Iwasawa invariants and class numbers of quadratic fields for the prime $3$

Let $d$ be a square-free integer with $d \equiv 1 \pmod{3}$ and $d > 0$. Put $k^{+}=\Bbb Q(\sqrt{d})$ and $k^{-}=\Bbb Q(\sqrt{-3d})$. For the cyclotomic $\Bbb Z_3$-extension $k^{+}_\infty$ of $k^{+}$, we denote by $k^{+}_n$ the $n$-th layer of $k^{+}_\infty$ over $k^{+}$. We prove that the $3$-Sylow subgroup of the ideal class group of $k^{+}_n$ is trivial for all integers $n \geq 0$ if and only if the class number of $k^{-}$ is not divisible by the prime $3$. This enables us to show that there exist infinitely many real quadratic fields in which $3$ splits and whose Iwasawa $\lambda _3$-invariant vanishes.