On the solutions for box differential equations under generalized Hukuhara derivative

Abstract

In this paper, the box differential equations under generalized Hukuhara derivatives are considered. We study the existence and uniqueness results for box integral equations employing the method of upper and lower solutions. Moreover, the approach is followed to prove the existence of solutions for the initial value problem of box differential equations. The method is illustrated by some examples.

1 Introduction

The set-valued differential and integral equations are an important part of the theory of set-valued analysis, and they have the high value for control theory and its application. They were first studied in 1969 by De Blasi and Iervolino [1]. Recently, set-valued differential equations have been studied by many scientists due to their applications in many areas. For many results in the theory of set-valued differential and integral equations, the readers are referred to the following books and papers [2–10] and references therein.

In many real-word problems, it is desirable to transforms the behavior of a special phenomenon into a deterministic initial value problem of linear systems of differential equations, namely,

$$X^{\prime }(t)=AX(t)+F(t),X(0)=X_0,$$

(1.1)

where $A=(a_{ij})\in M^n(\mathbb{R})$, $X=(x_1,x_2,\dots ,x_n),X_0=(x_{10},x_{20},\dots ,x_{n0})\in {\mathbb{R}}^n$, $t\in [0,T]$, and $F=(f_1,f_2,\dots ,f_n):[0,T]\to {\mathbb{R}}^n$. However, the model is not usually perfect due to the lack of certain information of the initial value $X_0$, matrix A or F, 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 special cases of set-valued analysis and set-valued differential equations, respectively. In many cases, 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 understanding of the full models. The interval-valued differential equations can be used to model dynamical systems subjected to uncertainties. The papers of Stefanini and Bede [11, 12], Malinowski [13, 14], and An et al. [15, 16] are focused on the interval-valued differential equations. These equations can be studied with a framework of the Hukuhara derivative [17]. However, because of this the solutions have increasing length of their values. Furthermore, we notice that this definition of Hukuhara derivative is very restrictive; for instance we can see that if $X(t)=C\cdot x(t)$, where C is an interval-valued constant and $x:[a,b]\to {\mathbb{R}}_{+}$ is a real-valued function with $x^{\prime }(t)<0$, then $X(t)$ is not differentiable. Recently, to avoid this difficulty, Stefanini and Bede [11] proposed to consider a so-called generalized derivative of interval-valued functions. The interval-valued differential equations with this derivative can have solutions with decreasing length of their values, but the uniqueness is lost. The paper of Stefanini and Bede was the starting point for the topic of interval-valued differential equations (see Malinowski [13, 14], An et al. [15]) and later also for fuzzy differential equations. The connection between the fuzzy analysis and the interval analysis is very well known (Moore and Lodwick [18]). 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. The main theoretical and practical results in the fields of fuzzy analysis and the interval analysis can be found in several works (Moore [19], Alefeld and Mayer [20], Kolev [21]). Furthermore, some very important extensions of the interval-valued differential equations are the set-valued differential equations [8, 9, 22], the fuzzy differential equations [23–32], the random fuzzy differential equations [33–37], and fuzzy stochastic differential equations [38–42]; and some methods for solving fuzzy differential equations are presented e.g. in [43–50].

Based on the extensions of (1.1) for the interval cases, in this paper, we study the initial value problem of interval-valued nonlinear systems of differential equations under generalized Hukuhara differentiability, i.e., box-valued differential equations of the form

$$D_{\mathrm{gH}}X(t)=F(t,X(t)),X(0)=X_0\in {\mathbb{I}}^n,t\in [0,T],$$

(1.2)

where $X=(X_1,X_2,\dots ,X_n),X_0=(X_{10},X_{20},\dots ,X_{n0})\in {\mathbb{I}}^n$, $t\in [0,T]$ and $F=(F_1,F_2,\dots ,F_n):[0,T]\times {\mathbb{I}}^n\to {\mathbb{I}}^n$ (, is the space of interval-valued functions). In [11, 13–15, 51–53] the authors studied box-valued differential equations under generalized Hukuhara differentiability for the case $n=1$. The existence and uniqueness of an initial value problem are then obtained under an assumption that the coefficients satisfy a condition with the Lipschitz constant (see [11]). The proof is based on the application of the Banach fixed point theorem. In [14], under the generalized Lipschitz condition, Malinowski obtained the existence and uniqueness of solutions to both kinds of IDEs. In this paper, we study the existence and uniqueness of solutions of two forms of the box integral equations. We employ a the method of upper and lower solutions. The first one leads to the solutions which possess the trajectories with nonincreasing diameter of their values. In the second form the solutions have the trajectories with nondecreasing diameter of their values. Moreover, the approach is applied to prove the existence of solutions for the box-valued differential equations under generalized Hukuhara derivative (1.2).

This paper is organized as follows: In Section 2, we present some needed concepts about differentiation and integration of box-valued functions. In Section 3, we study the existence and uniqueness theorems of solutions for two general forms of box integral equations by the method of upper and lower solutions. We use these results in order to investigate the existence results of solutions for the initial value problem of box differential equations. Finally, we give some examples being simple illustrations of the theory of box-valued differential equations.

2 Preliminaries

Let us denote by the set of any nonempty compact intervals of the real line ℝ. The addition and scalar multiplication in , are defined as usual, i.e. for $A,B\in \mathbb{I}$, $A=[\underline{A},\overline{A}]$, $B=[\underline{B},\overline{B}]$, where $\underline{A}\le \overline{A}$, $\underline{B}\le \overline{B}$, and $\lambda \ge 0$, then we have

$$A+B=[\underline{A}+\underline{B},\overline{A}+\overline{B}],\lambda A=[\lambda \underline{A},\lambda \overline{A}](-\lambda A=[-\lambda \overline{A},-\lambda \underline{A}]).$$

Furthermore, let $A\in \mathbb{I}$, ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4,\in R$ and ${\lambda }_3{\lambda }_4\ge 0$, then we have ${\lambda }_1({\lambda }_{2}A)=({\lambda }_1{\lambda }_2)A$ and $({\lambda }_3+{\lambda }_4)A={\lambda }_{3}A+{\lambda }_{4}A$. Let $A,B\in \mathbb{I}$ as above, the Hausdorff metric H in is defined as follows:

$$H[A,B]=max\{|\underline{A}-\underline{B}|,|\overline{A}-\overline{B}|\}.$$

(2.1)

We notice that $(\mathbb{I},H)$ is a complete, separable and locally compact metric space. The magnitude and the length of $A\in \mathbb{I}$ are defined by

$$H[A,\{0\}]=\parallel A\parallel =max\{|\underline{A}|,|\overline{A}|\},len(A)=\overline{A}-\underline{A},$$

respectively, where $\{0\}$ is the zero element of , which is regarded as one point.

The generalized Hukuhara difference (or gH-difference) of two intervals $A=[\underline{A},\overline{A}]$, $B=[\underline{B},\overline{B}]$ is defined as follows:

$$A{\ominus }_{\mathrm{gH}}B=[\underline{A},\overline{A}]{\ominus }_{\mathrm{gH}}[\underline{B},\overline{B}]=[min\{\underline{A}-\underline{B},\overline{A}-\overline{B}\},max\{\underline{A}-\underline{B},\overline{A}-\overline{B}\}].$$

(2.2)

In the present paper, we introduce the so-called boxes and box-valued functions. Let now ${\mathbb{I}}^n=\mathbb{I}\times \mathbb{I}\times \cdots \times \mathbb{I}$ (n times) (the Cartesian product) denote the family of axis-aligned boxes (boxes for short) of the ${\mathbb{R}}^n$. An element $A\in {\mathbb{I}}^n$ is of the form $A=[{\underline{A}}_1,{\overline{A}}_1]\times [{\underline{A}}_2,{\overline{A}}_2]\times \cdots \times [{\underline{A}}_n,{\overline{A}}_n]={\prod }_{i=1}^n{A}_i$, for short, where $A_i=[{\underline{A}}_i,{\overline{A}}_i]\in \mathbb{I}$ are real compact intervals.

If $A={\prod }_{i=1}^n[{\underline{A}}_i,{\overline{A}}_i]$, $B={\prod }_{i=1}^n[{\underline{B}}_i,{\overline{B}}_i]$ in ${\mathbb{I}}^n$ where ${\underline{A}}_i\le {\overline{A}}_i$, ${\underline{B}}_i\le {\overline{B}}_i$, then the operations on the boxes are defined by

$$A+B=\prod _{i=1}^n[{\underline{A}}_i+{\underline{B}}_i,{\overline{A}}_i+{\overline{B}}_i]\text{and}\lambda A=\{\begin{array}{ll}{\prod }_{i=1}^n[\lambda {\underline{A}}_i,\lambda {\overline{A}}_i],& \text{if }\lambda >0,\\ \{0\},& \text{if }\lambda =0,\\ {\prod }_{i=1}^n[\lambda {\overline{A}}_i,\lambda {\underline{A}}_i],& \text{if }\lambda <0.\end{array}$$

The metric structure is given by the Hausdorff distance $H_B:{\mathbb{I}}^n\times {\mathbb{I}}^n\to {\mathbb{R}}^{+}\cup \{0\}$,

$$H_B[A,B]={(\sum _{i=1}^n{H}^2[A_i,B_i])}^{1/2}={\left(\sum _{i=1}^n{\{max[|{\underline{A}}_i-{\underline{B}}_i|,|{\overline{A}}_i-{\overline{B}}_i|]\}}^2\right)}^{1/2}.$$

(2.3)

Based on the properties of $(\mathbb{I},H)$, see [11, 12], we deduce that $({\mathbb{I}}^n,H_B)$ is a complete, separable metric space. For the metric $H_B$ the following properties hold:

$$\begin{array}{c}{H}_B[A+B,C+D]\le H_B[A,C]+H_B[B,D],\hfill \\ H_B[\lambda A,\lambda B]=|\lambda |H_B[A,B]\hfill \end{array}$$

for every $A,B,C,D\in {\mathbb{I}}^n$, and every $\lambda \in \mathbb{R}$.

The gH-difference of two boxes $A={\prod }_{i=1}^n{A}_i$, $B={\prod }_{i=1}^n{B}_i$ is defined as follows:

$$A{\ominus }_{\mathrm{gH}}B=\{\begin{array}{ll}{\prod }_{i=1}^n[{\underline{A}}_i-{\underline{B}}_i,{\overline{A}}_i-{\overline{B}}_i],& \text{if }len(A_i)\ge len(B_i),\mathrm{\forall }i,\\ {\prod }_{i=1}^n[{\overline{A}}_i-{\overline{B}}_i,{\underline{A}}_i-{\underline{B}}_i],& \text{if }len(A_i)<len(B_i),\mathrm{\forall }i.\end{array}$$

If $A,B,C\in {\mathbb{I}}^n$ then it is easy to see that

$$A{\ominus }_{\mathrm{gH}}B=C\iff \{\begin{array}{ll}A=B+C,& \text{if }len(A_i)\ge len(B_i),\mathrm{\forall }i,\\ B=A+(-C),& \text{if }len(A_i)<len(B_i),\mathrm{\forall }i.\end{array}$$

Proposition 2.1 Let $A={\prod }_{i=1}^n{A}_i,B={\prod }_{i=1}^n{B}_i\in {\mathbb{I}}^n$. The gH-difference $A{\ominus }_{\mathrm{gH}}B$ exists if and only if one of the two conditions is satisfied:

• (i) $len(A_i)\ge len(B_i)$, $i=1,2,\dots ,n$,

or

• (ii) $len(A_i)\le len(B_i)$, $i=1,2,\dots ,n$.

Based on the gH-difference and the gH-derivative of interval-valued functions (see [11]) we propose the following.

Definition 2.1 Let $t_0\in [a,b]$ and h be such that $t_0+h\in [a,b]$, the gH-derivative of a function $X:[a,b]\to {\mathbb{I}}^n$ at $t_0$ is defined as

$$D_{\mathrm{gH}}X(t_0)=\underset{h\to 0}{lim}\frac{1}{h}[X(t_0+h){\ominus }_{\mathrm{gH}}X(t_0)].$$

(2.4)

If $D_{\mathrm{gH}}X(t_0)\in {\mathbb{I}}^n$ satisfying (2.4) exists, we say that X is generalized Hukuhara differentiable (gH-differentiable for short) at $t_0$.

Let us remark that the gH-derivative exists at $t_0$ if and only if the left and right derivatives at $t_0$ exist and they are equal. We say that X is gH-differentiable on $[a,b]$ if $D_{\mathrm{gH}}X(t)\in {\mathbb{I}}^n$ exists at each point $t\in [a,b]$. At the end points of $[a,b]$ we consider only the one sided gH-derivatives.

Definition 2.2 Let $X:[a,b]\to {\mathbb{I}}^n$ be such that $X(t)={\prod }_{i=1}^n{X}_i(t)={\prod }_{i=1}^n[{\underline{X}}_i(t),{\overline{X}}_i(t)]$ and gH-differentiable at $t_0\in [a,b]$. We say that X is ((i)-gH)-differentiable at $t_0$ if

• (i)
  $$D_{\mathrm{gH}}X(t_0)=\prod _{i=1}^n[\frac{d}{dt}{\underline{X}}_i(t_0),\frac{d}{dt}{\overline{X}}_i(t_0)]$$

  (2.5)

and that X is ((ii)-gH)-differentiable at $t_0$ if

• (ii)
  $$D_{\mathrm{gH}}X(t_0)=\prod _{i=1}^n[\frac{d}{dt}{\overline{X}}_i(t_0),\frac{d}{dt}{\underline{X}}_i(t_0)].$$

  (2.6)

Definition 2.3 Let $X:[a,b]\to {\mathbb{I}}^n$ be such that $X(t)={\prod }_{i=1}^n{X}_i(t)={\prod }_{i=1}^n[{\underline{X}}_i(t),{\overline{X}}_i(t)]$ and ${\underline{X}}_i(t)$, ${\overline{X}}_i(t)$, $i\in \{1,2,\dots ,n\}$ are measurable and Lebesgue integrable on $[a,b]$. Then the integral of box-valued function X is defined as usually:

$${\int }_a^{b}X(t)dt=\prod _{i=1}^n[{\int }_a^b{\underline{X}}_i(t)dt,{\int }_a^b{\overline{X}}_i(t)dt].$$

We say that X is Lebesgue integrable on $[a,b]$. It is easy to show that the box-valued function X is integrable on $[a,b]$ if and only if X is measurable and $H_B[X(t),\{0\}]$ is Lebesgue integrable on $[a,b]$.

In this paper, we use the following partial ordering (see [7]).

Definition 2.4 Suppose $A={\prod }_{i=1}^n[{\underline{A}}_i,{\overline{A}}_i]$, $B={\prod }_{i=1}^n[{\underline{B}}_i,{\overline{B}}_i]$. We say that $A\le B$ if and only if ${\underline{A}}_i\le {\underline{B}}_i$ and ${\overline{A}}_i\le {\overline{B}}_i$, for $i=1,\dots ,n$.

We can also define the interval $[A,B]=\{C\in {\mathbb{I}}^n:A\le C\le B\}$. Moreover, letting $X,Y\in C([a,b],{\mathbb{I}}^n)$ be two box-valued functions, we say that $X\le Y$ if $X(t)\le Y(t)$ for $t\in [a,b]$. We define $[X(t),Y(t)]=\{Z\in C([a,b],{\mathbb{I}}^n):X(t)\le Z(t)\le Y(t)\}$.

We recall some properties on the partial ordering ≤ the space of box-valued functions, which are useful for our procedure.

Proposition 2.2 Let $A,B,C,D\in {\mathbb{I}}^n$ and $c\in {\mathbb{R}}_{+}$,

• (i) $A=B$ if and only if $A\le B$ and $A\ge B$;

• (ii) if $A\le B$, then $A+C\le B+C$;

• (iii) if $A\le B$ and $C\le D$, then $A+C\le B+D$;

• (iv) if $A\le B$, then $cA\le cB$.

Lemma 2.1 ([26])

Let $F,G,H\in C([0,T],{\mathbb{I}}^n)$ and $X_0\in {\mathbb{I}}^n$ and also $G\le H$, then

• (i) ${\int }_0^{t}G(s)ds\le {\int }_0^{t}H(s)ds$, $t\in [0,T]$;

• (ii) $X_0\ominus (-1){\int }_0^{t}G(s)ds\le X_0\ominus (-1){\int }_0^{t}H(s)ds$, $t\in [0,T]$;

• (iii) if $G(t)\le F(t)\le H(t)$, $t\in [0,T]$, then ${\int }_0^{t}F(s)ds$ is well defined on $[0,T]$.

The following theorem is similar to the result proved in [11].

Theorem 2.1 Let $X:[a,b]\to {\mathbb{I}}^n$ be ((i)-gH)-differentiable or ((ii)-gH)-differentiable on $[a,b]$, and assume that the derivative $D_{\mathrm{gH}}X$ is integrable over $[a,b]$. We have

• (a) if X be ((i)-gH)-differentiable on $[a,b]$, then ${\int }_a^b{D}_{\mathrm{gH}}X(t)dt=X(b)\ominus X(a)$;

• (b) if X be ((ii)-gH)-differentiable on $[a,b]$, then ${\int }_a^b{D}_{\mathrm{gH}}X(t)dt=(-1)(X(a)\ominus X(b))$.

3 Box differential equations

In the sequel, we study the existence and uniqueness theorems of solutions for two general forms of box integral equations by the method of upper and lower solutions. We use these results in order to investigate the existence results of solutions for the initial value problem of box differential equations.

We consider two forms of the following box integral equations:

$$X(t)=X_0+{\int }_0^{t}F(s,X(s))ds,t\in [0,T]$$

(3.1)

and

$$Y(t)=Y_0\ominus (-1){\int }_0^{t}F(s,Y(s))ds,t\in [0,T],$$

(3.2)

where $F\in C([0,T]\times {\mathbb{I}}^n,{\mathbb{I}}^n)$ and the integral of F is integral from Definition 2.3.

The solutions of equations (3.1) and (3.2) have no crisp elements and different geometrical properties.

Remark 3.1 If $X:[0,T]\to {\mathbb{I}}^n$ is a solution to (3.1), then the function $t\mapsto diam(X(t))$ is nondecreasing, where $diam(Z)$ denotes the diameter of the box Z.

Remark 3.2 If $Y:[0,T]\to {\mathbb{I}}^n$ is a solution to (3.2), then the function $t\mapsto diam(Y(t))$ is nonincreasing.

Definition 3.1 Let $X_{\mathrm{U}},X_{\mathrm{L}},Y_{\mathrm{U}},Y_{\mathrm{L}}\in C([0,T],{\mathbb{I}}^n)$, and $F(t,X_{\mathrm{U}}(t))$, $F(t,X_{\mathrm{L}}(t))$, $F(t,Y_{\mathrm{U}}(t))$, $F(t,Y_{\mathrm{L}}(t))$ are integrable on $[0,T]$, we say that

• (i) $X_{\mathrm{U}}$, $X_{\mathrm{L}}$ are upper and lower solutions for (3.1), respectively, if

  $$X_{\mathrm{U}}(t)\ge X_0+{\int }_0^{t}F(s,X_{\mathrm{U}}(s))ds,t\in [0,T]$$

and

$$X_{\mathrm{L}}(t)\le X_0+{\int }_0^{t}F(s,X_{\mathrm{L}}(s))ds,t\in [0,T];$$

• (ii) $Y_{\mathrm{U}}$, $Y_{\mathrm{L}}$ are upper and lower solutions for (3.2), respectively, if

  $$Y_{\mathrm{U}}(t)\ge Y_0\ominus (-1){\int }_0^{t}F(s,Y_{\mathrm{U}}(s))ds,t\in [0,T]$$

and

$$Y_{\mathrm{L}}(t)\le Y_0\ominus (-1){\int }_0^{t}F(s,Y_{\mathrm{L}}(s))ds,t\in [0,T],$$

provided the involved Hukuhara differences are well defined.

Definition 3.2 Let $X_{\mathrm{L}}$, $X_{\mathrm{U}}$ be lower and upper solutions for the box integral equations (3.1), $[X_{\mathrm{L}},X_{\mathrm{U}}]=\{X\in C([a,b],{\mathbb{I}}^n):X_{\mathrm{L}}\le X\le X_{\mathrm{U}}\}$. We say that $X\in [X_{\mathrm{L}},X_{\mathrm{U}}]$ is a solution for equation (3.1), if X satisfies equation (3.1). Moreover, the solution $X_{\max }\in [X_{\mathrm{L}},X_{\mathrm{U}}]$ is a maximal solution of equation (3.1), if

$$X_{\max }(t)\ge X(t),t\in [0,T]$$

for any solution $X\in [X_{\mathrm{L}},X_{\mathrm{U}}]$. We define a minimal solution $X_{\min }\in [X_{\mathrm{L}},X_{\mathrm{U}}]$ as a function satisfying the reverse inequalities.

Definition 3.3 Let $Y_{\mathrm{L}}$, $Y_{\mathrm{U}}$ be lower and upper solutions for the set integral equations (3.2), $[Y_{\mathrm{L}},Y_{\mathrm{U}}]=\{Y\in C([a,b],{\mathbb{I}}^n):Y_{\mathrm{L}}\le Y\le Y_{\mathrm{U}}\}$. We say that $Y\in [Y_{\mathrm{L}},Y_{\mathrm{U}}]$ is a solution for equation (3.2), if Y satisfies equation (3.2). Moreover, the solution $Y_{\max }\in [Y_{\mathrm{L}},Y_{\mathrm{U}}]$ is a maximal solution of equation (3.2), if

$$Y_{\max }(t)\ge Y(t),t\in [0,T]$$

for any solution $Y\in [Y_{\mathrm{L}},Y_{\mathrm{U}}]$. We define a minimal solution $Y_{\min }\in [Y_{\mathrm{L}},Y_{\mathrm{U}}]$ as a function satisfying the reverse inequalities.

Theorem 3.1 Assume that

• (i) $X_{\mathrm{L}},X_{\mathrm{U}}\in C([0,T],{\mathbb{I}}^n)$ are coupled upper and lower solutions of the box integral equations (3.1) such that $X_{\mathrm{L}}\le X_{\mathrm{U}}$, $t\in [0,T]$;

• (ii) $F\in C([0,T]\times {\mathbb{I}}^n,{\mathbb{I}}^n)$, $F(t,X)$ is nondecreasing in $X\in [X_{\mathrm{L}},X_{\mathrm{U}}]$ for all $t\in [0,T]$.

Then there exist maximal and minimal solutions $X_{\max },X_{\min }\in [X_{\mathrm{L}},X_{\mathrm{U}}]$ for equation (3.1).

Proof Let us construct two sequences by

$$V_{n+1}(t)=X_0+{\int }_0^{t}F(s,V_n(s))ds,V_0=X_{\mathrm{L}},t\in [0,T]$$

(3.3)

and

$$W_{n+1}(t)=X_0+{\int }_0^{t}F(s,W_n(s))ds,W_0=X_{\mathrm{U}},t\in [0,T].$$

(3.4)

We claim that $V_n$ and $W_n$ are well defined, and

$$\begin{array}{rcl}{X}_{\mathrm{L}}(t)& =& V_0(t)\le V_1(t)\le \cdots \le V_n(t)\le \cdots \le W_n(t)\le \cdots \\ \le & W_1(t)\le W_0(t)=X_{\mathrm{U}}(t),t\in [0,T].\end{array}$$

(3.5)

Indeed, from equation (3.3) we get for $n=0$

$$V_1(t)=X_0+{\int }_0^{t}F(s,V_0(s))ds.$$

According to the integrability of F, $V_1$ is well defined. Since $V_0=X_{\mathrm{L}}$, by Definition 3.1, we get

$$V_0(t)\le V_1(t),t\in [0,T].$$

Moreover, $X_{\mathrm{L}}(t)\le X_{\mathrm{U}}(t)$, $F(t,X)$ is nondecreasing in X, we get $V_1(t)\le W_0(t)$.

As a result, we obtain

$$V_0(t)\le V_1(t)\le W_0(t),t\in [0,T].$$

We assume inductively

$$V_{n-1}(t)\le V_n(t)\le W_0(t),t\in [0,T],$$

(3.6)

and $V_k$ are well defined for $k\le n$. From equation (3.3), we find

$$V_{n+1}(t)=X_0+{\int }_0^{t}F(s,V_n(s))ds$$

and

$$V_n(t)=X_0+{\int }_0^{t}F(s,V_{n-1}(s))ds.$$

Employing the nondecreasing property of function F, along with the relation in equation (3.6), we get

$$F(t,X_{\mathrm{L}})\le F(t,V_n)\le F(t,X_{\mathrm{U}}).$$

By Lemma 2.1, $V_{n+1}$ is well defined and we get

$$V_n(t)\le V_{n+1}(t)\le W_0(t),t\in [0,T].$$

Similarly, we can prove that $V_0\le W_{n+1}\le W_n(t)\le W_0$ and $W_k$ are well defined for $k=1,2,\dots$ . Next, we prove $V_n\le W_n$, $\mathrm{\forall }n\in \mathbb{N}$. Indeed, $V_0=X_{\mathrm{L}}\le X_{\mathrm{U}}=W_0$ comes from the assumption. Now, we suppose inductively

$$V_n(t)\le W_n(t),t\in [0,T].$$

(3.7)

By equations (3.3), (3.4), and applying the fact that F is nondecreasing, equation (3.7), we deduce that

$$V_{n+1}(t)\le W_{n+1}(t),t\in [0,T].$$

Since the sequence of functions $V_n(t)$ is monotone nondecreasing and bounded from above, and the sequence of functions $W_n(t)$ is monotone nonincreasing and bounded from below, the pointwise limits exist, and these limits are denoted by $X_{\min }$ and $X_{\max }$ such that they satisfy

$$\underset{n\to \mathrm{\infty }}{lim}{V}_n(t)=X_{\min }(t),\underset{n\to \mathrm{\infty }}{lim}{W}_n(t)=X_{\max }(t),t\in [0,T].$$

Next, taking the limits as $n\to \mathrm{\infty }$ in equations (3.3), (3.4), using continuity and monotone nondecreasing of F, we infer that

$$X_{\min }(t)=X_0+{\int }_0^{t}F(s,X_{\min }(s))ds,t\in [0,T]$$

and

$$X_{\max }(t)=X_0+{\int }_0^{t}F(s,X_{\max }(s))ds,t\in [0,T].$$

Moreover, by equation (3.5), we get

$$V_0\le X_{\min }\le X_{\max }\le W_0,t\in [0,T].$$

(3.8)

Now, we claim that $X_{\min }$, $X_{\max }$ are minimal and maximal solutions of equation (3.1), that is, if $X(t)$ is any solution of equation (3.1) such that

$$V_0\le X(t)\le W_0,t\in [0,T],$$

then

$$V_0\le X_{\min }\le X(t)\le X_{\max }\le W_0,t\in [0,T].$$

(3.9)

Suppose that for some n,

$$V_n\le X(t)\le W_n,t\in [0,T].$$

By using the monotone nondecreasing property of F and Lemma 2.1, we get

$$V_{n+1}(t)=X_0+{\int }_0^{t}F(s,V_n(s))ds\le X_0+{\int }_0^{t}F(s,X(t)(s))ds=X(t),t\in [0,T].$$

Similarly $W_{n+1}\ge X(t)$, $t\in [0,T]$. Hence, by induction, equation (3.8) is true for all $n\in \mathbb{N}$. Now, taking the limit as $n\to \mathrm{\infty }$, we get equation (3.9). The proof is complete. □

Corollary 3.1 If, in addition to the assumptions of Theorem  3.1, F satisfies, whenever $Y\ge Z$; $Y,Z\in {\mathbb{I}}^n$,

$$F(t,Y)+LZ\le F(t,Z)+LY,L>0,$$

(3.10)

then $X_{\min }=X=X_{\max }$ is the unique solution of equation (3.1).

Proof We have $X_{\min }={\prod }_{i=1}^n[{\underline{X}}_{i,\min },{\overline{X}}_{i,\min }]$, $X_{\max }={\prod }_{i=1}^n[{\underline{X}}_{i,\max },{\overline{X}}_{i,\max }]$.

Since $X_{\min }\le X_{\max }$, by Definition 2.3, we get $X_{\max }=X_{\min }+R$, where $R={\prod }_{i=1}^n[{\underline{R}}_i,{\overline{R}}_i]$; ${\underline{R}}_i,{\overline{R}}_i\ge 0$, for $i=1,\dots ,n$. By assumption (3.10), we have

$$\begin{array}{c}\prod _{i=1}^n[{\underline{X}}_{i,\min }(t),{\overline{X}}_{i,\min }(t)]+\prod _{i=1}^n[{\underline{R}}_i(t),{\overline{R}}_i(t)]\hfill \\ =\prod _{i=1}^n[{\underline{X}}_{i,\max }(t),{\overline{X}}_{i,\max }(t)]\hfill \\ =\prod _{i=1}^n[{\underline{X}}_{i0},{\overline{X}}_{i0}]+\prod _{i=1}^n[{\int }_0^t{\underline{F}}_i(s,X_{i,\max }(s))ds,{\int }_0^t{\overline{F}}_i(s,X_{i,\max }(s))ds]\hfill \\ \le \prod _{i=1}^n[{\underline{X}}_{i0},{\overline{X}}_{i0}]+\prod _{i=1}^n[{\int }_0^t{\underline{F}}_i(s,X_{i,\min }(s))ds,{\int }_0^t{\overline{F}}_i(s,X_{i,\min }(s))ds]\hfill \\ +L\prod _{i=1}^n[{\int }_0^t{\underline{R}}_i(s)ds,{\int }_0^t{\overline{R}}_i(s)ds]\hfill \\ =\prod _{i=1}^n[{\underline{X}}_{i,\min },{\overline{X}}_{i,\min }]+L\prod _{i=1}^n[{\int }_0^t{\underline{R}}_i(s)ds,{\int }_0^t{\overline{R}}_i(s)ds],\hfill \end{array}$$

and hence

$${\underline{R}}_i(t)\le L{\int }_{t_0}^t{\underline{R}}_i(s)ds,{\overline{R}}_i(t)\le L{\int }_{t_0}^t{\overline{R}}_i(s)ds,i=1,\dots ,n.$$

The Gronwall inequality implies that ${\underline{R}}_i(t)\le 0$, ${\overline{R}}_i(t)\le 0$ and as a result, ${\underline{R}}_i(t)=0$, ${\overline{R}}_i(t)=0$ for $i=1,\dots ,n$. Therefore, $X_{\min }=X=X_{\max }$ is the unique solution. □

Theorem 3.2 Assume that

• (i) $Y_{\mathrm{U}},Y_{\mathrm{L}}\in C([0,T],{\mathbb{I}}^n)$ are coupled upper and lower solutions of the box integral equations (3.2) such that $Y_{\mathrm{L}}\le Y_{\mathrm{U}}$, $t\in [0,T]$;

• (ii) $F\in C([0,T]\times {\mathbb{I}}^n,{\mathbb{I}}^n)$, $F(t,Y)$ is nondecreasing in $Y\in [Y_{\mathrm{L}},Y_{\mathrm{U}}]$ for all $t\in [0,T]$;

• (iii) the sequences

  $$\begin{array}{c}{V}_0=Y_{\mathrm{L}},V_{n+1}(t)=X_0\ominus (-1){\int }_0^{t}F(s,V_n(s))ds,\hfill \\ W_0=Y_{\mathrm{U}},W_{n+1}(t)=X_0\ominus (-1){\int }_0^{t}F(s,W_n(s))ds\hfill \end{array}$$

for all $t\in [0,T]$ are well defined.

Then there exist maximal and minimal solutions $Y_{\max },Y_{\min }\in [Y_{\mathrm{L}},Y_{\mathrm{U}}]$ for equation (3.2).

Corollary 3.2 If, in addition to the assumptions of Theorem  3.2, F satisfies, whenever $Y\ge Z$, $Y,Z\in {\mathbb{I}}^n$,

$$F(t,Y)\le F(t,Z)+L(Y\ominus Z),L>0,$$

then $Y_{\min }=Y=Y_{\max }$ is the unique solution of equation (3.2).

Proof The proof is similar Theorem 3.1 and Corollary 3.1 by employing Lemma 2.1. □

In the sequel, we study the box-valued differential equations using the concept of generalized Hukuhara differentiability. In this setting, we prove the existence of two solutions, each one corresponding to a different type of differentiability.

We consider the following box-valued differential equation (BDE) with the initial condition:

$$D_{\mathrm{gH}}X(t)=F(t,X(t)),X(0)=X_0\in {\mathbb{I}}^n,t\in [0,T],$$

(3.11)

where $F:[0,T]\times {\mathbb{I}}^n\to {\mathbb{I}}^n$ is continuous box-valued function and $X_0\in {\mathbb{I}}^n$ is a nontrivial box. In this paper, we consider only ((i)-gH)-differentiable type and ((ii)-gH)-differentiable type solutions, i.e. such that there are no switching points in $[0,T]$. Let us remark here that the equation $D_{\mathrm{gH}}X(t)=F(t,X(t))$ is equivalent to $D_{\mathrm{gH}}X(t){\ominus }_{\mathrm{gH}}F(t,X(t))=\{0\}$ and also equivalent to $F(t,X(t)){\ominus }_{\mathrm{gH}}{D}_{\mathrm{gH}}X(t)=\{0\}$, but it is not equivalent to $D_{\mathrm{gH}}X(t)-F(t,X(t))=\{0\}$ (here the ‘−’ is in the usual interval arithmetic sense).

The following lemma is similar to the result proved in [11].

Lemma 3.1 The box-valued differential equation (3.11) is equivalent to the following integral equation:

$$X(t){\ominus }_{\mathrm{gH}}X(0)={\int }_0^{t}F(s,X(s))ds,t\in [0,T].$$

(3.12)

Two cases of existence of the gH-difference imply that the integral equation in Lemma 3.1 is actually a unified formulation for one the integral equations

$$X(t)\ominus X(0)={\int }_0^{t}F(s,X(s))ds\text{and}X(0)\ominus X(t)=(-1){\int }_0^{t}F(s,X(s))ds,$$

with ⊖ being the classical Hukuhara difference. Now, let X and Y be solutions of equation (3.11) in ((i)-gH)-differentiable type and ((ii)-gH)-differentiable type, respectively, then by using Lemma 3.1 and Theorem 2.1, we have

$$X(t)=X_0+{\int }_0^{t}F(s,X(s))ds,$$

(3.13)

$$Y(t)=X_0\ominus (-1){\int }_0^{t}F(s,Y(s))ds.$$

(3.14)

Definition 3.4 Let $X:I\to {\mathbb{I}}^n$ be a box-valued function which is ((i)-gH)-differentiable. If X and its derivative satisfy problem (3.11), we say that X is a (i)-solution of problem (3.11).

Definition 3.5 Let $Y:I\to {\mathbb{I}}^n$ be a box-valued function which is ((ii)-gH)-differentiable. If Y and its derivative satisfy problem (3.11), we say that Y is a (ii)-solution of problem (3.11).

In [14] Malinowski proved the existence and uniqueness of solutions to interval-valued differential equations by using generalized Lipschitz conditions. In [11] authors proved the existence and uniqueness of solutions to interval-valued differential equations by using the Banach fixed point theorem. In the sequel, we study the existence and uniqueness results for box integral and differential equations employing the method of upper and lower solutions.

Theorem 3.3 Assume that

• (i) $X_{\mathrm{U}},X_{\mathrm{L}}\in C([0,T],{\mathbb{I}}^n)$ are coupled upper and lower solutions of the box integral equations (3.13) such that $X_{\mathrm{L}}(t)\le X_{\mathrm{U}}(t)$, $t\in [0,T]$;

• (ii) $F\in C([0,T]\times {\mathbb{I}}^n,{\mathbb{I}}^n)$, $F(t,X)$ is nondecreasing in $X\in [X_{\mathrm{L}},X_{\mathrm{U}}]$ for all $t\in [0,T]$.

Then there exists at least one solution $X\in [X_{\mathrm{L}},X_{\mathrm{U}}]$ for equation (3.11), which is ((i)-gH)-differentiable.

Theorem 3.4 Assume that

• (i) $Y_{\mathrm{U}},Y_{\mathrm{L}}\in C([0,T],{\mathbb{I}}^n)$ are coupled upper and lower solutions of the box integral equations (3.14) such that $Y_{\mathrm{L}}(t)\le Y_{\mathrm{U}}(t)$, $t\in [0,T]$;

• (ii) $F\in C([0,T]\times {\mathbb{I}}^n,{\mathbb{I}}^n)$, $F(t,Y)$ is nondecreasing in $Y\in [Y_{\mathrm{L}},Y_{\mathrm{U}}]$ for all $t\in [0,T]$;

• (iii) the sequences

  $$\begin{array}{c}{V}_0=Y_{\mathrm{L}},V_{n+1}(t)=Y_0\ominus (-1){\int }_{t_0}^{t}F(s,V_n(s))ds,\hfill \\ W_0=Y_{\mathrm{U}},W_{n+1}(t)=Y_0\ominus (-1){\int }_{t_0}^{t}F(s,W_n(s))ds\hfill \end{array}$$

for all $t\in [0,T]$ are well defined.

Then there exists at least one solution $Y\in [Y_{\mathrm{L}},Y_{\mathrm{U}}]$ for equation (3.11), which is ((ii)-gH)-differentiable.

Proof Since all the conditions of Theorem 3.2 are fulfilled, there exists at least a solution, say X, for the integral (3.13). We wish to show that X is a solution for equation (3.11).

For small $h>0$, we have

$$\begin{array}{c}X(t+h)+(-1){\int }_t^{t+h}F(s,X(s))ds\hfill \\ ={\int }_0^{t+h}F(s,X(s))ds+(-1){\int }_t^{t+h}F(s,X(s))ds\hfill \\ ={\int }_0^{t}F(s,X(s))ds+{\int }_t^{t+h}F(s,X(s))ds+(-1){\int }_t^{t+h}F(s,X(s))ds\hfill \\ ={\int }_0^{t}F(s,X(s))ds=X(t).\hfill \end{array}$$

Hence, we obtain

$$X(t)\ominus X(t+h)=(-1){\int }_t^{t+h}F(s,X(s))ds.$$

Moreover, we observe that

$$\begin{array}{rl}{H}_B[\frac{1}{h}{\int }_t^{t+h}F(s,X(s))ds,F(t,X(t))]& =H_B[\frac{1}{h}{\int }_t^{t+h}F(s,X(s))ds,\frac{1}{h}{\int }_t^{t+h}F(t,X(t))]\\ \le \frac{1}{h}{\int }_t^{t+h}{H}_B[F(s,X(s)),F(t,X(t))]ds\\ \le \underset{|s-t|\le h}{sup}{H}_B[F(s,X(s)),F(t,X(t))]\end{array}$$

and since F is continuous, for $h\searrow 0$ the last term ↘0 which means that

$$\underset{h\searrow 0}{lim}\frac{X(t)\ominus X(t+h)}{-h}=F(t,X(t)).$$

Similarly,

$$\underset{h\searrow 0}{lim}\frac{X(t-h)\ominus X(t)}{-h}=\underset{h\searrow 0}{lim}\frac{1}{h}{\int }_{t-h}^{t}F(s,X(s))ds=F(t,X(t)).$$

It follows that X is ((ii)-gH)-differentiable and is a solution of equation (3.11). □

Example 3.5 Consider the box-valued differential equations with initial value:

$$D_{\mathrm{gH}}X(t)=(1+2t)X(t),X(0)=[-1,1]\in {\mathbb{I}}^1,t\in [0,1].$$

(3.15)

Case 1: X is ((i)-gH)-differentiable, equation (3.15) is equivalent to

$$\{\begin{array}{l}{\underline{X}}^{\prime }(t)=(1+2t)\underline{X}(t),\\ {\overline{X}}^{\prime }(t)=(1+2t)\overline{X}(t),\\ \underline{X}(0)=-1,\\ \overline{X}(0)=1.\end{array}$$

(3.16)

We get the lower solution $X_{\mathrm{L}}(t)=[-e^t,e^t]$ and upper solution $X_{\mathrm{U}}(t)=[-e^{3t},e^{3t}]$ for the integral equation (3.1) corresponding to problem (3.15), respectively. It is easy to see that all the conditions of Theorem 3.3 are fulfilled and conclusion, there exists a solution for this problem. By solving the above system, we obtain the following exact solution $X(t)=[-e^{t(t+1)},e^{t(t+1)}]$. This solution is shown in Figure 1.

Figure 1

Exact solution of Example 3.5 , Case 1 ( $\mathit{n}\mathbf{=}\mathbf{1}$ ).

Case 2: X is ((ii)-gH)-differentiable, equation (3.15) is equivalent to

$$\{\begin{array}{l}{\underline{X}}^{\prime }(t)=(1+2t)\overline{X}(t),\\ {\overline{X}}^{\prime }(t)=(1+2t)\underline{X}(t),\\ \underline{X}(0)=-1,\\ \overline{X}(0)=1.\end{array}$$

(3.17)

We get the lower solution $X_{\mathrm{L}}(t)=[-cosh(t)+sinh(t),cosh(t)-sinh(t)]$ and upper solution $X_{\mathrm{U}}(t)=[-cosh(3t)+sinh(3t),cosh(3t)-sinh(3t)]$ for the integral equation (3.2) corresponding to problem (3.15), respectively. It is easy to see that all the conditions of Theorem 3.4 are fulfilled and conclusion, there exists a solution for this problem. By solving the above system, we obtain the following exact solution: $X(t)=[-cosh(t^2+t)+sinh(t^2+t),cosh(t^2+t)-sinh(t^2+t)]$. This solution is shown in Figure 2.

Figure 2

Exact solution of Example 3.5 , Case 2 ( $\mathit{n}\mathbf{=}\mathbf{1}$ ).

Example 3.6 Consider the box-valued differential equations with initial value:

$$D_{\mathrm{gH}}X(t)=(1+sin(t))X(t),X(0)=\prod _{i=1}^2[0,1+\frac{1}{i}]\in {\mathbb{I}}^2,t\in [0,\pi ].$$

(3.18)

Case 1: X is ((i)-gH)-differentiable, equation (3.18) is equivalent to

$$\{\begin{array}{l}{\underline{X}}_i^{\prime }(t)=(1+sin(t)){\underline{X}}_i(t),i\in \{1,2\},\\ {\overline{X}}_i^{\prime }(t)=(1+sin(t)){\overline{X}}_i(t),i\in \{1,2\},\\ {\underline{X}}_i(0)=0,i\in \{1,2\},\\ {\overline{X}}_i(0)=1+\frac{1}{i},i\in \{1,2\}.\end{array}$$

(3.19)

Similarly as in Example 3.5 we get the lower solution $X_{\mathrm{L}}(t)={\prod }_{i=1}^2[0,(1+\frac{1}{i})e^t]$ and upper solution $X_{\mathrm{U}}(t)={\prod }_{i=1}^2[0,(1+\frac{1}{i})e^{2t}]$. Moreover, the conditions of Theorem 3.3 are fulfilled and conclusion, there exists a solution for this problem. By solving the above system, we obtain the following exact solution: $X(t)={\prod }_{i=1}^2[0,(1+\frac{1}{i})e^{t+1-cos(t)}]$. This solution is shown in Figures 3 and 4.

Figure 3

Exact solution of Example 3.6 , Case 1 ( $\mathit{n}\mathbf{=}\mathbf{2}$ ) ( $\mathit{i}\mathbf{=}\mathbf{1}$ ).

Figure 4

Exact solution of Example 3.6 , Case 1 ( $\mathit{n}\mathbf{=}\mathbf{2}$ ) ( $\mathit{i}\mathbf{=}\mathbf{2}$ ).

Case 2: X is ((ii)-gH)-differentiable, equation (3.18) is equivalent to

$$\{\begin{array}{l}{\underline{X}}_i^{\prime }(t)=(1+sin(t)){\overline{X}}_i(t),i\in \{1,2\},\\ {\overline{X}}_i^{\prime }(t)=(1+sin(t)){\underline{X}}_i(t),i\in \{1,2\},\\ {\underline{X}}_i(0)=0,i\in \{1,2\},\\ {\overline{X}}_i(0)=1+\frac{1}{i},i\in \{1,2\}.\end{array}$$

(3.20)

By solving the above system, we obtain the following exact solution:

$$X(t)=\prod _{i=1}^2[(1+\frac{1}{i})sinh(t+1-cos(t)),(1+\frac{1}{i})cosh(t+1-cos(t))].$$

This solution is shown in Figures 5 and 6.

Figure 5

Exact solution of Example 3.6 , Case 2 ( $\mathit{n}\mathbf{=}\mathbf{2}$ ) ( $\mathit{i}\mathbf{=}\mathbf{1}$ ).

Figure 6

Exact solution of Example 3.6 , Case 2 ( $\mathit{n}\mathbf{=}\mathbf{2}$ ) ( $\mathit{i}\mathbf{=}\mathbf{2}$ ).

4 Conclusion

Using the method of upper and lower solutions, an existence result of solutions for box integral and differential equations is studied. Also, the monotone iterative technique is developed and the existence results for maximal and minimal solutions are obtained.