The solvability of the theory $ T_f = \ Th (\ omega, 0, 1, <, +, f (x), F (x)) $

In the paper we generalize the result which was established in our previous article. We demonstrate the theory $T_f=\ Th(\omega, 0, 1, <, +, f(x), F(x))$ is decidable and every formula is equivalent to an existential one, if $f$ is an effectively addition-connected function, and $F$ is the effectively periodic hyperfunction for $f$.