On a binary Diophantine inequality involving primes of a special type

Abstract: Let $1<c<\frac{1787}{1502}$ and $N$ be a sufficiently large real number. In this paper, it is proved that for any arbitrarily large number $E\>0$ and for almost all real $R \in (N,2N]$, the Diophantine inequality $$|p_{1}^{c}+p_{2}^{c}-R|<(log N)^{-E}$$ is solvable in prime variables $p_1,p_2$ such that, each of the numbers $p_{1}+2,p_{2}+2$ has at most $[\frac{79606}{35740-30040c}]$ prime factors, counted with multiplicity. Moreover, we prove that the Diophantine inequality $$|p_{1}^{c}+p_{2}^{c}+p_{3}^{c}+p_{4}^{c}-N|<(log N)^{-E}$$ is solvable in prime variables $p_1,p_2,p_3,p_4$ such that, each of the numbers $p_{i}+2(i=1,2,3,4)$ has at most $[\frac{93801402}{35740000-30040000c}]$ prime factors, counted with multiplicity.