Singular cauchy problems with a large parameter for systems of non-linear ordinary differential equations

Abstract

Systems of non-linear ordinary differential equations are considered in the semi-infinite interval T 0⩽ t<∞. The coefficients of the equations can have infinite upper limits as t→∞. Theorems of the existence and uniqueness of the solutions of such singular Cauchy problems are given, and the continuous dependence of these solutions on the singularly large parameter occurring in the equations is investigated. For problems with a power “degeneracy” with respect to the parameter μ theorems are given on the asymptotic behaviour of the solutions with respect to the parameter, and the asymptotic forms obtained are dual: for fixed t and μ→∞ and for fixed μ and t→∞.