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Let $P_{r}$ denote an integer with at most $r$ prime factors counted withmultiplicity. In this paper we prove that for any $0<\lambda<\frac{1}{4}$, theinequality $\{\sqrt{p}\}<p^{-\lambda}$ has infinitely many solutions in primes$p$ such that $p+2=P_r$, where $r=\lfloor \frac{8}{1-4\lambda} \rfloor$. Thisgeneralizes the previous result of Cai.

A remark on the distribution of $\sqrt{p}$ modulo one involving primes of special type