On Chen's theorem, Goldbach's conjecture and almost prime twins III

Let $N$ denote a sufficiently large even integer and $x$ denote a sufficiently large integer, we define $D_{1,r}\left(N, c, \theta \right)$ as the number of primes $p$ such that $N - p$ has at most $r$ prime factors and $p \in \left[c N, c N+N^{\theta}\right]$. In this paper, we mention an important 1979 result of Lou and Yao, and use this result to give positive lower bounds for $D_{1,3}\left(N, \frac{1}{2}, 0.872\right)$ and $D_{1,3}\left(N, 0, 0.817\right)$, improving on the author's previous results. Similar methods can be used to give a simplified proof for the positive lower bounds for $D_{1,2}\left(N, \frac{1}{2}, 0.97\right)$ and $D_{1,2}\left(N, 0, 0.9409\right)$.