Fixed Points of Multivalued Mappings Useful in the Theory of Differential and Random Differential Inclusions

Fixed point theory is very useful in nonlinear analysis, diferential equations, differential and random differen- 
tial inclusions. It is well known that different types of fixed points implies the existence of specific solutions 
of the respective problem 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 continous mappings 
of subsets of the Euclidean space 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 as remark that the class of MANR-s is much larger 
as 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 onto the case of ANMR-s. Next 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 
di?erential inclusions see: [1], [2], [8]-[11], [16], [25], [26].