Resumen de Differentiating complementarity problems and fractional index convolution complementarity problems
David E. Stewart
Two functions a and b are said to be complementary if a has values in a closed convex cone K (such as the non-negative orthant) while b has values in its dual cone K* (which can also be the non-negative orthant), yet the inner product of a(t) and b(t) is zero for (almost) all t. In this paper we consider implications of the form: ``If a and b are complementary functions, then the inner product of a(t) with the derivative b'(t) is zero for (almost) all t''. This is proved, for example, where a is in Lp and b' is in Lq, 1/p+1/q=1, where a is continuous and b has bounded variation, and where a and b' lie in dual Sobolev spaces. Consequences for more than one derivative are also shown: the inner product of a'(t) with b'(t) being non-negative and the inner product of a(t) with b''(t) non-positive for almost all t provided a and b satisfy mild regularity conditions. These implications can be used to prove conservation of energy in impact systems as well as existence and regularity results for dynamic complementarity problems of various kinds. In particular, it is shown that solutions exist for a convolution complementarity problem where b = k*a + q in Rn with k(t) ~ k0 t c, 0 < c < 1, for small t and k0 positive definite. Such problems arise in connection with the impact of a viscoelastic rod.
