Learn More
Let T ⊂ R be a periodic time scale in shifts δ ± associated with the initial point t 0 ∈ 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(t 0) = x 0 has a periodic solution in shifts δ ±. We extend and unify periodic differential, difference, h-difference and especially q-difference(More)
Let T be any time scale such that 0, 1 be subset of T. The concept of dynamic equations on time scales can build bridges between differential and difference equations. This concept not only gives us unified approach to study the boundary value problems on discrete intervals with uniform step size and real intervals but also gives an extended approach to(More)
and Applied Analysis 3 and integrating over [ρ(a), T , we get p (r) θ Δ (r) = p (ρ (a)) θ Δ (ρ (a)) + ∫ r ρ(a) q (t) θ (t) ∇t. (22) Since p(ρ(a)) > 0, θΔ(ρ(a)) ≥ 0, q(t) > 0, and θ(t) > 0, we obtain p(r)θΔ(r) > 0. Thus, we determine θΔ(r) > 0. This contradiction shows that the solution θ(t) is strictly increasing and positive on [ρ(a), T as desired. Similar(More)
  • 1