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.

[1]  Jaqueline Godoy Mesquita,et al.  Almost automorphic solutions of dynamic equations on time scales , 2013 .

[2]  Saudi Arabia,et al.  Existence and Exponential Convergence of almost Periodic Solutions for a Discrete Nicholson’s Blowflies Model with Nonlinear Harvesting Term , 2013, Mathematical Sciences Letters.

[3]  Jaqueline Godoy Mesquita,et al.  A connection between almost periodic functions defined on timescales and ℝ , 2014 .

[4]  Lang Li,et al.  Almost Periodic Solutions for Second Order Dynamic Equations on Time Scales , 2015 .

[5]  Carlos Lizama,et al.  Almost automorphic solutions of non-autonomous difference equations , 2013 .

[6]  Yang Liu,et al.  Existence and global exponential stability of almost periodic solutions to Cohen-Grossberg neural networks with distributed delays on time scales , 2014, Neurocomputing.

[7]  A. Peterson,et al.  Dynamic Equations on Time Scales , 2001 .

[8]  Youssef N. Raffoul,et al.  Periodic solutions for a neutral nonlinear dynamical equation on a time scale , 2006 .

[9]  Yongkun Li,et al.  Almost Periodic Functions on Time Scales and Applications , 2011 .

[10]  Yan Wang,et al.  Almost Periodic Solutions for Second Order Dynamic Equations on Time Scales , 2013 .

[11]  Ke Wang,et al.  Translation properties of time scales and almost periodic functions , 2013, Math. Comput. Model..

[12]  Yongkun Li,et al.  Uniformly Almost Periodic Functions and Almost Periodic Solutions to Dynamic Equations on Time Scales , 2011 .

[13]  Yongkun Li,et al.  Pseudo almost periodic functions and pseudo almost periodic solutions to dynamic equations on time scales , 2012, Advances in Difference Equations.

[14]  Chao Wang,et al.  Weighted piecewise pseudo almost automorphic functions with applications to abstract impulsive ∇-dynamic equations on time scales , 2014 .

[15]  Billy J. Jackson,et al.  Partial dynamic equations on time scales , 2006 .

[16]  Yongkun Li,et al.  Weighted pseudo almost automorphic functions with applications to abstract dynamic equations on time scales , 2013 .

[17]  Yaşar Bolat,et al.  Almost periodic dynamics of a discrete Nicholson’s blowflies model involving a linear harvesting term , 2012, Advances in Difference Equations.

[18]  Chao Wang,et al.  Almost periodic solutions of impulsive BAM neural networks with variable delays on time scales , 2014, Commun. Nonlinear Sci. Numer. Simul..

[19]  Ravi P. Agarwal,et al.  Dynamic equations on time scales: a survey , 2002 .