Alternative conjectures in a Bertrand-Edgeworth model

In a model with increasing marginal costs the 'normal' Bertrand equilibrium does not exist, as is well known. We study a Bertrand duopoly where each firm has a given market share at equal prices that may differ from 1 2. Firms have conjectures about the way in which the competitor will react on a price change at a given price pair. These conjectures are different for price increases and price decreases. Undercutting is always matched. Price pairs are feasible only if either both prices are equal or they differ by at least α, α being a small positive number, by which α-equilibria are defined. It is proved that, given the assumptions, α-equilibria exist, among which appear both single price equilibria and equilibria where the prices differ by α.