Strongly continuous (i.e. having open fibers) multifunctions on compact or paracompact spaces with generalized convex values in

$G$-spaces are considered. The concept of such

$G$-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.