Unramified Iwasawa module of Z₂-extension of certain quadratic fields with a bounded quotient

We consider an infinite family of real quadratic fields k where the discriminant has three distinct odd prime factors, and the prime 2 splits. We show that the unramified Iwasawa module X (k_) associated with the Z₂-extension of k has a bounded quotient. Thus, we also verify Greenberg's conjecture on the vanishing of Iwasawa invariants for such fields and obtain a finer structure for X (k_).