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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 thateach 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.

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