Mixing and some integro-functional equations

Summary It is proved that the operatorP: L ¹ (0, ) L ¹(0, ), given byPg(z) = _z/c g(x)/cx]dx, is completely mixing, i.e.,P ⁿ g ₁ 0 forg L ¹(0, ) with g dx = 0. This implies that, forc (0, 1), each continuous and bounded solution of the equationf(x)= ₀ ^cx f(t)dt/(cx) (x (0, 1]) is constant.