Let $x(t),t\in [1,\infty)$ be a nondecreasing positive function satisfying the following inequality: $$ x'(t) \le \int_t^{+\infty} x(s)\frac{k(s)}{s^2}\,ds, $$ for any $t \ge 1$, where $k(t),t\in [1,\infty)$ is a nonincreasing positive function such that $$ \int_1^{+\infty}\frac{k(s)}{s}\,ds <\infty. $$

Can we prove that $x(t)$ is a bounded function?