Journal of Computational Information Systems 8: 20 (2012) 8341–8347 Available at http://www.Jofcis.com
Existence and Uniqueness Theorem for Uncertain Delay Differential Equations ? Xintong GE 1,2 , 1 Department 2 School
Yuanguo ZHU 1,∗
of Applied Mathematics, Nanjing University of Science and Technology, Jiangsu 210094, China
of Mathematics and Computation Sciences, Fuyang Normal College, Anhui 236041, China
Abstract Uncertain delay differential equation is a type of functional differential equations driven by canonical process. This paper presents a method to solve an uncertain delay differential equation, and proves an existence and uniqueness theorem of solution for uncertain delay differential equations under Lipschitz condition and linear growth condition by Banach fixed point theorem. Keywords: Uncertainty; Delay Differential Equation; Existence and Uniqueness Theorem; Banach Fixed Point Theorem
1
Introduction
Delay differential equation (DDE) derives from the fact that many of phenomena witnessed around us do not have an immediate effect from the moment of their occurrence. For example, in biology, engineering, physics and other sciences, system models with delay are often investigated. DDE is an important aspect of study and usually applied to model some hereditary characteristics such as after-effect, and to express the fact that the velocity of the system depends not only on the state of the system at a given instant but on the prehistory of the trajectory. There exists an extensive literature on functional differential equations and their applications. (see, for instance, Hale [9] 1997, and references cited therein). Because many practical problems contain some uncertainty and imprecise phenomena, so dynamical systems with delay are modeled by fuzzy delay differential equations (FDDEs) (see, for instance, Hsu, Ho, and Chou [10]) or stochastic delay (or functional) differential equations (for short, SDDEs or SFDEs) (see, for instance, Bakhtin [2, 1], Chang, Pang, and Pemy [4], Bauer, Rieder [3], Federico [6], Federico, Øksendal [7], and references cited therein.). Existence and uniqueness theorem for FDDEs has been established under local Lipschitz continuous conditions ?
Project supported by the National Natural Science Foundation of China (No. 61273009) and the Natural Science Foundation of Fuyang Normal College (No. 2007LZ01). ∗ Corresponding author. Email addresses:
[email protected] (Xintong GE),
[email protected] (Yuanguo ZHU).
1553–9105 / Copyright © 2012 Binary Information Press October 15, 2012
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on the coefficients (Lupulescu and Abbas [15]). Existence and uniqueness theorems for SDDEs have been established under global Lipschitz conditions on the coefficients (Mohammed [17]), under local Lipschitz conditions and linear growth conditions (Mao [16], Xu, Yang and Huang [19]) or under monotonicity of the coefficients conditions (von Renesse and Scheutzow [18]). The complexity of the world makes the events we face uncertain in various forms, such as “about 80km”, “approximately 100km”, “young”, “old”, etc. However, a lot of surveys showed that they behave neither like randomness nor like fuzziness. In order to deal with this uncertainty, uncertainty theory was founded by Liu [11] in 2007 and refined by Liu [14] in 2010. Based on uncertain calculus, uncertain differential equation proposed by Liu [11] is a type of differential equation driven by a canonical process Ct . Chen and Liu [5] proved an existence and uniqueness theorem of solution to uncertain differential equations under global Lipschitz condition and linear growth condition by a successive approximation method. Gao [8] proved similar theorem under local Lipschitz condition. Yao, Gao and Gao [20] gave some stability theorems for uncertain differential equations. Zhu [21] introduced and dealt with uncertain optimal control problems based on uncertain differential equations by using dynamic programming. The aim of this paper is to prove an existence and uniqueness theorem for uncertain delay differential equation driven by canonical process. The paper is organized as follows. In Section 2, we review some notations, concepts and lemmas. In Section 3, we introduce an uncertain delay differential equation and give an example. In Section 4, we prove an existence and uniqueness theorem of solution for uncertain delay differential equation under Lipschitz condition and linear growth condition.
2
Preliminary
In convenience, we give some notations, concepts and lemmas. Let Γ be a nonempty set, and L a σ-algebra over Γ. Each element Λ ∈ L is called an event. A set function M defined on the σ-algebra over L is called an uncertain measure if it satisfies that M{Γ} = 1 for the universal set ∞ Γ; M{Λ}+M{Λc } = 1 for any event Λ; M{∪∞ i=1 Λi } ≤ Σi=1 M{Λi } for Λi ∈ L. Then the triplet (Γ, L, M) is said to be an uncertainty space. An uncertain variable is a measurable function ξ from an uncertainty space (Γ, L, M) to the set of real numbers, i.e., for any Borel set of real numbers, the set {ξ ∈ B} = {γ ∈ Γ|ξ(γ) ∈ B} is an event. Let (Γk , Lk , Mk ) be uncertainty spaces for k = 1, 2, · · · , n. Then the product uncertain Qnmeasure M is an uncertain measure on the product σ-algebra L1 × L2 × · · · × Ln satisfying M { k=1 Λk } = min1≤k≤n Mk {Λk }. That is, for each event Λ ∈ L, we have M{Λ} = sup min Mk {Λk } if sup min Mk {Λk } > 0.5, or Λ1 ×Λ2 ×···×Λn ⊂Λ 1≤k≤n
1−
sup
min Mk {Λk } if
Λ1 ×Λ2 ×···×Λn ⊂Λc 1≤k≤n
sup
Λ1 ×Λ2 ×···×Λn ⊂Λ 1≤k≤n
min Mk {Λk } > 0.5, or 0.5 otherwise.
Λ1 ×Λ2 ×···×Λn ⊂Λc 1≤k≤n
The distribution Φ : R → [0, 1] of an uncertain variable ξ is defined Rby Φ(x) = M{γ ∈ +∞ Γ | ξ(γ) ≤ x} for x ∈ R. The expected value of ξ is defined by E[ξ] = 0 M{ξ ≥ r}dr − R0 M{ξ ≤ r}dr, provided that at least one of the two integrals is finite. The variance of ξ −∞ 2 is V [ξ] ( m = E[(ξ − ) E[ξ]) ]. The uncertain variables ξ1 , ξ2 , · · · , ξm are said to be independent if \ M {ξ ∈ Bi } = min M{ξ ∈ Bi } for any Borel sets B1 , B2 , · · · , Bm of real numbers. i=1
1≤i≤m
Definition 1 (Liu [12]) Let T be an index set and let (Γ, L, M) be an uncertainty space. An
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uncertain process is a measurable function from T × (Γ, L, M) to the set of real numbers, ie., for each t ∈ T and any Borel set B of real numbers, the set {Xt ∈ B} = {γ ∈ Γ|Xt (γ) ∈ B}. is an event. Definition 2 (Liu [13]) An uncertain process Xt is said to have independent increments if Xt1 − Xt0 , Xt2 − Xt1 , · · · , Xtk − Xtk−1 are independent uncertain variables for any times t0 < t1 < · · · < tk . An uncertain process Xt is said to have stationary increments if, for any given t > 0, the increments Xs+t −Xs are identically distributed uncertain variables for all s > 0. Definition 3 (Liu [13]) An uncertain process Ct is said to be a canonical process if (i) C0 = 0 and almost all sample paths are Lipschitz continuous, (ii) Ct has stationary and independent increments, (iii) every increment Cs+t − Cs is a normal uncertain variable with expected value 0 and variance t2 , whose uncertainty distribution is Φ(x) = 1 + exp
−πx √ 3t
−1 ,
x ∈ R.
(1)
In [14], Liu gave a proof of the existence of a canonical process. Based on canonical process, a new kind of uncertain differential was introduced by Liu [12]. The following concept of uncertain differential equation is important in theory and applications, and essential in the study of this paper. Definition 4 (Liu [12]) Suppose Ct is a canonical process, g1 and g2 are some given functions. Then dXt = g1 (Xt , t)dt + g2 (Xt , t)dCt (2) is called an uncertain differential equation. A solution is an uncertain process Xt that satisfies (2) identically in t. Chen and Liu [5] presented an existence and uniqueness theorem for an uncertain differential equation. Lemma 1 (Chen and Liu [5]) Suppose that C(t) is a canonical process, and X(t) is an integrable uncertain process on [a, b] with respect to t. Then the inequality Z b Z b Xt (γ)dC(t, γ) ≤ K(γ) |Xt (γ)|dt a
a
holds, where K(γ) is the Lipschitz constant of the sample path Xt (γ).
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Uncertain Delay Differential Equations
For an uncertain process X : [0, T ] × Γ → R, denote an uncertain segment process of Xt by Xt+θ (γ) for θ ∈ [−r, 0], r > 0, t ∈ [0, T ], and γ ∈ Γ, which is the history of Xt up to t, and called an uncertain process with delay (or memory). Consider the uncertain functional differential equation dXt = f (Xt+θ , t)dt + g(Xt+θ , t)dCt , t ∈ [0, T ], θ ∈ [−r, 0]
(3)
t ∈ [−r, 0]
X = ϕ(t), t
where Ct is a canonical process, and f (x, t), g(x, t) : R × [0, T ] → R are continuous maps. If r is finite, the equation is called the one with finite delay; otherwise, the one with infinite delay. The former is our main subject of study. To solve an uncertain delay differential equations with delay, we give an example. Consider the following uncertain delay differential equation: dXt = mXt−q dt + σdCt , t ∈ [0, T ] X = 1, t
(4)
t ∈ [−q, 0]
where q > 0, m and σ are constants, and Ct is a canonical process. For t ∈ [0, T ], there exists n ∈ N such that t ∈ [nq, (n + 1)q]. If n = 0, t ∈ [0, q], then t − q ∈ [−q, 0], and Xt−q = 1. So we have dXt = mdt + σdCt (5) X = 1. 0 Z
t
Z
σdCs
mds +
Xt = X0 + 0
= 1 + mt + σCt ,
t
(6)
0
t ∈ [0, q].
If n = 1, t ∈ [q, 2q], then t − q ∈ [0, q], and Xt−q = 1 + m(t − q) + σCt−q . So we have dXt = m[1 + m(t − q) + σCt−q ]dt + σdCt
(7)
X = 1 + mq + σC . q q Z
t
Z m[1 + m(s − q) + σCs−q ]ds +
Xt = Xq + q
t
σdCs q
2 (t
= 1 + mt + m
− q)2 + σCt + mσ 2
Z
t
(8)
Cs−q ds. q
We can continue this method, and find the expression for X(t) on each interval [nτ, (n + 1)τ ] with n ∈ N .
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4
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Existence and Uniqueness of the Solution
Assume C[0, T ] denotes the space of continuous R−valued functions on [0, T ]. Then C[0, T ] is a Banach space with the norm: kϕk[0,T ] = maxt∈[0,T ] |ϕ(t)|. Functions f, g : R × [0, T ] → R. Suppose that there exists some positive constant L such that C1. (Lipschitz condition) |f (x, t) − f (y, t)| + |g(x, t) − g(y, t)| ≤ L|x − y|,
∀x, y ∈ R, t ∈ [0, T ].
and C2. (linear growth condition) |f (x, t)| + |g(x, t)| ≤ L(1 + |x|),
∀x ∈ R, t ∈ [0, T ].
Now we introduce the following mapping Φ on C[0, T ]: for Xt ∈ C[0, T ], Z
t
Z
g(Xs+θ , s)dCs , t ∈ [0, T ], θ ∈ [−r, 0].
f (Xs+θ , s)ds +
Φ(Xt ) = ϕ(0) +
t
(9)
0
0
Lemma 2 For any γ ∈ Γ, and Xt (γ) ∈ C[0, T ], we have Φ(Xt (γ)) ∈ C[0, T ]. Proof Suppose s1 , s2 ∈ [0, T ] and s1 > s2 . We have |Φ(Xs1 (γ)) − Φ(Xs2 (γ))| Z s1 Z s1 = f (Xs+θ (γ), s)ds + g(Xs+θ (γ), s)dCs (γ) sZ2 s1 Z ss21 ≤ |f (Xs+θ (γ), s)|ds + g(Xs+θ (γ), s)dCs (γ) s2 Z Zs2s1 s1 ≤ |f (Xs+θ (γ), s)|ds + K(γ) |g(Xs+θ (γ), s)|ds (by Lemma 1) s2
s2
≤ L(1 + kXt (γ)k[−r,T ] )(1 + K(γ))(s1 − s2 ) (by the linear growth condition). Thus |Φ(Xs1 (γ))−Φ(Xs2 (γ))| → 0 as |s1 −s2 | → 0. So we have that Φ(Xt ) is sample-continuous on [0, T ]. The Lemma is proved. Lemma 3 There exists c > 0 such that for any t ∈ [0, T ), the uncertain delay differential equation (3) has a unique solution on the interval [t, t + c] (setting t + c = T if t + c > T ) if the coefficients f and g satisfy C1. (Lipschitz condition) and C2. (linear growth condition). Proof Let c > 0 such that ρ(γ) = L(1 + K(γ))c ∈ (0, 1). For t ∈ [0, T ), define Z τ Z τ Φ(Xτ ) = Xt + f (Xs+θ , s)ds + g(Xs+θ , s)dCs , τ ∈ [t, t + c], θ ∈ [−r, 0]. t
t
The proof of Lemma 2 implies that Φ(Xτ ) ∈ C[t, t + c] for Xτ ∈ C[t, t + c].
(10)
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For any τ ∈ [t, t + c], we have |Φ(Xτ (γ)) − Φ(Yτ (γ))| Z τ ≤ |f (Xs+θ (γ), s) − f (Ys+θ (γ), s)|ds t Z τ |g(Xs+θ (γ), s) − g(Ys+θ (γ), s)|ds + K(γ) t Z τ |Xs+θ (γ) − Ys+θ (γ)| ds ≤ L(1 + K(γ)) Zt τ max |Xs+θ (γ) − Ys+θ (γ)| ds ≤ L(1 + K(γ)) t
(11)
−r≤θ≤0,t≤s≤t+c
≤ L(1 + K(γ))c kXτ (γ) − Yτ (γ)k[t−r,t+c] If we set Φ(Xτ (γ)) = Xt for τ ∈ [t − r, t], then kΦ(Xτ (γ)) − Φ(Yτ (γ))k[t−r,t+c] ≤ ρ(γ)kXτ (γ) − Yτ (γ)k[t−r,t+c] . That is, Φ is a contraction mapping on C[t − r, t + c]. Thus by the well-known Banach fixed point theorem we have a unique fixed point Xt (γ) ∈ C[t − r, t + c] which satisfies (10) in [t, t + c]. Thus, (3) has a unique solution on the interval [t, t + c] Theorem 1 The uncertain delay differential equation (3) has a unique solution on the interval [0, T ] if the coefficients f (Xt+θ , t) and g(Xt+θ , t) satisfy C1. (Lipschitz condition) and C2. (linear growth condition). Proof Assume that [0, c], [c, 2c], · · · , [kc, T ] are the subsets of [0, T ] with kc < T ≤ (k + 1)c. For any γ, it follows from Lemma 3 that the equation (3) has a unique solution Xti (γ) on the interval [(i − 1)c, ic] for i = 1, 2, · · · , k + 1 and setting (k + 1)c = T . Therefore, the uncertain delay differential equation (3) has a unique solution Xt on the interval [0, T ] by setting
Xt (γ) =
5
Xt1 (γ), ···
t ∈ [0, c],
Xtk (γ), t ∈ [(k − 1)c, kc], X k+1 (γ), t ∈ [kc, T ]. t
Conclusion
In this paper we introduced uncertain delay differential equations. By Banach fixed point theorem, we proved an existence and uniqueness theorem for uncertain delay differential equations under Lipschitz condition and linear growth condition on the interval [t, t + c], then extended the conclusion to the interval [0, T ].
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