I'm interested in the condition for existence of a solution $\displaystyle (x_1, x_2)$ given

$\displaystyle |a-x_1|\leq C_1$

$\displaystyle |x_1-x_2|\leq C_2$

$\displaystyle |b-x_2|\leq C_3$

$\displaystyle x_1,x_2\leq C_4$

This is my conclusion. Let

$\displaystyle I_1 = [a-C_1, a+C_1]$

$\displaystyle I_2 = [b-C_3, b+C_3]$

Now, $\displaystyle x_2 \in [x_1-C_2, x_1+C_2]$ and $\displaystyle x_1 \in [x_2-C_2,x_2+C_2]$

So, if

$\displaystyle I_3 = [a-C_1-C_2, a+C_1+C_2]$

$\displaystyle I_4 = [b-C_3-C_2, b+C_3+C_2]$

Then a solution exist if $\displaystyle I_A = I_1 \cap I_3 \cap [0, C_4] \neq \emptyset$ and $\displaystyle I_B = I_2 \cap I_4 \cap [0, C_4] \neq \emptyset$ and $\displaystyle x_1 \in I_A, x_2 \in I_B$

Is this correct? Does anyone see a problem? Thanks