Remark on invariant curves for planar mappings

Bounded solutions for a functional equation were investigated by Ng and Zhang [C. T. Ng, W. Zhang, Invariant curves for planar mapping, J. Difference Eqn. Appl. 3 (1997) 147-168]. In this paper, we consider its solutions which may be unbounded. We give not only existence and uniqueness of its solutions but also the convex and concave solutions. Moreover, some examples are given.