Journal article
Differentiating complementarity problems and fractional index convolution complementarity problems
Houston journal of mathematics, Vol.33(1), pp.301-322
01/01/2007
Abstract
Two functions a and b axe 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 for each t 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 (a(t), b'(t)) = 0 for (almost) all t". This is proved, for example, where a is in L-p and b' is in L-q, 1/p + 1/q = 1, where a is continuous and b has bounded variation, and where a and V lie in dual Sobolev spaces. Consequences for more than one derivative are also shown: (a'(t), b'(t)) <= 0 and (a(t), b ''(t)) >= 0 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 u(t) >= 0, (k * u) (t) + q(t) >= 0, u(t)(T) [(k * u) (t) + q(t)] = 0 in R-n with k(t) similar to k(0) t(alpha), 0 < alpha < 1 as t down arrow 0 and k(0) positive definite. Such problems arise in connection with the impact of a viscoelastic rod.
Details
- Title: Subtitle
- Differentiating complementarity problems and fractional index convolution complementarity problems
- Creators
- David E Stewart
- Resource Type
- Journal article
- Publication Details
- Houston journal of mathematics, Vol.33(1), pp.301-322
- Publisher
- UNIV HOUSTON
- ISSN
- 0362-1588
- Number of pages
- 22
- Language
- English
- Date published
- 01/01/2007
- Academic Unit
- Mathematics
- Record Identifier
- 9984240863402771
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