Let T ⊂ R be a periodic time scale in shifts δ± associated with the initial point t0 ∈ T∗. We use Brouwer’s fixed point theorem to show that the initial value problem x(t) = p(t)x(t) + q(t), t ∈ T, x(t0) = x0 has a periodic solution in shifts δ±. We extend and unify periodic differential, difference, h-difference and especially q-difference equations and more by a new periodicity concept on tim...

In this paper, we define a variant of billiards in which the ball bounces around a square grid erasing walls as it goes. We prove that there exist periodic tunnels with arbitrarily large period from any possible starting point, that there exist non-periodic tunnels from any possible starting point, and that there are versions of the problem for which the same starting point and initial directio...

In this paper, sufficient criteria for the existence of multiple positive periodic solutions of a certain nonlinear dynamic system with feedback control are established. This is done by the Avery–Henderson fixed point theorem and the Leggett– Williams fixed point theorem. By using the method of coincidence degree, sufficient conditions are derived ensuring the existence of at least one periodic...