Boundedness of Solutions of Second Order Nonlinear Dynamic Equations

This paper deals with the boundedness of the solutions of the following dynamic equations $$(r(t)x^\Delta (t))^\Delta + a(t)f(x^\sigma (t)) + b(t)g(x^\sigma (t)) = 0$$ and $$(r(t)x^\Delta (t))^\Delta + a(t)x^\sigma (t) + b(t)f(x(t - \tau (t))) = e(t)$$ on a time scale \(\mathbb{T}\). By using the Bellman integral inequality, we establish some sufficient conditions for boundedness of solutions of the above equations. Our results not only unify the boundedness results for differential and difference equations but are also new for the q-difference equations.