Class Of Nonlinear Dynamic Equations Research Articles

A Further Study of Almost Periodic Time Scales with Some Notes and Applications

We introduce three equivalent concepts of almost periodic time scales as a further study of the corresponding concept proposed in Li and Wang (2011) and several examples of almost periodic time scales which are not periodic are provided. Furthermore, the concepts of almost periodic functions are redefined under the sense of this new timescale concept. Finally, almost periodicity of Cauchy matrix for dynamic equations is proved under these new definitions. Based on these results, the existence of almost periodic solutions to a class of nonlinear dynamic equations is investigated by the almost periodicity of Cauchy matrix on almost periodic time scales. Besides, as an application, we apply our results to a class of high-order Hopfield neural networks.

On the boundedness of a class of nonlinear dynamic equations of second order

Abstract In this paper, a generalized nonlinear dynamic integral inequality on time scales is established and then is used to study the boundedness of a class of nonlinear second-order dynamic equations on time scales. These theorems contain as special cases results for second-order differential equations, difference equations and q -difference equations.

Existence of Nonoscillatory Solutions of a Class of Nonlinear Dynamic Equations with a Forced Term

In this paper, we consider the following forced higher-order nonlinear neutral dynamic equation                 12 0 ,, m k xt p t x t f t x t x t x t q t t t           ,, ,  ,  on time scales. By using Banach contraction principle, we obtain sufficient conditions for the existence of nonoscillatory solutions for general   pt and which means that we allow oscillatory  qt   pt and . We give some ex amples to illustrate the obtained results.  qt

Stability of solutions of one class of nonlinear dynamic equations

We study the stability of the zero solution of a nonlinear dynamic equation on a time scale under certain assumptions on the right-hand side of this equation. In addition to conditions for the existence and uniqueness of a solution of the Cauchy problem, we also assume that the exponential function of the linear approximation is bounded, and the norms of the nonlinear part and its derivatives with respect to the components of the space variable are majorized by power functions of the norm of the space variable. Using the generalized method of Lyapunov functions, we obtain sufficient conditions for the stability of the zero solution of the nonlinear equation under consideration.