A nonuniqueness criterion for a singular system of two ordinary differential equations

This paper deals with the singular initial-value problem g1(x)y1′=α(y2),g2(x)y2′=β(y1),y1(0+)=y2(0+)=0.A nonuniqueness result is derived by a generalization of Ważewski's retract technique. In contrast to known nonuniqueness criteria which use comparison theory combined with Lyapunov functions, the presented method does not require the construction of auxiliary functions for concrete problems. The essential monotonicity assumptions can be easily verified. In addition, the type of nonuniqueness can be specified. The initial point of the given system is a special branching point. Some necessary existence conditions are analysed as well.