Fixed point theory is very useful in nonlinear analysis, differential equations, and differential and random differential inclusions. It is well known that different types of fixed points imply the existence of specific solutions to the respective problems concerning differential equations or inclusions. There are several classifications of fixed points for single-valued mappings. Recall that in 1949, M.K. Fort [19] introduced the notion of essential fixed points. In 1965, F.E. Browder [12], [13] introduced the notions of ejective and repulsive fixed points. In 1965, A.N. Sharkovsky [31] provided another classification of fixed points, but only for continuous mappings of subsets of the Euclidean space $\mathbb{R}^n$.

For more information, see also: [15], [18–22], [3], [25], [27], [31]. Note that for multivalued mappings, these problems were considered only in a few papers (see: [2–8], [14], [23], [24], [32]) — always for admissible multivalued mappings of absolute neighbourhood retracts (ANR-s).

In this paper, ejective, repulsive, and essential fixed points for admissible multivalued mappings of absolute neighbourhood multi retracts (ANMR-s) are studied. Let us remark that the class of MANR-s is much larger than the class of ANR-s (see: [32]). In order to study the above notions, we generalize the fixed point index from the case of ANR-s to the case of ANMR-s. Using the above fixed point index, we are able to prove several new results concerning repulsive, ejective, and essential fixed points of admissible multivalued mappings. Moreover, the random case is mentioned. For possible applications to differential and random differential inclusions, see: [1], [2], [8–11], [16], [25], [26].