The Upper and Lower Solution Method for a Class of Interval Boundary Value Problems

: In this paper, the upper and lower solution method is proposed in order to solve the second order interval boundary value problem. We study ﬁrst a class of linear interval boundary value problems and then investigate a class of nonlinear interval boundary value problems by the upper and lower solution method under the gH-derivative, and we prove that there exist at least two solutions.


Introduction
In the process of mathematical modeling for solving problems, the initial data or parameter values are often uncertain due to measurement error. People often express these data and parameters as an interval number or fuzzy number. 1979, Markov proposed the interval-valued calculus [1]. This paper remained essentially un-cited for more than 30 years and was "rediscovered" after the publication of [2][3][4]. Stefanini considered a generalization of the Hukuhara difference and division for interval arithmetic and generalized Hukuhara differentiability of interval-valued functions and interval differential equations.
Recently, the interest for this topic increased significantly, in particular after the implementation of specific tools and classes in the C++ and Julia (among others) programming languages, or in computational systems, such as MATLAB or Mathematica [5]. The research activity in the calculus for interval-valued or set-valued functions is now very extended, particularly in connection with the more general calculus for fuzzy-valued functions with applications to almost all fields of applied mathematics [6][7][8].
Interval-valued differential equations are introduced as a good tool to study nonprobabilistic uncertainty in real world phenomena. 2009, Stefanini and Bede studied several kinds of derivatives of an interval-valued function, and provided some properties of solutions to interval-valued differential equations under the gH-derivative [4]. 2011, Chalco-Cano et al. revisited the expression of the gH-derivative of an interval-valued function in terms of the endpoints functions [9]. In 2013, Lupulescu discussed the gHdifferentiability of interval-valued functions, and studied interval differential equations on time-scales [10]. In 2017, by using a Krasnoselskii-Krein-type condition, Hoa, Lupulescu and O'Regan studied the existence and uniqueness of the solutions to initial value problems of fractional interval-valued differential equations [11].
In 2018, by applying the monotone iterative technique, Hoa considered the extremal solutions to initial value problems of fractional interval-valued integro-differential equations [12]. These studies expanded the scope of the research on interval-valued differential equations.
It is well known that the upper and lower solution method is a powerful tool for the solvability of differential equation [13]. Rodríguez-López applied the upper and lower solution method to develop a monotone iterative technique to approximate extremal solutions for the initial value problem relative to a fuzzy differential equation in a fuzzy functional interval [14]. Motivated by this idea, in order to solve the nonlinear interval boundary value problem 1], we propose an upper and lower solution method and obtain at least four solutions similar to linear fuzzy boundary value problems.
In what follows, we introduce some preliminaries, in Section 3, we study a class of linear interval boundary value problems and give conditions that ensure that linear interval boundary value problems have solutions, and, in Section 4, we propose an upper and lower solution method for a class of nonlinear interval boundary value problems. In the last section, we give a example to illustrate the effectiveness of the results in this paper.

Preliminaries
In this section, we introduce some preliminaries that can be found in [7]. We denote by K C the family of all bounded closed intervals in R, i.e., The well-known midpoint-radius representation is very useful: for A = [a − , a + ], and we define the midpointsâ andã, respectively, bŷ so that a − =â −ã and a + =â +ã. We will denote the interval by A = [a − , a + ] or, in midpoint notation, by A = (â;ã); thus, The gH-difference of two intervals always exists and, in midpoint notation, is given by the gH-addition for intervals is defined by Endowed with the Pompeiu-Hausdorff distance d H : and γ − 0, γ + 0 (eventually γ − = −∞ and/or γ + = +∞), we define the following order relation, denoted γ − ,γ + , Limits and continuity can be characterized, in the Pompeiu-Hausdorff metric d H for intervals, by the gH-difference. For a function F : where the limits are in the metric d H . If, in addition, x 0 ∈ K, we have In midpoint notation, let F(x) = (f (x);f (x)) and L = (l;l); then, the limits and continuity can be expressed, respectively, as Let C I, K C be the set of all continuous interval-value functions.
Proof. It is similar to prove that

Definition 2.
Given two interval-value functions F(x) and G(x), we define the distance of F(x) and G(x) as ). It follows that f n (x) andf n (x) are two continuous functions. Similar to the proof of Theorem 1, it is easy obtain thatf n (x) andf n (x) are uniformly bounded. Hence, lim n→∞f n =f , lim n→∞f n =f .
Therefore, lim n→∞ F n = F. Definition 3. Let x 0 ∈ (a, b) and h be such that x 0 + h ∈ (a, b), and then the gH-derivative of a function F : (a, b) → K C at x 0 is defined as if the limit exists. The interval F gH (x 0 ) ∈ K C is called the generalized Hukuhara derivative of F (gH-derivative for short) at x 0 .
For a gH-differentiable function, higher-order gH-derivatives are defined analogously to the ordinary case using the gH-differences applied to the gH-derivatives of previous order.

Definition 4.
Let F : (a, b) → K C be gH-differentiable on (a, b) and x 0 ∈ (a, b) and h be such that x 0 + h ∈ (a, b). The second order gH-derivative of F(x) at x 0 is defined as if the limit exists. The interval F gH (x 0 ) ∈ K C is called the second order gH-derivative of F at x 0 .

Remark 1. In the case of an interval-valued function in the form
Let C i I, K C be the set of all i order continuous differentiable interval-value functions, i = 1, 2.

The Linear Interval Boundary Problems
In this section, we consider a class of linear interval boundary problems under the gH-derivative. Let where A, B ∈ K C , F(t) ∈ C I, K C .

Definition 5.
If U(t) ∈ C 2 I, K C and U(t) satisfies Problem (1), then we say that U(t) is a solution of Problem (1).
The following theorems concern the existence of solutions of a two-point boundary value problem of linear interval differential equation under the gH-derivative.

Proof. For convenience, assume a solution of Problem
It is clear that operator T is a continuous operator.

The Nonlinear Interval Boundary Value Problems
In what follows, we consider a class of nonlinear interval boundary problems where A, B ∈ K C , F(t, U) ∈ C I × K C , K C .

Definition 6.
If U(x) ∈ C 2 I, K C and U(x) satisfy Problem (3), then we say that U(x) is a solution of Problem (3).
We present the concept of upper and lower solution of a two-point boundary value problem of nonlinear interval differential equation under the gH-derivative in the following definition.

U(t) is said to be a solution of Problem (3) if U(t) is an upper solution and is also a lower solution of Problem (3).
The following theorems concern the existence of solutions of a two-point boundary value problem of a nonlinear interval differential equation under the gH-derivative. (3), and U(t) γ − ,γ + U(t). If F(t, U) ∈ C I × K C , K C , and F(x, V) γ − ,γ + F(x, W) when V γ − ,γ + W, then Problem (3) exists at least two solutions.

Proof. Since U(x) is a lower solution of Problem (3), then
By Theorem 4, we know that there exists V 1 (x) ∈ C 2 I, K C , which is the solution of the linear interval boundary value problem Hence, Similarly, we also find By similar reasoning, if U is an upper solution of Problem (3), and U 1 is a solution of the linear fuzzy boundary value problem We find U γ − ,γ + U 1 .
Therefore, T is a monotone operator. Since, and let V n = TV n−1 , we have A similar argument for U n = TU n−1 , we have In addition, from U γ − ,γ + U we conclude that V 1 = TU γ − ,γ + TU = U 1 , and Similarly, V n = TV n−1 γ − ,γ + TU n−1 = U n .
In conclusion, By Theorems 3 and 5, we have i.e., Problem (3) exists at least two solutions.

Example
In this section, we will give a example to illustrate the effectiveness of the results.

Conclusions
In this paper, we studied a class of linear interval boundary value problems and then investigated a class of nonlinear interval boundary value problems by the upper and lower solution method under the gH-derivative. We found that there are at least four solutions for linear interval boundary value problems and at least two solutions for nonlinear interval boundary value problems. In our next works, we will consider the interval boundary problem when F(x, U) is increasing for U.
Author Contributions: Conceptualization, Y.Y. and Z.G.; investigation, Z.X. All authors have read and agreed to the published version of the manuscript.