Assume that $u~ \colon \mathbb{R}\_+ \to [-M,M]$ is a bounded continuously differentiable function such that $u(0) = 0$ and $$u(t) \leq \int\_0^t \lambda(s)~u(s)~\mathrm{d}s + C \label{1}\tag{1}$$ where $\lambda ~ \colon \mathbb{R}\_+ \to [-\infty,-\alpha]$ is a strictly negative function with $\lambda(t) \leq -\alpha < 0$ for all $t \geq 0$, and $C > 0$ is a fixed constant. Moreover, suppose that $-u$ satisfies the same inequality \eqref{1} as $u$, i.e., $$-u(t) \leq -\int\_0^t \lambda(s)~u(s)~\mathrm{d}s + C \label{2}\tag{2}.$$ My question is, is it possible to deduce from \eqref{1} and \eqref{2} that $$\lvert u(t)\rvert \leq C~\mathrm{e}^{-\alpha t}~?$$ If not, is it possible to deduce (at least) that $u(t) \to 0$ as $t \to \infty~?$



On the validity of a certain Grönwall-type inequality
No. A counterexample is $C=3$, $\lambda(t)=-1$, and $u(t)=\sin t$ for all $t$.


