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A retarded Gronwall-like inequality and its applications. (English) Zbl 0974.26007

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 $u,f\in C\left(\left[{t}_{0},T\right),{R}_{+}\right)$. Moreover, let $w\in C\left({R}_{+},{R}_{+}\right)$ be nondecreasing with $w\left(u\right)>0$ on $\left(0,\infty \right)$ and $\alpha \in {C}^{1}\left(\left[{t}_{0},T\right),\left[{t}_{0},T\right)\right)$ be nondecreasing with $\alpha \left(t\right)\le t$ on $\left[{t}_{0},T\right)$. If

$u\left(t\right)\le k+{\int }_{\alpha \left({t}_{0}\right)}^{\alpha \left(t\right)}f\left(s\right)w\left(u\left(s\right)\right)ds,\phantom{\rule{1.em}{0ex}}{t}_{0}\le t

where $k$ is a nonnegative constant, then, for ${t}_{0}\le t<{t}_{1}$,

$u\left(t\right)\le {G}^{-1}\left(G\left(k\right)+{\int }_{\alpha \left({t}_{0}\right)}^{\alpha \left(t\right)}f\left(s\right)ds\right),$

where $G\left(r\right)={\int }_{1}^{r}\frac{ds}{w\left(s\right)}$, $r>0$, and ${t}_{1}\in \left({t}_{0},T\right)$ is chosen so that

$G\left(k\right)+{\int }_{\alpha \left({t}_{0}\right)}^{\alpha \left(t\right)}f\left(s\right)ds\in \text{Dom}\left({G}^{-1}\right),$

for all $t$ lying in the interval $\left[{t}_{0},{t}_{1}\right)$.

##### MSC:
 26D10 Inequalities involving derivatives, differential and integral operators
##### Keywords:
retarded Gronwall-like inequality