Functions defined below are bounded or not? I have a set of functions ${{x}_{i}}:\mathbb{R}_{0}^{+}\to \mathbb{R}_{0}^{+}$ ($i=1,2,...,n$). Denote the independent variable with $t$. Suppose, there are ${{\eta }_{1}},{{\eta }_{2}},\ldots ,{{\eta }_{n}}\in {{\mathbb{R}}^{+}}$ coefficients such that the sum $\sum\limits_{i=1}^{n}{{{\eta }_{i}}{{x}_{i}}\left( t \right)}=c$ for all $t\ge 0$, where $c\in \mathbb{R}_{0}^{+}$. I would like to ask if the functions ${{x}_{i}}$ are bounded in this case. Thank you for your help!
Yes, they are bounded by $c/min\{\eta_{1}, \cdots, \eta_{n}\}$...?
Suppose there exists $\eta_i\neq 0$. Since every $\eta$ is non-negative, the function $t\mapsto x_i(t)$ is bounded according to your assumption.

If $\eta_i =0$, then there's nothing you can say.