On the stability of solutions of fuzzy differential equations in the quotient space of fuzzy numbers

Abstract

In this paper, the stability theory for fuzzy differential equations in the quotient space of fuzzy numbers was essentially investigated with Lyapunov-like functions. Some sufficient criteria for the stability, uniformly stability and exponentially stability of the trivial solution of the fuzzy differential equations were obtained by using the differential inequalities and the comparison principle for Lyapunov-like functions.

Keywords

Fuzzy differential equations fuzzy numbers stability quotient space

1 Introduction

Recently, the field of fuzzy differential equations based on a well known and widely used Hukuhara difference [12] and the H-differentiability of Puri and Ralescu [19] has been widely studied. Latest, the existence and uniqueness of solutions of the initial value problems for fuzzy differential equations under kinds of conditions were studied in [3–5, 14] and the relationship between a solution and its approximate solutions to fuzzy differential equations were established in [26, 27]. It is worth noting that several new methods have been applied to study the theory of fuzzy differential equations, such as the variational iteration method [1], the power series method [4] and the Laplace transform method [23] and so on [2, 25]. Further, the stability theory which corresponds to Lyapunov stability theory for fuzzy differential equations were studied in [8 , 29]. In [8], Diamond studies Lyapunov stability of fuzzy differential equations and the periodicity of the fuzzy solution set for both the time-dependent and autonomous cases. By using scale equations and the comparison principle for Lyapunov-like functions, Hien gives sufficient criteria for the stability and asymptotic stability of solutions of fuzzy differential equations [10]. In [24], the asymptotic equilibrium and the stability properties of the trivial fuzzy solution of the perturbed semilinear fuzzy evolution equations are investigated by extending the Lyapunov’s direct method. In [29], practical stabilities of fuzzy differential equations with the second type of Hukuhara derivative are considered.

However in many applications the H-diffe-rentiability and its generalizations appear to have several limitations [6, 14]. In [17, 18], Mareš presented a natural equivalence relation between fuzzy quantities. This equivalence relation can be used to partition of the set of fuzzy quantities into equivalence classes having the desired group properties for the addition operation [13]. Hong and Do [11] defined a more refined equivalence relation than Mareš [17] and improved Mareš’s results. In [20], Qiu et al. showed that the method of finding the inverse operation of fuzzy numbers in the sense of Mareš is very intuitive. As an application of the main results, it is shown that if we identify every fuzzy number with the corresponding equivalence class, there wound be more differentiable fuzzy functions than what is found in the literature. After that, the fuzzy differential equations in the quotient space of fuzzy numbers were investigated [21, 22]. In this paper, the stability of the trivial solution of the fuzzy differential equations in the quotient space of fuzzy numbers will be studied by using Lyapunov’s second method.

2 Preliminaries

A fuzzy set $\tilde{x}$ of $ℝ$ is characterized by a membership function $μ_{\tilde{x}} : ℝ \to [0, 1]$ . For each such fuzzy set $\tilde{x}$ , we denote by ${[\tilde{x}]}^{α} = {x \in ℝ : μ_{\tilde{x}} (x) \geq α}$ for any α ∈ (0, 1], its α-level set. Define the set ${[\tilde{x}]}^{0}$ by ${[\tilde{x}]}^{0} = \bar{⋃_{α \in (0, 1]} {[\tilde{x}]}^{α}}$ , where $\bar{A}$ denotes the closure of a crisp set A. A fuzzy set $\tilde{x}$ is said to be a fuzzy number if it satisfies the following conditions [9]:

(1) $\tilde{x}$ is normal, i.e., there exists an $x_{0} \in ℝ$ such that $μ_{\tilde{x}} (x_{0}) = 1$ ;

(2) $\tilde{x}$ is convex;

(3) $\tilde{x}$ is upper semi-continuous;

(4) ${[\tilde{x}]}^{0}$ is compact.

Equivalently, a fuzzy number $\tilde{x}$ is a fuzzy set with non-empty bounded closed level sets ${[\tilde{x}]}^{α} = [{\tilde{x}}_{L} (α), {\tilde{x}}_{R} (α)]$ for all α ∈ [0, 1], where $[{\tilde{x}}_{L} (α), {\tilde{x}}_{R} (α)]$ denotes a closed interval with the left end point ${\tilde{x}}_{L} (α)$ and the right end point ${\tilde{x}}_{R} (α)$ . Denote the class of fuzzy numbers by 𝒻. Notice that the real numbers $ℝ$ can be embedded in 𝒻 by defining a fuzzy number $\tilde{a}$ as $μ_{\tilde{a}} (x) = {\begin{matrix} 1, if x = a, \\ 0, otherwise, \end{matrix}$ for each $a \in ℝ$ . Thus we will represent the singleton {a} by $\tilde{a}$ for any real number $a \in ℝ$ and in particular $\tilde{0}$ is just the usual zero.

For any $\tilde{x}, \tilde{y} \in 𝒻$ and $a \in ℝ$ , owing to Zadeh’s extension principle [28], addition and scalar multiplication are defined for each $x \in ℝ$ by $μ_{\tilde{x} + \tilde{y}} (x) = sup_{x_{1} . x_{2} : x_{1} + x_{2} = x} min {μ_{\tilde{x}} (x_{1}), μ_{\tilde{y}} (x_{2})}$ and $μ_{a \times \tilde{x}} (x) = μ_{a \tilde{x}} (x) = {\begin{matrix} μ (\frac{x}{a}), if a \neq 0, \\ \tilde{0}, if a = 0 . \end{matrix}$

For any $\tilde{x} \in 𝒻$ , we define the fuzzy number $- \tilde{x}$ ∈𝒻 by $- \tilde{x} = (- 1) \times \tilde{x}$ , i.e., $μ_{- \tilde{x}} (x) = μ_{\tilde{x}} (- x)$ , for each $x \in ℝ$ . It is well known that ${[\tilde{x} + \tilde{y}]}^{α} = [{\tilde{x}}_{L} (α) + {\tilde{y}}_{L} (α), {\tilde{x}}_{R} (α) + {\tilde{y}}_{R} (α)]$ and ${[a \tilde{x}]}^{α} = a {[\tilde{x}]}^{α} = {\begin{matrix} [a {\tilde{x}}_{L} (α), a {\tilde{x}}_{R} (α)], if a > 0, \\ {0}, if a = 0, \\ [a {\tilde{x}}_{R} (α), a {\tilde{x}}_{L} (α)], if a < 0, \end{matrix}$ for any $\tilde{x}, \tilde{y} \in 𝒻$ and $a \in ℝ$ . In particularly, ${[- \tilde{x}]}^{α} = - {[\tilde{x}]}^{α} = - [{\tilde{x}}_{L} (α), {\tilde{x}}_{R} (α)] = [- {\tilde{x}}_{R} (α), - {\tilde{x}}_{L} (α)]$ . We say that a fuzzy number $\tilde{s} \in 𝒻$ is symmetric [17], if $μ_{\tilde{s}} (x) = μ_{\tilde{s}} (- x),$ for all $x \in ℝ$ , i.e., $\tilde{s} = - \tilde{s}$ . The set of all symmetric fuzzy numbers will be denoted by §.

Definition 2.1. [11] Let $\tilde{x}, \tilde{y} \in 𝒻$ . We say that $\tilde{x}$ is equivalent to $\tilde{y}$ and write $\tilde{x} \sim \tilde{y}$ if and only if there exist symmetric fuzzy numbers $\tilde{s_{1}}, \tilde{s_{2}} \in §$ such that $\tilde{x} + \tilde{s_{1}} = \tilde{y} + \tilde{s_{2}} .$

The equivalence relation defined above is reflexive, symmetric and transitive [17]. Let $〈 \tilde{x} 〉$ denote the equivalence class containing the element $\tilde{x}$ and denote the set of equivalence classes by 𝒻/§.

Definition 2.2. [15] Let $f : [a, b] \to ℝ$ . f is said be of bounded variation if there exists a C > 0 such that $\sum_{i = 1}^{n} | f (x_{i - 1}) - f (x_{i}) | \leq C$ for every partition a = x₀ < x₁ < ⋯ < x_n = b on [a, b]. The total variation of f on [a, b] is defined by $V_{a}^{b} (f) = sup_{p} \sum_{i = 1}^{n} | f (x_{i - 1}) - f (x_{i}) |,$ where p represents all partitions of [a, b]. The set of all functions of bounded variation on [a, b] is denoted by BV [a, b].

Lemma 2.1. [15] For any constants $c, d \in ℝ$ , if f, g ∈ BV [a, b], then so are cf + dg, fg and $V_{a}^{b} (cf + dg) \leq | c | V_{a}^{b} (f) + | d | V_{a}^{b} (g), V_{a}^{b} (fg) \leq M V_{a}^{b} (f) + L V_{a}^{b} (g),$ where M = sup |g (x) |< ∞ , L = sup |f (x) | < ∞.

Lemma 2.2. [15] Every monotonic function f is of bounded variation, and $V_{a}^{b} (f) = | f (a) - f (b) |$ .

Definition 2.3. [13] $\tilde{x}$ , we define a function ${\tilde{x}}_{M} : [0, 1] \to ℝ$ by assigning the midpoint of each α-level set to ${\tilde{x}}_{M} (α)$ for all α ∈ [0, 1], i.e., ${\tilde{x}}_{M} (α) = \frac{{\tilde{x}}_{L} (α) + {\tilde{x}}_{R} (α)}{2} .$ Then the function ${\tilde{x}}_{M} : [0, 1] \to ℝ$ will be called the midpoint function of the fuzzy number $\tilde{x}$ .

Lemma 2.3. [20] For any $\tilde{x} \in 𝒻$ , the midpoint function ${\tilde{x}}_{M}$ is continuous from the right at 0 and continuous from the left on [0, 1]. Furthermore it is a function of bounded variation on [0, 1].

Definition 2.4. [18] Let $\tilde{x} \in 𝒻$ and let $\hat{x}$ be a fuzzy number such that $\tilde{x} = \hat{x} + \tilde{s}$ for some $\tilde{s} \in §$ , if $\hat{x} = \tilde{y} + {\tilde{s}}_{1}$ for some $\tilde{y} \in 𝒻$ and ${\tilde{s}}_{1} \in §$ , then ${\tilde{s}}_{1} = \tilde{0}$ . Then the fuzzy number $\hat{x}$ will be called the Mareš core of the fuzzy number $\tilde{x}$ .

From the results [20], it follows that the each equivalence class corresponds to each midpoint function. Thus the Mareš core $\hat{x}$ of $\tilde{x}$ is also called the Mareš core of the equivalence class $〈 \tilde{x} 〉$ that contains the fuzzy number $\tilde{x}$ . It is natural to define the midpoint function for an equivalence class as follows:

Definition 2.5. [21] For any $〈 \tilde{x} 〉 \in 𝒻 / §$ , we define a midpoint function $M_{〈 \tilde{x} 〉} : [0, 1] \to ℝ$ by $M_{〈 \tilde{x} 〉} (α) = {\hat{x}}_{M} (α)$ for all α ∈ [0, 1], where $\hat{x}$ is Mareš core of $〈 \tilde{x} 〉$ .

Definition 2.6. [21] For any $〈 \tilde{x} 〉, 〈 \tilde{y} 〉 \in 𝒻 / §$ , we define $〈 \tilde{x} 〉 + 〈 \tilde{y} 〉$ by $〈 \tilde{x} 〉 + 〈 \tilde{y} 〉 = 〈 \tilde{x} + \tilde{y} 〉 .$

It is obvious that $M_{〈 \tilde{x} 〉 + 〈 \tilde{y} 〉} (α) = M_{〈 \tilde{x} + \tilde{y} 〉} (α) = M_{〈 \tilde{x} 〉} (α) + M_{〈 \tilde{y} 〉} (α)$ , for all α ∈ [0, 1]. Moreover, it follows from Definition 2.1 and 2.6 that $〈 \tilde{x} 〉 + 〈 \tilde{y} 〉 \supseteq {\tilde{z} = \tilde{x} + \tilde{y} : \tilde{x} \in 〈 \tilde{x} 〉, \tilde{y} \in 〈 \tilde{y} 〉},$ for any $〈 \tilde{x} 〉, 〈 \tilde{y} 〉 \in 𝒻 / §$ . In [22] Qiu et al. have given an example to show that the above inclusion can become strict.

Remark 2.1. The addition operation defined by Definition 2.6 is a group operation over the set of equivalence classes 𝒻/§ up to the equivalence relation in Definition 2.1.

By Lemma 2.1 and 2.3, for any $〈 \tilde{x} 〉, 〈 \tilde{y} 〉 \in 𝒻 / §$ , the function $M_{〈 \tilde{x} 〉} M_{〈 \tilde{y} 〉}$ is also continuous from the right at 0 and continuous from the left on [0, 1], and $M_{〈 \tilde{x} 〉} M_{〈 \tilde{y} 〉}$ is a function of bounded variation on [0, 1]. Hence, $M_{〈 \tilde{x} 〉} M_{〈 \tilde{y} 〉}$ is the midpoint function for some fuzzy number equivalence class.

Definition 2.7. [21] Let $〈 \tilde{x} 〉, 〈 \tilde{y} 〉, 〈 \tilde{z} 〉 \in 𝒻 / §$ . If for each α ∈ [0, 1], $M_{〈 \tilde{z} 〉} (α) = M_{〈 \tilde{x} 〉} (α) \cdot M_{〈 \tilde{y} 〉} (α),$ then $〈 \tilde{z} 〉$ is called the product of $〈 \tilde{x} 〉$ and $〈 \tilde{y} 〉$ , i.e., $〈 \tilde{z} 〉 = 〈 \tilde{x} 〉 \cdot 〈 \tilde{y} 〉$ .

Definition 2.8. [22] For any $〈 \tilde{x} 〉 \in 𝒻 / §$ and $λ \in ℝ$ , define $λ \cdot 〈 \tilde{x} 〉 = λ 〈 \tilde{x} 〉$ by $λ 〈 \tilde{x} 〉 = 〈 λ \tilde{x} 〉 .$

It is obvious that $M_{λ 〈 \tilde{x} 〉} (α) = M_{〈 λ \tilde{x} 〉} (α) = λ M_{〈 \tilde{x} 〉} (α)$ , for each α ∈ [0, 1] and $λ \hat{x}$ is the Mareš core of $λ 〈 \tilde{x} 〉$ if $\hat{x}$ is the Mareš core of $〈 \tilde{x} 〉$ .

Definition 2.9. [21] Define $d_{sup} : 𝒻 / § \times 𝒻 / § \to ℝ^{+} \cup {0}$ by $d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{y} 〉) = sup_{α \in [0, 1]} | M_{〈 \tilde{x} 〉} (α) - M_{〈 \tilde{y} 〉} (α) |,$ for any $〈 \tilde{x} 〉, 〈 \tilde{y} 〉 \in 𝒻 / §$ .

It is well known that (𝒻/§ , d_sup) is a metric space [20]. From Definition 2.9, some simple properties of the metric d_sup can be obtained as follow:

$(1) d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{z} 〉) \leq d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{y} 〉) + d_{sup}$ $(〈 \tilde{y} 〉, 〈 \tilde{z} 〉)$ ;

(2) $d_{sup} (λ 〈 \tilde{x} 〉, λ 〈 \tilde{y} 〉) = | λ | d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{y} 〉)$ ;

$(3) d_{sup} (〈 \tilde{x} 〉 + 〈 \tilde{z} 〉, 〈 \tilde{y} 〉 + 〈 \tilde{z} 〉) = d_{sup} (〈 \tilde{x} 〉,$ $〈 \tilde{y} 〉),$

for all $〈 \tilde{x} 〉, 〈 \tilde{y} 〉, 〈 \tilde{z} 〉 \in 𝒻 / §$ and $λ \in ℝ$ , where the addition and scalar multiplication on 𝒻/§ are defined in Definition 2.6 and 2.8, respectively.

3 Main results

Definition 3.1. [22] For each $m (t) \in C [J, ℝ]$ , where J is a subinterval of (0, + ∞), define $d^{+} : C [J, ℝ] \to ℝ$ by $d^{+} m (t) = \underset{h \to 0^{+}}{lim^{¯}} \frac{1}{h} (m (t + h) - m (t)) .$

Definition 3.2. [21] A mapping F : J→ 𝒻/§ is differentiable at t₀ ∈ J if for small |h|>0, there exists an F′ (t₀)∈ 𝒻/§ such that $lim_{h \to 0} d_{sup} (\frac{F (t_{0} + h) - F (t_{0})}{h}, F^{'} (t_{0})) = 0 .$

Definition 3.3. [21] A mapping F : J→ 𝒻/§ is measurable if F is measurable with respect to d_sup.

A mapping F : J→ 𝒻/§ is called integrably bounded if there exists an integrable function $h : J \to ℝ^{+} \cup {0}$ such that |M_F(t) (α) | ≤ h (t) for all t ∈ J and α ∈ [0, 1]; a mapping F : J→ 𝒻/§ is said to be of uniformly bounded variation with respect to α ∈ [0, 1] (for short, of uniformly bounded variation) if there exists a constant K > 0 such that $V_{0}^{1} (M_{F (t)}) \leq K$ , for each t ∈ J [22].

Definition 3.4. [21] Let F : J→ 𝒻/§ be measurable. The integral of F over J, denoted ∫_JF (t) dt, is a mapping $M_{\int_{J} F (t) dt} : [0, 1] \to ℝ$ , which is defined by the equation $M_{\int_{J} F (t) dt} (α) = \int_{J} M_{F (t)} (α) dt$ for each α ∈ [0, 1]. The mapping F is said to be integrable over J if there exists an $〈 {\tilde{x}}_{0} 〉 \in 𝒻 / §$ such that $M_{\int_{J} F (t) d t} = M_{〈 {\tilde{x}}_{0} 〉}$ . In this case, denote the integral by $\int_{J} F (t) dt = 〈 {\tilde{x}}_{0} 〉 .$

Assume that $f : ℝ_{+} \times S (ρ) \to 𝒻 / §$ is continuous and of uniformly bounded variation, where $S (ρ) = {〈 \tilde{x} 〉 \in 𝒻 / § : d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉) < ρ}$ . Consider the initial value problem for the fuzzy differential equation

$x^{'} (t) = f (t, x (t)), x (t_{0}) = x_{0} .$ (1)

Suppose that $f (t, 〈 \tilde{0} 〉) = 〈 \tilde{0} 〉$ so that we have the trivial solution $x (t) = 〈 \tilde{0} 〉$ for (1).

The stability results of solutions of (1) by Lyapunov’s second method will be shown. First, some notions of concerning the stability of the trivial solution of (1) need to define. Let x (t) = x (t, t₀, x₀) be any solution of (1) existing on [t₀, + ∞). Denote $K = {ω \in C [R_{+}, R_{+}], ω (0) = 0, ω (\cdot)$ is increasing }.

Definition 3.5. The trivial solution $x (t) = 〈 \tilde{0} 〉$ of (1) is said to be

(S1) stable, if for any ɛ > 0 and $t_{0} \in ℝ_{+}$ , there exists a δ = δ (t₀, ɛ) >0 such that $d_{sup} (x_{0}, 〈 \tilde{0} 〉) < δ$ implies $d_{sup} (x (t), 〈 \tilde{0} 〉) < ɛ, t \geq t_{0};$

(S2) uniformly stable, if δ in (S1) is independent of t₀;

(S3) exponentially asymptotically stable ( forshort, exponentially stable), if for any ɛ > 0, thereexists λ > 0 and δ = δ (ɛ) >0 such that if d_sup $(x_{0}, 〈 \tilde{0} 〉) < δ$ then $d_{sup} (x (t), 〈 \tilde{0} 〉) < ɛ e^{- λ (t - t_{0})}, t \geq t_{0} .$

Lemma 3.1. [16] Suppose that g (t, φ) be a continuous function on $ℝ_{+}^{2}$ and r (t) = r (t, t₀, φ₀), φ (t₀) = φ₀ be the maximal solution of the scalar differential equation:

$\frac{d φ}{dt} = g (t, φ), φ (t_{0}) = φ_{0} \geq 0,$ (2) existing on [t₀, + ∞). Let m (t) be a continuous function on $ℝ_{+}$ satisfies $d^{+} m (t) = \underset{h \to 0^{+}}{lim^{¯}} \frac{m (t + h) - m (t)}{h} \leq g (t, m (t)),$ whenever t ≥ t₀ . Then m (t) ≤ r (t), for each t ≥ t₀ if m (t₀) ≤ φ₀.

Let $V (t, 〈 \tilde{x} 〉) : ℝ_{+} \times S (ρ) \to ℝ$ be a given function. Then we define $\begin{matrix} D_{f}^{+} V (t, 〈 \tilde{x} 〉) \\ = \underset{h \to 0^{+}}{lim^{¯}} \frac{1}{h} (V (t + h, 〈 \tilde{x} 〉 + hf (t, 〈 \tilde{x} 〉)) - V (t, 〈 \tilde{x} 〉)), \end{matrix}$ where f (·) is the right-hand side of (1). Note that, if V (t, x) is Lipchitzian in x, then $d^{+} V (t, x (t)) \leq D_{f}^{+} V (t, x (t)) .$

Lemma 3.2. Suppose that (1) $| V (t, 〈 \tilde{x} 〉) - V$ $(t, 〈 \tilde{y} 〉) | \leq L (t) d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{y} 〉)$ , $V (\cdot, \cdot) \in C [ℝ_{+} \times S$ $(ρ), ℝ_{+}]$ and $L (\cdot) \in C [ℝ_{+}, ℝ_{+}]$ ; (2) $D_{f}^{+} V (t, 〈 \tilde{x} 〉)$ $\leq g (t, V (t, 〈 \tilde{x} 〉))$ , $g (\cdot, \cdot) \in C [ℝ_{+}^{2}, ℝ]$ If x (t) = x (t, t₀, x₀) is any solution of (1) through (t₀, x₀) existing on [t₀, + ∞) such that V (t₀, x₀) ≤ φ₀, then $V (t, x (t)) \leq r (t, t_{0}, φ_{0}), t \geq t_{0},$ where r (t, t₀, φ₀) is the maximal solution of the scalar differential equation (2) existing on [t₀, + ∞).

Proof. Let m (t) = V (t, x (t)), for each t ≥ t₀. Then m (t₀) = V (t₀, x₀) ≤ φ₀ and for small h > 0, $\begin{matrix} m (t + h) - m (t) \\ = V (t + h, x (t + h)) - V (t, x (t)) \\ = V (t + h, x (t + h)) - V (t + h, x (t) + hf (t, x (t))) \\ + V (t + h, x (t) + hf (t, x (t))) - V (t, x (t)) \\ \leq L (t + h) d_{sup} (x (t + h), x (t) + hf (t, x (t))) \\ + V (t + h, x (t) + hf (t, x (t))) - V (t, x (t)) . \end{matrix}$ Thus, $\begin{matrix} d^{+} m (t) \\ = \underset{h \to 0^{+}}{lim^{¯}} \frac{m (t + h) - m (t)}{h} \leq D_{f}^{+} V (t, x (t)) \\ + L (t) \underset{h \to 0^{+}}{lim^{¯}} \frac{1}{h} d_{sup} (x (t + h), x (t) + hf (t, x (t))) \\ = D_{f}^{+} V (t, x (t)) + L (t) d_{sup} (x^{'} (t), f (t, x (t))) \\ = D_{f}^{+} V (t, x (t)) \leq g (t, V (t, x (t))) = g (t, m (t)), \end{matrix}$ for each t ≥ t₀. By Lemma 3.1, we obtain $V (t, x (t)) = m (t) \leq r (t, t_{0}, φ_{0}), t \geq t_{0} .$

Corollary 3.1. Suppose that (1) $| V (t, 〈 \tilde{x} 〉)$ $- V (t, 〈 \tilde{y} 〉) | \leq L (t) d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{y} 〉)$ , $V (\cdot, \cdot) \in C [ℝ_{+}$ $\times S (ρ), ℝ_{+}]$ and $L (\cdot) \in C [ℝ_{+}, ℝ_{+}]$ ; (2) $D_{f}^{+} V$ $(t, 〈 \tilde{x} 〉) \leq 0$ . If x (t) = x (t, t₀, x₀) is any solution of (1) through (t₀, x₀) existing on [t₀, + ∞), then $V (t, x (t)) \leq V (t_{0}, x_{0}), t \geq t_{0} .$

Proof. Let the function g (t, φ) ≡0, $(t, φ) \in ℝ_{+}^{2}$ and φ₀ = V (t₀, x₀) in Lemma 3.2. Then we know that r (t, t₀, φ₀) ≡ V (t₀, x₀) is the unique solution of the scalar differential equation (2). Thus, by Lemma 3.2, we obtain $V (t, x (t)) \leq V (t_{0}, x_{0}), t \geq t_{0} .$

Theorem 3.1. Suppose that there exists a function $V (t, 〈 \tilde{x} 〉)$ satisfies the following conditions: (1) $| V (t, 〈 \tilde{x} 〉) - V (t, 〈 \tilde{y} 〉) | \leq L (t) d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{y} 〉)$ , $V (\cdot, \cdot) \in C [ℝ_{+} \times S (ρ), ℝ_{+}]$ and $L (\cdot) \in C [ℝ_{+}, ℝ_{+}]$ ; (2) $ω (d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉)) \leq V (t, 〈 \tilde{x} 〉)$ , $V (t, 〈 \tilde{0} 〉) = 0$ , $ω (\cdot) \in K$ ; (3) $D_{f}^{+} V (t, 〈 \tilde{x} 〉) \leq g (t, V (t, 〈 \tilde{x} 〉))$ , g (t, 0)= 0, where $g (\cdot, \cdot) \in C [ℝ_{+}^{2}, ℝ]$

If the solution φ (t) =0 of (2) is stable, then the trivial solution $x (t) = 〈 \tilde{0} 〉$ of (1) is stable.

Proof. Since the solution φ (t) =0 of (2) is stable, for any 0 < ɛ < ρ and $t_{0} \in ℝ_{+}$ , there exists a δ₀ = δ₀ (t₀, ɛ) >0 such that if 0 ≤ φ₀ < δ₀, then |φ (t, t₀, φ₀) | < ω (ɛ) for each t ≥ t₀. Since $V (t, 〈 \tilde{0} 〉) = 0$ , we have $\begin{matrix} V (t_{0}, 〈 \tilde{x} 〉) & = & | V (t_{0}, 〈 \tilde{x} 〉) - V (t_{0}, 〈 \tilde{0} 〉) | \\ \leq & L (t_{0}) d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉), \end{matrix}$ for each $〈 \tilde{x} 〉 \in S (ρ)$ . Thus, there exists δ = δ (t₀, ɛ) such that if $d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉) < δ$ , then $V (t_{0}, 〈 \tilde{x} 〉) < δ_{0}$ .

Let x (t) = x (t, t₀, x₀), $t_{0} \in ℝ_{+}$ be any solution of (1) through (t₀, x₀) existing on [t₀, + ∞). Next, we shall show that if $d_{sup} (x_{0}, 〈 \tilde{0} 〉) < δ$ then $d_{sup} (x (t), 〈 \tilde{0} 〉) < ɛ$ for each t ≥ t₀. By the conditions (1), (3) and Lemma 3.2, we get $V (t, x (t)) \leq r (t, t_{0}, V (t_{0}, x_{0})), t \geq t_{0},$ where r (t, t₀, V (t₀, x₀)) is the maximal solution of the scalar differential equation (2) existing on [t₀, + ∞). Since V (t₀, x₀) < δ₀, we have r (t, t₀, V (t₀, x₀)) < ω (ɛ) for each t ≥ t₀ and therefore $V (t, x (t)) \leq r (t, t_{0}, V (t_{0}, x_{0})) < ω (ɛ), t \geq t_{0} .$

By the condition (2), we get $ω (d_{sup} (x (t), 〈 \tilde{0} 〉)) \leq V (t, x (t)) < ω (ɛ), t \geq t_{0} .$

By the monotonicity of ω, we have $d_{sup} (x (t), 〈 \tilde{0} 〉) < ɛ, t \geq t_{0} .$

Hence, the trivial solution $x (t) = 〈 \tilde{0} 〉$ of (1) is stable.

Example 3.1. Define $F : ℝ_{+} \to 𝒻 / §$ by the α-level sets of the fuzzy mapping ${[\hat{F (t)}]}^{α} = [- \frac{2 e^{- α}}{{(1 + t)}^{2}}, 0], α \in [0, 1],$ where $\hat{F (t)}$ is the Mareš core of F (t), for each $t \in ℝ_{+}$ . Thus, we have $M_{F (t)} (α) = - \frac{e^{- α}}{{(1 + t)}^{2}}, α \in [0, 1],$ for each $t \in ℝ_{+}$ . It is obvious that M_F(t) (α) is continuous from the right at 0 and continuous from the left on [0, 1] with respect to α. Since M_F(t) (α) is increasing with respect to α and by Lemma 2.2, we get $V_{0}^{1} (M_{F (t)}) = \frac{1 - e^{- 1}}{{(1 + t)}^{2}} \leq 1 - e^{- 1}, t \in ℝ_{+} .$

Thus, we obtain that F (t) is of uniformly bounded variation. Since M_F(t) (α) is uniformly continuous with respect to $t \in ℝ_{+}$ , we get that F (t) is continuous with respect to d_sup. Define $f : ℝ_{+} \times 𝒻 / § \to 𝒻 / §$ by $f (t, 〈 \tilde{x} 〉) = F (t) 〈 \tilde{x} 〉,$ where the multiplication in 𝒻/§ is defined by Definition 2.7. It is obvious that f is continuous with respect to d_sup and of uniformly bounded variation.

Consider a Lyapunov function $V (t, 〈 \tilde{x} 〉) = d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉) .$

Then $V (t, 〈 \tilde{0} 〉) = d_{sup} (〈 \tilde{0} 〉, 〈 \tilde{0} 〉) = 0$ and $\begin{matrix} | V (t, 〈 \tilde{x} 〉) - V (t, 〈 \tilde{y} 〉) | \\ = & | d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉) - d_{sup} (〈 \tilde{y} 〉, 〈 \tilde{0} 〉) | \\ \leq & d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{y} 〉), \end{matrix}$ for any $(t, 〈 \tilde{x} 〉), (t, 〈 \tilde{y} 〉) \in ℝ_{+} \times 𝒻 / §$ . By Definition 2.9, for a small h > 0, we have $\begin{matrix} V (t + h, 〈 \tilde{x} 〉 + hf (t, 〈 \tilde{x} 〉)) \\ = & d_{sup} (〈 \tilde{x} 〉 + hf (t, 〈 \tilde{x} 〉), 〈 \tilde{0} 〉) \\ = & d_{sup} (〈 \tilde{x} 〉 + hF (t) 〈 \tilde{x} 〉, 〈 \tilde{0} 〉) \\ = & sup_{α \in [0, 1]} | M_{〈 \tilde{x} 〉} (α) + h M_{F (t)} (α) M_{〈 \tilde{x} 〉} (α) | \\ \leq & sup_{α \in [0, 1]} | M_{〈 \tilde{x} 〉} (α) | (1 + h sup_{α \in [0, 1]} | M_{F (t)} (α) |) \\ = & (1 + \frac{h}{(1 + t)^{2}}) d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉) . \end{matrix}$

Hence, $\begin{matrix} D_{f}^{+} V (t, 〈 \tilde{x} 〉) \\ = \underset{h \to 0^{+}}{lim^{¯}} \frac{1}{h} (V (t + h, 〈 \tilde{x} 〉 + hf (t, 〈 \tilde{x} 〉)) - V (t, 〈 \tilde{x} 〉)) \\ \leq \frac{1}{(1 + t)^{2}} d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉) . \end{matrix}$

Let $g (t, φ) = \frac{1}{(1 + t)^{2}} φ$ . Then, we have $D_{f}^{+} V (t, 〈 \tilde{x} 〉) \leq g (t, d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉)) = g (t, V (t, 〈 \tilde{x} 〉)) .$

It’s easy to show that the solution φ = 0 of (2) is stable. Hence, by Theorem 3.1, the trivial solution $x (t) = 〈 \tilde{0} 〉$ of (1) is stable.

Theorem 3.2. Suppose that there exists a function $V (t, 〈 \tilde{x} 〉)$ satisfies the following conditions: (1) $| V (t, 〈 \tilde{x} 〉) - V (t, 〈 \tilde{y} 〉) | \leq L (t) d_{sup}$ $(〈 \tilde{x} 〉, 〈 \tilde{y} 〉)$ , $V (\cdot, \cdot) \in C [ℝ_{+} \times S (ρ), ℝ_{+}]$ and L (·) ∈ $C [ℝ_{+}, ℝ_{+}]$ ; (2) $ω_{1} (d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉)) \leq V (t, 〈 \tilde{x} 〉)$ $\leq ω_{2} (t, d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉))$ , $ω_{1} (\cdot), ω_{2} (t, \cdot) \in K$ ; (3) $D_{f}^{+} V (t, 〈 \tilde{x} 〉) \leq 0$ .

Then the trivial solution $x (t) = 〈 \tilde{0} 〉$ of (1) is stable.

Proof. Since $ω_{1} (\cdot), ω_{2} (\cdot) \in K$ , for any 0 < ɛ < ρ and $t_{0} \in ℝ_{+}$ is given, there exist a δ = δ (t₀, ɛ) such that ω₂ (t₀, δ) < ω₁ (ɛ). Let x (t) = x (t, t₀, x₀) be any solution of (1) through (t₀, x₀) existing on [t₀, + ∞). By the conditions (1), (3) and Corollary 3.1, we get V (t, x (t)) ≤ V (t₀, x₀) , t ≥ t₀ . By the condition (2), we get $\begin{matrix} ω_{1} (d_{sup} (x (t), 〈 \tilde{0} 〉) \leq V (t, x (t)) \\ \leq V (t_{0}, x_{0}) \leq ω_{2} (t_{0}, d_{sup} (x_{0}, 〈 \tilde{0} 〉)), \end{matrix}$ for each t ≥ t₀. Thus, if $d_{sup} (x_{0}, 〈 \tilde{0} 〉) < δ$ , then $\begin{matrix} ω_{1} (d_{sup} (x (t), 〈 \tilde{0} 〉) \\ \leq & ω_{2} (t_{0}, d_{sup} (x_{0}, 〈 \tilde{0} 〉)) \leq ω_{2} (t_{0}, δ) < ω_{1} (ɛ), \end{matrix}$ which implies that $d_{sup} (x (t), 〈 \tilde{0} 〉) < ɛ, t \geq t_{0} .$ Hence, the trivial solution $x (t) = 〈 \tilde{0} 〉$ of (1) is stable.

Example 3.2. Define $F : ℝ_{+} \to 𝒻 / §$ by the α-level sets of the fuzzy mapping ${[\hat{F (t)}]}^{α} = [\frac{- 2 α}{1 + t}, 0], α \in [0, 1],$ where $\hat{F (t)}$ is the Mareš core of F (t), for each $t \in ℝ_{+}$ . Thus, we have $M_{F (t)} (α) = \frac{- α}{1 + t}, α \in [0, 1],$ for each $t \in ℝ_{+}$ . Define $f : ℝ_{+} \times 𝒻 / § \to 𝒻 / §$ by $f (t, 〈 \tilde{x} 〉) = F (t) 〈 \tilde{x} 〉,$ where the multiplication in 𝒻/§ is defined by Definition 2.7.

Consider a Lyapunov function $V (t, 〈 \tilde{x} 〉) = - d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉) .$

Then $\begin{matrix} | V (t, 〈 \tilde{x} 〉) - V (t, 〈 \tilde{y} 〉) | \\ = & | d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉) - d_{sup} (〈 \tilde{y} 〉, 〈 \tilde{0} 〉) | \\ \leq & d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{y} 〉), \end{matrix}$ for any $(t, 〈 \tilde{x} 〉), (t, 〈 \tilde{y} 〉) \in ℝ_{+} \times 𝒻 / §$ . By Definition 2.9, for a small h with 0 < h < 1, $\begin{matrix} V (t + h, 〈 \tilde{x} 〉 + hf (t, 〈 \tilde{x} 〉)) \\ = & - d_{sup} (〈 \tilde{x} 〉 + hf (t, 〈 \tilde{x} 〉), 〈 \tilde{0} 〉) \\ = & - d_{sup} (〈 \tilde{x} 〉 + hF (t) 〈 \tilde{x} 〉, 〈 \tilde{0} 〉) \\ = & - sup_{α \in [0, 1]} | M_{〈 \tilde{x} 〉} (α) + h M_{F (t)} (α) M_{〈 \tilde{x} 〉} (α) | \\ = & - sup_{α \in [0, 1]} | M_{〈 \tilde{x} 〉} (α) | | 1 + h sup_{α \in [0, 1]} M_{F (t)} (α) | \\ \leq & 0 . \end{matrix}$

By Theorem 3.2, the trivial solution $x (t) = 〈 \tilde{0} 〉$ of (1) is stable.

Theorem 3.3. Suppose that there exists a function $V (t, 〈 \tilde{x} 〉)$ satisfies the following conditions: (1) $| V (t, 〈 \tilde{x} 〉) - V (t, 〈 \tilde{y} 〉) | \leq L (t) d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{y} 〉)$ , $V (\cdot, \cdot) \in C [ℝ_{+} \times S (ρ), ℝ_{+}]$ and $L (\cdot) \in C [ℝ_{+}, ℝ_{+}]$ ; (2) $ω_{1} (d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉)) \leq V (t, 〈 \tilde{x} 〉) \leq ω_{2} (d_{sup} (〈 \tilde{x} 〉,$ $〈 \tilde{0} 〉))$ , $ω_{1} (\cdot), ω_{2} (\cdot) \in K$ ; (3) $D_{f}^{+} V (t, 〈 \tilde{x} 〉) \leq g (t, V$ $(t, 〈 \tilde{x} 〉))$ , g (t, 0) =0, where $g (\cdot, \cdot) \in C [ℝ_{+}^{2}, ℝ]$ .

If the solution φ (t) =0 of (2) is uniformly stable, then the trivial solution $x (t) = 〈 \tilde{0} 〉$ of (1) is uniformly stable.

Proof. Since the solution φ (t) =0 of (2) is uniformly stable, for any 0 < ɛ < ρ, there exists a δ₀ = δ₀ (ɛ) >0 such that if $t_{0} \in ℝ_{+}$ and 0 ≤ φ₀ < δ₀, then |φ (t, t₀, φ₀) | < ω₁ (ɛ) for each t ≥ t₀. Since $ω_{1} (\cdot), ω_{2} (\cdot) \in K$ , there exist a δ = δ (ɛ) >0 such that ω₂ (δ) < ω₁ (δ₀).

Let x (t) = x (t, t₀, x₀), $t_{0} \in ℝ_{+}$ be any solution of (1) through (t₀, x₀) existing on [t₀, + ∞). Next, we shall show that if $d_{sup} (x_{0}, 〈 \tilde{0} 〉) < δ$ then $d_{sup} (x (t), 〈 \tilde{0} 〉) < ɛ$ for each t ≥ t₀. By the conditions (1), (3) and Lemma 3.2, we get $V (t, x (t)) \leq r (t, t_{0}, ω_{1}^{- 1} (V (t_{0}, x_{0}))), t \geq t_{0},$ where $r (t, t_{0}, ω_{1}^{- 1} (V (t_{0}, x_{0})))$ is the maximal solution of the scalar differential equation (2) existing on [t₀, + ∞). By the condition (2), $V (t_{0}, x_{0}) \leq ω_{2} (d_{sup} (x_{0}, 〈 \tilde{0} 〉)) \leq ω_{2} (δ) < ω_{1} (δ_{0}) .$

Thus, by the monotonicity of ω₁, $ω_{1}^{- 1} (V (t_{0}, x_{0}))$ ≤δ₀, which implies that $r (t, t_{0}, ω_{1}^{- 1} (V (t_{0}, x_{0}))) < ω_{1} (ɛ), t \geq t_{0}$

and therefore $V (t, x (t)) \leq r (t, t_{0}, ω_{1}^{- 1} (V (t_{0}, x_{0}))) < ω_{1} (ɛ), t \geq t_{0} .$

By the condition (2), we get $ω_{1} (d_{sup} (x (t), 〈 \tilde{0} 〉)) \leq V (t, x (t)) < ω_{1} (ɛ), t \geq t_{0} .$

By the monotonicity of ω₁, we have $d_{sup} (x (t), 〈 \tilde{0} 〉) < ɛ, t \geq t_{0} .$

Hence, the trivial solution $x (t) = 〈 \tilde{0} 〉$ of (1) is uniformly stable.

Example 3.3. Define $F : ℝ_{+} \to 𝒻 / §$ by the α-level sets of the fuzzy mapping ${[\hat{F (t)}]}^{α} = [0, \frac{2 e^{- α}}{1 + t^{2}}], α \in [0, 1],$ where $\hat{F (t)}$ is the Mareš core of F (t), for each $t \in ℝ_{+}$ . Thus, we have $M_{F (t)} (α) = \frac{e^{- α}}{1 + t^{2}}, α \in [0, 1],$ for each $t \in ℝ_{+}$ . It is obvious that M_F(t) (α) is continuous from the right at 0 and continuous from the left on [0, 1] with respect to α. Since M_F(t) (α) is decreasing with respect to α and by Lemma 2.2, we get $V_{0}^{1} (M_{F (t)}) = \frac{1 - e^{- 1}}{1 + t^{2}} \leq 1 - e^{- 1}, t \in ℝ_{+} .$

Thus, we obtain that F (t) is of uniformly bounded variation. Since M_F(t) (α) is uniformly continuous with respect to t, we get that F (t) is continuous with respect to d_sup. Define $f : ℝ_{+} \times 𝒻 / § \to 𝒻 / §$ by $f (t, 〈 \tilde{x} 〉) = F (t) 〈 \tilde{x} 〉,$ where the multiplication in 𝒻/§ is defined by Definition 2.7. It is obvious that f is continuous with respect to d_sup and of uniformly bounded variation.

Consider a Lyapunov function $V (t, 〈 \tilde{x} 〉) = d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉) .$

Then $\begin{matrix} \frac{2}{3} d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉) \leq V (t, 〈 \tilde{x} 〉) \leq \frac{3}{2} d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉) \end{matrix}$ and $\begin{matrix} | V (t, 〈 \tilde{x} 〉) - V (t, 〈 \tilde{y} 〉) | \\ = & | d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉) - d_{sup} (〈 \tilde{y} 〉, 〈 \tilde{0} 〉) | \\ \leq & d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{y} 〉), \end{matrix}$ for any $(t, 〈 \tilde{x} 〉), (t, 〈 \tilde{y} 〉) \in ℝ_{+} \times 𝒻 / §$ . By Definition 2.9, for a small h > 0, $\begin{matrix} V (t + h, 〈 \tilde{x} 〉 + hf (t, 〈 \tilde{x} 〉)) \\ = & d_{sup} (〈 \tilde{x} 〉 + hf (t, 〈 \tilde{x} 〉), 〈 \tilde{0} 〉) \\ = & d_{sup} (〈 \tilde{x} 〉 + hF (t) 〈 \tilde{x} 〉, 〈 \tilde{0} 〉) \\ = & sup_{α \in [0, 1]} | M_{〈 \tilde{x} 〉} (α) + h M_{F (t)} (α) M_{〈 \tilde{x} 〉} (α) | \\ \leq & sup_{α \in [0, 1]} | M_{〈 \tilde{x} 〉} (α) | (1 + h sup_{α \in [0, 1]} | M_{F (t)} (α) |) \\ = & (1 + \frac{h}{1 + t^{2}}) d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉) . \end{matrix}$

Hence, $\begin{matrix} D_{f}^{+} V (t, 〈 \tilde{x} 〉) \\ = \underset{h \to 0^{+}}{lim^{¯}} \frac{1}{h} (V (t + h, 〈 \tilde{x} 〉 + hf (t, 〈 \tilde{x} 〉)) - V (t, 〈 \tilde{x} 〉)) \\ \leq \frac{1}{1 + t^{2}} d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉) . \end{matrix}$

Let $g (t, φ) = \frac{1}{1 + t^{2}} φ$ . Then, we have $D_{f}^{+} V (t, 〈 \tilde{x} 〉) \leq g (t, d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉)) = g (t, V (t, 〈 \tilde{x} 〉)) .$

It’s easy to show that the solution φ = 0 of (1) is uniformly stable. Hence, by Theorem 3.3, the trivial solution $x (t) = 〈 \tilde{0} 〉$ of (1) is uniformly stable.

Theorem 3.4. Suppose that there exists a function $V (t, 〈 \tilde{x} 〉)$ satisfies the following conditions:(1) $| V (t, 〈 \tilde{x} 〉) - V (t, 〈 \tilde{y} 〉) | \leq L (t) d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{y} 〉)$ , $V (\cdot, \cdot) \in C [ℝ_{+} \times S (ρ), ℝ_{+}]$ and $L (\cdot) \in C [ℝ_{+}, ℝ_{+}]$ ;(2) $ω_{1} (d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉)) \leq V (t, 〈 \tilde{x} 〉) \leq ω_{2} (d_{sup} (〈 \tilde{x} 〉,$ $〈 \tilde{0} 〉))$ , $ω_{1} (\cdot), ω_{2} (\cdot) \in K$ ; (3) $D_{f}^{+} V (t, 〈 \tilde{x} 〉) \leq 0$ .

Then the trivial solution $x (t) = 〈 \tilde{0} 〉$ of (1) is uniformly stable.

Proof. Let x (t) = x (t, t₀, x₀), $t_{0} \in ℝ_{+}$ be any solution of (1) through (t₀, x₀) existing on [t₀, + ∞). By the conditions (1), (3) and Corollary 3.1, we get $V (t, x (t)) \leq V (t_{0}, x_{0}), t \geq t_{0} .$

By the condition (2), $\begin{matrix} ω_{1} (d_{sup} (x (t), 〈 \tilde{0} 〉) \leq V (t, x (t)) \leq V (t_{0}, x_{0}) \\ \leq ω_{2} (d_{sup} (x_{0}, 〈 \tilde{0} 〉)), \end{matrix}$ for each t ≥ t₀. Since $ω_{1} (\cdot), ω_{2} (\cdot) \in K$ , for any 0 < ɛ < ρ, there exist a δ = δ (ɛ) such that ω₂ (δ) < ω₁ (ɛ). Thus, if $d_{sup} (x_{0}, 〈 \tilde{0} 〉) < δ$ , then $ω_{1} (d_{sup} (x (t), 〈 \tilde{0} 〉) \leq ω_{2} (d_{sup} (x_{0}, 〈 \tilde{0} 〉)) < ω_{1} (ɛ),$ which implies that $d_{sup} (x (t), 〈 \tilde{0} 〉) < ɛ, t \geq t_{0} .$ Hence, the trivial solution $x (t) = 〈 \tilde{0} 〉$ of (1) is uniformly stable.

Example 3.4. Define $F : ℝ_{+} \to 𝒻 / §$ as the one in Example 3.3. Define $f : ℝ_{+} \times 𝒻 / § \to 𝒻 / §$ by $f (t, 〈 \tilde{x} 〉) = F (t) 〈 \tilde{x} 〉,$ where the multiplication in 𝒻/§ is defined by Definition 2.7.

Consider a Lyapunov function $V (t, 〈 \tilde{x} 〉) = - d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉) .$

Then $- d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉) \leq V (t, 〈 \tilde{x} 〉) \leq d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉)$ and $\begin{matrix} | V (t, 〈 \tilde{x} 〉) - V (t, 〈 \tilde{y} 〉) | \\ = & | d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉) - d_{sup} (〈 \tilde{y} 〉, 〈 \tilde{0} 〉) | \\ \leq & d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{y} 〉), \end{matrix}$ for any $(t, 〈 \tilde{x} 〉), (t, 〈 \tilde{y} 〉) \in ℝ_{+} \times 𝒻 / §$ . By Definition 2.9, for a small h with 0 < h < 1, we have $\begin{matrix} V (t + h, 〈 \tilde{x} 〉 + hf (t, 〈 \tilde{x} 〉)) \\ = & - d_{sup} (〈 \tilde{x} 〉 + hf (t, 〈 \tilde{x} 〉), 〈 \tilde{0} 〉) \\ = & - d_{sup} (〈 \tilde{x} 〉 + hF (t) 〈 \tilde{x} 〉, 〈 \tilde{0} 〉) \\ = & - sup_{α \in [0, 1]} | M_{〈 \tilde{x} 〉} (α) + h M_{F (t)} (α) M_{〈 \tilde{x} 〉} (α) | \\ \leq & 0, \end{matrix}$ which implies that $D_{f}^{+} V (t, 〈 \tilde{x} 〉) \leq 0 .$ Hence, by Theorem 3.4, the trivial solution $x (t) = 〈 \tilde{0} 〉$ of (1) is uniformly stable.

Theorem 3.5. Suppose that there exists a function $V (t, 〈 \tilde{x} 〉)$ satisfies the following conditions: (1) $| V (t, 〈 \tilde{x} 〉) - V (t, 〈 \tilde{y} 〉) | \leq L (t) d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{y} 〉)$ , $V (\cdot, \cdot) \in C [ℝ_{+} \times S (ρ), ℝ_{+}]$ and $L (\cdot) \in C [ℝ_{+}, ℝ_{+}]$ ; (2) $a (d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉))^{p} \leq V (t, 〈 \tilde{x} 〉) \leq b (d_{sup} (〈 \tilde{x} 〉,$ $〈 \tilde{0} 〉))^{q}$ ; (3) $D_{f}^{+} V (t, 〈 \tilde{x} 〉) \leq - γ (d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉))^{q}$ , where a, b, p, q, γ are positive numbers. Then the trivial solution $x (t) = 〈 \tilde{0} 〉$ of (1) is exponentially stable.

Proof. Let x (t) = x (t, t₀, x₀) be any solution of (1) through (t₀, x₀) existing on [t₀, + ∞), $M = \frac{γ}{b}$ and m (t) = V (t, x (t)) e^M(t-t₀) for each t ≥ t₀. Then, for small h > 0, $\begin{matrix} m (t + h) - m (t) \\ = V (t + h, x (t + h)) e^{M (t + h - t_{0})} - V (t, x (t)) e^{M (t - t_{0})} \\ = V (t + h, x (t + h)) (e^{M (t + h - t_{0})} - e^{M (t - t_{0})}) \\ + (V (t + h, x (t + h)) - V (t, x (t))) e^{M (t - t_{0})} . \end{matrix}$

Thus, we get $\begin{matrix} d^{+} m (t) \\ = \underset{h \to 0^{+}}{lim^{¯}} \frac{m (t + h) - m (t)}{h} \\ = \underset{h \to 0^{+}}{lim^{¯}} \frac{V (t + h, x (t + h)) (e^{M (t + h - t_{0})} - e^{M (t - t_{0})})}{h} \\ + e^{M (t - t_{0})} \underset{h \to 0^{+}}{lim^{¯}} \frac{(V (t + h, x (t + h)) - V (t, x (t)))}{h} \\ = MV (t, x (t)) e^{M (t - t_{0})} + e^{M (t - t_{0})} d^{+} V (t, x (t)) \\ \leq MV (t, x (t)) e^{M (t - t_{0})} + e^{M (t - t_{0})} D_{f}^{+} V (t, x (t)) \\ \leq MV (t, x (t)) e^{M (t - t_{0})} + e^{M (t - t_{0})} - γ {(d_{sup} (x (t), 〈 \tilde{0} 〉))}^{q} \\ \leq MV (t, x (t)) e^{M (t - t_{0})} - MV (t, x (t)) e^{M (t - t_{0})} = 0 . \end{matrix}$

By Lemma 3.1, we get m (t) ≤ m (t₀) for each t ≥ t₀. Since $m (t_{0}) = V (t_{0}, x_{0}) \leq b (d_{sup} (x_{0}, 〈 \tilde{0} 〉))^{q}$ , we have $m (t) \leq b (d_{sup} (x_{0}, 〈 \tilde{0} 〉))^{q}, t \geq t_{0},$ which implies that $V (t, x (t)) \leq b (d_{sup} (x_{0}, 〈 \tilde{0} 〉))^{q} e^{- M (t - t_{0})}, t \geq t_{0} .$

By the condition (2), $a (d_{sup} (x (t), 〈 \tilde{0} 〉))^{p} \leq b (d_{sup} (x_{0}, 〈 \tilde{0} 〉))^{q} e^{- M (t - t_{0})} .$

Let $λ = \frac{M}{p}$ . Then, we obtain $d_{sup} (x (t) 〈 \tilde{0} 〉) \leq {(\frac{b}{a})}^{\frac{1}{p}} {(d_{sup} (x_{0}, 〈 \tilde{0} 〉))}^{\frac{q}{p}} e^{- λ (t - t_{0})} .$

Consequently, the trivial solution $x (t) = 〈 \tilde{0} 〉$ of (1) is exponentially stable.

Example 3.5. Define $F : ℝ_{+} \to 𝒻 / §$ by the α-level sets of the fuzzy mapping ${[\hat{F (t)}]}^{α} = [\frac{- 2 α}{(1 + t)^{1 / 2}}, 0], α \in [0, 1],$ where $\hat{F (t)}$ is the Mareš core of F (t), for each $t \in ℝ_{+}$ . Thus, we have $M_{F (t)} (α) = \frac{- α}{(1 + t)^{1 / 2}}, α \in [0, 1],$ for each $t \in ℝ_{+}$ . Define $f : ℝ_{+} \times 𝒻 / § \to 𝒻 / §$ by $f (t, 〈 \tilde{x} 〉) = F (t) 〈 \tilde{x} 〉,$ where the multiplication in 𝒻/§ is defined by Definition 2.7.

Consider a Lyapunov function $V (t, 〈 \tilde{x} 〉) = d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉)^{1 / 2} .$

Let a = 1/2, b = 2, p = q = 1/2 and γ = 1. By some routine calculations we obtain $| V (t, 〈 \tilde{x} 〉) - V (t, 〈 \tilde{y} 〉) | \leq L (t) d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{y} 〉),$ $V (\cdot, \cdot) \in C [ℝ_{+} \times S (ρ), ℝ_{+}]$ and $L (\cdot) \in C [ℝ_{+}, ℝ_{+}]$ ; $1 / 2 (d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉))^{1 / 2} \leq V (t, 〈 \tilde{x} 〉) \leq 2 (d_{sup} (〈 \tilde{x} 〉,$ $〈 \tilde{0} 〉))^{1 / 2};$ $D_{f}^{+} V (t, 〈 \tilde{x} 〉) \leq - (d_{sup} (〈 \tilde{x} 〉, 〈 \tilde{0} 〉))^{1 / 2} .$ By Theorem 3.5, the trivial solution $x (t) = 〈 \tilde{0} 〉$ of (1) is exponentially stable.

4 Conclusions

In this paper, the stability of the trivial solution of the fuzzy differential equations in the quotient space of fuzzy numbers have been researched with Lyapunov’s second method. Some sufficient conditions for the stability, uniformly stability and exponentially stability of the trivial solution of the fuzzy differential equations have been given by using the differential inequalities and the comparison principle for Lyapunov-like functions. In [20, 21], some comparative studies have done on our method with the well known H-derivative [14] and gH-derivative [7] (see Theorem 5.7, Remark 5.1, Example 5.1, Theorem 6.2, Remark 6.1 and Example 6.1 of [21] and Theorem 5.2 of [20]). From those results, it follows that the theory of fuzzy differential equations in the quotient space of fuzzy numbers can be regarded as a generalization of the ones in the metric space of fuzzy numbers.

Footnotes

Acknowledgments

The authors are grateful to the reviewers for their valuable comments. This work was supported by The National Natural Science Foundation of China (Grant no. 61472056), The Graduate Teaching Reform Research Program of Chongqing Municipal Education Commission (No.YJG143010), and The Fundamental Research Funds for Chongqing University of Arts and Sciences (Grant No.Y2013SC41)

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