Fixed points and stability of neutral stochastic delay differential equations

Abstract

In this paper we consider a linear scalar neutral stochastic differential equation with variable delays and give conditions to ensure that the zero solution is asymptotically mean square stable by means of fixed point theory. These conditions do not require the boundedness of delays, nor do they ask for a fixed sign on the coefficient functions. An asymptotic mean square stability theorem with a necessary and sufficient condition is proved. Some well-known results are improved and generalized.

Stability of stochastic partial differential equations with infinite delays

“…Very recently, Appleby [1] and Luo [11] have used the fixed point theory to deal with the stability for stochastic differential equations. More precisely, Appleby [1] studied the almost sure stability for a classical equation by splitting the stochastic differential equation into two equations, one being a fixed stochastic problem and the other a deterministic stability problem with forcing function.…”

Stability Analysis of Impulsive Stochastic Reaction-Diffusion Cellular Neural Network with Distributed Delay via Fixed Point Theory

“…Since our CNN model involves pulse and Laplacian operators, our model is different from the previous model ([ [15][16][17][18][19][20][21][22]), which also implies some difficulties in mathematical techniques. Motivated by the previous literature related to fixed point theory ( [15][16][17][18][19][20][21][22][25][26][27][28][29][30][31]), the authors employed Banach fixed point theorem, Hö lder inequality, Burkholder-Davis-Gundy inequality, and the continuous semigroup of Laplace operators to derive the stochastically exponential stability criterion of impulsive stochastic reaction-diffusion cellular neural networks with distributed delay.…”

“…Different methods lead to different criteria for stability criteria which may imply innovations. Fixed point theory and method is one of the alternative methods ( [15][16][17][18][19][20][21][22]). Unlike the known literature, we try to employ Banach fixed point theory in this paper to derive the stability of impulsive stochastic reactiondiffusion cellular neural networks with distributed delay.…”

Existence and asymptotic behavior of solutions for neutral stochastic partial integrodifferential equations with infinite delays

“…The existence, uniqueness and asymptotic behavior of solutions of stochastic partial differential equations have been considered by many authors (see for example [1,2,3,8,9,10,11,14,17,26,27]). Caraballo and Liu [8], Liu and Mao [19] and Taniguchi [26] discussed the exponential stability of the strong and mild solutions, by imposing some kind of coercivity condition, using the Lyapunov method and by a direct estimate of solutions, respectively.…”