We show that generalized eventually narrow sequences on a strongly inaccessible cardinal $\kappa$ are preserved under the Cummings-Shaleh non-linear iterations of the higher Hechler forcing on $\kappa$. Moreover assuming GCH, $\kappa^{ <\kappa}=\kappa$, we show that: (1) if $\kappa$ is strongly unfoldable, $\kappa^+\leq\beta=\hbox{ cf}(\beta)\leq \hbox{ cf}(\delta)\leq\delta\leq\mu$ and $\hbox{ cf}(\mu)>\kappa$,then there is a cardinal preserving generic extension in which $$\mathfrak{ s}(\kappa)=\kappa^+\leq\mathfrak{ b}(\kappa)=\beta\leq\mathfrak{ d}(\kappa)=\delta\leq 2^\kappa=\mu.$$ (2) if $\kappa$ is strongly inaccessible, $\lambda>\kappa^+$, then in the generic extension obtained as the $<\kappa$-support iteration of $\kappa$-Hechler forcing of length $\lambda$ there are no $\kappa$-towers of length $\lambda$.
Categories: math.LO