A weak form of Greenberg's conjecture for the cyclotomic Z₂-extension of real quadratic fields

In this paper, we show that a weak form of Greenberg's conjecture for the cyclotomic Z₂-extension of Q(√p) is true, where p is an odd prime number satisfying certain arithmetic conditions. As an application, we obtain an upper bound of the Iwasawa λ-invariant of the cyclotomic Z₂-extension of Q(√p) under some additional conditions on Q(√p).