Existence and Hyers-Ulam stability of random impulsive stochastic functional differential equations with finite delays

ABSTRACT

In this paper, we investigate the existence and Hyers-Ulam stability for random impulsive stochastic functional differential equations with finite delays. Firstly, we prove the existence of mild solutions to the equations by using Krasnoselskii's fixed point. Then, we investigate the Hyers-Ulam stability results under the Lipschitz condition on a bounded and closed interval. Finally, an example is given to illustrate our results.     

Modelling traffic flow with a time‐dependent fundamental diagram

We model traffic flow with a time‐dependent fundamental diagram. A time‐dependent fundamental diagram arises naturally from various factors such as weather conditions, traffic jam or modern traffic congestion managements, etc. The model is derived from a car‐following model which takes into account the situation changes over the time elapsed time. It is a system of non‐concave hyperbolic conservation laws with time‐dependent flux and the sources. The global existence and uniqueness of the solution to the Cauchy problem is established under the condition that the variation in time of the fundamental diagram is bounded. The zero relaxation limit of the solutions is found to be the unique entropy solution of the equilibrium equation. Copyright © 2004 John Wiley & Sons, Ltd.     

Razumikhin type boundedness theorems in terms of two measures for impulsive integro-differential systems

Some new direct criteria of boundedness in terms of two measures for impulsive integro-differential systems with fixed moments of impulse effects are established by Lyapunov functions coupled with Razumikhin techniques.     

Complete controllability of nonlinear stochastic impulsive functional systems

This paper is concerned with the controllability of a kind of nonlinear stochastic impulsive functional systems. Sufficient conditions for the complete controllability of the nonlinear stochastic impulsive functional systems are obtained by using Schauder fixed-point theorem. A numerical example is given to illustrate the effectiveness of the theoretical results.     

Existence and stability of periodic solution of a stage‐structured model with state‐dependent impulsive effects

A single population growth model with stage‐structured and state‐dependent impulsive control is proposed. By using the Poincar'e map and the analogue of Poincaré's criterion, we prove the existence and the stability of positive order‐1 or order‐2 periodic solution. Moreover, we show that there is no periodic solution with order greater than or equal to three. Numerical results are carried out to illustrate the feasibility of our main results and the superiority of state feedback control strategy is also discussed. Copyright © 2011 John Wiley & Sons, Ltd.     

Stability properties of nonlinear stochastic impulsive systems with time delay

This article establishes mean square stability and stabilization for stochastic delay systems with impulses. Using Razumikhin methodology, two approaches, classical Lyapunov-based method and comparison principle, are proposed to develop sufficient conditions that guarantee the stability and stabilization properties. It is shown that if the continuous system is stable and the impulses are destabilizing, the impulses should not be applied frequently. On the other hand, if the continuous system is unstable, but the impulses are stabilizing, the impulses should occur frequently to compensate the continuous state growth. Numerical examples are also presented to clarify the proposed theoretical results.     

Existence and stability results for semilinear systems of impulsive stochastic differential equations with fractional Brownian motion

Some results on the existence and uniqueness of mild solution for a system of semilinear impulsive differential equations with infinite fractional Brownian motions are proved. The approach is based on Perov's fixed point theorem and a new version of Schaefer's fixed point theorem in generalized Banach spaces. The relationship between mild and weak solutions and the exponential stability of mild solutions are investigated as well. The abstract theory is illustrated with an example.     

Unique solvability for the density‐dependent non‐Newtonian compressible fluids with vacuum

The paper is devoted to the existence and uniqueness of local solutions for the density‐dependent non‐Newtonian compressible fluids with vacuum in one‐dimensional bounded intervals. The important points in this paper are that the initial density may vanish in an open subset and the viscosity coefficient is nonlinearly dependent of density and shear rate.     

Existence and Uniqueness of Solutions to Random Impulsive Differential Systems

The existence and uniqueness in mean square of solutions to certain random impulsive differentialsystems is discussed in this paper.Cauchy-Schwarz inequality,Lipschtiz condition and techniques in stochasticanalysis are employed in achieve the desired results.     

Existence of solutions to Schrödinger‐Poisson systems with critical and supercritical nonlinear terms

In this paper, the existence of nontrivial solutions to a class of Schrödinger‐Poisson systems with critical and supercritical nonlinear terms is obtained via variational methods. By using the potential function, a compactness imbedding result is obtained. The properties of potential function play an important role for insuring variational setting.     

Existence,uniqueness and boundedness results for impulsive delay differential equations

This paper considers general impulsive delay differential equations. By utilizing a non-classical approach, the theory of existence and uniqueness of solutions are developed. Criteria on boundedness of solutions are also established through the use of Lyapunov functionals.     

Approximate controllability of impulsive semilinear stochastic system with delay in state

Many practical systems in physical and biological sciences have impulsive dynamical behaviors during the evolution process that can be modeled by impulsive differential equations. This article studies the approximate controllability of impulsive semilinear stochastic system with delay in state in Hilbert spaces. Assuming the conditions for the approximate controllability of the corresponding deterministic linear system, we obtain the sufficient conditions for the approximate controllability of the impulsive semilinear stochastic system with delay in state. The results are obtained by using Banach fixed point theorem. Finally, two examples are given to illustrate the developed theory.     

Stability of neutral stochastic delayed systems with switching and distributed-delay dependent impulses

This paper is devoted to discuss the exponential stability in mean square of neutral stochastic delayed systems (NSDDs) with switching and distributed-delay dependent impulses. By using multiple Lyapunov functions and average dwell time (ADT), we provide some sufficient conditions for the exponential stability in mean square for NSDDs with switching and distributed-delay dependent impulses. Compared with the existing related works, we consider not only the influences of switches and neutral type on the stability of NSDDs with switching and distributed-delay dependent impulses but also the influences of both the stable continuous dynamics case and the stable discrete dynamics case. Finally, we provide two examples to illustrate the effectiveness of the theory.     

Robust stability results for nonlinear Markovian jump systems with mode‐dependent time‐varying delays and randomly occurring uncertainties

This article is concerned with the robust stability analysis for Markovian jump systems with mode‐dependent time‐varying delays and randomly occurring uncertainties. Sufficient delay‐dependent stability results are derived with the help of stability theory and linear matrix inequality technique using direct delay‐decomposition approach. Here, the delay interval is decomposed into two subintervals using the tuning parameter η such that , and the sufficient stability conditions are derived for each subintervals. Further, the parameter uncertainties are assumed to be occurring in a random manner. Numerical examples are given to validate the derived theoretical results. © 2015 Wiley Periodicals, Inc. Complexity 21: 50–60, 2016     

Mean square exponential synchronization for impulsive coupled neural networks with time‐varying delays and stochastic disturbances

In this article, the mean square exponential synchronization of a class of impulsive coupled neural networks with time‐varying delays and stochastic disturbances is investigated. The information transmission among the systems can be directed and lagged, that is, the coupling matrices are not needed to be symmetrical and there exist interconnection delays. The dynamical behaviors of the networks can be both continuous and discrete. Specially, the time‐varying delays are taken into consideration to describe the impulsive effects of the system. The control objective is that the trajectories of the salve system by designing suitable control schemes track the trajectories of the master system with impulsive effects. Consequently, sufficient criteria for guaranteeing the mean square exponential convergence of the two systems are obtained in view of Lyapunov stability theory, comparison principle, and mathematical induction. Finally, a numerical simulation is presented to show the verification of the main results in this article. © 2015 Wiley Periodicals, Inc. Complexity 21: 190–202, 2016     

Age‐dependent single‐species population dynamics with delayed argument

The model of age‐dependent population dynamics was for the first time described by McKendrick (1926). This model was based on the first‐order partial differential equation with the standard initial condition and the non‐local boundary condition in integral form. Gurtin and MacCamy in their paper (1974) analyzed a more general model, where the progress of the population depends on its number. They proved the existence of the unique solution to their model for all time. In our paper the results of Gurtin and MacCamy will be generalized on the case, when the dependence on a number of a population is delayed. Copyright © 2009 John Wiley & Sons, Ltd.     

Existence results for impulsive neutral stochastic evolution inclusions in Hilbert space

This paper is concerned with the existence of mild solutions of a class of impulsive neutral stochastic evolution inclusions in Hilbert space in the case where the right hand side is convex or nonconvex-valued. The results are obtained by using two fixed point theorems for multivalued mappings and evolution system theory.     

On a class of PDEs with nonlinear distributed in space and time state‐dependent delay terms

A new class of nonlinear partial differential equations with distributed in space and time state‐dependent delay is investigated. We find appropriate assumptions on the kernel function which represents the state‐dependent delay and discuss advantages of this class. Local and long‐time asymptotic properties, including the existence of global attractor and a principle of linearized stability, are studied. Copyright © 2008 John Wiley & Sons, Ltd.