First order boundary value problem $u'=f(t, u),~ au(t_0) + bu(t_1) = 0$, existence condition

Question

Consider $$ u' = f(t, u), \quad a u(t_0) + bu(t_1) = 0 $$ where $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ is globally Lipschitz in $u$ (with Lipschitz constant $L$ independent of $t$) and $-\infty< t_0 < t_1 < \infty$, $a \neq -b$. I read that with the help of Green's function it can be shown that if $$ \lvert t_0 - t_1 \rvert L \max\lbrace a, b \rbrace \lvert a+b\rvert^{-1} < 1, $$ then this problem has a unique solution. I want to get this (or a better) estimate without Green's function. The fixed point equation $$ x(t) = -\frac{b}{a}x(t_1) + \int^t_{t_0} f(s, x(s))~\mathrm{d}s $$ leads to the condition $$ \frac{\lvert b \rvert}{\lvert a \rvert} + L \lvert t -t_0 \rvert < 1 $$ for a contraction which is not quite the one that i listed above. A colleague of mine proposed to look at the fixed point formulation $$ x(t) = \frac{1}{1 + \frac{a}{b}}\int^t_{t_0} f(s, x(s))~\mathrm{d}s $$ but from which I cannot reconstruct a solution to the above BVP. Can anyone help me out?