Abstract. We shall present current results concerning Browder--Gupta-type theorems, Banach principle for multivalued mappings and the inverse limit method including its applications to ordinary differential equations and differential inclusions. Consequently, Aronszajn's-type topological characterization of the set of solutions for differential equations and inclusions is considered. Note that some new results of the above mentioned type will be discussed.
AMSclassification. 59B99, 54C60, 47H10, 54H25, 55M20, 34A60, 46A04, 55M25
Keywords. Fixed points, multivalued maps, inverse systems, acyclicity, topological structure, limit map, topological degree, admissible maps, differential inclusions, Fr\'echet spaces, Cauchy problem for ODE's.