Abstract
This chapter summarizes the modern theory of fixed point and equilibrium. Based on the concept of ɛ-approximate minimum, Ekeland variational principle is established first, along with Caristi fixed point theorem, and Nadler contraction theorem for set-valued maps. The next central result is a general equilibrium theorem from which virtually all important fixed point and equilibrium propositions can be derived in a simple unified manner: Ky Fan inequality, Brouwer and Kakutani fixed point theorems, Nash equilibrium theorem, and also the basic solvability theorem for variational inequalities.
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Tuy, H. (2016). Fixed Point and Equilibrium. In: Convex Analysis and Global Optimization. Springer Optimization and Its Applications, vol 110. Springer, Cham. https://doi.org/10.1007/978-3-319-31484-6_3
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DOI: https://doi.org/10.1007/978-3-319-31484-6_3
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