A ternary diophantine inequality by primes near to squares

Let $c$ be fixed with $1<c<35\mathrm{/}34$. In this paper we prove that for every sufficiently large real number $N$ and a small constant $\epsilon >0$, the diophantine inequality \begin{equation*} |p_1^c+p_2^c+p_3^c-N|<\varepsilon \end{equation*} is solvable in primes $p_1,p_2,p_3$ near to squares.