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During the first year of the project a) Sufficient conditions ensuring the covering properties of the composition of two set-valued mappings were obtained in terms of covering mappings on a collection of balls. The sufficiency of these conditions was illustrated with several examples. Moreover, covering criteria for a certain type of smooth mappings were obtained. b) The theorem on the Lipschitz perturbation of conditionally covering set-valued mappings was obtained. This result gives sufficient conditions for solvability of inclusions of a certain type. This result was applied to the Cauchy problem for differential inclusions. The local solvability conditions for the problem were obtained. c) Both local and global types of covering for mappings in generalized metric spaces were investigated. Coincidence points theorems for single-valued and set-valued mappings acting in generalized metric spaces were proved. The stability of coincidence points in the considered problems is proved. d) This task concerns control systems with mixed constraints and control constraints, that is, constraints that are defined by functions that depend both in the control and state variables. The Solvability conditions for control systems with mixed constraints and control constraints were derived in two cases:
e) Certain properties of some functional generalized metric spaces were investigated yielding results on coincidence points of mappings acting in generalized metric space which were applied to obtain solvability conditions for functional equations in the space of continuous functions. |