Non-unique utilities in a competitive equilibrium
This answer is in response to the edited (2015.10.16 ca. 12:30 CET) question. 

I think the concept of equilibrium means that given the circumstances you have no incentive to alter your behavior. (This is pretty much based on the Nash-equilibrium.) 
 In this situation equilibrium consists of a price and an allocation of goods. Given the price everyone makes a trade decision. The decision has to be feasible both with respect to the budget and the physical availability of the goods, i.e. you cannot buy goods that do not exist. We are in equilibrium if everyone's decision was optimal (for themselves). 
In your situation for any $p_x \in [0,2]$ there exists an equilibrium allocation, where consumer 2 bought the good $x$ from consumer 1 for $p_x$ units of good $y$. There is no better feasible decision for either of them.

If you were to adhere to the strict/dogmatic interpretation there would be no equilibrium at all, because for any $p_x &lt;3$ consumer 2 wants to buy an infinite amount of good $x$ and for any $p_x &gt; 2$ consumer 1 wants to sell an infinite amount of good $x$. Since there is no $p_x$ such that $p_x \leq 2, \ p_x \geq 3$ you could not have an equilibrium with finite amounts.