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1.
The main aim of this paper is to develop some basic theories of stochastic functional differential equations (SFDEs). Firstly, we establish stochastic versions of the well-known Picard local existence-uniqueness theorem given by Driver and continuation theorems given by Hale and Driver for functional differential equations (FDEs). Then, we extend the global existence-uniqueness theorems of Wintner for ordinary differential equations (ODEs), Driver for FDEs and Taniguchi for stochastic ordinary differential equations (SODEs) to SFDEs. These show clearly the power of our new results. 相似文献
2.
The aim of this paper is to develop some basic theories of stochastic functional differential equations (SFDEs) under the local Lipschitz condition in continuous functions space ?. Firstly, we establish a global existence-uniqueness lemma for the SFDEs under the global Lipschitz condition in ? without the linear growth condition. Then, under the local Lipschitz condition in ?, we show that the non-continuable solution of SFDEs still exists if the drift coefficient and diffusion coefficient are square-integrable with respect to t when the state variable equals zero. And the solution of the considered equation must either explode at the end of the maximum existing interval or exist globally. Furthermore, some more general sufficient conditions for the global existence-uniqueness are obtained. Our conditions obtained in this paper are much weaker than some existing results. For example, we need neither the linear growth condition nor the continuous condition on the time t. Two examples are provided to show the effectiveness of the theoretical results. 相似文献
3.
Existence-Uniqueness Problems For Infinite Dimensional Stochastic Differential Equations With Delays 下载免费PDF全文
The main aim of this paper is to develop the basic theory of a class of infinite dimensional stochastic differential equations with delays (IDSDEs) under local Lipschitz conditions. Firstly, we establish a global existence-uniqueness theorem for the IDSDEs under the global Lipschitz condition in \(C\) without the linear growth condition. Secondly, the non-continuable solution for IDSDEs is given under the local Lipschitz condition in \(C\). Then, the classical Itô's formula is improved and a global existence theorem for IDSDEs is obtained. Our new theorems give better results while conditions imposed are much weaker than some existing results. For example, we need only the local Lipschitz condition in \(C\) but neither the linear growth condition nor the continuous condition on the time \(t\). Finally, two examples are provided to show the effectiveness of the theoretical results. 相似文献
4.
The main aim of this article is to deal with the almost-sure stability of stochastic differential delay equations. Our improved theorems give better results while conditions imposed on the Lyapunov function are much weaker, thus, it is easier to find a right Lyapunov function in application. 相似文献
5.
Xiao-Li Ding Yao-Lin Jiang 《Communications in Nonlinear Science & Numerical Simulation》2012,17(12):5143-5150
Semilinear fractional differential equations (SFDEs) often arise in some dynamical systems. In most of the existing literature, the fixed point theorems are used to prove the existence and uniqueness of the solutions of SFDEs. In this paper, we give a new way to prove the existence and uniqueness of the solutions by introducing a new integral operator associated with the Mittag–Leffler function. 相似文献
6.
7.
Approximation properties for solutions to non‐Lipschitz stochastic differential equations with Lévy noise 下载免费PDF全文
In this paper, we consider the non‐Lipschitz stochastic differential equations and stochastic functional differential equations with delays driven by Lévy noise, and the approximation theorems for the solutions to these two kinds of equations will be proposed respectively. Non‐Lipschitz condition is much weaker condition than the Lipschitz one. The simplified equations will be defined to make its solutions converge to that of the corresponding original equations both in the sense of mean square and probability, which constitute the approximation theorems. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
8.
Svetlana Jankovi? 《Journal of Mathematical Analysis and Applications》2009,355(2):811-6134
The paper discusses both pth moment and almost sure exponential stability of solutions to neutral stochastic functional differential equations and neutral stochastic differential delay equations, by using the Razumikhin-type technique. The main goal is to find sufficient stability conditions that could be verified more easily then by using the usual method with Lyapunov functionals. The analysis is based on paper [X. Mao, Razumikhin-type theorems on exponential stability of neutral stochastic functional differential equations, SIAM J. Math. Anal. 28 (2) (1997) 389-401], referring to mean square and almost sure exponential stability. 相似文献
9.
The main aim of this paper is to study the stability of the stochastic functional differential equations with infinite delay.
We establish several Razumikhin-type theorems on the exponential stability for stochastic functional differential equations
with infinite delay. By applying these results to stochastic differential equations with distributed delay, we obtain some
sufficient conditions for both pth moment and almost surely exponentially stable. Finally, some examples are presented to illustrate our theory. 相似文献
10.
Numerical solutions of stochastic differential delay equations under the generalized Khasminskii-type conditions 总被引:2,自引:0,他引:2
Xuerong Mao 《Applied mathematics and computation》2011,217(12):5512-5524
The classical existence-and-uniqueness theorem of the solution to a stochastic differential delay equation (SDDE) requires the local Lipschitz condition and the linear growth condition (see e.g. [11], [12] and [20]). The numerical solutions under these conditions have also been discussed intensively (see e.g. [4], [10], [13], [16], [17], [18], [21], [22] and [24]). Recently, Mao and Rassias [14] and [15] established the generalized Khasminskii-type existence-and-uniqueness theorems for SDDEs, where the linear growth condition is no longer imposed. These generalized Khasminskii-type theorems cover a wide class of highly nonlinear SDDEs but these nonlinear SDDEs do not have explicit solutions, whence numerical solutions are required in practice. However, there is so far little numerical theory on SDDEs under these generalized Khasminskii-type conditions. The key aim of this paper is to close this gap. 相似文献
11.
Existence and Uniqueness of Solutions to Time-delays Stochastic Fractional Differential Equations with Non-Lipschitz Coefficients 下载免费PDF全文
In this paper, we consider the existence and uniqueness of solutions to time-varying delays stochastic fractional differential equations (SFDEs) with non-Lipschitz coefficients. By using fractional calculus and stochastic analysis, we can obtain the existence result of solutions for stochastic fractional differential equations. 相似文献
12.
Bapurao C. Dhage 《Nonlinear Analysis: Theory, Methods & Applications》2009,71(12):e319
In this paper, an existence result for perturbed abstract measure differential equations is proved via hybrid fixed point theorems of Dhage [B.C. Dhage, On some nonlinear alternatives of Leray-Schauder type and functional integral equations, Arch. Math. (Brno) 42 (2006) 11-23] under the mixed generalized Lipschitz and Carathéodory conditions. The existence of the extremal solutions is also proved under certain monotonicity conditions and using a hybrid fixed point theorem of Dhage given in the above-mentioned reference, on ordered Banach spaces. Our existence results include the existence results of Sharma [R.R. Sharma, An abstract measure differential equation, Proc. Amer. Math. Soc. 32 (1972) 503-510], Joshi [S.R. Joshi, A system of abstract measure delay differential equations, J. Math. Phy. Sci. 13 (1979) 497-506] and Shendge and Joshi [G.R. Shendge, S.R. Joshi, Abstract measure differential inequalities and applications, Acta Math. Hung. 41 (1983) 53-54] as special cases under weaker continuity condition. 相似文献
13.
Sheng Jun FAN 《数学学报(英文版)》2007,23(8):1427-1434
In this paper, under the most elementary conditions on a backward stochastic differential equation (BSDE for short) introduced by Peng, a new relationship between the conditional g-evaluation system and the generator g of BSDE is obtained in the sense of "process", based on some recent results of Jiang. Moreover, as applications, two converse comparison theorems and two uniqueness theorems on the generators of BSDEs are proved. 相似文献
14.
In this paper, we establish a local representation theorem for generators of reflected backward stochastic differential equations (RBSDEs), whose generators are continuous with linear growth. It generalizes some known representation theorems for generators of backward stochastic differential equations (BSDEs). As some applications, a general converse comparison theorem for RBSDEs is obtained and some properties of RBSDEs are discussed. 相似文献
15.
Homoclinic solutions for fourth order differential equations with superlinear nonlinearities 下载免费PDF全文
In this paper we investigate the existence of homoclinic solutions for a class of fourth order differential equations with superlinear nonlinearities. Under some superlinear conditions weaker than the well-known (AR) condition, by using the variant fountain theorem, we establish one new criterion to guarantee the existence of infinitely many homoclinic solutions. 相似文献
16.
Huang Xiao-Qin Wang Mian-Sen Jia Jun-Guo 《Journal of Applied Mathematics and Computing》2007,24(1-2):377-385
In this paper, by the equations of Mao [9] and Peng [5], we use the martingale method to establish the comparison theorems of backward stochastic differential equations (BSDEs). We generalize the results of Cao-Yan [1]. 相似文献
17.
This paper aims at solving a multidimensional backward stochastic differential equation (BSDE) whose generator g satisfies a weak monotonicity condition and a general growth condition in y. We first establish an existence and uniqueness result of solutions for this kind of BSDEs by using systematically the technique of the priori estimation, the convolution approach, the iteration, the truncation and the Bihari inequality. Then, we overview some assumptions related closely to the monotonieity condition in the literature and compare them in an effective way, which yields that our existence and uniqueness result really and truly unifies the Mao condition in y and the monotonieity condition with the general growth condition in y, and it generalizes some known results. Finally, we prove a stability theorem and a comparison theorem for this kind of BSDEs, which also improves some known results. 相似文献
18.
本文对中立型随机泛函微分方程建立了Khasminskii型定理,这个定理显示在局部Lipschitz条件但是不要求线性增长的条件下,中立型随机泛函微分方程存在一个全局解.本文的这个解存在性条件可以包含更广的一类非线性中立型随机泛函微分方程.最后,本文给出一个例子来阐述我们的思想. 相似文献
19.
Generalization for Amann's and Leggett-Williams' three-solution theorems and applications 总被引:1,自引:0,他引:1
In this paper, some new three-solution theorems are obtained. Moreover, the famous Amann's and Leggett-Williams' three-solution theorems in nonlinear functional analysis can be seen as their special cases, namely these two famous theorems are united. So they are both improved. And that some new three-solution theorems are applied to ordinary differential equations, and some new existence results are obtained. 相似文献
20.
Qian Lin 《Applied mathematics and computation》2011,217(22):9322-9333
In this paper, we deal with one dimensional backward doubly stochastic differential equations (BDSDEs). We obtain existence theorems and comparison theorems for solutions of BDSDEs with weak assumptions on the coefficients. 相似文献