Existence of solutions of piecewise differentiable systems of equations
A. NEUMAmR · 1986
Interval arithmetic is an ideal tool for the verification of the existence of a solution x* of a system of equations F(x*) = 0; see e.g. Alefeld [1], Kahan [2], Moore [7], Neumaier [9], Nickel [10], Qi [11], Rump [12]. A number of tests based on Krawczyk type operators were discussed in the literature after Moore's paper [7] appeared; but implicitly, Moore's result is already in Krawczyk [3; Satz 1]. These tests are known to work under the assumption that F satisfies an interval Lipschitz condition. However, tests based on Newton type operators required up to now the assumption that F is continuously differentiable. Hence the latter were not applicable to systems of equations involving the absolute value or the positive part of numbers. The present paper removes this restrictive assumption from the test. The notation used follows Neumaier [8], [9]: liN, Ht, denote the set of real intervals, n-dimensional interval vectors, and n x n interval matrices, and liD:= {x ~ liN Ix __ D} for D ~ N. The interval hull of a bounded subset S ~ N is defined as D Z : = [infZ, sup S]. An interval matrix A ~ ~N is called regular if all/T ~ A are nonsingular; in this case Anb is defined as the interval hull of the solution set of the system of linear interval equations AY = ~(./i e A, b ~ b). In particular ,7t-lb~Anb for all .~A, b~b. Note that A n is not an interval matrix but a sublinear mapping from liN to ]IN; Anb is generally a tighter enclosure for the solution set than the matrix-vector product A - a b with an optimal enclosure A- 1 = D {A- 1 [/i ~ A}. For example, if