Searching problems above arithmetical transfinite recursion

We investigate some Weihrauch problems between ${\mathsf{A}\mathsf{T}\mathsf{R}}_2$ and ${\mathsf{C}}_{{\omega }^{\omega }}$ . We show that the fixed point theorem for monotone operators on the Cantor space (a weaker version of the Knaster-Tarski theorem) is not Weihrauch reducible to ${\mathsf{A}\mathsf{T}\mathsf{R}}_2$. Furthermore, we introduce the $\omega$-model reflection ${\mathsf{A}\mathsf{T}\mathsf{R}}_2^{\mathrm{r}\mathrm{f}\mathrm{n}}$ of $\mathsf{A}\mathsf{T}\mathsf{R}$ and show that it is an upper bound for problems provable from the axiomatic system ${\mathrm{A}\mathrm{T}\mathrm{R}}_0$ which are of the form $\mathrm{\forall }X(\theta (X)\to \mathrm{\exists }Y\eta (X,Y))$ with arithmetical formulas $\theta ,\eta$. We also show that Weihrauch degrees of relativized least fixed point theorem for monotone operators on the Cantor space forms a linear hierarchy between ${\mathsf{A}\mathsf{T}\mathsf{R}}^{\mathrm{r}\mathrm{f}\mathrm{n}}$ and ${\mathsf{C}}_{{\omega }^{\omega }}$.