Theorem 1 in [Sugiyama, Shohei. On the existence and uniqueness theorems of difference-differential equations. Kōdai Math. Sem. Rep.  12  1960 179--190. [MR0121552][1]] (which you can probably get from [here][2]) gives an existence and uniqueness theorem which provides non-linear solutions *on finite intervals*.

 [1]: https://mathscinet.ams.org/mathscinet-getitem?mr=MR0121552
 [2]: https://projecteuclid.org/journals/kodai-mathematical-journal/volume-12/issue-4/On-the-existence-and-uniqueness-theorems-of-difference-differential-equations/10.2996/kmj/1138844324.full