Fractional index convolution complementarity problems

Convolution complementarity problems have the form: given a kernel function k and a function q , find a function u such that u ( t ) ≥ 0 , ( k ∗ u ) ( t ) + q ( t ) ≥ 0 for (almost) all t , and where ∫ 0 T u ( t ) T [ ( k ∗ u ) ( t ) + q ( t ) ] d t = 0 . A fractional index problem of this kind has k ( t ) ∼ K 0 t α − 1 for t small, with 0 < α < 1 . Such problems are shown to have unique solutions under mild conditions.