Mixing and some integro-functional equations

Summary It is proved that the operatorP: L¹ (0, ) L¹(0, ), given byPg(z) = _z/c[g(x)/cx]dx, is completely mixing, i.e.,Pⁿg₁ 0 forg L¹(0, ) with g dx = 0. This implies that, forc (0, 1), each continuous and bounded solution of the equationf(x)=₀^cxf(t)dt/(cx) (x (0, 1]) is constant.