A retarded Gronwall-like inequality and its applications (Q1840807)

In the present paper a retarded Gronwall-like inequality is proved and some applications are given to show its usefulness. The main result given here can be stated as follows. Let \(u,f\in C([t_0,T), R_+)\). Moreover, let \(w\in C(R_+, R_+)\) be nondecreasing with \(w(u)> 0\) on \((0,\infty)\) and \(\alpha\in C^1([t_0, T),[t_0,T))\) be nondecreasing with \(\alpha(t)\leq t\) on \([t_0, T)\). If \[ u(t)\leq k+ \int^{\alpha(t)}_{\alpha(t_0)} f(s) w(u(s)) ds,\quad t_0\leq t< T, \] where \(k\) is a nonnegative constant, then, for \(t_0\leq t< t_1\), \[ u(t)\leq G^{-1}\Biggl(G(k)+ \int^{\alpha(t)}_{\alpha(t_0)} f(s) ds\Biggr), \] where \(G(r)= \int^r_1{ds\over w(s)}\), \(r>0\), and \(t_1\in (t_0, T)\) is chosen so that \[ G(k)+ \int^{\alpha(t)}_{\alpha(t_0)} f(s) ds\in \text{Dom}(G^{-1}), \] for all \(t\) lying in the interval \([t_0,t_1)\).