Decider for the family of Turing machines that move infinitly to the right on some input
Given a Turing machine $M$ for which such an input exists, what does $M$'s state graph look like?


   For some input $w$, there is a computation of the form $q_0 \leadsto q (\leadsto q)^\omega$ whose transitions all have movement $R$ (or $N$, depending on the meaning of "only"). Therefore, the state graph has to have an "$R$-path" from $q_0$ to some $q$ which lies on an "$R$-cycle".


Is this also a sufficient criterion? The answers to both questions lead you towards a decision procedure.