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A dynamic two-sided matching market is considered. We examine two existing notions of stability—the core and recursive core—for this multi-period market and argue that they both have limitations. We define two new notions of stability and label them, self-sustaining stability and strict self-sustaining stability. Both concepts can be viewed as the recursive core with more stringent conditions for when deviating coalitions are effective. We show that these concepts overcome some of the weaknesses of the core and the recursive core. We also provide conditions for the existence of our concepts.