Solvability of Runge-Kutta and block-BVMs systems applied to scalar ODEs
by
Felice Iavernaro
Dipartimento di Matematica, Università di Bari
Coauthors: Giovanni Di Lena (Università di Bari)

A square, real matrix A=(\alpha_ij) in R^{n ×n} is said to be a P₀ matrix if all its principal minors are nonnegative. A characterization of P₀ matrices is introduced and used to derive simple necessary and sufficient conditions for the unique solvability of a class of nonlinear systems of equations depending on a parameter. As application, we consider the problem of existence and uniqueness of the solutions of systems arising from one-step implicit schemes such as Runge-Kutta methods or block Boundary Value Methods, applied to scalar Initial Value Problems

ì í î y'(t)   =  f(t, y),          t in [t0, tf], y(t0)   =   y0,

where f satisfies a one-sided Lipschitz condition with one-sided Lipschitz constant \mu in R

f(t, x) - f(t, y) x - y <= \mu              for allx, y in R,   x =/= y.

In the vector case this study has been conducted by a number of authors and related to the property of Lyapunov diagonally semi-stability of suitable matrices. Since semi-stable matrices form a proper subset of the wider class of P₀ matrices, we realize that weaker restrictions on the product h \mu (h stands for the stepsize of integration) occur in this case.

Date received: February 8, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caen-11.