e^{(s - t)} X^{L} (s - \frac{π}{12}) ds, t \in [0, π / 12] \\ X^{L} (t) = φ_{1} (t) = [1 - ɛ, 2 - t - ɛ] \leq φ (t), t \in [- \frac{π}{12}, 0], \end{matrix}$ (3.25) and ${\begin{matrix} D_{H}^{g} X^{U} (t) \geq X^{U} (t - \frac{π}{12}) \\ + \int_{0}^{t} e^{(s - t)} X^{U} (s - \frac{π}{12}) ds, t \in [0, π / 12] \\ X^{U} (t) = φ_{2} (t) = [1 + ɛ, 2 - t + ɛ] \geq φ (t), t \in [- \frac{π}{12}, 0], \end{matrix}$ (3.26) Next, let us construct the sequences by $\begin{matrix} {\begin{matrix} D_{H}^{g} V^{n + 1} (t) = V^{n} (t - \frac{π}{12}) \\ + \int_{0}^{t} e^{(s - t)} V^{n} (s - \frac{π}{12}) ds - \frac{\hat{ɛ}}{n + 1}, t \in [0, π / 12] \\ V^{n + 1} (t) = φ (t) - \frac{\hat{ɛ}}{n + 1}, t \in [- \frac{π}{12}, 0], \end{matrix} \end{matrix}$ and $\begin{matrix} {\begin{matrix} D_{H}^{g} W^{n + 1} (t) = α W^{n} (t - \frac{π}{12}) \\ + α \int_{0}^{t} e^{(s - t)} W^{n} (s - \frac{π}{12}) ds + \frac{\hat{ɛ}}{n + 1}, t \in [0, π / 12] \\ W^{n + 1} (t) = φ (t) + \frac{\hat{ɛ}}{n + 1}, t \in [- \frac{π}{12}, 0], \end{matrix} \end{matrix}$ for all n = 0, 1, . . . and $\hat{ɛ} > 0$ . We verify that monotone sequences of the above contructions satisfy

(Wⁿ (t)) xrightarrown → ∞ X^max (t) and X^max is a maximal of (3.24).

(Vⁿ (t)) xrightarrown → ∞ X^min (t) and X^min is a minimal of (3.24).

First, we prove (a). Indeed, let $0 < {\hat{ɛ}}_{2} < {\hat{ɛ}}_{1} ⩽ \hat{ɛ}$ , then for each positive integer n we obtain $W^{n + 1} (t, {\hat{ɛ}}_{2}) ⩽ W^{n + 1} (t, {\hat{ɛ}}_{1}), t \in [- \frac{π}{12}, 0]$ and $\begin{matrix} D_{H}^{g} W^{n + 1} (t, {\hat{ɛ}}_{2}) ⩽ α W^{n} (t - \frac{π}{12}) \\ + α \int_{0}^{t} e^{(s - t)} W^{n} (s - \frac{π}{12}) ds + \frac{{\hat{ɛ}}_{2}}{n + 1} \end{matrix}$ $\begin{matrix} D_{H}^{g} W^{n + 1} (t, {\hat{ɛ}}_{1}) ⩾ α W^{n} (t - \frac{π}{12}) \\ + α \int_{0}^{t} e^{(s - t)} W^{n} (s - \frac{π}{12}) ds + \frac{{\hat{ɛ}}_{2}}{n + 1} . \end{matrix}$

Hence we apply Corollary 3.1 and conclude that $\begin{matrix} W^{n + 1} (t, {\hat{ɛ}}_{2}) ⩽ W^{n + 1} (t, {\hat{ɛ}}_{1}) \end{matrix}$ for all $t \in [- \frac{π}{12}, \frac{π}{12}]$ 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 $[0, \frac{π}{12}]$ , 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 $[0, \frac{π}{12}]$ . Since $\begin{matrix} {\begin{matrix} W^{{(n + 1)}_{k}} (t, \hat{ɛ}) = φ (0) + \frac{\hat{ɛ}}{(n + 1)_{k}} \\ + α \int_{0}^{t} W^{{(n + 1)}_{k}} (s - \frac{π}{12}) ds \\ + α \int_{0}^{t} \int_{0}^{t} e^{(s - t)} W^{(n + 1)_{k}} (s - \frac{π}{12}) ds ds, t \in [0, \frac{π}{12}] \\ W^{(n + 1)_{k}} (t, \hat{ɛ}) = φ (t) + \frac{\hat{ɛ}}{(n + 1)_{k}}, t \in [- \frac{π}{12}, 0], \end{matrix} \end{matrix}$ we obtain the maximal (i)-solution of (3.24) on [0, π/12] as below $\begin{matrix} X^{max} (t) = [1 + 2 α t + α (e^{t} - 1), \\ 2 + 2 α t (2 + π / 12) - α t^{2} + α (e^{t} - 1) (2 + π / 12)] . \end{matrix}$

Next we shall show that X^max (t) is a required maximal solution of (3.24) on $[0, \frac{π}{12}]$ . For this purpose, we observe that $X (t) = φ (t) < φ (t) + \frac{\hat{ɛ}}{(n + 1)} = W^{n + 1} (t, \hat{ɛ}), t \in [- π / 12, 0]$ and F, G are nondecreasing, hence we get $D_{H}^{g} X (t)$ $\begin{matrix} < α W^{n} (t - \frac{π}{12}, \hat{ɛ}) \\ + α \int_{0}^{t} e^{(s - t)} W^{n} (s - \frac{π}{12}, \hat{ɛ}) ds + \frac{\hat{ɛ}}{n + 1} \\ ⩽ D_{H}^{g} W^{n + 1} (t, \hat{ɛ}) . \end{matrix}$ By using Corollary 3.1, then we get X (t) < Wⁿ⁺¹ $(t, \hat{ɛ})$ on $[- \frac{π}{12}, \frac{π}{12}]$ . The uniqueness of maximal 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 (3.24) with $X^{max} (t) = lim_{k \to \infty} W^{{(n + 1)}_{k}} (t, \hat{ɛ}) .$ Next, we shall prove (b). Proceeding similarly as above, let $0 < {\hat{ɛ}}_{2} < {\hat{ɛ}}_{1} ⩽ \hat{ɛ}$ , then for each positive integer n we obtain $V^{n + 1} (t, {\hat{ɛ}}_{1}) ⩽ V^{n + 1} (t, {\hat{ɛ}}_{2}), t \in [- \frac{π}{12}, 0]$ and $\begin{matrix} D_{H}^{g} V^{n + 1} (t, {\hat{ɛ}}_{1}) ⩽ V^{n} (t - \frac{π}{12}) \\ + \int_{0}^{t} e^{(s - t)} V^{n} (s - \frac{π}{12}) ds - \frac{{\hat{ɛ}}_{1}}{n + 1} \end{matrix}$ $\begin{matrix} D_{H}^{g} V^{n + 1} (t, {\hat{ɛ}}_{2}) ⩾ V^{n} (t - \frac{π}{12}) \\ + \int_{0}^{t} e^{(s - t)} V^{n} (s - \frac{π}{12}) ds - \frac{{\hat{ɛ}}_{1}}{n + 1} . \end{matrix}$ Hence we apply Corollary 3.1 and conclude that $\begin{matrix} V^{n + 1} (t, {\hat{ɛ}}_{1}) ⩽ V^{n + 1} (t, {\hat{ɛ}}_{2}) \end{matrix}$ for all $t \in [- \frac{π}{12}, \frac{π}{12}]$ 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 $[0, \frac{π}{12}]$ , 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 $[0, \frac{π}{12}]$ . Since $\begin{matrix} {\begin{matrix} V^{{(n + 1)}_{k}} (t, \hat{ɛ}) = φ (0) - \frac{\hat{ɛ}}{(n + 1)_{k}} \\ + \int_{0}^{t} V^{{(n + 1)}_{k}} (s - \frac{π}{12}) ds \\ + \int_{0}^{t} \int_{0}^{t} e^{(s - t)} V^{(n + 1)_{k}} (s - \frac{π}{12}) ds ds, t \in [0, \frac{π}{12}] \\ V^{(n + 1)_{k}} (t, \hat{ɛ}) = φ (t) - \frac{\hat{ɛ}}{(n + 1)_{k}}, t \in [- \frac{π}{12}, 0], \end{matrix} \end{matrix}$ we obtain the minimal (i)-solution of (3.24) on [0, π/12] as below $\begin{matrix} X^{min} (t) = [2 t + e^{t}, 2 + (4 + π / 12) \\ t - t^{2} + (e^{t} - 1) (2 + π / 12)] . \end{matrix}$

Next we shall show that X^min (t) is a required minimal solution of (3.24) on $[0, \frac{π}{12}]$ . For this purpose, we observe that $X (t) = φ (t) > φ (t) - \frac{\hat{ɛ}}{(n + 1)} = V^{n + 1} (t, \hat{ɛ}), t \in [- π / 12, 0]$ and F, G are nondecreasing, hence we obtain $\begin{matrix} D_{H}^{g} X (t) > (1 + sin (t)) X (t - \frac{π}{12}) \\ + \int_{0}^{t} e^{(s - t)} (1 + sin (s)) X (s - \frac{π}{12}) ds - \frac{\hat{ɛ}}{(n + 1)} \\ \geq V^{n} (t - \frac{π}{12}, \hat{ɛ}) \\ + \int_{0}^{t} e^{(s - t)} V^{n} (s - \frac{π}{12}, \hat{ɛ}) ds - \frac{\hat{ɛ}}{n + 1} \\ \geq D_{H}^{g} V^{n + 1} (t, \hat{ɛ}) . \end{matrix}$

By using Corollary 3.1, then we get $V^{n + 1} (t, \hat{ɛ}) < X (t)$ on $[- \frac{π}{12}, \frac{π}{12}]$ . The uniqueness of minimal 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 (3.24) with $X^{min} (t) = lim_{k \to \infty} V^{{(n + 1)}_{k}} (t, \hat{ɛ}) .$ The maximal and minimal (i)-solutions of (3.24) on [- π/12, π/12] is illustrated in Fig. 3 and Fig. 4 respectively. Case 2: In this case, we show that (3.24) exits extremal (ii)-solutions on $[0, \frac{π}{12}] .$ Let $Y^{L} (t) = [{\underline{Y}}^{L} (t), {\bar{Y}}^{L} (t)]$ be the lower (ii)-solution on $[0, \frac{π}{12}]$ , where ${\underline{Y}}^{L} (t) = (2 + ɛ) + 2 (1 + ɛ) t + (1 + ɛ) (e^{t} - 1),$ $\begin{matrix} {\bar{Y}}^{L} (t) & = (1 + ɛ) + (2 (2 + ɛ + π / 12) + ɛ) t \\ - t^{2} + (e^{t} - 1) (2 + 2 ɛ + π / 12) \end{matrix}$ and let $Y^{U} (t) = [{\underline{Y}}^{U} (t), {\bar{Y}}^{U} (t)]$ be the upper (ii)-solution on $[0, \frac{π}{12}]$ , where $\begin{matrix} {\underline{Y}}^{U} (t) & = (2 + ɛ) + 2 α (1 + ɛ) t + α (1 + ɛ) (e^{t} - 1), \\ {\bar{Y}}^{U} (t) & = (1 + ɛ) + (2 α (2 + ɛ + π / 12) + ɛ) t \\ - α t^{2} + (e^{t} - 1) (2 (2 + 2 ɛ + π / 12) + ɛ), \end{matrix}$ where we put α = (1 + sin(π/12)) and ɛ > 0. Moreover, we also obtain (3.25) and (3.26) for the case of (ii)-differentiability. Proceeding similarly asCase 1, we obtain $Y^{min} (t) = lim_{k \to \infty} V^{{(n + 1)}_{k}} (t, \hat{ɛ}),$ $^{max} (t) = lim_{k \to \infty} W^{{(n + 1)}_{k}} (t, \hat{ɛ}),$ where $\begin{matrix} {\begin{matrix} V^{{(n + 1)}_{k}} (t, \hat{ɛ}) = φ (0) - \frac{\hat{ɛ}}{(n + 1)_{k}} \\ ⊖ (- 1) \int_{0}^{t} V^{{(n + 1)}_{k}} (s - \frac{π}{12}) ds \\ ⊖ (- 1) \int_{0}^{t} \int_{0}^{t} e^{(s - t)} V^{(n + 1)_{k}} (s - \frac{π}{12}) ds ds, t \in [0, \frac{π}{12}] \\ V^{(n + 1)_{k}} (t, \hat{ɛ}) = φ (t) - \frac{\hat{ɛ}}{(n + 1)_{k}}, t \in [- \frac{π}{12}, 0], \end{matrix} \end{matrix}$ and $\begin{matrix} {\begin{matrix} W^{{(n + 1)}_{k}} (t, \hat{ɛ}) = φ (0) + \frac{\hat{ɛ}}{(n + 1)_{k}} \\ ⊖ (- 1) α \int_{0}^{t} W^{{(n + 1)}_{k}} (s - \frac{π}{12}) ds \\ ⊖ (- 1) α \int_{0}^{t} \int_{0}^{t} e^{(s - t)} W^{(n + 1)_{k}} (s - \frac{π}{12}) ds ds, t \in [0, \frac{π}{12}] \\ W^{(n + 1)_{k}} (t, \hat{ɛ}) = φ (t) + \frac{\hat{ɛ}}{(n + 1)_{k}}, t \in [- \frac{π}{12}, 0] . \end{matrix} \end{matrix}$

Therefore, we obtain the maximal and minimal (ii)-solutions of (3.24) on [0, π/12] as below $\begin{matrix} X^{min} (t) = [2 + 2 t + (e^{t} - 1), \\ 1 + 2 (2 + π / 12) t - t^{2} + (e^{t} - 1) (2 + π / 12)] \end{matrix}$

$\begin{matrix} X^{max} (t) = [2 + 2 α t + α (e^{t} - 1), \\ 1 + 2 α (2 + π / 12) t - α t^{2} + 2 (e^{t} - 1) (2 + π / 12)] \end{matrix}$ and they are illustrated in Figure 5 and Figure 6 respectively.

Footnotes

Acknowledgments

The authors would like to express deep gratitude to the Editor-in-Chief Professor Reza Langari and the anonymous referees for their valuable comments and suggestions which have greatly improve this paper. This research is funded by Foundation for Science and Technology Development of Ton Duc Thang University (FOSTECT), website: , under Grant: FOSTECT.2015.BR.19.

References

Agarwal

R.P.

and O’Regan

, Existence for set differential equations via mul-tivalued operator equations, Differential equations and Applications5 (2007), 1–5.

Agarwal

R.P.

, Arshad

, O’Regan

and Lupulescu

, A Schauder fixed point theorem in semilinear spaces and applications, Fixed Point Theory and Applications2013 (2013), 306.

Agarwal

R.P.

, Arshad

, O’Regan

and Lupulescu

, Fuzzy fractional integral equations under compactness type condition, Fract Calc Appl Anal15 (2012), 572–590.

T.V.

, Phu

N.D.

and Hoa

N.V.

, A note on solutions of interval-valued Volterra integral equations, Journal of Integral Equations and Applications26 (2014), 1–16.

Arshad

and Lupulescu

, On the fractional differential equations with uncertainty, Nonlinear Analysis: TMA74(11) (2011), 3685–3693.

Allahviranloo

, Abbasbandy

, Sedaghatfar

and Darabi

, A New method for solving fuzzy integro-differential equation under generalized differentiability, J Neural Computing and Applications21(1) (2012), 191–196.

Allahviranloo

, Amirteimoori

, Khezerloo

and Khezerloo

, A new method for solving fuzzy Volterra integrodifferential equations, Australian Journal of Basic and Applied Sciences54 (2011), 154–164.

Allahviranloo

, Hajighasemi

, Khezerloo

, Khorasany

and Salahshour

, Existence and uniqueness of solutions of fuzzy Volterra integro-differential equations, IPMU II81 (2010), 491–500.

Allahviranloo

and Gholami

, Note on “Generalized Hukuhara differentiability of interval-valued functions and interval differential equations”, Journal of Fuzzy Set Valued Analysis2012 (2012). DOI: 10.5899/2012/jfsva-00135.

10.

Bhaskar

T.G.

, Lakshmikantham

and Devi

J.V.

, Monotone iterative technique for functional differential equations with retardation and anticipation, Nonlinear Analysis: TMA66, 2237–2242.

11.

Bede

and Gal

S.G.

, Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems151 (2005), 581–599.

12.

Bede

, Rudas

I.J.

and Bencsik

A.L.

, First order linear fuzzy differential equations under generalized differentiability, Information Sciences177 (2007), 1648–1662.

13.

Bede

and Stefanini

, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems230 (2013), 119–141.

14.

Bhaskar

T.G.

and Lakshmikantham

, Set differential equations and flow invariance, J Appl Anal82(2) (2003), 357–368.

15.

Chalco-Cano

, Rufián-Lizana

, Román-Flores

and Jiménez-Gamero

M.D.

, Calculus for interval-valued functions using generalized Hukuhara derivative and applications, Fuzzy Sets and Systems219 (2013), 49–67.

16.

De Blasi

F.S.

and Iervolino

, Equazioni differenziali consoluzioni a valore compatto convesso. Journal of integral equations and applications, Boll Unione Mat Ital4(2) (1969), 194–501.

17.

De Blasi

F.S.

, Lakshmikantham

and Bhaskar

T.G

, An existence theorem for set differential inclusions in a semilinear metric space, Control and Cybernetics36(3) (2007), 571–582.

18.

Devi

J.V.

, Generalized monotone iterative technique for set differential equations involving causal operators with memory, Int J Adv Eng Sci Appl Math8(3) (2011), 74–83.

19.

Devi

J.V.

, Sreedhar

Ch.V.

and Nagamani

, Monotone iterative technique for integro differential equations with retardation and anticipation, Commun Appl Anal14 (2010), 325–336.

20.

Hale

J.K.

, Theory of Functional Differential Equations, Springer, NewYork, 1977.

21.

Hoa

N.V.

and Phu

N.D.

, On maximal and minimal solutions for set-valued differential equations with feedback control, J Abstract and Applied Analysis2012 (2012), 1–11.

22.

Hoa

N.V.

, Phu

N.D.

, Tung

T.T.

and Quang

L.T.

, Interval-valued functional integro-differential equations, Advcances in Difference Equations2014 (2014), 177.

23.

Hoa

N.V.

and Phu

N.D.

, Fuzzy functional integro-differential equations under generalized H-differentiability, Journal of Intelligent and Fuzzy Systems26 (2014), 2073–2085.

24.

Hoa

N.V.

, Fuzzy fractional functional differential equations under Caputo gH-differentiability, Communications in Nonlinear Science and Numerical Simulation22 (2015), 1134–1157.

25.

Hoa

N.V.

, Fuzzy fractional functional integral and differential equations, Fuzzy Sets and Systems280 (2015), 58–90. (SCI).

26.

Hoa

N.V.

, The initial value problem for interval-valued second-order differential equations under generalized H-differentiability, Information Sciences311 (2015), 119–148.

27.

Hoa

N.V.

, Tri

P.V.

and Dao

T.T.

, and I’van Zelinka, Some global existence results and stability theorem for fuzzy functional differential equations, Journal of Intelligent and Fuzzy Systems28 (2015), 393–409.

28.

Hukuhara

, Inteǵration des applications mesurables dont la valeur est un compact convex, Funkcial Ekvac10 (1967), 205–229.

29.

Kolmanovskii

V.B.

and Myshkis

, Applied Theory of Functional Differential Equations. Kluwer Academic Publishers, Dordrecht, 1992.

30.

Kuang

, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993.

31.

Lakshmikantham

, Bhaskar

T.G.

and Devi

J.V.

, Theory of set differential equations in metric spaces, Cambridge Scientific Publisher, UK, 2006.

32.

Ladde

G.S.

, Lakshmikantham

and Vatsala

A.S.

, Monotone Iterative Technique for Nonlinear Differential equations, Pittman Publishing Inc, London, 1985.

33.

Lupulescu

, On a class of functional differential equations in Banach spaces, Electronic Journal of Qualitative Theory of Differential Equations64 (2010), 1–17.

34.

Lupulescu

, On a class of fuzzy functional differential equations, Fuzzy Sets and Systems160 (2009), 1547–1562.

35.

Lupulescu

, Initial value problem for fuzzy differential equations under dissipative conditions, Information Sciences178(23) (2008), 4523–4533.

36.

Lupulescu

, Hukuhara differentiability of intervalvalued functions and interval differential equations on time scales, Information Sciences248 (2013), 50–67.

37.

Lupulescu

, Dong

L.S.

, Hoa

N.V.

, Existence and uniqueness of solutions for random fuzzy fractional integral and differential equations, Journal of Intelligent and Fuzzy Systems29 (2015), 27–42.

38.

Malinowski

M.T.

, Interval Cauchy problem with a second type Hukuhara derivative, Information Sciences213 (2012), 94–105.

39.

Malinowski

M.T.

, Random fuzzy differential equations under generalized Lipschitz condition, Nonlinear Analysis RWA13(2) (2012), 860–881.

40.

Malinowski

M.T.

, On set differential equations in Banach spaces - a second type Hukuhara differentiability approach, Applied Mathematics and Computation219(1) (2012), 289–305.

41.

Malinowski

M.T.

, Second type Hukuhara differentiable solutions to the delay set-valued differential equations, Applied Mathematics and Computation218(18) (2012), 9427–9437.

42.

Malinowski

M.T.

, Interval differential equations with a second type Hukuhara derivative, Applied Mathematics Letters24 (2011), 2118–2123.

43.

Malinowski

M.T.

, Existence theorems for solutions to random fuzzy differential equations, Nonlinear Analysis TMA73(6) (2010), 1515–1532.

44.

Malinowski

M.T.

, Onrandom fuzzy differential equations, Fuzzy Sets and Systems160(21) (2009), 3152–3165.

45.

Moore

and Lodwick

, Interval analysis and fuzzy set theory, Fuzzy Sets and Systems135(1) (2003), 5–9.

46.

Rodríguez-López

, Monotone method for fuzzy differential equations, Fuzzy Sets Systems159 (2008), 2047–2076.

47.

Stefanini

, A generalization of Hukuhara difference In: Dubois

, Lubiano

M.A

, rade

, Gil

M.A

, Grzegorzewski

and Hryniewicz

, (Eds), Soft Methods for Handling Variability and Imprecision, Series on Advances in Soft Computing, (vol 48), Springer, 2008.

48.

Stefanini

and Bede

, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Analysis TMA71 (2009), 1311–1328.