Lemma 2.10 . [00PU]

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Lemma 2.10.

(cf. [15, Lemma 2.4 and Remark 2.5]) Let $f:[t_{0},\infty)\to[0,\infty)$ be a nonincreasing right-continuous function, such that

\begin{cases}f(t_{0})<\frac{1}{2B},\\ tf(\tau+t)\leq Bf(\tau)^{2},\quad\forall\tau\geq 0,\quad 0\leq t\leq 1,\\ \lim_{t\to\infty}f(t)=0.\end{cases}

Then $f(t)=0$ for $t\geq t_{0}+4Bf(t_{0})$ .