Fixed Points of Multivalued Mappings Useful in the Theory of Differential and Random Differential Inclusions
Keywords:
fixed point index, essential ejective and repulsive fixed points, multivalued mappings, compact absorbing contractions, absolute neighbourhood multi retracts differential inclusions, random differential inclusionsAbstract
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 solutionsof the respective problem concerning differential equations or inclusions. There are several classifications offixed points for single valued mappings. Recall that in 1949 M.K. Fort [19] introduced the notion of essentialfixed points. In 1965 F.E. Browder [12], [13] introduced the notions of ejective and repulsive fixed points. In1965 A.N. Sharkovsky [31] provided another classification of fixed points but only for continous mappingsof 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 absoluteneighbourhood multi retracts (ANMR-s) are studied. Let as remark that the class of MANR-s is much largeras the class of ANR-s (see: [32]). In order to study the above notions we generalize the fixed point indexfrom the case of ANR-s onto the case of ANMR-s. Next using the above fixed point index we are able toprove several new results concerning repulsive ejective and essential fixed points of admissible multivaluedmappings. Moreover, the random case is mentioned. For possible applications to differential and randomdi?erential inclusions see: [1], [2], [8]-[11], [16], [25], [26].
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