I have a sequence of vectors $\{v_n\} (\ v_n\in \mathbb{R}^N,\ n\ge 0),$ evolving in such a way that they satisfy the following inequalities, $$\|v_{n+1}\|_2\le \left\|a_n+\sum_{0\le k\le n}g_{nk}v_k\right\|_2,\quad n\ge 0$$ with $v_0=0$. Here $\|\cdot\|_2$ denotes the Euclidean 2-norm. My question is
How to get a "good" upper bound on $\|v_n\|_2$, i.e. where the upper bound would only consist of the vectors $a_n$ and matrices $g_{nk}$?
One method that I could follow is that I could use the triangle inequality and the matrix norm inequality to get a discrete Gronwall type inequality as below $$\|v_{n+1}\|_2\le\|a_n\|_2+\sum_{0\le k\le n}\|g_{nk}\|\|v_k\|_2,\quad n\ge 0$$ where $\|\cdot\|$ is the matrix operator norm induced by Euclidean norm. But in many cases taking triangle inequality can make the bounds loose.
So I was wondering is there any alternative better method to get an upper bound of $v_n$ from the original set of inequalities without taking the triangle inequality? Thanks in advance.