On an equation involving fractional powers with prime numbers of a special type

We consider the equation $[p_1^c]+[p_2^c]+[p_3^c]=N$, where $N$ is a sufficiently large integer, and prove that if $1<c<\frac{17}{16}$, then it has a solution in prime numbers $p_1$, $p_2$, $p_3$ such that each of the numbers $p_1+2$, $p_2+2$, $p_3+2$ has at most $\left [ \frac{95}{17 - 16c} \right ]$ prime factors, counted with the multiplicity.