Abstract
In this paper, we establish a new proof of the existence and uniqueness of solution to stochastic functional differential equations with infinite delay at phase space BC((∞, 0]: R^{d}) which denotes the family of bounded continuous R^{d}valued functions φ defined on (∞, 0] with norm φ = sup_{∞ < θ≤0 }φ(θ).
Mathematics Subject Classification (2000): 60H05, 60H10.
Keywords:
existence; uniqueness; stochastic functional differential equations; infinite delay.1. Introduction
Stochastic differential equations (SDEs) are well known to model problems from many areas of science and engineering. For instance, in 2006, Henderson and Plaschko [1] published the SDEs in Science and Engineering, in 2007, Mao [2] published the SDEs, in 2010, Li and Fu [3] considered the stability analysis of stochastic functional differential equations with infinite delay and its application to recurrent neural networks.
In recent years, there is an increasing interest in stochastic functional differential equations(SFDEs) with finite and infinite delay under less restrictive conditions than Lipschitz condition. For instance, in 2007, Wei and Wang [4] discussed the existence and uniqueness of the solution for stochastic functional differential equations with infinite delay, in 2008, Mao et al. [5] discussed almost surely asymptotic stability of neutral stochastic differential delay equations with Markovian switching, in 2008, Ren et al. [6] considered the existence and uniqueness of the solutions to SFDEs with infinite delay, and in 2009, Ren and Xia [7], discussed the existence and uniqueness of the solution to neutral SFDEs with infinite delay. Furthermore, on this topic, one can see Halidias [8], Henderson and Plaschko [1], Kim [9,10], Ren [11], Ren and Xia [7], Taniguchi [12] and references therein for details.
On the other hand, Mao [2] discussed ddimensional stochastic functional differential equations with finite delay
where x_{t }= {x(t + θ): τ ≤ θ ≤ 0} could be considered as a C([τ, 0]; R^{d})value stochastic process. The initial value of (1.1) was proposed as follows:
Furthermore, Ren et al. [6] also considered the stochastic functional differential equations with infinite delay at phase space BC((∞, 0]; R^{d}) to be described below.
where x_{t }= {X(t + θ): ∞ ≤ θ ≤ 0} could be considered as a BC((∞, 0]; R^{d})value stochastic process. The initial value of (1.2) was proposed as follows:
Following this way, now we recall the existence and uniqueness of the solutions to the Equation 1.2 with initial data (1.3) under the nonLipschitz condition and the weakened linear growth condition. In this paper, we will give some new proof of the existence and uniqueness of the solutions to ISDEs under an alternative way.
2. Preliminary
Let  ·  denote Euclidean norm in R^{n}. If A is a vector or a matrix, its transpose is denoted by A^{T}; if A is a matrix, its trace norm is represented by
With all the above preparation, consider a ddimensional stochastic functional differential equations:
where x_{t }= {x(t + θ): ∞ < θ ≤ 0} can be considered as a BC((∞, 0]; R^{d})value stochastic process, where
be Borel measurable. Next, we give the initial value of (2.1) as follows:
To find out the solution, we give the definition of the solution of the Equation 2.1 with initial data (2.2).
Definition 2.1. [6]R^{d}value stochastic process x(t) defined on ∞ < t ≤ T is called the solution of (2.1) with initial data (2.2), if x(t) has the following properties:
(i) x(t) is continuous and
(ii)
(iii)
A solution x(t) is called as a unique if any other solution
□
The integral inequalities of Gronwall type have been applied in the theory of SDEs to prove the results on existence, uniqueness, and stability etc. [10,1316]. Naturally, Gronwall's inequality will play an important role in next section.
Lemma 2.1. (Gronwall's inequality). Let u(t) and b(t) be nonnegative continuous functions for t ≥ α, and let
where a ≥ 0 is a constant. Then
Lemma 2.2. (Bihari's inequality). Let u and b be nonnegative continuous functions defined on R_{+}. Let g(u) be a nondecreasing continuous function on R_{+ }and g(u) > 0 on (0, ∞). If
for t ∈ R_{+}, where k ≥ 0 is a constant. Then for 0 ≤ t ≤ t_{1},
where
and G^{1 }is the inverse function of G and t_{1 }∈ R_{+ }is chosen so that
for all t ∈ R_{+ }lying in the interval 0 ≤ t ≤ t_{1}.
The following two lemmas are known as the moment inequality for stochastic integrals which will play an important role in next section.
Lemma 2.3. [2]. If p ≥ 2,
then
In particular, for p = 2, there is equality.
Lemma 2.4. [2]. If p ≥ 2,
then
3. Existence and Uniqueness of the Solutions
In order to obtain the existence and uniqueness of the solutions to (2.1) with initial
data (2.2), we define
Now we begin to establish the theory of the existence and uniqueness of the solution. We first show that the nonLipschitz condition and the weakened linea growth condition guarantee the existence and uniqueness.
Theorem 3.1. Assume that there exist a positive number K such that
(i) For any φ, ψ ∈ BC((∞, 0]; R^{d}) and t ∈ [t_{0}, T], it follows that
where κ(·) is a concave nondecreasing function from ℝ_{+ }to ℝ_{+ }such that κ(0) = 0, κ(u) > 0 for u > 0 and
(ii) For any t ∈ [t_{0}, T], it follows that f(0, t), g(0, t) ∈ L^{2 }such that
Then the initial value problem (2.1) has a solution x(t). Moreover,
Lemma 3.2. Let the assumption (3.1) and (3.2) of Theorem 3.1 hold. If x(t) is a solution of equation (2.1) with initial data (2.2), then
where c_{2 }= c_{1 }+ Eǁ ξ ǁ^{2}, c_{1 }= 3E ǁξ ǁ^{2 }+ 6(T  t_{0 }+ 1)(T  t_{0})[K + a]. In particular, x(t) belong to
Proof. For each number n ≥ 1, define the stopping time
Obviously, as n → ∞, τ_{n }↑ T a.s. Let x^{n}(t) = x(t ∧ τ_{n}), t ∈ (∞, T]. Then, for t_{0 }≤ t ≤ T, x^{n}(t) satisfy the following equation
Using the elementary inequality
By the Hölder's inequality and the moment inequality, we have
Hence, by the condition (3.1) and (3.2) one can further show that
Given that κ(·) is concave and κ(0) = 0, we can find a positive constants a and b such that κ(u) ≤ a + bu for u ≥ 0. So, we obtains that
where c_{1 }= 3Eξ^{2 }+ 6(T  t_{0 }+ 1)(T  t_{0})[K + a]. Noting the fact that
where c_{2 }= c_{1 }+ Eξ^{2}. So, by the Gronwall inequality yields that
Letting t = T, it then follows that
Thus
Consequently the required result follows by letting n → ∞. □
Proof of Theorem 3.1. Let x(t) and
By the elementary inequality (u + v)^{2 }≤ 2(u^{2 }+ v^{2}), one then gets
By the Hölder's inequality, the moment inequality, and (3.1) we have
Since κ(·) is concave, by the Jensen inequality, we have
Consequently, for any ϵ > 0,
By the Bihari inequality, one deduces that, for all sufficiently small ϵ > 0,
where
on r > 0, and G^{1}(·) be the inverse function of G(·). By assumption
Therefore, by letting ϵ → 0 in (3.3), gives
This implies that
Next to check the existence. Define
for t_{0 }≤ t ≤ T. Obviously,
Taking the expectation on both sides and using the Hölder inequality and moment inequality, we have
Using the elementary inequality (u + v)^{2 }≤ 2u^{2 }+ 2v^{2}, (3.1), and (3.2), we have
Given that κ(·) is concave and κ(0) = 0, we can find a positive constants a and b such that κ(u) ≤ a + bu for u ≥ 0. So, we have
where c_{1 }= 3Eξ^{2 }+ 6(T  t_{0})(T  t_{0}+ 1)[K + a]. It also follows from the inequality that for any k ≥ 1,
where c_{2 }= c_{1 }= 6b(T  t_{0})(T  t_{0}+ 1) Eξ^{2}. The Gronwall inequality implies
Since k is arbitrary, we must have
for all t_{0 }≤ t ≤ T, n ≥ 1.
Next, we that the sequence {x^{n}(t)} is Cauchy sequence. For all n ≥ 0 and t_{0 }≤ t ≤ T, we have
Next, using an elementary inequality
On the other hand, by Hölder's inequality, Lemma (2.4), and the condition, one can show that
where β = (T  t_{0})/α + 4/(1  α). Let
From (3.6), for any ϵ > 0, we get
By the Bihari inequality, one deduces that, for all sufficiently small ϵ > 0,
where
on r > 0, and G^{1}(·) be the inverse function of G(·). By assumption, we get Z(t) = 0. This shows the sequence {x^{n}(t), n ≥ 0} is a Cauchy sequence in L^{2}. Hence, as n → ∞, x^{n}(t)  x(t), that is E x^{n}(t)  x(t)^{2 }→ 0. Letting n → ∞ in (3.5) then yields that
for all t_{0 }≤ t ≤ T. Therefore,
Noting that sequence x^{n}(t) is uniformly converge on (∞, T ], it means that
as n → ∞, further
as n → ∞. Hence, taking limits on both sides in the Picard sequence, we obtain that
on t_{0 }≤ t ≤ T. The above expression demonstrates that x(t) is the solution of (2.3). So, the existence of theorem is complete.
Remark 3.1. In the proof of Theorem 3.1, the solution is constructed by the successive approximation. It shows that how to get the approximate solution of (2.1) and how to construct Picard sequence x^{n}(t). For SFDEs, we know that a weakened linear growth condition imposed on Theorem 3.1 is rigorous for our discussion. In papers [6,7], the proofs of the assertions are based on some function inequalities. Recall that the procedures has become more and more complicated in those proofs. For this reason, although analogous problem is studied here, the proof of the assertion in the Theorem is completely different with respect to the ones from [7]. Moreover, our new proof in this paper is completed by Bihari's inequality and more simple than that reported in [7].
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors read and approved the final manuscript.
Acknowledgements
The authors wish to thank the anonymous referees for their endeavors and valuable comments. Also, this research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(20110023547)
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