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A nonuniqueness criterion for a singular system of two ordinary differential equations. (English) Zbl 1111.34007

The authors consider the singular initial value problem

g 1 (x)y 1 ' =α(y 2 ),g 2 (x)y 2 ' =β(y 1 ),y 1 (0 + )=y 2 (0 + )=0·

Some necessary conditions for solvability of the above problem are derived. Moreover (as a main result), the authors prove that under certain conditions, if there exists a solution to the above problem, then there exist infinitely many solutions of this problem. To prove it the authors apply an extension of the Wazewski retract technique. The results are illustrated by some examples.

34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions