Theorem 1 in [Sugiyama, Shohei. On the existence and uniqueness theorems of difference-differential equations. Kōdai Math. Sem. Rep.  12  1960 179--190. [MR0121552](http://www.ams.org/mathscinet-getitem?mr=MR0121552)] (which you can probably get from [here](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.kmj/1138844324)) gives an existence and uniqueness theorem which provides non-linear solutions *on finite intervals*.