Abstract
ABSTRACTIn this paper, we consider a non-linear stochastic differential delay equation (SDDE) of second order. We derive new sufficient conditions which guarantee stochastically stability and stochastically asymptotically stability of the zero solution of that SDDE. Here, the technique of the proof is based on the definition of a suitable Lyapunov-Krasovskii functional, which gives meaningful results for the problem under consideration. The derived results extend and improve some result of in the relevant literature, which are related to the qualitative properties of solutions of a SDDE of second order. The results of this paper are new and have novelty, and they do a contribution to the topic and relevant literature. As an application, an example is given to show the effectiveness and applicability of the obtained results. Finally, by the results of this paper, we extend and improve some recent results that can be found in the relevant literature.
Highlights
Since 1950s the qualitative properties of solutions, such as stability analysis, convergence analysis, asymptotic analysis, chaotic behaviour, oscillation, globally existence of solutions, existence of periodic solutions and so on, for linear and non-linear delay differential equations (DDEs) of second order have been extensively investigated since such delay differential equations and in addition ordinary differential equations have been successfully applied in many fields such as physical, biological, control theory, engineering, medical, social sciences, economics, finance and so on
When DDEs are subject to environmental disturbances, they can be characterized by stochastic differential delay equation (SDDE), see [21]
It is thought that the obtained results may be useful for researchers working in the various fields of sciences and engineering, for instance, in biology, mechanics, economy, control theory, population dynamics, medicine, engineering and so on
Summary
Since 1950s the qualitative properties of solutions, such as stability analysis, convergence analysis, asymptotic analysis, chaotic behaviour, oscillation, globally existence of solutions, existence of periodic solutions and so on, for linear and non-linear delay differential equations (DDEs) of second order have been extensively investigated since such delay differential equations and in addition ordinary differential equations have been successfully applied in many fields such as physical, biological, control theory, engineering, medical, social sciences, economics, finance and so on. The motivation for considering SDDE [2] and studying the qualitative properties of solutions of this equation come from the paper of Abou-El-Ela et al [2] and the works that can found in the references of this paper, see [1–58]. Definition 1: The zero solution of SDDE [4] is said to be stochastically stable or stable in the probability if for every pair ε ∈ (0, 1) and r > 0, there exists δ = δ(ε, r) > 0 such that. Definition 2: The zero solution of SDDE [4] is said to be stochastically asymptotically stable if it is stochastically stable, and for every pair ε ∈ (0, 1) and r > 0, there exists δ0 = δ0(ε) > 0 such that. The zero solution of SDDE [4] is stochastically asymptotically stable
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