Weakly contractive mappings in partially ordered metric space and solutions of delay interval-valued differential equations under generalized Hukuhara differentiability

Abstract

In this paper, by weakly contractive mappings in partially ordered metric space, we have proved problem of existence and uniqueness solution for delay interval-valued differential equations under generalized Hukuhara differentiability (DIDEs). We give also some examples being simple illustrations of this DIDEs.

Keywords

Partially ordered space The metric Hausdorff metric space Generalized Hukuhara differentiability Delay interval-valued differential equations Weakly contractive mapping

1 Introduction

1.1 Background

The interval-valued differential equations (IDEs) are important parts of the interval-valued analysis (see [7 , 24–31]). As we have known that there are many approaches to the IDEs. In case, when modeling real-world phenomena, information about the behavior of a dynamical system is uncertain and one has to consider these uncertainties to gain better meaning of full models. The interval-valued differential equations can be used for modeling dynamical systems subject to uncertainties. The papers [21 , 28] are focused on the interval-valued differential equations. These equations can be studied with a framework of the Hukuhara derivative [21]. The first method was based on Hukuhara difference notation and was further investigated by Markov [27], Moore [28] and other authors. The approach based on Hukuhara derivative [21] has the disadvantage that it leads to solutions which have an increased length of their support. Recently, Stefanini and Bede (see [31]) and Chalco-Cano (see [15]) studied IDEs under generalized Hukuhara differentiability of interval-valued functions. In this case, the derivative exists and the solution of the IDEs may have a decreasing length of the support, but the uniqueness is lost. Therefore, our point is that the generalization of the concept of Hukuhara differentiability can be of great help in the dynamic study of interval-valued differential equations. The paper of Stefanini and Bede was the starting point for the topic of interval-valued differential equations under generalized Hukuhara differentiability (can see [7 , 24–26]) and this studies of the interval-valued differential equations are very beneficial to survey the fuzzy differential equations under generalized Hukuhara differentiability - FDEs (can see for example, Allahviranloo [4, 5], or Bede [13, 14]). In the papers [7 , 29], we can find the studies on interval-valued differential equations under generalized Hukuhara differentiability (IDEs), i.e. the IVP for IDEs in the form:

${\begin{matrix} D_{H}^{g} X (t) = F (t, X (t)), \\ X (t_{0}) = X_{0} \in K_{C} (ℝ), t \in [t_{0}, t_{0} + p], \end{matrix}$ (1.1) where $D_{H}^{g}$ is understood as the classical Hukuhara derivative, then as the second type Hukuhara derivative - the generalized Hukuhara derivative. The existence and uniqueness of solution to the IVP for IDEs is obtained under an assumption that the coefficients satisfy a condition with the Lipschitz constant (see [24]). In [26], under the generalized Lipschitz condition, Malinowski obtained the existence and uniqueness of solutions to both kinds of IVP for IDEs (1.1) by the Banach fixed point theorem.

1.2 Approach and Objective

In this paper, we consider the initial valued problem for delay interval-valued differential equations under generalized Hukuhara differenytiability (IVP for DIDEs): ${\begin{matrix} D_{H}^{g} X (t) = F (t, X (t), X (t - τ)), \\ X_{t_{0}} (τ) = φ (t_{0} - τ) = φ_{0}, τ \in [0, σ], \end{matrix}$ (1.2) where $X_{t} (τ) = X (t - τ) \in K_{C} (ℝ)$ is delay interval-valued process, $F \in C ([t_{0}, t_{0} + p] \times K_{C} (ℝ) \times C ([t_{0} - σ, t_{0}], K_{C} (ℝ)), K_{C} (ℝ))$ is continuous interval-valued function and the symbol $D_{H}^{g}$ denotes the generalized Hukuhara derivative. The existence and uniqueness of solution to IVP for DIDEs (ref eq1.2) are obtained by weakly contractive mappings and by the extremal solutions in partially ordered metric space.

1.3 Outline

The paper is organized as follows. In section 2, it is devoted to the preliminaries of interval-valued analysis and interval-valued differential equations. In Section 3, we consider the weakly contractive mappings in partially ordered metric space and formulate existence and uniqueness of solution of IVP for DIDEs. In the last section, we give some examples being simple illustrations of existence and uniqueness of solution to this IVP for DIDEs.

2 Preliminaries and notation

Let $K_{C} (ℝ)$ be the family all nonempty, compact and bounded subsets of $ℝ .$ The addition and scalar multiplication in $K_{C} (ℝ)$ , we define 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 we have λ₁ (λ₂A) = (λ₁λ₂) A and (λ₃ + λ₄) A = λ₃A + λ₄A . Let $A, B \in K_{C} (ℝ)$ as above. Then 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 $H [A, {0}] = | | A | | = m a x {| \underline{Α} |, | \bar{Α} |},$ $l e n (A) = \bar{Α} - \underline{Α},$ , respectively, where {0} is the zero element of $K_{C} (ℝ)$ , which is regarded as one point. The Hausdorff metric (2.1) satisfies the following properties: $\begin{matrix} H [A + C, B + C] & = H [A, B], H [A, B] = H [B, A], \\ H [A + B, C + D] & \leq H [A, C] + H [B, D], \\ H [λ A, λ B] & = | λ | H [A, B], \end{matrix}$ for all $A, B, C, D \in K_{C} (ℝ)$ and λ ∈ R.

Definition 2.1. (see [27]) 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 that we can see the following properties for $A, B, C, D \in K_{C} (ℝ)$ (see [27]):

A ⊖ A = {0};

If A ⊖ B, A ⊖ C exist, then H [A⊖ B, A ⊖ C] = H [B, C] ;

If A ⊖ B, C ⊖ D exist, then H [A⊖ B, C ⊖ D] = H [A + D, B + C] ;

If A ⊖ B, A ⊖ (B + C) exist, then there exist (A ⊖ B) ⊖ C and (A⊖ B) ⊖ C = A ⊖ (B + C) ;

If A ⊖ B, A ⊖ C, C ⊖ B exist, then there exist (A ⊖ B) ⊖ (A ⊖ C) and (A ⊖ B) ⊖ (A ⊖ C) = C ⊖ B .

Definition 2.2.(see [31]) Let $A, B \in K_{C} (ℝ) .$ If there exists an interval $C \in K_{C} (ℝ)$ such that $\begin{matrix} A \underset{gH}{⊖} B = C \Leftrightarrow {\begin{matrix} (i) A = B + C, \\ or (ii) B = A + (- 1) C . \end{matrix} \end{matrix}$ we call C the generalized Hukuhara difference of A and B. We denote the interval C by $A \underset{gH}{⊖} B$ . Besides, we can see the following properties for $A, B, C, D \in K_{C} (ℝ)$ (see [31]):

the generalized Hukuhara is unique and $[\underline{A}, \bar{A}] \underset{gH}{⊖} [\underline{B}, \bar{B}] = [\underline{C}, \bar{C}],$ where $\begin{matrix} \underline{C} = \min {\underline{A} - \underline{B}, \bar{A} - \bar{B}}, \bar{C} = \max {\underline{A} - \underline{B}, \bar{A} - \bar{B}}, \end{matrix}$

$A \underset{gH}{⊖} A = {0}$ ,

if $A \underset{gH}{⊖} B$ exists then $B \underset{gH}{⊖} A$ exists.

$(A + B) \underset{gH}{⊖} B = A$ ,

${0} \underset{gH}{⊖} (A \underset{gH}{⊖} B) = (- B) \underset{gH}{⊖} (- A)$ ,

$A \underset{gH}{⊖} B = B \underset{gH}{⊖} A = C$ if and only if C = {0} and A = B.

Definition 2.3.We say that the interval-valued mapping $X : [t_{0}, t_{0} + p] \subset ℝ^{+} \to K_{C} (ℝ)$ is continuous at the point t ∈ [t₀, t₀ + p] if for every ɛ > 0 there exists δ = δ (t, ɛ) >0 such that H [X (t) , X (s)] ≤ ɛ, for all s ∈ [t₀, t₀ + p] and |t - s| < δ .

The generalized differentiability was introduced in [31] and studied by Bede in [13 , 31], by Chalco-Cano, Román-Flores, Rodrguez-Lpez, and Jimnez-Gamero in [15, 16], by Lupulescu in [24] and by Malinowski in [25, 26].

Definition 2.4. Let $X : (t_{0}, t_{0} + p) \to K_{C} (ℝ)$ and t ∈ (t₀, t₀ + p). We say that X is generalized differentiable at t if there exists $D_{H}^{g} X (t) \in K_{C} (ℝ)$ such that

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

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

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

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

In this definition, case (H^g1)-differentiable corresponds to the classical H-derivative, so this differentiability concept is a generalization of the Hukuhara derivative. In this paper, we consider only the first two case of Definition 2.4. In the other cases, the derivative is trivial because it is reduced to a crisp element (more precisely, $D_{H}^{g} X (t) \in K_{C} (ℝ)$ ). Further, we say that X is (H^g1)-differentiable or (H^g2)-differentiable on (t₀, t₀ + p) if it is differentiable in the sense (H^g1) or (H^g2) of Definition 2.4, respectively.

Corollary 2.1. Let p > 0 . Assume that $D_{H}^{g} X : [t_{0}, t_{0} + p] \to K_{C} (ℝ)$ is continuous. The interval-valued function X (t) is equivalent to one of the following integral equations: $\begin{matrix} X (t) = X (t_{0}) + \int_{t_{0}}^{t} D_{H}^{g} X (s) ds, \forall t \in [t_{0}, t_{0} + p], \end{matrix}$ if X is (H^g1)-differentiable, and $\begin{matrix} X (t) = X (t_{0}) ⊖ (- 1) \int_{t_{0}}^{t} D_{H}^{g} X (s) ds, \forall t \in [t_{0}, t_{0} + p] \end{matrix}$ if X is (H^g2)-differentiable, provided that the H-difference exists.

Proof. Can see [7].

Corollary 2.2. Let $X : [t_{0}, t_{0} + p] \to K_{C} (ℝ),$ we denote $X (t) = [\underline{X} (t), \bar{X} (t)]$ with t ∈ [t₀, t₀ + p], where $\underline{X}, \bar{X} : [t_{0}, t_{0} + p] \to ℝ .$

If the mapping X is (H^g1)-differentiable (i.e. classical Hukuhara differentiable) at t ∈ [t₀, t₀ + p], then the real-valued functions $\underline{X}$ and $\bar{X}$ are differentiable at t and $D_{H}^{g} X (t) = [{\underline{X}}^{'} (t), {\bar{X}}^{'} (t)] .$

If the mapping X is (H^g2)-differentiable at t ∈ [t₀, t₀ + p], then the real-valued functions $\underline{X}$ and $\bar{X}$ are differentiable at t and $D_{H}^{g} X (t) = [{\bar{X}}^{'} (t), {\underline{X}}^{'} (t)] .$

Proof. Can see [27].

Definition 2.5. We say that the real function $lent (X (t)) = \bar{X} (t) - \underline{X} (t)$ is the width of the interval-valued function X (t) .

Corollary 2.3. Let the interval-valued function X (t) with t ∈ (t₀, t₀ + p):

If the lent (X (t)) is increasing then X (t) is (H^g1)-differentiable and $D_{H}^{g} X (t) = [{\underline{X}}^{'} (t), {\bar{X}}^{'} (t)] .$

If the lent (X (t)) is decreasing then X (t) is (H^g2)-differentiable and $D_{H}^{g} X (t) = [{\bar{X}}^{'} (t), {\underline{X}}^{'} (t)] .$

Proof. Can see [7, 20].

Definition 2.6. [19] We say that the real function ψ : [0, + ∞) → [0, + ∞) is distance translation function if it is a continuous increasing function and ψ (0) =0 only when t = 0 .

Definition 2.7. [19] The real function f : (ℵ , d) → (ℵ , d) in complete metric space is called a weakly contractive mapping if existence the distance translation functions ψ, γ : [0, + ∞) → [0, + ∞) such that satisfies: $ψ (d [f (x), f (y)]) \leq ψ (d [x, y]) - γ (d [x, y]),$ $\forall x, y \in (ℵ, d)$

Theorem 2.1. Assume that the real continuous function f : (ℵ , d) → (ℵ , d) in complete metric space satisfies: $ψ (d [f (x), f (y)]) \leq ψ (d [x, y]) - γ (d [x, y]), x \geq y,$ (2.2) with ψ, γ are distance translation functions and ${(x_{k})}_{k \in ℵ} \Rightarrow x \leq x_{k}$ (2.3) if exists x₀∈ ℵ such that x₀ ≤ f (x₀) then x₀ = f (x₀), that means x₀ is fixed point

Proof. Can see [19].

Theorem 2.2. In addition to the assumptions in Theorem 2.1, suppose that f :ℵ → ℵ is a not decreasing monotonic function in metric space ℵ. If the sequence (x_k) _k∈N does not increase and convergence ${(x_{k})}_{k \in ℵ} \Rightarrow x \leq x_{k}$ (2.4) or f :ℵ → ℵ is continuous, then if exists x₀∈ ℵ such that x₀ ≤ f (x₀) then x₀ = f (x₀), that means x₀ is fixed point.

Proof. Can see [19].

Corollary 2.4. Assume that the real continuous function f : (ℵ , d) → (ℵ , d) in partially ordered complete metric space ℵ satisfies conditions in Theorem 2.1, Theorem 2.2 and $lim_{k \to \infty} f^{k} (x) = x_{0}$ (2.5) then x₀ is uniqueness fixed point.

Proof. Can see [19].

3 Main results

3.1 The initial valued problems for delay interval-valued differential equations (IVP for DIDEs) under generalized Hukuhara differentiability

Definition 3.1. In $K_{C} (ℝ),$ we consider the initial valued problems for delay interval-valued differential equations (IVP for DIDEs) under generalized Hukuhara differentiability: ${\begin{matrix} D_{H}^{g} X (t) = F (t, X (t), X (t - τ)), t_{0} - τ \leq t_{0} \leq t \\ X_{t_{0}} (τ) = φ (t_{0} - τ) = φ_{0}, τ \in [0, σ] \end{matrix}$ (3.1) where for σ > 0, p > 0, let $C_{σ} = C ([t_{0} - σ, t_{0}], K_{C} (ℝ))$ denote the space of continuous mappings from [t₀ - σ, t₀] to $K_{C} (ℝ),$ the function $F : [t_{0}, t_{0} + p] \times K_{C} (ℝ) \times C_{σ} \to K_{C} (ℝ), φ \in C_{σ} .$ In addition, we denote J = [t₀ - σ, t₀] ∪ [t₀, t₀ + p] = [t₀ - σ, t + p]

Lemma 3.1. Assume that $X \in C (J, K_{C} (ℝ))$ is a local solution of (3.1) on [t₀, t₀ + p] then it is expressed as interval-valued integral equations as follows: ${\begin{matrix} X_{t_{0}} (τ) = φ (t_{0} - τ) = {|φ}_{0}, τ \in [0, σ], \\ X_{t} (τ) = φ_{0} + \int_{t_{0}}^{t} F (s, X (s), X_{s}) ds, \\ t_{0} - τ \leq t_{0} \leq t, \end{matrix}$ (3.2) if $X \in C ([t_{0}, t_{0} + p], K_{C} (ℝ))$ is (H^g1)-differentiable and ${\begin{matrix} X_{t_{0}} (τ) = φ (t_{0} - τ) = φ_{0}, τ \in [0, σ], \\ X_{t} (τ) = φ_{0} ⊖ (- 1) \int_{t_{0}}^{t} F (s, X (s), X_{s}) ds, \\ t_{0} - τ \leq t_{0} \leq t, \end{matrix}$ (3.3) if $X \in C ([t_{0}, t_{0} + p], K_{C} (ℝ))$ is (H^g2)-differentiable.

Proof. It is obtained by the generalized Hukuhara diffrfrentiability (H^g1) and (H^g2) of interval-valued function $X \in C (J, K_{C} (ℝ)) .$ For example, for the case, when interval-valued function $X \in C (J, K_{C} (ℝ))$ is (H^g2)-differentiable, then $D_{H}^{g} X (t)$ is integrable and $X (t_{0}) = X (t) + (- 1) \int_{t_{0}}^{t} D_{H}^{g} X (s) ds,$ from 3.1, we get $X_{t} (τ) = φ (0) ⊖ (- 1) \int_{t_{0}}^{t} F (s, X (s), X_{s}) ds, t \in [t_{0}, t_{0} + p],$ with X_{t
₀} (τ) = φ (t₀ - τ) = φ₀, t₀ - τ ≤ t₀ ≤ t .

Definition 3.2. Let X (t) is a solution of the initial valued problems for delay interval-valued differential equations (IVP for DIDEs) under generalized Hukuhara differentiability. From Lemma 3.1, a/ if X (t) is (H^g1)-differentiable then X (t) is a solution of interval-valued integral equation: ${\begin{matrix} X_{t_{0}} (τ) = φ (t_{0} - τ) = φ_{0}, τ \in [0, σ], \\ X_{t} (τ) = φ_{0} + \int_{t_{0}}^{t} F (s, X (s), X_{s}) ds, \\ t_{0} - τ \leq t_{0} \leq t, \end{matrix}$ (3.4) b/ if X (t) is (H^g2)-differentiable then X (t) is a solution of interval-valued integral equation: ${\begin{matrix} X_{t_{0}} (τ) = φ (t_{0} - τ) = φ_{0}, τ \in [0, σ], \\ X_{t} (τ) = φ_{0} ⊖ (- 1) \int_{t_{0}}^{t} F (s, X (s), X_{s}) ds, \\ t_{0} - τ \leq t_{0} \leq t, \end{matrix}$ (3.5)

Definition 3.3. With p > 0, σ > 0 assume that $X (t), Y (t) \in C ([t_{0} - σ, t_{0} + p], K_{C} (ℝ))$ , we denote a metric: $H [X, Y] = \max_{t \in [t_{0} - σ, t_{0} + p]} {| \underline{Χ} (t) - \underline{Y} (t) |, | \bar{X} (t) - \bar{Y} (t) |}$ (3.6)

Definition 3.4. The local solution of IVP for DIDEs (3.1) $X (t) \in C ([t_{0} - σ, t_{0} + p], K_{C} (ℝ))$ is unique on [t₀ - σ, t₀ + p] if for any the other solution $Y (t) \in C ([t_{0} - σ, t_{0} + p], K_{C} (ℝ))$ satisfies H [X (t) , Y (t)] = 0 on [t₀ - σ, t₀ + p] .

Definition 3.5. Let $X (t), Y (t) \in C ([t_{0}, t_{0} + p], K_{C} (ℝ)),$ we denote a metric: $H_{C} [X, Y] = \max_{t \in [t_{0}, t_{0} + p]} {| \underline{Χ} (t) - \underline{Y} (t) |, | \bar{X} (t) - \bar{Y} (t) |}$ (3.7)

Definition 3.6. Assume that $X (t), Y (t) \in C_{σ} = C (t_{0} - σ, t_{0}], K_{C} (ℝ))$ , we denote metric: $H_{σ} [X, Y] = \max_{t \in [t_{0} - σ, t_{0}]} {| \underline{Χ} (t) - \underline{Y} (t) |, | \bar{X} (t) - \bar{Y} (t) |}$ (3.8)

Definition 3.7. Assume that $X (t), Y (t) \in C ([t_{0} - σ, t_{0} + p], K_{C} (ℝ))$ , we denote new metric: $H_{k} [X, Y] = \max_{t \in [t_{0} - σ, t_{0} + p]} {H [X (t) - Y (t)] \exp [- k (t + σ)]}$ (3.9)

Corollary 3.1. For all $X, Y \in C ([t_{0} - σ, t_{0} + p], K_{C} (ℝ))$ we have $H [X (t), Y (t)] \leq H_{k} [X, Y] {\exp [k (t + σ)]}, \forall t \geq t_{0} - σ$ and $H [X (0), Y (0)] \leq H_{k} [X, Y] {\exp [k (σ)]}, t_{0} - σ \leq 0$

Proof. We have these results from Definition 3.7

3.2 The weakly contractive mappings in partially ordered metric space and existence of a unique delay solution

Definition 3.8. The interval-valued functions $X, Y : [t_{0}, t_{0} + p] \to K_{C} (ℝ)$ are called partially ordered X ⪯ Y or (X≥Y) , if and only if satisfies one of the followings:

X ⪯ Y if $\underline{Y} (t) \leq \underline{X} (t), \bar{X} (t) \leq \bar{Y} (t), \forall t \in [t_{0}, t_{0} + p];$

X ⪰Y if $\underline{Y} (t) \geq \underline{X} (t), \bar{X} (t) \geq \bar{Y} (t), \forall t \in [t_{0}, t_{0} + p] .$

with metric between two interval-valued functions

X (t), Y (t) \in C ([t_{0}, t_{0} + p], K_{C} (ℝ))

denoted by

H_{C} [X (t), Y (t)] = {∥ X (t) \underset{gH}{⊖} Y (t) ∥}_{C}

(3.10) where

{∥ X (t) ∥}_{C} = sup_{t \in [t_{0}, t_{0} + p]} H [X (t), 0] .

Definition 3.9. We denote distance translation function for the functions ψ, γ : [0, + ∞) → [0, + ∞) by $\begin{matrix} ψ (H_{C} [X, Y]) = sup_{t \in [t_{0}, t_{0} + p]} {L . \int_{t_{0}}^{t} H_{C} [X (s), Y (s)] ds}, \\ γ (H_{σ} [X_{t}, Y_{t}]) = sup_{t \in [t_{0} - σ, t_{0}]} {L . \int_{t_{0} - σ}^{t} H_{σ} [X_{s}, Y_{s}] ds} . \end{matrix}$ (3.11)

Remark. The real functions ψ (H_C [X, Y]) and γ (H_σ [X, Y]) are the distance translation functions because ψ, γ : [0, + ∞) → [0, + ∞) in Definition 3.9 are two continuous increasing function and ψ (0) =0, γ (0) =0 only when H_C [X, Y] =0, H_σ [X, Y] =0 .

Theorem 3.1. Assume that $F \in C (ℝ_{+} \times K_{C} (ℝ) \times K_{C} (ℝ), K_{C} (ℝ))$ and F (t, A, B) is monotone increasing interval-valued function with t ∈ [t₀, t₀ + p] such that if A≥C and B≥D then F (t, A, B)≥F (t, C, D) . Furthermore, F (t, A, B) is a weakly contractive mapping in partially ordered metric space such that satisfies: $ψ (H [F (t, A, B), F (t, C, D)]) \leq {ψ (H_{C} [A, C]) +$ $ψ (H_{σ} [B, D])} - {γ (H_{C} [A, C]) + γ (H_{σ} [B, D])}, (*)$ where ψ and γ are distance translation functions in Definition 2.6 and Definition 3.3 for interval-valued function F (t, A, B) then:

H [F (t, X (t) , X_t) , F (t, Y (t) , Y_t)] ≤ L (H_C [X (t) , Y (t)] + H_σ [X_t, Y_t]) ,

the delay interval-valued differential equations under generalized Hukuhara differentiability - DIDEs (3.1) has a unique solution on [t₀ - σ, t₀ + p] .

Proof. a/ if the interval-valued functions X and Y belong to $C ([t_{0} - σ, t_{0} + p], K_{C} (ℝ))$ . We have: ${ψ (H_{C} [X (t), Y (t)] + ψ (H_{σ} [X_{t}, Y_{t}])} -$ ${γ (H_{C} [X (t), Y (t)] + γ (H_{σ} [X_{t}, Y_{t}])}$ $\leq ψ (H_{C} [X (t), Y (t)] + ψ (H_{σ} [X_{t}, Y_{t}]) +$ $γ (H_{C} [X (t), Y (t)] + γ (H_{σ} [X_{t}, Y_{t}])$ $\leq L . H_{C} [X (t), Y (t)] + L . H_{σ} [X_{t}, Y_{t}],$ then for every X ⪰ Y on [t₀ - σ, t₀ + p] . We have $ψ (H [F (t, X (t), X_{t}), F (t, Y (t), Y_{t})]) \leq$ $L (H_{C} [X (t), Y (t)] + H_{σ} [X_{t}, Y_{t}]) .$ Assume that the formula a/ is wrong then for X ⪰ Y on [t₀ - σ, t₀ + p] that means: $H [F (t, X (t), X_{t}), F (t, Y (t), Y_{t})] \geq$ $H_{C} [X (t), Y (t)] + H_{σ} [X_{t}, Y_{t}],$ such that implies $ψ (H [F (t, X (t), X_{t}), F (t, Y (t), Y_{t})]) \geq$ $ψ (H_{C} [X (t), Y (t)]) + ψ (H_{σ} [X_{s}, Y_{s}]) .$ From (*) we have ψ (H_C [X (t) , Y (t)]) + ψ (H_σ [X_t, Y_t]) = ψ (H [F (t, X (t) , X_t) , F (t, Y (t) , Y_t)]) for every X ⪰ Y . From (*) and 0 ≤ - γ (H_C [X (t) , Y (t)]) - γ (H_σ [X_t, Y_t]) , implies that $γ (H_{C} [X (t), Y (t)] + H_{σ} [X_{t}, Y_{t}]) = 0 .$ If γ is a distance translation function by Definition 3.9, then we have H [X (t) , Y (t)] = H [X_t, Y_t] =0 for all X ⪰ Y, that implies H [F (t, X (t) , X_t) , F (t, Y (t) , Y_t)] =0 and $ψ (H_{C} [X (t), Y (t)]) + γ (H_{σ} [X_{t}, Y_{t}])$ $\leq ψ (H [F (t, X (t), X_{t}), F (t, Y (t), Y_{t})]),$ what contradicts the notion of distance change function ψ, γ . Finally, we have H [F (t, X (t) , X_t) , F (t, Y (t) , Y_t)] ≤ L (H_C [X (t) , Y (t)] + H_σ [X_t, Y_t]) .

b/ if the interval-valued functions X and Y belong to $C ([t_{0} - σ, t_{0} + p], K_{C} (ℝ))$ . We prove that (3.1) exists the unique solution.

Putting $𝕏 (t) : = X (t) \underset{gH}{⊖} φ_{0}, t \in [t_{0} - σ, t_{0} + p],$ we denote a operator $ℚ : C ([t_{0} - σ, t_{0} + p], K_{C} (ℝ)) \to C ([t_{0} - σ, t_{0} + p], K_{C} (ℝ)),$ which is $(ℚ 𝕏) (t) = φ (t - t_{0}) \underset{gH}{⊖} φ_{0}, t \in [t_{0} - σ, t_{0}),$ $(ℚ 𝕏) (t) = \int_{t_{0}}^{t} F (s, X (s), X_{s}) ds, t \in [t_{0}, t_{0} + p] .$ Choosing Y ⪰ X on [t₀ - σ, t₀ + p] (Y_s ⪰ X_s, ∀ s ∈ [t₀, t₀ + p]) such that for all t ∈ [t₀ - σ, t₀] , we have $(ℚ 𝕏) (t) = φ (t - t_{0}) \underset{gH}{⊖} φ_{0} = (ℚ 𝕐) (t),$ then $ℚ 𝕐 ⪰ ℚ 𝕏$ with Y ⪰ X on [t₀ - σ, t₀ + p] , implies that operator $ℚ$ is not decreased. On the other hand, we have $H [(ℚ 𝕏) (t), (ℚ 𝕐) (t)] =$ $H [φ (t - t_{0}) \underset{gH}{⊖} φ_{0}, φ (t - t_{0}) \underset{gH}{⊖} φ_{0}] = 0$ for all X ⪰ Y on [t₀ - σ, t₀] . We consider all X ⪰ Y on [t₀, t₀ + p] : $H [(ℚ 𝕏) (t), (ℚ 𝕐) (t)] =$ $H [\int_{t_{0}}^{t} F (s, X (s), X_{s}) ds, \int_{t_{0}}^{t} F (s, Y (s), Y_{s}) ds]$ $\leq L . \int_{t_{0}}^{t} (H_{C} [X (s), Y (s)] + H_{σ} [X_{s}, Y_{s}]) ds$ $= ψ (H_{C} [X (t), Y (t)]) + γ (H_{σ} [X_{t}, Y_{t}]) .$ With L > 0, we have the operator $ℚ$ is contractive that means exists a fixed point $𝕏$ in $C ([t_{0} - σ, t_{0} + p], K_{C} (ℝ)),$ thus the delay interval-valued differential equations under generalized Hukuhara differentiability - DIDEs (3.1) has a unique solution on [t₀ - σ, t₀ + p] .

3.3 Existence and uniqueness of solution by the extremal solutions and the weakly contractive mappings in partially ordered metric space

Definition 3.10. The monotone increasing interval-valued functions $X^{L} (t) \in C ([t_{0}, t_{0} + p], K_{C} (ℝ))$ with $len (X^{L}) = {\bar{X}}^{L} (t) - {\underline{X}}^{L} (t)$ is called a lower solution on [t₀, t₀ + p] to IVP for DIDEs (3.1) if and only if it is ${\begin{matrix} X_{t_{0}}^{L} (τ) = ξ (t_{0} - τ) ⪯ φ (t_{0} - τ) = φ_{0}, t_{0} - τ \leq t_{0} \leq t \\ X_{t}^{L} (τ) \underset{gH}{⊖} φ_{0} ⪯ \int_{t_{0}}^{t} F (s, X (s), X_{s}) ds, t \in [t_{0}, t_{0} + p], \end{matrix}$ (3.12) where ξ (t₀ - τ) ∈ C_σ, t₀ ≥ τ ≥ t₀ - σ .

Definition 3.11.The monotone increasing interval-valued functions $X^{U} (t) \in C ([t_{0}, t_{0} + p], K_{C} (ℝ))$ with $len (X^{U}) = {\bar{X}}^{U} (t) - {\underline{X}}^{U} (t)$ is called a upper solution on [t₀, t₀ + p] to IVP for DIDEs (3.1), if and only if it is ${\begin{matrix} X_{t_{0}}^{U} (τ) = ξ (t_{0} - τ) ⪰ φ (t_{0} - τ) = φ_{0}, t_{0} - τ \leq t_{0} \leq t \\ X_{t}^{U} (τ) \underset{gH}{⊖} φ_{0} ⪰ \int_{t_{0}}^{t} F (s, X (s), X_{s}) ds, t \in [t_{0}, t_{0} + p], \end{matrix}$ (3.13) where ξ (t₀ - τ) ∈ C_σ, t₀ ≥ τ ≥ t₀ - σ .

Remark 3.1. We can repeat this proof for the case, when existence a upper solution X^U (t) $\in C ([t_{0}, t_{0} + p], K_{C} (ℝ))$ on [t₀, t₀ + p] with monotone $len (X^{U}) = {\bar{X}}^{U} (t) - {\underline{X}}^{U} (t),$ then we prove that $𝕏^{U} ⪰ ℚ 𝕏^{U} .$

Definition 3.12. The interval-value function $X^{L} (t) \in C ([t_{0} - σ, t_{0} + p], K_{C} (ℝ))] \cap C ([t_{0}, t_{0} + p], K_{C} (ℝ))$ with monotone $len (X^{L}) = {\bar{X}}^{L} (t) - {\underline{X}}^{L} (t)$ is the lower solution of IVP for DIDEs (3.1) if ${\begin{matrix} X_{t}^{L} (τ) \underset{gH}{⊖} φ_{0} ⪯ \int_{t_{0}}^{t} F (s, X (s), X_{s}) ds, t \in [t_{0}, t_{0} + p] \\ X_{t_{0}}^{L} (τ) = ξ (t_{0} - τ) ⪯ φ (t_{0} - τ) = φ_{0}, t_{0} \geq τ \geq t_{0} - σ \end{matrix}$ (3.14) where ξ (t₀ - τ) ∈ C_σ, t₀ ≥ τ ≥ t₀ - σ .

Definition 3.13. The interval-value function $X^{U} (t) \in C [[t_{0} - σ, t_{0} + p], K_{C} (ℝ)] \cap C^{1, F} [[t_{0}, t_{0} + p], K_{C} (ℝ)]$ with monotone $len (X^{U}) = {\bar{X}}^{U} (t) - {\underline{X}}^{U} (t)$ is the upper solution of IVP for DIDEs (3.1), if: ${\begin{matrix} X_{t}^{U} (τ) \underset{gH}{⊖} φ_{0} ⪰ \int_{t_{0}}^{t} (F (s, X (s), X_{s}) ds, t \in [t_{0}, t_{0} + p] \\ X_{t_{0}}^{U} (τ) = ξ (t_{0} - τ) ⪰ φ (t_{0} - τ) = φ_{0}, t_{0} \geq τ \geq t_{0} - σ \end{matrix}$ (3.15) where ξ (t₀ - τ) ∈ C_σ, t₀ ≥ τ ≥ t₀ - σ .

Theorem 3.2 Assume that $F \in C (ℝ_{+} \times K_{C} (ℝ) \times K_{C} (ℝ) \to K_{C} (ℝ))$ and F (t, A, B) is monotone increasing interval-valued functions on t ∈ [t₀, t₀ + p] such that if A ⪰ C and B ⪰ D then F (t, A, B) ⪰ F (t, C, D) . Furthermore, F (t, A, B) is weakly contractive mapping in partially ordered metric space and hold the followings:

if existence a (H^g1) lower and a (H^g1) upper solutions: $X^{L} (t), X^{U} (t) \in C ([t_{0} - σ, t_{0} + p], K_{C} (ℝ)) \cap C^{1, F} ([t_{0}, t_{0} + p], K_{C} (ℝ))$ on [t₀, t₀ + p] with monotone $len (X (t)) = \bar{X} (t) - \underline{X} (t)$ ;

or if existence a (H^g2) lower and a (H^g2) upper solutions: $Y^{L} (t), Y^{U} (t) \in C ([t_{0} - σ, t_{0} + p], K_{C} (ℝ)) \cap C^{1, F} ([t_{0}, t_{0} + p], K_{C} (ℝ))$ on [t₀, t₀ + p] with monotone $len (Y (t)) = \bar{Y} (t) - \underline{Y} (t)$ ,

then the delay interval-valued differential equations under generalized Hukuhara differentiability - DIDEs (3.1) has a unique solution on [t₀ - σ, t₀ + p] that it is one of the type in (3.2)-(3.3).

Proof. It is obtained by generalized Hukuhara diffrfrentiability (H^g1) and (H^g2) of interval-valued function $X \in C (J, K_{C} (ℝ)) .$ For example, we consider condition (A2) for the case, when interval-valued function $X \in C (J, K_{C} (ℝ))$ is (H^g2)-differentiable then $D_{H}^{g} X (t)$ is integrable and $X (t_{0}) = X (t) + (- 1) \int_{t_{0}}^{t} D_{H}^{g} X (s) ds,$ and by Lemma 3.1 it becomes $X_{t} (τ) = φ_{0} ⊖ (- 1) \int_{t_{0}}^{t} (F (s, X (s), X_{s}) ds t \in [t_{0}, t_{0} + p],$ with X_{t
₀} (τ) = φ (t₀ - τ) = φ₀, t₀ ≥ τ ≥ t₀ - σ .

Similarly to prove Theorem 3.1, putting $ℙ : C ([t_{0} - σ, t_{0} + p], K_{C} (ℝ)) \to$ $C ([t_{0} - σ, t_{0} + p], K_{C} (ℝ))$ which is $(ℙ 𝕐) (t) = φ (t - t_{0}) \underset{gH}{⊖} φ_{0}, t \in [t_{0} - σ, t_{0}],$ $(ℙ 𝕐) (t) = \int_{t_{0}}^{t} F (s, Y (s), Y_{s}) ds, t \in [t_{0}, t_{0} + p] .$ such that exists a generalized Hukuhara difference on [t₀, t₀ + p] . If we have Y ⪯ Z, we obtain: $(ℙ Y) (t) = φ_{0} ⊖ (- 1) \int_{t_{0}}^{t} F (s, Y (s), Y_{s}) ds \leq$ $φ_{0} ⊖ (- 1) \int_{t_{0}}^{t} F (s, Z (s), Z_{s}) ds = (ℙ Z) (t), t \in [t_{0}, t_{0} + p],$ implies that operator $ℙ$ is not decreased. By condition (*), we have $H [F (t, Y (t), Y_{t}), F (t, Z (t), Z_{t})]$ $\leq H_{C} [Y (t), Z (t)] + H_{σ} [Y_{t}, Z_{t}] .$ Because Y ⪯ X, then $H_{k} [ℙ Y, ℙ Z] \leq (1 / k^{α})$ H_k [Y, Z] and implies $ψ (H_{k} [ℙ Y, ℙ Z]) \leq ψ (H_{k} [ℙ Y, ℙ Z]) - γ (H_{k} [ℙ Y, ℙ Z]),$ where $γ (H_{k} [ℙ Y, ℙ Z]) = ψ (H_{k} [ℙ Y, ℙ Z]) - ψ ((1 / k^{α}) H_{k} [ℙ Y, ℙ Z]) .$ Finally, we have existence of extremal (H^g2)-solution: $\begin{matrix} Y^{L} (t) = ψ_{0} ⊖ (- 1) \int_{t_{0}}^{t} {D_{H}}^{g} Y^{L} (s) ds \\ \leq φ_{0} ⊖ \int_{t_{0}}^{t} F (s, Y^{L} (s), Y_{s}^{L}) ds = (ℙ Y^{L}) (t), \\ t \in [t_{0}, t_{0} + p] . \end{matrix}$ It implies that $Y^{L} \leq ℙ Y^{L}$ and $ℙ$ is weakly contractive mapping in partially ordered metric space $C ([t_{0} - σ, t_{0} + p], K_{C} (ℝ))$ that means exists a fixed point Y and it is an unique solution (H^g2) of DIDEs (3.1).

4 Illustrations

Example 4.1. we consider the initial valued problem for delay interval-valued differential equation under generalized Hukuhara differentialbility (IVP for DIDEs): ${\begin{matrix} D_{H}^{g} X (t) = λ X (t - τ), t \in [0, 1] \\ X (0) = φ (- τ) = [4 - τ, 6 - τ], τ \in [0, 1], λ \in ℝ^{+} \end{matrix}$ (4.1)

In this example, we solve the Problem (4.1) on [0, 1].

a/ Case 1, by detailed calculation, we get a solution, which is (H^g1)-differentiable, of problem (4.1) on [0, 1], $X_{t} (τ) = [λ t [4 - \frac{t}{2} + τ] + 4, λ t [6 - \frac{t}{2} + τ] + 6],$ with X_{t
₀} (τ) = [4 - τ, 6 - τ] , τ ∈ [0, 1] and its the picture illustrations presented in Fig. 1.

b/ Case 2, we get a solution, which is (H^g2)-differentiable, of problem (4.1) on [0, 1], $X_{t} (τ) = [λ t [6 - \frac{t}{2} + τ] + 4, λ t [4 - \frac{t}{2} + τ] + 6],$ with X_{t
₀} (τ) = [4 - τ, 6 - τ] , τ ∈ [0, 1] and its the picture illustrations presented in Fig. 2.

Fig. 1.

(H^g1)-solution to IVP for DIDEs (4.1) (when τ = 1, λ = 0, 5, t ∈ [-1, 1]).

Fig. 2.

(H^g2)-solution to IVP for DIDEs (4.1) (when τ = 1, λ = 0.5, t ∈ [-1, 1]).

Example 4.2. we consider the initial valued problem for delay interval-valued differential equation under generalized Hukuhara differentialbility (IVP for DIDEs):

${\begin{matrix} D_{H}^{g} X (t) = - \frac{2 λ}{3} X (\frac{t}{2}) - \frac{λ}{3} X (t - τ), t \in [0, 2], λ \in ℝ \\ X (t_{0}) = φ (t_{0} - τ), τ \in [0, 1], φ (t_{0} - τ) = [- 1, 1] . \end{matrix}$ (4.2)

For ease of calculation for this example, we give τ = 1 and t₀ = 0 in the problem (4.2). Then the problem is rewritten as follows

${\begin{matrix} D_{H}^{g} X (t) = - \frac{2 λ}{3} X (\frac{t}{2}) - \frac{λ}{3} X (t - 1), t \in [0, 2], \\ X (0) = φ (- 1), φ (- 1) = [- 1, 1], λ \in ℝ \end{matrix}$ (4.3)a/Case 1, when coefficient λ < 0 : A solution of problem (4.3) is interval-valued function, which is (H^g1)-differentiable. We find the (H^g1)-solution with delay condition X (0) = [-1, 1] , λ < 0 is $\begin{matrix} X_{t} (τ = 1) = \\ {\begin{matrix} [- 1 - \frac{λ}{2} t, 1 + \frac{λ}{2} t], t \in [0, 1] \\ [- 1 - \frac{λ}{2} t - \frac{λ {(t - 1)}^{2}}{3}, 1 + \frac{λ}{2} t + \frac{λ {(t - 1)}^{2}}{3}], \\ t \in [1, 2], \end{matrix} \end{matrix}$ its the picture illustrations presented in Fig. 3 (with τ = 1, λ = -1). b/Case 2, when coefficient λ > 0 : A solution of problem (4.3) is interval-valued function, which is H^g2-differentiable. We find the H^g2-solution with delay condition X (0) = [-1, 1] , λ < 0 is $\begin{matrix} X_{t} (τ = 1) = \\ {\begin{matrix} [- 1 + \frac{λ}{2} t, 1 - \frac{λ}{2} t], t \in [0, 1] \\ [- 1 + \frac{λ}{2} t - \frac{λ {(t - 1)}^{2}}{3}, 1 - \frac{λ}{2} t + \frac{λ {(t - 1)}^{2}}{3}], \\ t \in [1, 2], \end{matrix} \end{matrix}$ its the picture illustrations presented in Fig. 4 (with τ = λ = 1).

Fig. 3.

(H^g1)-solution to IVP for DIDEs (4.3) (with τ = 1, λ = -1, t₀ = 0, t ∈ [-1, 2]).

Fig. 4.

(H^g2)-solution to IVP for DIDEs (4.3) (when τ = λ = 1, t₀ = 0, t ∈ [-1, 2]).

5 Conclusion

There is always a close relationship between the class of the fuzzy differential equations (FDEs) - the fuzzy systems (FSs) and interval-valued differential equations (IDEs), so the mathematicians draw the attention to study the class of IDEs. The initial valued problem for delay interval-valued differential equations under generalized Hukuhara differentiability (IVP for DIDEs) (3.1) is very most different from IVP for interval-valued differential equations under generalized Hukuhara differentiability (IVP for IDEs) (1.1). This paper presented developed the concept of partially ordered metric space of interval-valued functions and the weakly contractive mappings in partially ordered metric space. The existence and uniqueness of the solution to IVP for DIDEs (1.2) are obtained by weakly contractive mappings and by the extremal solutions in partially ordered metric space.

Footnotes

Acknowledgements

The authors would like to express their gratitude to the anonymous referees for their helpful comments and suggestions.

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