On the validity of certain Gronwall-type inequality

Question

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 $$|u(t)| \leq C~\mathrm{e}^{-\alpha t}~?$$ If not, is it possible to deduce (at least) that $u(t) \to 0$ as $t \to \infty~?$