In this paper we study estimation, continuous dependence and Hyers–Ulam stability for continuous solutions of a second order iterative equation. First we give an estimate for a bound of its continuous solutions. Then we give a Lipschitz estimation with the Lipschitz conditions (i.e., Hölder exponent = 1) to given functions, which implies a continuous dependence of solutions and encourages a further discussion on the Hyers–Ulam stability. We similarly give a Hölder estimation with the Hölder conditions (i.e., Hölder exponent < 1) to given functions, in a weaker sense than the above Lipschitz one, but it does not imply a continuous dependence. This actually suggests a question: Is the continuous dependence of solutions really critical for the Hölder exponent between < 1 and = 1 ?