Strongly continuous (i.e. having open fibers) multifunctions on compact or paracompact spaces with generalized convex values in
-spaces are considered. The concept of such
-spaces is the author’s very general nice extension of the generalized convexity in the spirit of R. Bielawski
[J. Math. Anal. Appl. 127, No. 4, 155-171 (1987; Zbl 0638.52002
)]. The existence of global or local continuous selections, fixed points and equilibria is investigated in the above framework. Large and exhaustive comparison with the existing results by Ben-El-Mechaiekh, Horváth, Kim, Yannelis and Prabhakar, Pasicki and many others is enclosed.