Some Theorems on Properties of DC Equations of Nonlinear Networks

For each positive integer n let 5 n denote that collection of mappings of the real n-dimensional Euclidean space J5T onto itself, defined by: F t if and only if there exist, for i = 1, · · · , n, strictly monotone increasing functions /, mapping E l onto E l such that, for each x = , *,)' * En, F(x) = (/,(»,), · · · , 1M)'. The main purpose of this paper is to report on some results concerning l 2 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1969 properties of the equation F(x) + Ax = B, (1) where A is an n X n matrix of real numbers, F maps En into En, and B e En. In particular, a condition to be satisfied by A is given which is both necessary and sufficient to guarantee that for each F e and each B e FT there exists a unique solution of equation (1). We also study the problem of obtaining bounds on the solution of equation (1). These bounds show that (if F t £Fn and our condition on A is satisfied) the solution depends continuously on B. The bounds are often of use in computing the solution by standard iteration methods such as the Newton-Raphson method. By appealing to a theorem of R. S. Palais it is shown that the bounds can also be used to obtain a theorem essentially the same as, but somewhat weaker than, our principal result. Several results can be found in the literature which specify sufficient conditions for the existence of a unique solution of equation (1). For example, if A is positive semidefinite then a special case of a theorem of Ref. 1 guarantees the existence of a unique solution of equation (1) for all those F z 5n which have the property that the slope of each /,· is bounded from above and below by positive constants, and for all B e En.