almost sure exponential stability of numerical solutions to stochastic delay hopfield neural networks stability of numerical solutions to stochastic delay differential equations have received an increasing attention but there has been so far little work on the stability analysis of numerical solutions to stochastic delay hopfield neural networks the aim of this paper is to study the almost sure exponential stability of numerical solutions to stochastic delay hopfield neural networks by using two approaches the euler method and the backward euler method under some reasonable conditions both the euler scheme and the backward euler scheme are proved to be almost sure exponential stability in particular the euler method and the backward euler method are mainly based on the semimartingale convergence theorem

exponential stability of stochastic reaction diffusion cohen grossberg neural networks with delays abstract in this paper stochastic cohen grossberg neural network with time varying delays and reaction diffusion terms is considered the sufficient conditions are obtained to guarantee the mean value exponential stability of an equilibrium solution

global robust stability for stochastic interval neural networks with continuously distributed delays of neutral type in this paper by using the lyapunov krasovskii functional method we investigate the global robust stability for stochastic interval neural networks with continuously distributed delays of neutral type some new stability criteria are presented in terms of linear matrix inequality lmi two numerical examples are also given to show the effectiveness of the obtained results using lmi control toolbox in matlab

heat kernels and analyticity of non symmetric jump diffusion semigroups let d geqslant 1 and alpha in 0 2 consider the following non local and non symmetric levy type operator on mathbb r d begin aligned fancyscript l kappa alpha f x hbox p v int mathbb r d f x z f x frac kappa x z z d alpha mathord mathrm d z end aligned where 0 kappa 0 leqslant kappa x z leqslant kappa 1 kappa x z kappa x z and kappa x z kappa y z leqslant kappa 2 x y beta for some beta in 0 1 using levi s method we construct the fundamental solution also called heat kernel p kappa alpha t x y of fancyscript l kappa alpha and establish its sharp two sided estimates as well as its fractional derivative and gradient estimates we also show that p kappa alpha t x y is jointly holder continuous in t x the lower bound heat kernel estimate is obtained by using a probabilistic argument the fundamental solution of fancyscript l kappa alpha gives rise a feller process x mathbb p x x in mathbb r d on mathbb r d we determine the levy system of x and show that mathbb p x solves the martingale problem for fancyscript l kappa alpha c 2 b mathbb r d furthermore we show that the c 0 semigroup associated with fancyscript l kappa alpha is analytic in l p mathbb r d for every p in 1 infty a maximum principle for solutions of the parabolic equation partial t u fancyscript l kappa alpha u is also established as an application of the main result of this paper sharp two sided estimates for the transition density of the solution of mathord mathrm d x t a x t mathord mathrm d y t is derived where y is a rotationally symmetric stable process on mathbb r d and a x is a holder continuous d times d matrix valued function on mathbb r d that is uniformly elliptic and bounded

transition probabilities for symmetric jump processes we consider symmetric markov chains on the integer lattice in d dimensions where a 0 2 and the conductance between x and y is comparable to x y d we establish upper and lower bounds for the transition probabilities that are sharp up to constants

mean square stability of two classes of theta methods for numerical computation and simulation of delayed stochastic hopfield neural networks abstract recently the investigation on the stability of the numerical solutions to delayed stochastic differential equations has received an increasing attention but there has been little work on the stability analysis of the numerical solutions to delayed stochastic hopfield neural networks dshnns so far the aim in this paper is to study the mean square exponential stability of the split step theta sst method and the stochastic linear theta slt method for the underlying model it is proved that for any 0 1 2 there exists a constant 0 depending on such that the numerical schemes produced by the sst method and the slt method are mean square exponentially stable for 0 under the same assumptions as those to guarantee the mean square exponential stability of the underlying continuous model for the case 1 2 1 we show the same stability conclusion for all 0 to carry out the required conclusion a novel technique for the stability analysis of discrete numerical schemes with multi time delays namely the weighted sum lyapunov functional method is proposed finally a numerical example is given to illustrate the application of the suggested methods and to verify the stability conclusions obtained

mean square stability and almost sure exponential stability of two step maruyama methods of stochastic delay hopfield neural networks abstract in this paper the two step maruyama methods of stochastic delay hopfield neural networks are studied we have found that under what choices of step size the two step maruyama methods of stochastic delay hopfield networks maintain the stability of exact solutions the mean square stability of two step maruyama methods of stochastic delay hopfield neural networks is investigated under suitable conditions also the almost sure exponential stability of two step maruyama methods of stochastic delay hopfield networks is proved using the semi martingale convergence theorem further the comparisons of stability conditions to the previous results in liu and zhu 2015 rathinasamy 2012 and ronghua et al 2010 are given numerical experiments are provided to illustrate our theoretical results

heat kernels for non symmetric diffusion operators with jumps abstract for d 2 we prove the existence and uniqueness of heat kernels to the following time dependent second order diffusion operator with jumps l t 1 2 i j 1 d a i j t x i j 2 i 1 d b i t x i l t where a a i j is a uniformly bounded elliptic and holder continuous matrix valued function b belongs to some suitable kato s class and l t is a non local stable type operator with bounded kernel moreover we establish sharp two sided estimates gradient estimate and fractional derivative estimate for the heat kernel under some mild conditions

improved global robust stability of interval delayed neural networks via split interval generalizations abstract the problem of global robust stability of hopfield type delayed neural networks with the intervalized network parameters is revisited recently a computationally tractable i e linear matrix inequality lmi based global robust stability criterion derived from an earlier criterion based on dividing the given interval into more that two intervals has been presented in the present paper generalizations i e division of the given interval into m intervals where m is an integer greater than or equal to 2 is considered and some new lmi based global robust stability criteria are derived it is shown that in some cases m 2 may not suffice i e m 2 may be needed to realize the improvement an example showing the effectiveness of the proposed generalization is given the paper also provides a complete and systematic explanation of the split interval idea

heat kernel estimates for jump processes of mixed types on metric measure spaces in this paper we investigate symmetric jump type processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces the class of metric measure spaces includes the alfors d regular sets which is a class of fractal sets that contains geometrically self similar sets a typical example of our jump type processes is the symmetric jump process with jumping intensity e c 0 x y x y int alpha 1 alpha 2 frac c alpha x y x y d alpha nu d alpha where is a probability measure on alpha 1 alpha 2 subset 0 2 c x y is a jointly measurable function that is symmetric in x y and is bounded between two positive constants and c 0 x y is a jointly measurable function that is symmetric in x y and is bounded between 1 and 2 where either 2 1 0 or 1 2 0 this example contains mixed symmetric stable processes on mathbb r n as well as mixed relativistic symmetric stable processes on mathbb r n we establish parabolic harnack principle and derive sharp two sided heat kernel estimate for such jump type processes

brief paper growth rate of switched homogeneous systems we consider discrete time homogeneous systems under arbitrary switching and study their growth rate the analogue of joint spectral radius for switched linear systems we show that a system is asymptotically stable if and only if its growth rate is less than unity we also provide an approximation algorithm to compute growth rate with arbitrary accuracy

robust exponential stability of markovian jump impulsive stochastic cohen grossberg neural networks with mixed time delays this paper is concerned with the problem of exponential stability for a class of markovian jump impulsive stochastic cohen grossberg neural networks with mixed time delays and known or unknown parameters the jumping parameters are determined by a continuous time discrete state markov chain and the mixed time delays under consideration comprise both time varying delays and continuously distributed delays to the best of the authors knowledge till now the exponential stability problem for this class of generalized neural networks has not yet been solved since continuously distributed delays are considered in this paper the main objective of this paper is to fill this gap by constructing a novel lyapunov krasovskii functional and using some new approaches and techniques several novel sufficient conditions are obtained to ensure the exponential stability of the trivial solution in the mean square the results presented in this paper generalize and improve many known results finally two numerical examples and their simulations are given to show the effectiveness of the theoretical results

stability of asymmetric hopfield networks in this paper we discuss dynamical behaviors of recurrently asymmetrically connected neural networks in detail we propose an effective approach to study global and local stability of the networks many of well known existing results are unified in our framework which gives much better test conditions for global and local stability sufficient conditions for the uniqueness of the equilibrium point and its stability conditions are given too

global exponential stability and periodic solutions of cohen grossberg neural networks with continuously distributed delays abstract convergence dynamics of cohen grossberg neural networks cgnns with continuously distributed delays are discussed without assuming the differentiability and monotonicity of activation functions the differentiability of amplification functions and the symmetry of synaptic interconnection weights by skilfully constructing suitable lyapunov functionals and employing inequality technique three sets of easily verifiable delay independent criteria to guarantee the global exponential stability of a unique equilibrium point are given and moreover by constructing poincare mapping other three sets of easily verifiable delay independent criteria to assure the existence and globally exponential stability of periodic solutions are obtained six examples are given to illustrate the theoretical results

on some stability problems of impulsive stochastic cohen grossberg neural networks with mixed time delays abstract this paper covers the topic of both the p th moment p 2 and almost sure stability of impulsive stochastic cohen grossberg neural networks with mixed time delays we partially use a known result on exponential stability of impulsive stochastic functional differential systems based on the razumikhin type technique and extend it to the case of stochastic neural networks using the lyapunov function method and a gronwall type inequality additionally we consider the stability with respect to a general decay function which includes exponential but also more general lower rate decay functions as the polynomial and the logarithmic ones this fact gives us the opportunity to study general decay almost sure stability even when the exponential one cannot be discussed suitable examples which support the theory are also presented

laplacian perturbed by non local operators suppose that d ge 1 and 0 beta 2 we establish the existence and uniqueness of the fundamental solution q b t x y to the operator mathcal l b delta mathcal s b where begin aligned mathcal s bf x int mathbb r d left f x z f x nabla f x cdot z 1 1 z le 1 right frac b x z z d beta dz end aligned and b x z is a bounded measurable function on mathbb r d times mathbb r d with b x z b x z for x z in mathbb r d we show that if for each x in mathbb r d b x z ge 0 for a e z in mathbb r d then q b t x y is a strictly positive continuous function and it uniquely determines a conservative feller process x b which has strong feller property furthermore sharp two sided estimates on q b t x y are derived

on a general decay stability of stochastic cohen grossberg neural networks with time varying delays abstract to the best of our knowledge there are only few results on general decay stability applied to stochastic neural networks for stochastic cohen grossberg neural networks with time varying delays we study in the present paper both the pth moment p 2 and almost sure stability on a general decay rate and partly generalize and improve some known results referring to the exponential stability we also extend the usual notion on a general decay function which allows us to study both the pth moment and almost sure stability even if the exponential stability cannot be shown some examples are presented to support and illustrate the theory

computation issue of pointwise controls for diffusion hopfield neural network system this paper consider the pointwise controls problem of hopfield neural network equation with diffusion term subject to quadratic criteria in the framework of variational method based on established optimal control theory to pointwise case numerical study is carried out by constructing a semi discrete algorithm with continuous time furthermore for given point control inputs the optimal pointwise controls are obtained to achieve the minimization experiments demonstration is implemented for verifying proposed algorithm

exponential stability of stochastic delayed hopfield neural networks the exponential stability for stochastic delayed hopfield neural networks hnns is considered the sufficient conditions to guarantee the almost surely exponential stability and exponential stability in mean square of an equilibrium solution are given one example is also given to demonstrate our results

an analysis of global asymptotic stability of delayed cellular neural networks in this paper a new sufficient condition is given for the uniqueness and global asymptotic stability of the equilibrium point for delayed cellular neural networks dcnns this condition imposes constraints on the feedback and delayed feedback matrices of a dcnn independently of the delay parameter this result is also compared with the previous results derived in the literature