Existence and uniqueness theoremfor uncertain spring vibration equation

Abstract

Uncertain spring vibration equation is a type of uncertain differential equations, whose external force is affected by an uncertain interference. The solution and inverse uncertainty distribution of solution of uncertain spring vibration equation in different cases have been derived, respectively. This paper proves an existence and uniqueness theorem of solution for general uncertain spring vibration equation in different cases under linear growth condition and Lipschitz condition.

Keywords

Spring vibration equation uncertain differential equation uncertain process uncertainty theory

1 Introduction

For modelling belief degree, uncertainty theory as a mathematical system was founded by Liu [4] in 2007, and perfected by Liu [6] in 2009 based on normality, duality, subadditivity and product axioms. As a fundamental concept in uncertainty theory, uncertain process was proposed by Liu [5] for modelling the evolution of uncertain phenomena. An uncertain process is essentially a sequence of uncertain variables indexed by time, that is, it is an uncertain variable at each time. As a canonical process, Liu process was proposed by Liu [6] as a supplement of Wiener process. It is a stationary and independent normal increment process and almost all sample paths are Lipschitz continuous. Based on Liu process, Liu [6] proposed uncertain calculus to deal with the integral and differential of an uncertain process. Nowadays, uncertain renewal process [5], uncertain renewal reward process [7], uncertain alternating renewal process [16], and uncertain contour process [17] have been established.

As a type of differential equations, uncertain differential equation was introduced by Liu [5] in 2008 driven by Liu process. In theory, the existence and uniqueness theorem for an uncertain differential equation was first verified by Chen and Liu [1] in 2010. More importantly, Yao and Chen [15] showed that the solution of an uncertain differential equation can be represented by a family of solutions of ordinary differential equations. In practice, uncertain differential equation has been widely applied in many fields such as finance (Liu [8]), optimal control (Zhu [18]), differential game (Yang and Gao [9, 10]), heat conduction (Yang and Yao [11], Yang and Ni [12], Yang [13]), string vibration (Gao [2]), and spring vibration (Jia and Dai [3]). Nowadays, the theory aspect and practice aspect of uncertain differential equation have achieved fruitful results, interested readers may consult Yao’s book[14].

Uncertain spring vibration equation was first proposed by Jia and Dai [3], and they studied the external force is affected by an uncertain interference. They also gave the solution of the uncertain spring vibration equation and inverse uncertainty distribution of the solution in different cases. Based on their works, this paper will prove an existence and uniqueness theorem of solution for general uncertain spring vibration equation in different cases. The remainder of the paper is arranged as follows. Section 1, we reviews some basic concepts and theorems in uncertainty theory. Section 1 introduces uncertain spring vibration equation. Section 1 proves some existence and uniqueness theorems in different cases, and Section 1 conclusion gives some conclusions.

2 Preliminaries

In this section, we review some basic concepts including uncertain variable, uncertain process and uncertain calculus.

For modelling belief degrees, Liu [4] defined an uncertain measure by the following three axioms:

Axiom 1. (Normality Axiom) M {Γ} =1 for the universal set Γ.

Axiom 2. (Duality Axiom) M {Λ} + M {Λ^c} =1 for any event Λ.

Axiom 3. (Subadditivity Axiom) For every countable sequence of events Λ₁, Λ₂, ⋯, we have $M [⋃_{i = 1}^{\infty} Λ_{i}] \leq \sum_{i = 1}^{\infty} M {Λ_{i}} .$

Furthermore, Liu [6] defined a product uncertain measure by the fourth axiom:

Axiom 4. (Product Axiom) Let (Γ_k, L_k, M_k) be uncertainty spaces for k = 1, 2, ⋯. The product uncertain measure M is an uncertain measure satisfying $M [\prod_{k = 1}^{\infty} Λ_{k}] = ⋀_{k = 1}^{\infty} M_{k} {Λ_{k}}$ where Λ_k are arbitrarily chosen events form L_k for k = 1, 2, ⋯, respectively.

Liu [4] in 2007 proposed some important concepts of uncertainty space, uncertain variable, and uncertainty distribution in uncertainty theory. An uncertainty space is a triplet (Γ, L, M), where Γ is a nonempty set, L is a σ-algebra over Γ, and M is an uncertain measure. An uncertain variable ξ is a function from an uncertainty space (Γ, L, M) to the set of real numbers such that for any Borel set B of real numbers, the set {ξ ∈ B} = {γ ∈ Γ|ξ (γ) ∈ B} is an event. The uncertainty distribution Φ of an uncertain variable ξ is defined by Φ (x) = M {ξ ≤ x} for any real number x.

Definition 2.1. (Liu [4]) The uncertain variables ξ₁, ξ₂, ⋯, ξ_n are said to be independent if $M [⋂_{i = 1}^{n} (ξ_{i} \in B_{i})] = ⋀_{i = 1}^{n} M {ξ_{i} \in B_{i}}$ for any borel sets B₁, B₂, ⋯, B_n of real numbers.

An uncertain process is essentially a sequence of uncertain variables indexed by time. A formal definition of uncertain process is as follows.

Definition 2.2. (Liu [5]) Let T be a totally ordered set (e.g. time) and let (Γ, L, M) be an uncertainty space. An uncertain process is a function X_t (γ) from T × (Γ, L, M) to the set of real numbers such that {X_t ∈ B} is an event for any Borel set B of real numbers at each time t.

An uncertain process X_t is said to have independent increments if, for any given t > 0, the increments X_s+t - X_s are independent uncertain variables for all s > 0. An uncertain process X_t is said to have stationary increments if, for any given t > 0, the increments X_s+t - X_s are identically distributed uncertain variables for all s > 0.

Definition 2.3. (Liu [6]) An uncertain process C_t is said to be a Liu process if (i) C₀ = 0 and almost all sample paths are Lipschitz continuous, (ii) C_t has stationary and independent increments, (iii) every increment C_t+s - C_s is a normal uncertain variable with an uncertainty distribution $Φ_{t} (x) = {(1 + exp (- \frac{π x}{\sqrt{3} t}))}^{- 1}, x \in R .$

Definition 2.4. (Liu [6]) Let X_t be an uncertain process and let C_t be a Liu process. For any partition of closed interval [a, b] with a = t₁ < t₂ < ⋯ < t_k+1 = b, the mesh is written as $Δ = max_{1 \leq i \leq k} | t_{i + 1} - t_{i} | .$ Then Liu integral of X_t with respect to C_t is $\int_{a}^{b} X_{t} d C_{t} = lim_{Δ \to 0} \sum_{i = 1}^{k} X_{t_{i}} \cdot (C_{t_{i + 1}} - C_{t_{i}}),$ provided that the limit exists almost surely and is finite. In this case, the uncertain process X_t is said to be integrable.

Definition 2.5. (Liu [5]) Suppose C_t is a Liu process, and f and g are two given functions. Then $d X_{t} = f (t, X_{t}) d t + g (t, X_{t}) d C_{t}$ (1) is called an uncertain differential equation. A solution is an uncertain process that satisfies (1) identically in t.

Theorem 2.1. (Chen and Liu [1]) Let X_t be an integrable uncertain process on [a, b]. Then for a sample path C_t (γ) with a Lipschitz constant K (γ), we have $| \int_{a}^{b} X_{t} (γ) d C_{t} (γ) | \leq K (γ) \int_{a}^{b} | X_{t} (γ) | d t .$

3 Uncertain spring vibration equation

As a classic type of differential equations, vibration equation describes the vibration of an object subjected to a spring and a time varying external force acting on it. Nevertheless, the external force is affected by an interference of noise in practice. For modelling the noise, two processes are used, one is Wiener process based on probability theory, another is Liu process based on uncertainty theory. If we consider the noise as Wiener process, then the vibration turns into stochastic spring vibration equation. However, Jia and Dai [3] pointed out that it is irrational to model the vibration via stochastic spring vibration equation. Thus, Jia and Dai [3] proposed an uncertain spring vibration equation whose noise of external force is described by Liu process as follows, ${\begin{matrix} \frac{d^{2} X_{t}}{d t^{2}} + 2 δ \frac{d X_{t}}{d t} + ω^{2} X_{t} = f (t) + σ (t) \dot{C_{t}}, \\ t > 0, X_{t} \in [l^{s} - l^{e}, l^{l} - l^{e}] \\ X_{t} = X_{0} |_{t = 0}, {\frac{d X_{t}}{d t} |}_{t = 0} = V_{0} \end{matrix}$ where 2δ = c/m, c is a damping constant, m is a mass of the vibration object, ω² = K/m, K is a Hooke’s constant, X_t is displacement, $\dot{C_{t}} = d C_{t} / d t$ denotes the time white noise, C_t is a Liu process, f (t) is an external force, σ (t) is a diffusion term of external force, l^e, l^l, l^s represent the lengths of static equilibrium position with external force is zero, longest, and shortest of the spring, respectively, X₀ and V₀ are two given initial displacement and velocity at time t = 0, respectively. They proved that the solution of uncertain spring vibration equation in different cases.

4 Existence and uniqueness theorem

Let us consider the general uncertain spring vibration equation as follows, ${\begin{matrix} \frac{d^{2} X_{t}}{d t^{2}} + 2 δ \frac{d X_{t}}{d t} + ω^{2} X_{t} = f (t, X_{t}) + σ (t, X_{t}) \dot{C_{t}}, \\ t > 0, X_{t} \in [l^{s} - l^{e}, l^{l} - l^{e}] \\ X_{t} = X_{0} |_{t = 0}, {\frac{d X_{t}}{d t} |}_{t = 0} = V_{0} . \end{matrix}$ (2)

Theorem 4.1. Assume δ > ω. Then the uncertain spring vibration equation (2) has a unique solution if the functions f (t, x) and σ (t, x) satisfy linear growth condition $\begin{matrix} | f (t, x) | + | σ (t, x) | \leq L (1 + | x |), \\ \forall x \in [l^{s} - l^{e}, l^{l} - l^{e}], t > 0 \end{matrix}$ and Lipschitz condition $\begin{matrix} | f (t, x) - f (t, y) | + | σ (t, x) - σ (t, y) | \leq L | x - y |, \\ \forall x, y \in [l^{s} - l^{e}, l^{l} - l^{e}], t > 0 \end{matrix}$ for some constants L. Moreover, the solution is sample-continuous.

Proof: In order to prove the existence of solution, a successive approximation method will be proposed to construct a solution of the uncertain spring vibration equation (2). Define $\begin{matrix} X_{t}^{(0)} (γ) = (\frac{1}{2} X_{0} + \frac{V_{0} + δ X_{0}}{2 \sqrt{δ^{2} - ω^{2}}}) e^{- δ t + \sqrt{δ^{2} - ω^{2}} t} \\ + (\frac{1}{2} X_{0} - \frac{V_{0} + δ X_{0}}{2 \sqrt{δ^{2} - ω^{2}}}) e^{- δ t - \sqrt{δ^{2} - ω^{2}} t} . \end{matrix}$ Then we define $\begin{matrix} X_{t}^{(n + 1)} (γ) \\ = X_{t}^{0} (γ) + \frac{1}{2 \sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} (e^{(- δ + \sqrt{δ^{2} - ω^{2}}) (t - s)} \\ - e^{(- δ - \sqrt{δ^{2} - ω^{2}}) (t - s)}) f (s, X_{s}^{(n)} (γ)) d s \\ + \frac{1}{2 \sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} (e^{(- δ + \sqrt{δ^{2} - ω^{2}}) (t - s)} \\ - e^{(- δ - \sqrt{δ^{2} - ω^{2}}) (t - s)}) σ (s, X_{s}^{(n)} (γ)) d C_{s} (γ), \\ n = 0, 1, 2, \dots . \end{matrix}$ For any sample γ, we define $\begin{matrix} D_{t}^{(n)} (γ) = max_{0 \leq s \leq t} | X_{s}^{(n + 1)} (γ) - X_{s}^{(n)} (γ) |, \\ n = 0, 1, 2, \dots \end{matrix}$ We claim that $\begin{matrix} D_{t}^{(n)} (γ) \leq (1 + | X_{0} |) \frac{L_{1}^{n + 1} (1 + K (γ))^{n + 1}}{(n + 1)!} t^{n + 1}, \\ n = 0, 1, 2, \dots, 0 \leq t \leq T \end{matrix}$ where T is a constant, K (γ) is the Lipschitz constant of the sample path C_t (γ) (see Theorem 2.1), $L_{1} = \frac{L}{\sqrt{δ^{2} - ω^{2}}}$ . Indeed for n = 0, we get $\begin{matrix} D_{t}^{(0)} (γ) = max_{0 \leq s \leq t} | \frac{1}{2 \sqrt{δ^{2} - ω^{2}}} \int_{0}^{s} (e^{(- δ + \sqrt{δ^{2} - ω^{2}}) (s - v)} \\ - e^{(- δ - \sqrt{δ^{2} - ω^{2}}) (s - v)}) f (v, X_{0}) d v \\ + \frac{1}{2 \sqrt{δ^{2} - ω^{2}}} \int_{0}^{s} (e^{(- δ + \sqrt{δ^{2} - ω^{2}}) (s - v)} \\ - e^{(- δ - \sqrt{δ^{2} - ω^{2}}) (s - v)}) σ (v, X_{0}) d C_{v} (γ) | \\ \leq \frac{1}{2 \sqrt{δ^{2} - ω^{2}}} max_{0 \leq s \leq t} \int_{0}^{s} | (e^{(- δ + \sqrt{δ^{2} - ω^{2}}) (s - v)} \\ - e^{(- δ - \sqrt{δ^{2} - ω^{2}}) (s - v)}) f (v, X_{0}) | d v \\ + \frac{K (γ)}{2 \sqrt{δ^{2} - ω^{2}}} max_{0 \leq s \leq t} \int_{0}^{s} | (e^{(- δ + \sqrt{δ^{2} - ω^{2}}) (s - v)} \\ - e^{(- δ - \sqrt{δ^{2} - ω^{2}}) (s - v)}) σ (v, X_{0}) | d v \\ \leq \frac{1}{\sqrt{δ^{2} - ω^{2}}} max_{0 \leq s \leq t} \int_{0}^{s} | f (v, X_{0}) | d v \\ + \frac{K (γ)}{\sqrt{δ^{2} - ω^{2}}} max_{0 \leq v \leq t} \int_{0}^{s} | σ (v, X_{0}) | d v \\ \leq \frac{1}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} | f (v, X_{0}) | d v \\ + \frac{K (γ)}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} | σ (v, X_{0}) | d v \\ \leq \frac{(1 + X_{0}) L (1 + K (γ)) t}{\sqrt{δ^{2} - ω^{2}}} \\ = (1 + X_{0}) L_{1} (1 + K (γ)) t . \end{matrix}$ This confirms the claim for n = 0. Next we assume the claim is true for some n - 1, then we obtain $\begin{matrix} D_{t}^{(n)} (γ) \\ = max_{0 \leq s \leq t} | \frac{1}{2 \sqrt{δ^{2} - ω^{2}}} \int_{0}^{s} (e^{(- δ + \sqrt{δ^{2} - ω^{2}}) (s - v)} \\ - e^{(- δ - \sqrt{δ^{2} - ω^{2}}) (s - v)}) (f (v, X_{v}^{(n + 1)} (γ)) \\ - f (v, X_{v}^{(n)} (γ))) d v \\ + \frac{1}{2 \sqrt{δ^{2} - ω^{2}}} \int_{0}^{s} (e^{(- δ + \sqrt{δ^{2} - ω^{2}}) (s - v)} \\ - e^{(- δ - \sqrt{δ^{2} - ω^{2}}) (s - v)}) (σ (v, X_{v}^{(n + 1)} (γ)) \\ - σ (v, X_{v}^{(n)} (γ))) d C_{v} (γ) | \\ \leq \frac{1}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} | (f (v, X_{v}^{(n + 1)} (γ)) \\ - f (v, X_{v}^{(n)} (γ))) | d v \\ + \frac{K (γ)}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} | (σ (v, X_{v}^{(n + 1)} (γ)) \\ - σ (v, X_{v}^{(n)} (γ))) | d v \\ \leq \frac{L}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} | (X_{v}^{(n + 1)} (γ) - X_{v}^{(n)} (γ) | d v \\ + \frac{K (γ) L}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} | X_{v}^{(n + 1)} (γ) - X_{v}^{(n)} (γ) | d v \\ = L_{1} (1 + K (γ)) \int_{0}^{t} | (X_{v}^{(n + 1)} (γ) - X_{v}^{(n)} (γ) | d v \end{matrix}$ $\begin{matrix} \leq L_{1} (1 + K (γ)) \int_{0}^{t} (1 + | X_{0} |) \frac{L_{1}^{n} (1 + K (γ))^{n} v^{n}}{n!} d v \\ \leq (1 + | X_{0} |) \frac{L_{1}^{n + 1} (1 + K (γ))^{n + 1}}{(n + 1)!} t^{n + 1} . \end{matrix}$ Thus, the claim is verified. It follows from Weierstrass’ criterion that, for each sample γ, $\begin{matrix} \sum_{n = 0}^{+ \infty} (1 + | X_{0} |) \frac{L_{1}^{n + 1} (1 + K (γ))^{n + 1}}{(n + 1)!} t^{n + 1} \\ \leq \sum_{n = 0}^{+ \infty} (1 + | X_{0} |) \frac{L_{1}^{n + 1} (1 + K (γ))^{n + 1}}{(n + 1)!} T^{n + 1} \\ < + \infty . \end{matrix}$ Then $X_{t}^{(n)} (γ)$ converges uniformly in t ∈ [0, T]. We denote the limits by $X_{t} (γ) = lim_{n \to \infty} X_{t}^{(n)} (γ), γ \in Γ, t \in [0, T] .$ Then $\begin{matrix} X_{t} = (\frac{1}{2} X_{0} + \frac{V_{0} + δ X_{0}}{2 \sqrt{δ^{2} - ω^{2}}}) e^{- δ t + \sqrt{δ^{2} - ω^{2}} t} \\ + (\frac{1}{2} X_{0} - \frac{V_{0} + δ X_{0}}{2 \sqrt{δ^{2} - ω^{2}}}) e^{- δ t - \sqrt{δ^{2} - ω^{2}} t} \\ + \frac{1}{2 \sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} (e^{(- δ + \sqrt{δ^{2} - ω^{2}}) (t - s)} \\ - e^{(- δ - \sqrt{δ^{2} - ω^{2}}) (t - s)}) f (s, X_{s}) d s \\ + \frac{1}{2 \sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} (e^{(- δ + \sqrt{δ^{2} - ω^{2}}) (t - s)} \\ - e^{(- δ - \sqrt{δ^{2} - ω^{2}}) (t - s)}) σ (s, X_{s}) d C_{s} . \end{matrix}$

Thus if δ > ω, then X_t is the solution of (2) for all t ≥ 0 since T is arbitrary.

Next, we will prove that the solution of uncertain spring vibration equation (2) is unique when δ > ω. Assume that both of X_t and $X_{t}^{*}$ are solutions of (2) with the same initial value X₀. Then for each γ ∈ Γ, we have $\begin{matrix} | X_{t} (γ) - X_{t}^{*} (γ) | \\ \leq | \frac{1}{2 \sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} (e^{(- δ + \sqrt{δ^{2} - ω^{2}}) (t - v)} \\ - e^{(- δ - \sqrt{δ^{2} - ω^{2}}) (t - v)}) (f (v, X_{v} (γ)) \\ - f (v, X_{v}^{*} (γ))) d v \\ + \frac{1}{2 \sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} (e^{(- δ + \sqrt{δ^{2} - ω^{2}}) (t - v)} \\ - e^{(- δ - \sqrt{δ^{2} - ω^{2}}) (t - v)}) ((σ (v, X_{v} (γ)) \\ - (σ (v, X_{v}^{*} (γ))) d C_{v} (γ) | \\ \leq \frac{1}{2 \sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} | (e^{(- δ + \sqrt{δ^{2} - ω^{2}}) (t - v)} \\ - e^{(- δ - \sqrt{δ^{2} - ω^{2}}) (t - v)}) (f (v, X_{v} (γ)) \\ - f (v, X_{v}^{*} (γ))) | d v \\ + | \frac{1}{2 \sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} (e^{(- δ + \sqrt{δ^{2} - ω^{2}}) (t - v)} \\ - e^{(- δ - \sqrt{δ^{2} - ω^{2}}) (t - v)}) ((σ (v, X_{v} (γ)) \\ - (σ (v, X_{v}^{*} (γ))) d C_{v} (γ) | \\ \leq \frac{1}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} | f (v, X_{v} (γ)) - f (v, X_{v}^{*} (γ)) | d v \\ + \frac{K (γ)}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} | (σ (v, X_{v} (γ)) - (σ (v, X_{v}^{*} (γ)) | d v \\ \leq \frac{L}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} | X_{v} (γ) - X_{v}^{*} (γ) | d v \\ + \frac{K (γ) L}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} | X_{v} (γ) - X_{v}^{*} (γ) | d v \\ = L_{1} (1 + K (γ)) \int_{0}^{t} | X_{v} (γ) - X_{v}^{*} (γ) | d v . \end{matrix}$

It follows from Gronwall inequality that $| X_{t} (γ) - X_{t}^{*} (γ) | \leq 0 \cdot exp (L_{1} (1 + K (γ)) t) = 0$ for all γ. Hence $X_{t} = X_{t}^{*}$ .

At last, we will prove the sample-continuity of X_t. By the above proof, we get $\begin{matrix} X_{t} (γ) \leq \sum_{n = 0}^{\infty} (1 + | X_{0} |) \frac{(L_{1} (1 + K (γ) t))^{n}}{n!} \\ = (1 + | X_{0} |) exp (L_{1} (1 + K (γ)) t) . \end{matrix}$ Suppose t > s > 0. We have $\begin{matrix} | X_{t} (γ) - X_{s} (γ) | \\ \leq | \frac{1}{2 \sqrt{δ^{2} - ω^{2}}} \int_{s}^{t} (e^{(- δ + \sqrt{δ^{2} - ω^{2}}) (t - v)} \\ - e^{(- δ - \sqrt{δ^{2} - ω^{2}}) (t - v)}) f (v, X_{v} (γ)) d v \\ + \frac{1}{2 \sqrt{δ^{2} - ω^{2}}} \int_{s}^{t} (e^{(- δ + \sqrt{δ^{2} - ω^{2}}) (t - v)} \\ - e^{(- δ - \sqrt{δ^{2} - ω^{2}}) (t - v)}) (σ (v, X_{v} (γ))) d C_{v} (γ) | \\ \leq \frac{1}{2 \sqrt{δ^{2} - ω^{2}}} \int_{s}^{t} | (e^{(- δ + \sqrt{δ^{2} - ω^{2}}) (t - v)} \\ - e^{(- δ - \sqrt{δ^{2} - ω^{2}}) (t - v)}) f (v, X_{v} (γ)) | d v \\ + | \frac{1}{2 \sqrt{δ^{2} - ω^{2}}} \int_{s}^{t} (e^{(- δ + \sqrt{δ^{2} - ω^{2}}) (t - v)} \\ - e^{(- δ - \sqrt{δ^{2} - ω^{2}}) (t - v)}) (σ (v, X_{v} (γ))) d C_{v} (γ) | \\ \leq \frac{1}{\sqrt{δ^{2} - ω^{2}}} \int_{s}^{t} | f (v, X_{v} (γ)) | d v \end{matrix}$ $\begin{matrix} + \frac{K (γ)}{\sqrt{δ^{2} - ω^{2}}} \int_{s}^{t} | (σ (v, X_{v} (γ)) | d v \\ \leq (1 + K (γ)) (1 + | X_{0} |) (t - s) exp \frac{(L (1 + K (γ)) t)}{\sqrt{δ^{2} - ω^{2}}} \\ \leq (1 + K (γ)) (1 + | X_{0} |) (t - s) exp (L_{1} (1 + K (γ)) t) . \end{matrix}$ Thus |X_t - X_s|→0, as s → t. Hence X_t is sample-continuous. The theorem is thus proved. □

Theorem 4.2. Assume δ = ω. Then the uncertain spring vibration equation (2) has a unique solution if the functions f (t, x) and σ (t, x) satisfy linear growth condition $\begin{matrix} | f (t, x) | + | σ (t, x) | \leq L (1 + | x |), \\ \forall x \in [l^{s} - l^{e}, l^{l} - l^{e}], t > 0 \end{matrix}$ and Lipschitz condition $\begin{matrix} | f (t, x) - f (t, y) | + | σ (t, x) - σ (t, y) | \leq L | x - y |, \\ \forall x, y \in [l^{s} - l^{e}, l^{l} - l^{e}], t > 0 \end{matrix}$ for some constants L. Moreover, the solution is sample-continuous.

Proof: In order to prove the existence of solution, a successive approximation method will be proposed to construct a solution of the uncertain spring vibration equation (2). Define $X_{t}^{(0)} (γ) = X_{0} e^{- δ t} + (V_{0} + δ X_{0}) {te}^{- δ t} .$ Then we define $\begin{matrix} X_{t}^{(n + 1)} (γ) = X_{t}^{(0)} (γ) \\ + \int_{0}^{t} (t - s) e^{- δ (t - s)} f (s, X_{s}^{(n)} (γ)) d s \\ + \int_{0}^{t} (t - s) e^{- δ (t - s)} σ (s, X_{s}^{(n) (γ)}) d C_{s} (γ) . \end{matrix}$ For any sample γ, we define $\begin{matrix} D_{t}^{(n)} (γ) = max_{0 \leq s \leq t} | X_{s}^{(n + 1)} (γ) - X_{s}^{(n)} (γ) |, \\ n = 0, 1, 2, \dots . \end{matrix}$ We claim that $\begin{matrix} D_{t}^{(n)} (γ) \leq (1 + | X_{0} |) \frac{L^{n + 1} (1 + K (γ))^{n + 1}}{(2 n + 2)!} t^{2 n + 2}, \\ n = 0, 1, 2, \dots, 0 \leq t \leq T \end{matrix}$ where T is a constant, K (γ) is the Lipschitz constant of the sample path C_t (γ) (see Theorem 2.1). Indeed for n = 0, we get $\begin{matrix} D_{t}^{(0)} (γ) \\ = max_{0 \leq s \leq t} | \int_{0}^{s} (s - v) e^{- δ (s - v)} f (v, X_{0}) d v \\ + \int_{0}^{s} (s - v) e^{- δ (s - v)} σ (v, X_{0}) d C_{v} (γ) | \\ \leq max_{0 \leq s \leq t} \int_{0}^{s} | (s - v) e^{- δ (s - v)} f (v, X_{0}) | d v \\ + max_{0 \leq s \leq t} | \int_{0}^{s} (s - v) e^{- δ (s - v)} σ (v, X_{0}) d C_{v} (γ) | \\ \leq max_{0 \leq s \leq t} \int_{0}^{s} | (s - v) f (v, X_{0}) | d v \\ + K (γ) max_{0 \leq s \leq t} \int_{0}^{s} | (s - v) σ (v, X_{0}) | d v \\ \leq (1 + | X_{0} |) L \int_{0}^{t} | t - v | d v \\ + (1 + | X_{0} |) LK (γ) \int_{0}^{t} | t - v | d v \\ = \frac{1}{2} (1 + X_{0}) L (1 + K (γ)) t^{2} . \end{matrix}$ This confirms the claim for n = 0. Next we assume the claim is true for some n - 1, then we obtain $\begin{matrix} D_{t}^{(n)} (γ) \\ = max_{0 \leq s \leq t} | \int_{0}^{s} (s - v) e^{- δ (s - v)} (f (v, X_{t}^{(n + 1)} (γ)) \\ - f (v, X_{t}^{(n)} (γ))) d v \\ + \int_{0}^{s} (s - v) e^{- δ (s - v)} (σ (v, X_{t}^{(n + 1)} (γ)) \\ - σ (v, X_{t}^{(n)} (γ))) d C_{v} (γ) | \\ \leq max_{0 \leq s \leq t} \int_{0}^{s} | (s - v) e^{- δ (s - v)} (f (v, X_{t}^{(n + 1)} (γ)) \\ - f (v, X_{t}^{(n)} (γ))) | d v \\ + max_{0 \leq s \leq t} | \int_{0}^{s} (s - v) e^{- δ (s - v)} (σ (v, X_{t}^{(n + 1)} (γ)) \\ - σ (v, X_{t}^{(n)} (γ))) d C_{v} (γ) | \end{matrix}$ $\begin{matrix} \leq max_{0 \leq s \leq t} \int_{0}^{s} | (s - v) (f (v, X_{t}^{(n + 1)} (γ)) \\ - f (v, X_{t}^{(n)} (γ))) | d v \\ + K (γ) max_{0 \leq s \leq t} \int_{0}^{s} | (s - v) (σ (v, X_{t}^{(n + 1)} (γ)) \\ - σ (v, X_{t}^{(n)} (γ))) | d v \\ \leq L \int_{0}^{t} | (t - v) | | X_{v}^{(n)} (γ) - X_{v}^{(n - 1)} (γ) | d v \\ + K (γ) L \int_{0}^{t} | (t - v) | | X_{v}^{(n)} (γ) - X_{v}^{(n - 1)} (γ) | d v \\ \leq L (1 + K (γ)) \int_{0}^{t} (1 + | X_{0} |) \frac{L^{n} (1 + K (γ))^{n} v^{2 n}}{(2 n)!} d v \\ \leq (1 + | X_{0} |) \frac{L^{n + 1} (1 + K (γ))^{n + 1}}{(2 n + 2)!} t^{2 n + 2} . \end{matrix}$ Thus, the claim is verified. It follows from Weierstrass’ criterion that, for each sample γ, $\begin{matrix} \sum_{n = 0}^{+ \infty} (1 + | X_{0} |) \frac{L^{n + 1} (1 + K (γ))^{n + 1}}{(2 n + 2)!} t^{2 n + 2} \\ \leq \sum_{n = 0}^{+ \infty} (1 + | X_{0} |) \frac{L^{n + 1} (1 + K (γ))^{n + 1}}{(2 n + 2)!} T^{2 n + 2} \\ < + \infty . \end{matrix}$ Then $X_{t}^{(n)} (γ)$ converges uniformly in t ∈ [0, T]. We denote the limits by $X_{t} (γ) = lim_{n \to \infty} X_{t}^{(n)} (γ), γ \in Γ, t \in [0, T] .$ Then $\begin{matrix} X_{t} = X_{0} e^{- δ t} + (V_{0} + δ X_{0}) {te}^{- δ t} \\ + \int_{0}^{t} (t - s) e^{- δ (t - s)} f (s, X_{s}) d s \\ + \int_{0}^{t} (t - s) e^{- δ (t - s)} σ (s, X_{s}) d C_{s} . \end{matrix}$ Thus if δ = ω, then X_t is the solution of (2) for all t ≥ 0 since T is arbitrary.

Next, we will prove that the solution of uncertain spring vibration equation (2) is unique when δ = ω. Assume that both of X_t and $X_{t}^{*}$ are solutions of (2) with the same initial value X₀. Then for each γ ∈ Γ, we have $\begin{matrix} | X_{t} (γ) - X_{t}^{*} (γ) | \\ \leq | \int_{0}^{t} (t - v) e^{- δ (t - v)} (f (v, X_{v} (γ)) \\ - f (v, X_{v}^{*} (γ))) d v \\ + \int_{0}^{t} (t - v) e^{- δ (t - v)} ((σ (v, X_{v} (γ)) \\ - (σ (v, X_{v}^{*} (γ))) d C_{v} (γ) | \\ \leq \int_{0}^{t} | (t - v) e^{- δ (t - v)} (f (v, X_{v} (γ)) \\ - f (v, X_{v}^{*} (γ))) | d v \\ + | \int_{0}^{t} (t - v) e^{- δ (t - v)} ((σ (v, X_{v} (γ)) \\ - (σ (v, X_{v}^{*} (γ))) d C_{v} (γ) | \\ \leq \int_{0}^{t} | (t - v) f (v, X_{v} (γ)) - f (v, X_{v}^{*} (γ)) | d v \\ + K (γ) \int_{0}^{t} | (t - v) (σ (v, X_{v} (γ)) \\ - (σ (v, X_{v}^{*} (γ)) | d v \\ \leq L \int_{0}^{t} | (t - v) (X_{v} (γ) - X_{v}^{*} (γ)) | d v \\ + K (γ) L \int_{0}^{t} | (t - v) (X_{v} (γ) - X_{v}^{*} (γ)) | d v \\ = L (1 + K (γ)) \int_{0}^{t} | (t - v) (X_{v} (γ) - X_{v}^{*} (γ)) | d v . \end{matrix}$ It follows from Gronwall inequality that $\begin{matrix} | X_{t} (γ) - X_{t}^{*} (γ) | \\ \leq 0 \cdot exp (L (1 + K (γ)) \int_{0}^{t} | t - s | d s) \\ = 0 \end{matrix}$ for all γ. Hence $X_{t} = X_{t}^{*}$ .

At last, we will prove the sample-continuity of X_t. By the above proof, we get $\begin{matrix} X_{t} (γ) \leq \sum_{n = 0}^{\infty} (1 + | X_{0} |) \frac{(L (1 + K (γ)))^{n} t^{2 n}}{(2 n)!} \\ \leq \sum_{n = 0}^{\infty} (1 + | X_{0} |) \frac{(L (1 + K (γ)) t^{2})^{n}}{n!} \\ = (1 + | X_{0} |) exp (L (1 + K (γ)) t^{2}) . \end{matrix}$ Suppose t > s > 0. We have $\begin{matrix} | X_{t} (γ) - X_{s} (γ) | \\ \leq | \int_{s}^{t} (t - v) e^{- δ (t - v)} f (v, X_{v} (γ)) d v \\ + (t - v) e^{- δ (t - v)} (σ (v, X_{v} (γ))) d C_{v} (γ) | \\ \leq \int_{s}^{t} | (t - v) e^{- δ (t - v)} f (v, X_{v} (γ)) | d v \\ + | \int_{s}^{t} (t - v) e^{- δ (t - v)} σ (v, X_{v} (γ)) d C_{v} (γ) | \\ \leq \int_{s}^{t} | (t - v) f (v, X_{v} (γ)) | d v \\ + K (γ) \int_{s}^{t} | (t - v) (σ (v, X_{v} (γ)) | d v \\ \leq \frac{1}{2} (1 + K (γ)) (1 + | X_{0} |) (t - s)^{2} \\ exp (L (1 + K (γ)) t) . \end{matrix}$ Thus |X_t - X_s|→0, as s → t. Hence X_t is sample-continuous. The theorem is thus proved. □

Theorem 4.3. Assume δ < ω. Then the uncertain spring vibration equation (2) has a unique solution if the functions f (t, x) and σ (t, x) satisfy linear growth condition $\begin{matrix} | f (t, x) | + | σ (t, x) | \leq L (1 + | x |), \\ \forall x \in [l^{s} - l^{e}, l^{l} - l^{e}], t > 0 \end{matrix}$ and Lipschitz condition $\begin{matrix} | f (t, x) - f (t, y) | + | σ (t, x) - σ (t, y) | \leq L | x - y |, \\ \forall x, y \in [l^{s} - l^{e}, l^{l} - l^{e}], t > 0 \end{matrix}$ for some constants L. Moreover, the solution is sample-continuous.

Proof: In order to prove the existence of solution, a successive approximation method will be proposed to construct a solution of the uncertain spring vibration equation (2). Define $\begin{matrix} X_{t}^{(0)} (γ) = X_{0} e^{- δ t} cos (t \sqrt{ω^{2} - δ^{2}}) \\ + \frac{V_{0} + δ X_{0}}{\sqrt{ω^{2} - δ^{2}}} e^{- δ t} sin (t \sqrt{ω^{2} - δ^{2}}) . \end{matrix}$ Then we define $\begin{matrix} X_{t}^{(n + 1)} (γ) \\ = X_{t}^{0} (γ) \\ + \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{t} e^{- δ (t - s)} sin ((t - s) \sqrt{ω^{2} - δ^{2}}) \\ f (s, X_{s}^{(n)}) d s \\ + \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{t} e^{- δ (t - s)} sin ((t - s) \sqrt{ω^{2} - δ^{2}}) \\ σ (s, X_{s}^{(n)}) d C_{s} (γ) . \end{matrix}$ For any sample γ, we define $\begin{matrix} D_{t}^{(n)} (γ) = max_{0 \leq s \leq t} | X_{s}^{(n + 1)} (γ) - X_{s}^{(n)} (γ) |, \\ n = 0, 1, 2, \dots . \end{matrix}$ We claim that $\begin{matrix} D_{t}^{(n)} (γ) \leq (1 + | X_{0} |) \frac{L_{1}^{n + 1} (1 + K (γ))^{n + 1}}{(n + 1)!} t^{n + 1}, \\ n = 0, 1, 2, \dots, 0 \leq t \leq T \end{matrix}$ where T is a constant, K (γ) is the Lipschitz constant of the sample path C_t (γ) (see Theorem 2.1). Indeed for n = 0, we get $\begin{matrix} D_{t}^{(0)} (γ) \\ = max_{0 \leq s \leq t} | \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{s} e^{- δ (s - v)} sin ((s - v) \\ \sqrt{ω^{2} - δ^{2}}) f (v, X_{0}) d v \\ + \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{s} e^{- δ (s - v)} sin ((s - v) \\ \sqrt{ω^{2} - δ^{2}}) σ (v, X_{0}) d C_{v} (γ) | \\ \leq max_{0 \leq s \leq t} \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{s} | e^{- δ (s - v)} sin ((s - v) \\ \sqrt{ω^{2} - δ^{2}}) f (v, X_{0}) | d v \\ + | max_{0 \leq s \leq t} \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{s} e^{- δ (s - v)} sin ((s - v) \\ \sqrt{ω^{2} - δ^{2}}) σ (v, X_{0}) d C_{v} (γ) | \\ \leq \frac{1}{\sqrt{ω^{2} - δ^{2}}} max_{0 \leq s \leq t} \int_{0}^{t} | f (v, X_{0}) | d v \\ + \frac{K (γ)}{\sqrt{ω^{2} - δ^{2}}} max_{0 \leq s \leq t} \int_{0}^{t} | σ (v, X_{0}) | d v \\ \leq \frac{(1 + X_{0}) L (1 + K (γ)) t}{\sqrt{ω^{2} - δ^{2}}} \\ \leq (1 + X_{0}) L_{1} (1 + K (γ)) t . \end{matrix}$ This confirms the claim for n = 0. Next we assume the claim is true for some n - 1, then we obtain $\begin{matrix} D_{t}^{(n)} (γ) \\ = max_{0 \leq s \leq t} | \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{s} e^{- δ (s - v)} sin ((s - v) \\ \sqrt{ω^{2} - δ^{2}}) (f (v, X_{t}^{(n + 1)} (γ)) - f (v, X_{t}^{(n)} (γ))) d v \\ + \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{s} e^{- δ (s - v)} sin ((s - v) \sqrt{ω^{2} - δ^{2}}) \\ (σ (v, X_{t}^{(n + 1)} (γ)) - σ (v, X_{t}^{(n)} (γ))) d C_{v} (γ) | \\ \leq max_{0 \leq s \leq t} \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{s} | e^{- δ (s - v)} sin ((s - v) \\ \sqrt{ω^{2} - δ^{2}}) (f (v, X_{t}^{(n + 1)} (γ)) - f (v, X_{t}^{(n)} (γ))) | d v \\ + | max_{0 \leq s \leq t} \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{s} e^{- δ (s - v)} sin ((s - v) \\ \sqrt{ω^{2} - δ^{2}}) (σ (v, X_{t}^{(n + 1)} (γ)) - σ (v, X_{t}^{(n)} (γ))) \\ d C_{v} (γ) | \\ \leq \frac{1}{\sqrt{ω^{2} - δ^{2}}} max_{0 \leq s \leq t} \int_{0}^{s} | (f (v, X_{t}^{(n + 1)} (γ)) \\ - f (v, X_{t}^{(n)} (γ))) | d v \\ + \frac{K (γ)}{\sqrt{ω^{2} - δ^{2}}} max_{0 \leq s \leq t} \int_{0}^{s} | (σ (v, X_{t}^{(n + 1)} (γ)) \\ - σ (v, X_{t}^{(n)} (γ))) | d v \\ \leq \frac{L}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{t} | X_{v}^{(n)} (γ) - X_{v}^{(n - 1)} (γ) | d v \\ + \frac{K (γ) L}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{t} | X_{v}^{(n)} (γ) - X_{v}^{(n - 1)} (γ) | d v \\ = \frac{L (1 + K (γ))}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{t} | X_{v}^{(n)} (γ) - X_{v}^{(n - 1)} (γ) | d v \\ \leq L_{1} (1 + K (γ)) \int_{0}^{t} (1 + | X_{0} |) \frac{L_{1}^{n} (1 + K (γ))^{n} v^{n}}{n!} d v \\ \leq (1 + | X_{0} |) \frac{L_{1}^{n + 1} (1 + K (γ))^{n + 1}}{(n + 1)!} t^{n + 1} . \end{matrix}$ Thus, the claim is verified. It follows from Weierstrass’ criterion that, for each sample γ, $\begin{matrix} \sum_{n = 0}^{+ \infty} (1 + | X_{0} |) \frac{L_{1}^{n + 1} (1 + K (γ))^{n + 1}}{(n + 1)!} t^{n + 1} \\ \leq \sum_{n = 0}^{+ \infty} (1 + | X_{0} |) \frac{L_{1}^{n + 1} (1 + K (γ))^{n + 1}}{(n + 1)!} T^{n + 1} \\ < + \infty . \end{matrix}$ Then $X_{t}^{(n)} (γ)$ converges uniformly in t ∈ [0, T]. We denote the limits by $X_{t} (γ) = lim_{n \to \infty} X_{t}^{(n)} (γ), γ \in Γ, t \in [0, T] .$ Then $\begin{matrix} X_{t} = X_{0} e^{- δ t} cos (t \sqrt{ω^{2} - δ^{2}}) \\ + \frac{V_{0} + δ X_{0}}{\sqrt{ω^{2} - δ^{2}}} e^{- δ t} sin (t \sqrt{ω^{2} - δ^{2}}) \\ + \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{t} e^{- δ (t - s)} sin ((t - s) \sqrt{ω^{2} - δ^{2}}) \\ f (s, X_{s}) d s \end{matrix}$ $\begin{matrix} + \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{t} e^{- δ (t - s)} sin ((t - s) \sqrt{ω^{2} - δ^{2}}) \\ σ (s, X_{s}) d C_{s} . \end{matrix}$ Thus if δ < ω, then X_t is the solution of (2) for all t ≥ 0 since T is arbitrary.

Next, we will prove that the solution of uncertain spring vibration equation (2) is unique when δ < ω. Assume that both of X_t and $X_{t}^{*}$ are solutions of (2) with the same initial value X₀. Then for each γ ∈ Γ, we have $\begin{matrix} | X_{t} (γ) - X_{t}^{*} (γ) | \\ \leq | \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{t} e^{- δ (t - v)} sin ((t - v) \sqrt{ω^{2} - δ^{2}}) \\ (f (v, X_{v} (γ)) - f (v, X_{v}^{*} (γ))) d v \\ + \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{t} e^{- δ (t - v)} sin ((t - v) \sqrt{ω^{2} - δ^{2}}) \\ (σ (v, X_{v} (γ)) - σ (v, X_{v}^{*} (γ))) d C_{v} (γ) | \\ \leq \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{t} | e^{- δ (t - v)} sin ((t - v) \sqrt{ω^{2} - δ^{2}}) \\ (f (v, X_{v} (γ)) - f (v, X_{v}^{*} (γ))) | d v \\ + | \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{t} e^{- δ (t - v)} sin ((t - v) \sqrt{ω^{2} - δ^{2}}) \\ (σ (v, X_{v} (γ)) - σ (v, X_{v}^{*} (γ))) d C_{v} (γ) | \\ \leq \frac{1}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} | f (v, X_{v} (γ)) - f (v, X_{v}^{*} (γ)) | d v \\ + \frac{K (γ)}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} | (σ (v, X_{v} (γ)) - (σ (v, X_{v}^{*} (γ)) | d v \\ \leq \frac{L}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} | X_{v} (γ) - X_{v}^{*} (γ) | d v \\ + \frac{K (γ) L}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} | X_{v} (γ) - X_{v}^{*} (γ) | d v \\ \leq L_{1} (1 + K (γ)) \int_{0}^{t} | X_{v} (γ) - X_{v}^{*} (γ) | d v . \end{matrix}$ It follows from Gronwall inequality that $\begin{matrix} | X_{t} (γ) - X_{t}^{*} (γ) | \leq 0 \cdot exp (L_{1} (1 + K (γ)) t) \\ = 0 \end{matrix}$ for all γ. Hence $X_{t} = X_{t}^{*}$ .

At last, we will prove the sample-continuity of X_t. By the above proof, we get $\begin{matrix} X_{t} (γ) \leq \sum_{n = 0}^{\infty} (1 + | X_{0} |) \frac{(L_{1} (1 + K (γ) t))^{n}}{n!} \\ = (1 + | X_{0} |) exp (L_{1} (1 + K (γ)) t) . \end{matrix}$ Suppose t > s > 0. We have $\begin{matrix} | X_{t} (γ) - X_{s} (γ) | \\ \leq | \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{s}^{t} e^{- δ (t - v)} sin ((t - v) \sqrt{ω^{2} - δ^{2}}) \\ f (v, X_{v} (γ)) d v \\ + \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{s}^{t} e^{- δ (t - v)} sin ((t - v) \sqrt{ω^{2} - δ^{2}}) \\ σ (v, X_{v} (γ)) d C_{v} (γ) | \\ \leq \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{s}^{t} | e^{- δ (t - v)} sin ((t - v) \sqrt{ω^{2} - δ^{2}}) \\ f (v, X_{v} (γ)) | d v \\ + | \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{s}^{t} e^{- δ (t - v)} sin ((t - v) \sqrt{ω^{2} - δ^{2}}) \\ σ (v, X_{v} (γ)) d C_{v} (γ) | \\ \leq \frac{1}{\sqrt{δ^{2} - ω^{2}}} \int_{s}^{t} | f (v, X_{v} (γ)) | d v \\ + \frac{K (γ)}{\sqrt{δ^{2} - ω^{2}}} \int_{s}^{t} | σ (v, X_{v} (γ)) | d v \\ \leq (1 + K (γ)) (1 + | X_{0} |) (t - s) exp (\frac{L (1 + K (γ)) t}{\sqrt{δ^{2} - ω^{2}}}) \\ \leq (1 + K (γ)) (1 + | X_{0} |) (t - s) exp (L_{1} (1 + K (γ)) t) . \end{matrix}$ Thus |X_t - X_s|→0, as s → t. Hence X_t is sample-continuous.

The theorem is thus proved. □

Remark 4.1. No matter what the relationship between δ and ω, the uncertain spring vibration equation (2) has a unique solution only if the coefficient function f (t, x) and σ (t, x) satisfy linear growth condition and Lipschitz condition. And the solution is sample-continuous. An existence and uniqueness theorem stating that equation has one and only one solution is of fundamental importance, because it says that problem can be solved uniquely. Once you know that solution exists, you can go for numerical methods to approximate it.

Example 4.1. Consider an uncertain spring vibration equation as follows, ${\begin{matrix} \frac{d^{2} X_{t}}{d t^{2}} + 2 δ \frac{d X_{t}}{d t} + ω^{2} X_{t} = \dot{C_{t}}, \\ t > 0, X_{t} \in [l^{s} - l^{e}, l^{l} - l^{e}] \\ X_{t} |_{t = 0} = 0, {\frac{d X_{t}}{d t} |}_{t = 0} = 0 . \end{matrix}$ (3) Based on Theorems 4.1–4.3, we conclude that equation (3) has a unique solution as follows. Assume δ > ω. Then the solution of (3) is $\begin{matrix} X_{t} = \frac{1}{2 \sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} (e^{(- δ + \sqrt{δ^{2} - ω^{2}}) (t - s)} \\ - e^{(- δ - \sqrt{δ^{2} - ω^{2}}) (t - s)}) d C_{s} . \end{matrix}$ Assume δ = ω. Then the solution of (3) is $\begin{matrix} X_{t} = e^{- δ t} \int_{0}^{t} (t - s) e^{δ s} d C_{s} . \end{matrix}$ Assume δ < ω. Then the solution of (3) is $\begin{matrix} X_{t} = \\ \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{t} e^{- δ (t - s)} sin ((t - s) \sqrt{ω^{2} - δ^{2}}) d C_{s} . \end{matrix}$

5 Conclusion

Uncertain spring vibration equation is an important part of differential equations in uncertain environments. For a class of linear uncertain spring vibration equations, the analytic solution and the inverse uncertainty distribution of solutions in different cases were derived. Nevertheless, it is difficult to obtain the analytic solution of general uncertain spring vibration equations. The contribution of this paper was first to prove an existence and uniqueness theorem under linear growth condition and Lipschitz condition in different cases.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China Grant No.61573210 and the Fundamental Research Funds for the Central Universities in UIBE No.17QD12.

References

Chen

and Liu

, Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optimization and Decision Making 9(1) (2010), 69–81.

Gao

, Uncertain wave equation with infinite half-boundary, Applied Mathematics and Computation 304 (2017), 28–40.

Jia

and Dai

, Uncertain spring vibration equation, http://orsc.edu.cn/online/171205.pdf, 2017

Liu

, Uncertainty Theory, 2nd ed., Berlin Springer-Verlag, 2007.

Liu

, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems 2(1) (2008), 3–16.

Liu

, Some research problems in uncertainty theory, Journal of Uncertain Systems 3(1) (2009), 3–10.

Liu

, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010.

Liu

, Toward ucertain finance theory, Journal of Uncertainty Analysis and Applications 1 (2013), Ariticle 1.

Yang

and Gao

, Uncertain differential games with application to capitalism, Journal of Uncertainty Analysis and Applications 1 (2013), Article 17.

10.

Yang

and Gao

, Linear-quadratic uncertain differential games with application to resource extraction problem, IEEE Transactions on Fuzzy Systems 24(4) (2016), 819–826.

11.

Yang

and Yao

, Uncertain partial differential equation with application to heat conduction, Fuzzy Optimization and Decision Making 16(3) (2017), 379–403.

12.

Yang

and Ni

, Existence and uniqueness theorem for uncertain heat equation, Journal of Ambient Intelligence and Humanized Computing 8(5) (2017), 717–725.

13.

Yang

, Solving uncertain heat equation via numerical method, Applied Mathematics and Computation 329 (2018), 92–104.

14.

Yao

, Uncertainty Differential Equation, Springer-Verlag, Berlin, 2016.

15.

Yao

and Chen

, A numerical method for solving uncertain differential equations, Journal of Intelligent and Fuzzy Systems 25(3) (2013), 825–832.

16.

Yao

and Li

, Uncertain alternating renewal process and its applications, IEEE Transactions on Fuzzy Systems 20(6) (2012), 1154–1160.

17.

Yao

, Uncertain contour process and its application in stock model with floating interest rate, Fuzzy Optimization and Decision Making 14(4) (2015), 399–424.

18.

Zhu

, Uncertain optimal control with application to a portfolio selection model, Cybernetics and Systems 41(7) (2010), 535–547.