Construction of continuous solutions and stability for the polynomial-like iterative equation

Most of known results such as existence, uniqueness and stability for polynomial-like iterative equations were given under the assumption that the coefficient of the first order iteration term does not vanish. The existence with a non-zero leading coefficient was therefore raised as an open problem. It was positively answered for local C1 solutions later. In this paper this problem is answered further by constructing C0 solutions. Moreover, we discuss the stability of those C0 solutions, which consequently implies a result of the stability for iterative roots.