Some sublinear dynamic integral inequalities on time scales (Q608020)

The author studies some nonlinear dynamic inequalities on time scale in this form \[ \begin{aligned} x(t) &\leq a(t)+b(t)\int_{t_0}^{t}[g(s)x(s)+h(s)x^{}(s)]\Delta s,\quad t\in T^{k}, \\ x(t) &\leq a(t)+b(t)\int_{t_0}^{t}w(t,s)[g(s)x(s)+h(s)x^{}(s)]\Delta s,\quad t\in T^{k}, \\ x(t) &\leq a(t)+b(t)\int_{t_0}^{t}f(s,x^{}(s))\Delta s,\quad t\in T^{k}, \end{aligned} \] where \(a, b, g, h, x:T^{k}\to \mathbb R^+\) are rd-continuous functions, \(w:T\times T^{k}\to \mathbb R^+\) is continuous and \(f:T^{k}\to \mathbb R^+\) is continuous. The results include many existing ones in the literature as special cases and can be used as tools in the qualitative theory of certain classes of dynamic equations on time scales. Some applications are considered for the special cases \(T=\mathbb R\) and \(T=\mathbb Z\).