On P- and S-functions and related classes of n-dimensional nonlinear mappings

Abstract

This paper introduces and analyzes certain classes of mappings on R n which represent nonlinear generalizations of the P- and S-matrices of Fiedler and Pták, and of several closely related types of matrices. As in the case of the corresponding matrices, these nonlinear P- and S-functions arise frequently in applications. Basic properties of the different functions and of their inverses and subfunctions are established, and then a number of theorems are proved about the interrelationships between the various mappings. In particular, it is shown that the well-known monotone mappings, as well as the M-functions and certain of the strictly diagonally dominant mappings recently analyzed by Rheinboldt and Moré, respectively,are special cases of the P-functions. In turn, these P-functions and also the inverse isotone mappings are subclasses of the S-functions. In a final section, a series of characterization theorems for the different functions are presented in terms of conditions on their derivatives.