This article offers an excellent exposition of two of the most fundamental results in topological fixed point theory for set-valued maps: the Browder-Ky Fan fixed point theorem (Theorem 1 of F.E. Browder [Math. Ann. 177, 283-301 (1968; Zbl 0176.45204)] and Theorem 2 of K. Fan [in: Inequalities III, Proc. 3rd Symp., Los Angeles 1969, 103–113 (1972; Zbl 0302.49019)]). These theorems “are central existence results in nonlinear functional analysis as they lie at the foundation of solvability problems in a wide array of areas.”
The subject matter is organized into the following main sections: Ky Fan and Kakutani maps; Continuous selections and approximations; The Browder-Ky Fan and the Kakutani-Ky Fan fixed point theorems and their consequences; Relaxing compactness and related results; Systems of nonlinear inequalities and applications. According to the author, the arguments in this well written article “are kept elementary” and thus may be used as “a first course in topological fixed point theory and its applications.”
For the entire collection see [Zbl 1278.47002].