Bochner Definition Of Stepanov-like Almostautomorphic Functions On Time Scales And An Application To Cellular Neural Networks With Delays

As the development of time scale theory,the study of almost periodic class functions on time scale has become a hot topic for many mathematicians.The generalization of the concept of almost periodic functions on time scale is an important aspect of the research.In literature[1],Dhama and Abbas gave the definition of Stepanov pseudo alomst automorphic function on time scale,but there were some errors in the results involving Bochner transformation in the paper.In this paper,we give the Bochner definition of Stepanov almost automorphic function on time scale by Bochner like transformation.This result corrects the previous mistakes,improves the definition of Stepanov-like almost automorphic function,and proves that a sufficient and necessary condition for a function to be a Stepanov almost automorphic function on a time scale is that it satisfies the Bochner definition of Stepanov almost automorphic function on a time scale.As an application,we first obtain the existence and uniqueness of the alomst automorphic solutions for a class of dynamic equations u~?(s)=A(s)u(s)+?(s),s?T(3-1)where A?R(T,Rn×Rn),? is a continuous Stepanov almost automorphic function.In addition,we also discuss the almost automorphic of a class of time delay cellular neural networks(?) where (?) is Stepanov-like on time scale.