In this talk, we establish a new mean value theorem of Bombieri-Vinogradov's type over Piatetski-Shapiro sequence. As an application of the mean value theorem, for $\gamma$ satisfying

\[ 1-\frac{0.03208}{2^{38}+17}<\gamma<1, \]

we prove that there exist infinitely many primes p such that p+2=P_2 with P_2 being Piatetski-Shapiro almost-primes of type $\gamma$, and there exist infinitely many Piatetski-Shapiro primes p of type $\gamma$ such that p+2=P_2. These results generalize the result of Pan and Ding, and constitute an improvement upon a series of previous results of Li-Zhang-Xue, Wang-Cai, Peneva, etc.