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Inequalities for solutions of singular initial problems for Carathéodory systems via Ważewski’s principle. (English) Zbl 1170.34008
Summary: We give sufficient conditions for solvability of a singular initial problem formulated for Carathéodory systems of ordinary differential equations. The existence of solutions is proved by the supposition that corresponding auxiliary lower and upper singular problems have solutions. The proof technique uses a notion of a regular polyfacial subset which is developed for Carathéodory systems of ordinary differential equations and a modification of the topological method for such systems given by Palamides, Sficas and Staikos. An application concerning the existence of positive solutions for a special class of singular problems is given as well.
MSC:
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions