Existence of continuous solutions and stability for a general iterative functional equation

•We investigate the existence of solutions for a general form iterative equation.•The stability of those solutions is also discussed.•We generalize the work from a linear combination of iterates to nonlinear.•An algorithm is presented to compute the piecewise linear solutions.

In this paper, C0C0 solutions for a general form of iterative functional equation are considered when the coefficient of the first order iterate can be zero, which is called the leading coefficient problem for polynomial-like iterative equations. Moreover, the stability of those C0C0 solutions is also discussed. Finally, as an application, we give an algorithm to compute the piecewise linear solutions of polynomial-like iterative equations.