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

Abstract

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].