The no-Ponzi condition is a technical condition that needs to be imposed to deal with the case of a time-varying value like capital
k diverging at infinity. Even if penalised by an exponential discount, if
k increases too fast, its integral will have a non-zero value/undefined value in the limit.
In this particular case, I cannot be sure what it is being asked of you. Technically, the budget constraint you wrote implies something about the behaviour of the functions
c(t) at infinity. (You cannot write the integral if they diverge faster than the exponential discounting can handle!) But, most likely, the answer that you need is that there is no no-Ponzi condition in there, and it needs to be factored in later.
Look in the paper/textbook for its correct statement.
Also, your definition of
R(t) is wrong.