Comparison theorems for nonlinear iterative functional inequality (Q1895186)

Consider the functional inequality \((*)\) \(\psi (x) \geq h(x, \psi (f(x))\), \(x \in [a,b)\), and the associated equation \((**)\) \(\varphi (x) = h(x, \varphi (f(x))\), \(x \in [a,b)\), where \(\psi\) and \(\varphi\) are unknown functions. Assume that the function \(f : [a,b) \to [a,b)\) is continuous, strictly increasing and \(a < f(x) < x\) in \((a,b)\); suppose \(h : D \subset \mathbb{R}^2 \to \mathbb{R}\) is continuous, strictly increasing with respect to the second variable, \(D_x \neq \emptyset\) and \(h(f(x), D_{f(x)}) \subset D_x\) for \(x \in [a,b)\); moreover there exists \(d\) such that \(h(a,d) = d\). Let \(\psi\) be a continuous solution of \((*)\) with \(\psi (a) = d\). If there exists a continuous solution \(\overline \varphi\) of \((**)\) such that \(\overline \varphi (x) \leq \psi (x)\), then the sequence \(\{\psi_n\}\) given by \(\psi_0 (x) = \psi (x)\), \(\psi_{n + 1} (x) = h(x, \psi_n (f(x)))\), converges in \([a,b)\) to a solution \(\varphi_0\) of \((**)\) and this function is the maximal solution of \((**)\) with \(\varphi_0 (x) \leq \psi (x)\).