This paper addresses a competitive situation of price discrimination between a firm's own subscribers and a competitor's subscribers. In such competition, consumers can pursue switching discounts by repeatedly switching their subscribed firm. However, existing literature has either failed to consider this or has analyzed it under restrictive assumptions. This paper derives a general equilibrium that accounts for consumers' repeated switching under assumptions of arbitrary time periods and consumers' multi-period utility maximization, and identifies the conditions on initial market shares that yield price equilibria sustained over multiple periods. Only such multi-period sustained equilibria can be meaningful as rational predictions of the market. Additionally, the paper examines that the conditions on initial market shares encompass possible market situations prior to the introduction of price discrimination. In the derived equilibrium, consumers' repeated switching renders firms' efforts to acquire competitors' consumers futile in the subsequent period. Moreover, discounts for switching subscribers can, under repeated switching conditions, induce not only switching by competitor subscribers but also switching by the firm's own subscribers. Therefore, the consideration of repeated switching has the effect of mitigating price discount competition.