Local Lipschitz Condition Research Articles

Existence and Stability of Solutions to Highly Nonlinear Stochastic Differential Delay Equations Driven by G-Brownian Motion

Under linear expectation (or classical probability), the stability for stochastic differential delay equations (SDDEs), where their coefficients are either linear or nonlinear but bounded by linear functions, has been investigated intensively. Recently, the stability of highly nonlinear hybrid stochastic differential equations is studied by some researchers. In this paper, by using Peng’s G-expectation theory, we first prove the existence and uniqueness of solutions to SDDEs driven by G-Brownian motion (G-SDDEs) under local Lipschitz and linear growth conditions. Then the second kind of stability and the dependence of the solutions to G-SDDEs are studied. Finally, we explore the stability and boundedness of highly nonlinear G-SDDEs.

Strong convergence rates of modified truncated EM methods for neutral stochastic differential delay equations

The aim of this paper is to investigate strong convergence of modified truncated Euler–Maruyama method for neutral stochastic differential delay equations introduced in Lan (2018). Strong convergence rates of the given numerical scheme to the exact solutions at fixed time T are obtained under local Lipschitz and Khasminskii-type conditions. Moreover, convergence rates over a time interval [0,T] are also obtained under additional polynomial growth condition on g without the weak monotonicity condition (which is usually the standard assumption to obtain the convergence rate). Two examples are presented to interpret our conclusions.

Asymptotic stability of nonlinear hybrid stochastic systems driven by linear discrete time noises

Author in the reference Mao (2016) designed a linear discrete time noise to exponentially stabilize an unstable differential system with the global Lipschitz condition, which essentially opened a new field of stochastic systems driven by discrete time noises. However, the global Lipschitz condition is over strict and the result of exponential convergence is excessively rapid for many systems. Hence we see the necessity to loosen it to the local Lipschitz condition and develop some new stability results. In the present paper, by the method of Lyapunov functionals, we not only investigate the exponential stability for hybrid stochastic differential systems driven by linear discrete time noises with the local Lipschitz condition, but also study the H∞ stability and asymptotic stability using some new techniques.

Convergence rate of the truncated Milstein method of stochastic differential delay equations

This paper is concerned with the strong convergence of highly nonlinear stochastic differential delay equations (SDDEs) without the linear growth condition. On the one hand, these nonlinear SDDEs do not have explicit solutions, therefore implementable numerical methods for such SDDEs are required. On the other hand, the implicit Euler methods are known to converge strongly to the exact solution of such SDDEs. However, they require additional computational efforts. In this article, we propose the truncated Milstein method which is an explicit method under the local Lipschitz condition plus Khasminskii-type condition, study its pth monent boundedness (p is a parameter in Khasminskii-type condition) and show that its rate of strong convergence is close to one.

Fixed points of multivalued maps under local Lipschitz conditions and applications

AbstractIn this work we are concerned with the existence of fixed points for multivalued maps defined on Banach spaces. Using the Banach spaces scale concept, we establish the existence of a fixed point of a multivalued map in a vector subspace where the map is only locally Lipschitz continuous. We apply our results to the existence of mild solutions and asymptotically almost periodic solutions of an abstract Cauchy problem governed by a first-order differential inclusion. Our results are obtained by using fixed point theory for the measure of noncompactness.

Connectivity enhancing coordinated tracking control of multi-agent systems with a state-dependent jointly-connected dynamic interaction topology

We revisit the problem of virtual leader tracking by a group of agents of second order dynamics when only a portion of the agents are informed of the information of the virtual leader, under a state-dependent interaction topology. By proposing a simpler controller structure, we obtain results on coordinated tracking control under a jointly connected topology, both in the presence and in the absence of the virtual leader velocity information. The simplification in control protocols also enables us to arrive at our second contribution on connectivity enhancing coordinated control. In comparison with the existing connectivity preserving control algorithms, our proposed connectivity enhancing mechanism is effective in maintaining both the initially existent topology edges and those newly formed topology edges that were not initially existent. Besides, our proposed algorithms are designed in such a way that the local Lipschitz condition is satisfied and thus technical issues on solution uniqueness of the closed-loop system are avoided. Our proposed technique applies to both consensus tracking and flocking.

The Asymptotic Behavior of Solutions for 3D Globally Modified Bénard Problem with Delay

In this paper, we mainly study the existence and uniqueness of solutions and the asymptotic behavior of solutions for three-dimensional globally modified Benard systems with delays under local Lipschitz conditions.

Almost sure stability with general decay rate of neutral stochastic pantograph equations with Markovian switching

This paper focuses on the general decay stability of nonlinear neutral stochastic pantograph equations with Markovian switching (NSPEwMSs). Under the local Lipschitz condition and non-linear growth condition, the existence and almost sure stability with general decay of the solution for NSPEwMSs are investigated. By means of M-matrix theory, some sufficient conditions on the general decay stability are also established for NSPEwMSs.

一类随机泛函微分方程带随机步长的EM逼近的渐近稳定

The Euler-Maruyama (EM) approximation to a class of stochastic functional differential equations was studied. First, a numerical approximation with the EM method with random variable stepsizes was defined, then two characteristics of the random variable stepsizes were got: the summation of finite stepsizes is a stopping time and the summation of countably infinite stepsizes diverges. Finally, with the theory of non-negative semi-martingale convergence in discrete time, it was proved that the numerical approximation converges to zero almost surely if the coefficients satisfy the local Lipschitz condition and the monotonic condition. The results generalize the corresponding results of MAO Xuerong in a previous literature, where the EM approximation to a class of stochastic differential equations was studied and the numerical solution was proved to converge to zero almost surely.

The pth moment exponential stability and almost surely exponential stability of stochastic differential delay equations with Poisson jump

This paper is devoted to study a class of nonlinear stochastic differential delay equations with Poisson jump. In comparison to the Brownian motion, the jump leads to the discontinuity of sample paths, which makes the analysis more difficult. We first introduce the local Lipschitz condition and a new nonlinear growth condition, which is weaker than those in the previous literature. Then by virtue of Lyapunov function and semi-martingale convergence theorem, we prove that the considered stochastic system has a unique global solution. Moreover, we also investigate the pth moment exponential stability and the almost surely exponential stability of solutions. Finally, an example is given to illustrate the effectiveness of our theoretical results.

Basic theory and stability analysis for neutral stochastic functional differential equations with pure jumps

This paper investigates the existence and uniqueness of solutions to neutral stochastic functional differential equations with pure jumps (NSFDEwPJs). The boundedness and almost sure exponential stability are also considered. In general, the classical existence and uniqueness theorem of solutions can be obtained under a local Lipschitz condition and linear growth condition. However, there are many equations that do not obey the linear growth condition. Therefore, our first aim is to establish new theorems where the linear growth condition is no longer required whereas the upper bound for the diffusion operator will play a leading role. Moreover, the pth moment boundedness and almost sure exponential stability are also obtained under some loose conditions. Finally, we present two examples to illustrate the effectiveness of our results.

Viability for stochastic functional differential equations in Hilbert spaces driven by fractional Brownian motion

In this paper, we consider a class of stochastic functional differential equations in Hilbert spaces driven by a fractional Brownian motion with Hurst parameter 1/2 < H < 1. By using pathwise approach, we prove a global existence and uniqueness result of the mild solution for the equations considered under some local Lipschitz conditions. Subsequently, by establishing some new estimates, we also prove some viability results to the stochastic systems under investigation.

Stability Analysis in a Class of Markov Switched Stochastic Hopfield Neural Networks

Recently, a new class of stochastic systems induced by linear discrete time noises was proposed and studied. Up to now, the existing literatures mainly investigated the exponential stability of such stochastic systems under the global Lipschitz condition. Our aim here is to weaken the strictly global Lipschitz condition and explore new stability theory for a new class of Markov switched stochastic Hopfield neural networks induced by nonlinear discrete time noises. In the present paper, we propose such Markov switched stochastic Hopfield neural networks, and creatively introduce a new class of Lyapunov functionals to investigate the $$H_{\infty }$$ stability, asymptotic stability and exponential stability for such systems under the local Lipschitz condition using some novel skills. Furthermore, we specially study the case induced by linear discrete time noises.

The projected explicit Itô–Taylor methods for stochastic differential equations under locally Lipschitz conditions and polynomial growth conditions

Although the numerical methods to stochastic differential equations with the coefficients of locally Lipschitz and polynomial growth have been discussed commonly by some authors, there are few works on the high strong order numerical methods. In this paper, the mean-square convergence of the general projected explicit Itô–Taylor methods is considered under the assumption that drift and diffusion coefficient functions of stochastic differential equations satisfy the global monotonicity condition, suitable local Lipschitz conditions and polynomial growth conditions. Our analysis follows the idea of stochastic C-stability and stochastic B-consistency. After giving the selection strategy of optimal parameters, we prove that the projected explicit Itô–Taylor methods, owning optimal parameters, share the same mean-square convergence orders with non-projected ones. Finally, two numerical experiments are presented to show the effectiveness of theoretical results. • The general projected explicit Itô–Taylor methods for SDEs are constructed. • A simple selection strategy for proper projection parameters is introduced. • The methods run well for SDEs under locally Lipschitz, polynomial growth condition. • The methods hold the same mean-square convergence order as non-projection ones.

Strong convergence of a tamed theta scheme for NSDDEs with one-sided Lipschitz drift

This paper is concerned with strong convergence of a tamed theta scheme for neutral stochastic differential delay equations with one-sided Lipschitz drift. Strong convergence rate is revealed under a global one-sided Lipschitz condition, while for a local one-sided Lipschitz condition, the tamed theta scheme is modified to ensure the well-posedness of implicit numerical schemes, then we show the convergence of the numerical solutions.

Almost sure convergence rate of θ-EM scheme for neutral SDDEs

This paper is concerned with almost sure convergence of θ -EM scheme for neutral stochastic differential delay equations under local Lipschitz conditions with θ ∈ 1 2 , 1 . It is revealed that the strong convergence rate under local Lipschitz condition is 1/2, while almost sure convergence rate is less than 1/2, provided the local Lipschitz constant, valid on balls of radius R , grows not faster than log R .

Backward semi-linear parabolic equations with time-dependent coefficients and local Lipschitz source

Let H be a Hilbert space with the inner product and the norm , a positive self-adjoint unbounded time-dependent operator on H and . We establish stability estimates of Hölder type and propose a regularization method with error estimates of Hölder type for the ill-posed backward semi-linear parabolic equation with the source function f satisfying a local Lipschitz condition.

Distributional van der Pol equation with state-dependent impulses∗

We investigate the distributional differential van der Pol equation with impulsive forcing term. We prove the existence of a periodic solution without global or local Lipschitz conditions for impulsive functions and find a simple assumption on the parameter of the van der Pol equation guaranteeing its solvability.

Regularity properties of mild solutions for a class of Volterra equations with critical nonlinearities

We study a class of abstract nonlinear integral equations of convolution type defined on a Banach space. We prove the existence of a unique, locally mild solution and an extension property when the nonlinear term satisfies a local Lipschitz condition. Moreover, we guarantee the existence of the global mild solution and blow up profiles for a large class of kernels and nonlinearities. If the nonlinearity has critical growth, we prove the existence of the local $\epsilon $-mild solution. Our results improve and extend recent results for special classes of kernels corresponding to nonlocal in time equations. We give an example to illustrate the application of the theorems so obtained.

Asymptotic exponential stability of modified truncated EM method for neutral stochastic differential delay equations

The aim of this paper is to investigate the asymptotic mean–square and almost sure exponential stability of modified truncated Euler–Maruyama method for neutral stochastic differential delay equations under non linear conditions. It is proved that the so called modified truncated Euler–Maruyama method is mean–square (almost surely) exponentially stable under suitable (local Lipschitz and Khasminskii-type) conditions. An example is provided to support our conclusions.