Solutions of linear iterative functional equations behaving like the forcing term in the equation (Q1592819)

The author considers the functional equation \[ \varphi \circ f=g\cdot \varphi+h \tag{*} \] where \(\varphi\) is the unknown function, and studies continuous solutions which are asymptotically comparable at the origin with the function \(h\). The general assumptions are the following. 1) \(I\) is an interval with \(0\) as left-end point (belonging or not to \(I\)), \(f:I \to I\) is continuous, strictly increasing and \(0<f(x)<x\), for \(x\in I\), \(x>0\); 2) \(h,g:I \to \mathbb R\) are continuous, \(h(0)=0\), \(h(x)\) and \(g(x)\) are different from zero for \(x\in I\), \(x>0\). Theorem: If the previous assumptions hold, the limit \[ q:=\lim_{x \to 0^+} \frac{h(f(x))}{h(x)} \] exists and \(0<q<|g(0)|\), then equation (*) has a unique continuous solution \(\varphi\) such that the limit \[ L_{\varphi}:=\lim_{x \to 0^+} \frac{\varphi(x)}{h(x)} \] exists and \(L_{\varphi}=(q-g(0))^{-1}\). The previous theorem is applied to determine a unique solution to the iterative functional inequality \[ \varphi(f^2(x))\leq (p(x)+q(f(x)))\psi(f(x))+p(x)q(x)\psi(x) . \]