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Zbl 1170.34008
Dibl{\'\i}k, Josef; Ruužičková, Miroslava
Inequalities for solutions of singular initial problems for Carathéodory systems via Ważewski's principle. (English)
[J] Nonlinear Anal., Theory Methods Appl. 69, No. 12, A, 4482-4495 (2008). ISSN 0362-546X

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 2000:: *34A12 Initial value problems for ODE

Keywords: Carathéodory system; singular problem; lower and upper problems; inequalities for solutions; positive solutions

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