Almost automorphic solutions of Volterra equations on time scales. (English) Zbl 1413.35354

Lizama, Carlos; Mesquita, Jaqueline G.; Ponce, Rodrigo; Toon, Eduard

Differ. Integral Equ. 30, No. 9-10, 667-694 (2017).

This paper focuses on the qualitative properties of Volterra integral equations on time scales. The main result is the conditional theorem of the existence of the unique asymptotically almost automorphic solution for the class of the following linear Volterra integral equation on time scales \[ u(t)=\int\limits_{-\infty}^ta(t,\sigma(s))[u(s)+g(s)]\Delta s, \] where \(a\), \(g\) are almost automorphic functions with respect to both variables. In addition, the sufficient condition for the existence of asymptotically almost automorphic solutions of the semilinear Volterra delta integral equation is derived.

Reviewer: Denis Sidorov (Irkutsk)

MSC:

35N05	Overdetermined systems of PDEs with constant coefficients
45D05	Volterra integral equations
43A60	Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions

Keywords:

almost automorphic solutions; asymptotic behavior; delta-derivative; generalized exponential function