“…, u n ) = n k=1 u α k k , where G is given and g is unknown. Unlike those [6,9,11,13,17,18] on compact intervals, our work to (1.2) is concentrated to solving (1.1) on the whole R. Our strategy is to restrict our discussion of (1.2) on R + := (0, +∞) and use an exponential function to reduce in conjugation to the well-known form of polynomial-like iterative equation (1.1) on the whole R. Note that all found results on the polynomial-like iterative equation are given either on a compact interval or near a fixed point, none of which can be applied to our case. In this paper we generally discuss a polynomial-like iterative equation on the whole R and use obtained result to give solutions of equation (1.2) on R + and R − := (−∞, 0).…”