Boundary value problems for interval-valued differential equations on unbounded domains

doi:10.1016/j.fss.2021.03.019

Fuzzy Sets and Systems

Volume 436, 30 May 2022, Pages 102-127

https://doi.org/10.1016/j.fss.2021.03.019 Get rights and content

Abstract

By using the Banach fixed point theorem and Schauder fixed-point theorem for semilinear spaces, we study the existence of solutions to some class of boundary value problems for interval-valued differential equations on unbounded domains. Some sufficient conditions are provided in order to deduce the existence of solutions without switching points, and also for mixed solutions with a unique switching point. The influences of the range of the parameter in the boundary value condition has on the existence of solutions is also discussed. Finally, two examples are given to demonstrate the feasibility of the theorems.

Introduction

In this paper, we investigate a class of boundary value problems for the first-order interval-valued differential equations on an unbounded domain, or infinite intervals, that is, a half-line.

Interval-valued differential equations are introduced as a good tool to study non-probabilistic uncertainty in real-world phenomena. Some early studies on interval analysis and computation can be found in [10][14][15].

In 2009, L. Stefanini and B. Bede [16] 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. Then, in 2011, Y. Chalco-Cano et al. [4] revisited the expression of the gH-derivative of an interval-valued function in terms of the endpoints functions.

In 2013, V. Lupulescu [12] discussed gH-differentiability of interval-valued functions, and studied interval differential equations on time-scales. By using a Krasnoselskii-Krein-type condition, N.V. Hoa, V. Lupulescu and D. O'Regan [8] studied in 2017 the existence and uniqueness of the solutions to initial value problems of fractional interval-valued differential equations. In 2018, by applying the monotone iterative technique, N.V. Hoa [7] considered the extremal solutions to initial value problems of fractional interval-valued integro-differential equations. These studies expanded the scope of the research on interval-valued differential equations.

On the other hand, boundary value problems on unbounded domains have been extensively studied in the last twenty years. In this sense, some applications and early results can be found in [2] and the references therein. The interest of considering boundary value problems on infinite interval arises from the study of many real world problems, for example, nonlinear elliptic equation models, gas pressure analysis in a semi-infinite porous medium, etc.

In 2001, B. Yan [20] studied boundary value problems for differential equations with impulses and infinite delay on $[0, + \infty)$ . In 2003, the existence of nonnegative solutions to boundary value problems on $[0, + \infty)$ was considered in [13]. We can also find some results about singular boundary value problems in [5], [11], [17], [19] and fractional integro-differential equations on unbounded domains in [21].

Motivated by the results on boundary value problems for differential equations on unbounded domains, we consider in this paper the following boundary value problem for interval-valued differential equations. ${\begin{matrix} x^{'} (t) = f (t, x (t)), t \in (0, + \infty), \\ x (0) = α \lim_{t \to + \infty} x (t), \end{matrix}$ where $f : [0, + \infty) \times K_{C} \to K_{C}$ is continuous, being $K_{C}$ the set of all non-empty compact and convex subsets of $R$ . By applying the Banach fixed point theorem, and Schauder fixed-point theorem for semilinear spaces, we provide some sufficient conditions for the existence of solutions to the boundary value problem (1).

The paper is organized as follows. In section 2, some basic definitions and facts about intervals and interval-valued functions are included. In section 3, we consider the problem with right-hand side independent of the state $x^{'} (t) = g (t)$ , $x (0) = α \lim_{t \to + \infty} x (t)$ , where $α \in R ∖ {0, 1}$ . In section 4, the existence of solutions to the boundary value problem (1) is discussed. In section 5, some examples are provided to illustrate the theorems. Finally, conclusions section is included.

Section snippets

Preliminaries

Let $R$ be the set of real numbers, $K_{C}$ be the family of all non-empty compact and convex subsets of $R$ [16]. For every $u, v \in K_{C}$ , we set $u + v : = {a + b | a \in u, b \in v}$ , $u - v = u + (- 1) v = {a - b | a \in u, b \in v}$ , and $k u = {k a | a \in u}$ , $k \in R$ .

Definition 1

[9] Let $u, v, w \in K_{C}$ , $w = u ⊖ v$ is called the Hukuhara difference of u and v if $u = v + w$ .

Let

u = [u_{-}, u_{+}]

v = [v_{-}, v_{+}]

, then

u ⊖ v = [u_{-} - v_{-}, u_{+} - v_{+}]

u ⊖ v

exists. It is well-known that

u ⊖ v

exists if and only if

l e n (u) \geq l e n (v)

, where

l e n (u) = u_{+} - u_{-}

In the following lemma, we include some important properties of interval

Existence of solutions to $x^{'} (t) = g (t), x (0) = α \lim_{t \to + \infty} x (t)$

In this section, we consider the boundary value problem ${\begin{matrix} x^{'} (t) = g (t), t \in (0, + \infty), \\ x (0) = α \lim_{t \to + \infty} x (t), \end{matrix}$ where $g : [0, + \infty) \to K_{C}$ , and $α \in R ∖ {0}$ .

The expressions of the (I)-gH-differentiable solution, (II)-gH-differentiable solution and (I-II), (II-I) mixed solutions to this problem are obtained. These results are applied to construct the operator of integral type corresponding to the boundary value problem (1), which is fundamental for the subsequent developments.

Lemma 11

Suppose that $g : [0, + \infty) \to K_{C}$ is continuous and

Existence of solutions to $x^{'} (t) = f (t, x (t)), x (0) = α \lim_{t \to + \infty} x (t)$

In this section, we consider the existence of gH-differentiable solutions to the boundary value problem (1). Let us start with an important lemma, which introduces two conditions to guarantee the convergence of $\int_{0}^{+ \infty} f (t, x (t)) d t$ , for $x \in E$ .

Lemma 14

Suppose that $f : [0, + \infty) \times K_{C} \to K_{C}$ is continuous, and that the following conditions are satisfied.

(H1)
$γ : = \int_{0}^{+ \infty} H (f (t, {0}), {0}) d t < + \infty$ .
(H2)
There exists $p \in C ([0, + \infty), [0, + \infty))$ satisfying $\int_{0}^{+ \infty} p (t) (1 + t) d t < + \infty$ , such that $H (f (t, u), f (t, v)) \leq p (t) H (u, v), t \in [0, + \infty), u, v \in K_{C} .$

Then

\int_{0}^{+ \infty} f (t, x (t)) d t

Examples

In this section, we show some examples to verify our results.

Example 1

Consider the following interval-valued differential equation $x^{'} (t) = \frac{0.3}{{(1 + t)}^{3}} x (t) + \frac{1}{{(1 + t)}^{3}} [1, 2], t \in [0, + \infty)$ with boundary value condition $x (0) = α \lim_{t \to + \infty} x (t) .$

Let $p (t) = \frac{0.3}{{(1 + t)}^{3}}$ , $f (t, u) = p (t) \cdot u + \frac{1}{{(1 + t)}^{3}} [1, 2]$ . Then we have $f (t, {0}) = \frac{1}{{(1 + t)}^{3}} [1, 2]$ and $H (f (t, u), f (t, v)) = p (t) H (u, v)$ , for $t \in [0, + \infty)$ . Moreover, $\begin{matrix} γ = \int_{0}^{+ \infty} H (f (t, {0}), {0}) d t = \int_{0}^{+ \infty} \frac{2}{{(1 + t)}^{3}} d t = 1 < + \infty, \\ \int_{0}^{+ \infty} p (t) (1 + t) d t = \int_{0}^{+ \infty} \frac{0.3}{{(1 + t)}^{2}} d t = 0.3 < + \infty . \end{matrix}$ Hence, (H1) and (H2) hold.

(i) (I)-gH-differentiable solution.

Let $x$

Conclusions

Boundary value problems on $[0, + \infty)$ occur naturally in the study of radially symmetric solutions of nonlinear elliptic equations. In general, researchers need to define a proper Banach space and prove the existence of solutions by using fixed point theory on the Banach space. However, the same method is not effective for boundary value problems of interval or fuzzy-valued differential equations on $[0, + \infty)$ . In interval or fuzzy cases, the continuity of the operator, and the convergence of the

Declaration of Competing Interest

There are no conflicts of interest.

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^☆: This work was supported by National Key Research and Development Program of China (No. 2020YFC2006200).

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Boundary value problems for interval-valued differential equations on unbounded domains☆

Abstract

Introduction

Section snippets

Preliminaries

Existence of solutions to x′(t)=g(t),x(0)=αlimt→+∞⁡x(t)

Existence of solutions to x′(t)=f(t,x(t)),x(0)=αlimt→+∞⁡x(t)

Examples

Conclusions

Declaration of Competing Interest

Inf. Sci.

J. Math. Anal. Appl.

Fuzzy Sets Syst.

Appl. Math. Comput.