On some coupled quasi-fixed points theorems

Some existence theorems are known for coupled quasi-fixed points of nonlinear mixed monotone operators from a k-fold conical segment [u₀, u₀] into a real Banach space, where k = 1 or 2. In this note, we extend those results to an arbitrary but finite k by a simple but nontrivial iterative mechanism. Interestingly enough, for mixed monotone operators with k ≥ 3 folds, the Lipschitz condition ensuring the uniqueness of their fixed points is as simple as the cases of k ≤ 2.