On the example of Ostrów Tumski in Wrocław (Silesia, Polish lands) an attempt of identification of places associated with trade and exchange in early medieval strongholds was made. The alleged marketplace existed in the stronghold from the end of the 11th century to the end of the 12th century. Its existence would be confirmed by the empty square between buildings, the accumulation of finds related to trade and exchange as well as spending free time. The connection of some buildings form Ostrów Tumski with commercial activities is also not excluded. On the basis of analogies from the main centers of the Piast state and selected sites from the Bohemian and Moravian territories, the potential location of the marketplaces was analyzed. Attention was drawn to the relationship between finds related to trade and exchange with other manifestations of economic activity (e.g. non-ferrous metallurgy). An attempt was also made to answer the question when and in what political and economic conditions such places could be created.
We use Brouwer degree to prove existence and multiplicity results for the solutions of some nonlinear second order difference equations with Dirichlet boundary conditions. We obtain in particular upper and lower solutions theorems, Ambrosetti-Prodi type results, and sharp existence conditions for nonlinearities which are bounded from below or from above.