i am currently stuck and need some approval / disapproval. Suppose i am given a strictly increasing function $f:\mathbb{R}\longrightarrow \mathbb{R}$, $a_i$ between 0 and 1 and looking for a solution x of \begin{equation*} 0=\sum_{i=1}^n a_i [f(v_i+x)-f(v)], \qquad v,v_i \in \mathbb{R}. \end{equation*} I procced by applying Taylor expansion around v for $f(v_i +x)$ \begin{align*} f(v_i+x)-f(v) = f'(\xi_i)(v_i+x-v) \end{align*} for some $\xi_i$ between both points and bound $f'(\xi)$ from below and above by constant $C^+,C^-$ globally for all $i$. Hence, \begin{align*} 0 &\leq C^+ \sum_{i=1}^n a_i(v_i+x-v) \\ 0 &\geq C^- \sum_{i=1}^n a_i(v_i+x-v) \end{align*} yielding \begin{align*} x &\geq v-A^{-1}\sum_{i=1}^n a_iv_i \\ x &\leq v-A^{-1}\sum_{i=1}^n a_iv_i, \end{align*} where A is the sum of all $a_i$ and thus \begin{equation*} x = v-A^{-1}\sum_{i=1}^n a_iv_i. \end{equation*}
So, yeah. It somehow feels strange. Did I overlook something? I am pretty sure I did but I am stuck for hours...Suppose, that everything is OK with Taylor and finiteness of $f'$ and that the derivatives exist...
Thanks so much for helping!