Can a DFA have an empty string as input? Given string w such that every odd position in w is a 1

My solution is



The book's solution is 



from my understanding $\epsilon$ should not be an accepted state, however in the book's solution it is. How is it handling this case?
Yes, an empty string is a valid input to a DFA. If this were not the case, DFAs would not be closed under all of the Kleene algebra operations.

In this case, the argument is correct. Think of the problem statement as an implication. If $i$ is an odd position, then $S_i$ is 1.