Monotone continuous solutions of an equation in linear combination of alternative iterates

Iteration is one of the most important topics in computer science and attentions are paid to functional equations involving iterates, one of which is the linear combination of alternative iterates. Recently existence, uniqueness and dependence for increasing Lipschitzian solutions were obtained for linear combination of alternative iterates on $ [0, 1] $ [0,1] and $ \mathbb {R} $ R in the case that all given functions are strictly increasing. In this paper we work in various cases of monotonicity (increasing and decreasing) among given functions and the unknown function.