On the Stochastic Stability and Boundedness of Solutions for Stochastic Delay Differential Equation of the Second Order

Abstract

We present two qualitative results concerning the solutions of the following equation: x¨(t)+g(x˙(t))+bx(t-h)+σx(t)ω˙(t)=p(t,x(t),x˙(t),x(t-h)); the first result covers the stochastic asymptotic stability of the zero solution for the above equation in case p≡0, while the second one discusses the uniform stochastic boundedness of all solutions in case p≢0. Sufficient conditions for the stability and boundedness of solutions for the considered equation are obtained by constructing a Lyapunov functional. Two examples are also discussed to illustrate the efficiency of the obtained results.

On the asymptotic stability of solutions of stochastic differential delay equations of second order

“…We establish a new theorem and corollary on the stochastically stability and stochastically asymptotically stability of zero solution of SDDE (2). The theorem and the corollary extend, improve and complement the result of Abou-El-Ela et al [2] and that can be found in [1,3,4,23,27]. In addition, we give an example, which satisfies our assumptions and shows the applicability of them.…”

“…Despite the existence of a lot of papers on the qualitative analysis of solutions of DDEs second order and SDDEs of first order, to the best of our information, we observe only a few papers from the literature on the qualitative analysis of SDDEs of second order, see [1][2][3][4]23,27] In these works, based on the suitable Lyapunov-Krasovskii functionals, the qualitative analysis of solutions have been proceeded in that papers.…”

Permutation Groups and Periodicity of Systems of Difference Equations

“…With respect to our observation, up to this moment, the corresponding problem for the stability (S) and boundedness (B) of solutions of higher order (SDDEs) have been studied in the literature by only a few researches, see, for instance, [28], [29], [30], [31], [32] and [33]. Therefore, it is worth discussing the (QBSs)of (SDDEs).…”

“…(3) The obtained results in [28] , [29], [31] and [33] are on (SDDEs) of second-order, but the obtained results in [30] and [32] are only on (SDDEs)of third order in the previous literature.…”

On the behaviour of solutions to a kind of third order neutral stochastic differential equation with delay

“…In addition, outstanding papers on properties of solutions of nonlinear second and third order neutral and stochastic differential equations, using various techniques, have been discussed by researchers, see for example the works of Abou-El-Ela et al [1][2][3], Ademola [4], Ademola et al [5][6][7], Adesina et al [8], Bohner et al [11], Bouakkaz et al [12], Cahlon and Schmidt [15], Chen et al [16], Mahmoud and Tunç [32], Oudjedi et al [42], Panigrahi and Basu [43], Philos and Purnaras [44], Tripathy et al [47], Yeniçerioğlu and Demir [49] and the references cited therein.…”