Smooth solutions of a class of iterative functional equations (Q6590663)

This paper is concerned with the functional equation\N\[\N\phi ^{2}(x)=h(\phi (f(x)))+g(x) , \tag{1}\N\]\Nwhere \(h,f\) and \(g\) are given functions and \(\phi \) is the unknown function.\NAn example is\N\[\N\phi ^{2}(x)=\sin (\phi (e^{x}+5x))+4\phi (e^{x}+5x)+\cos x.\N\]\N\NThe main result is the following. Assume that \(h,f,g\) are real continuously differentiable functions on \(\mathbb{R}\) such that:\N\[\N\inf_{x\in \mathbb{R}}\left\vert h^{\prime }(x)\right\vert \geq K,\text{ }\N\inf_{x\in \mathbb{R}}\left\vert f^{\prime }(x)\right\vert \geq \alpha ,\N\]\N\[\N\sup_{x\in \mathbb{R}}\left\vert g(x)\right\vert <+\infty ,\text{ }\N\sup_{x\in \mathbb{R}}\left\vert g^{\prime }(x)\right\vert \leq \beta ,\N\]\Nwhere \(K>1,\alpha >0,\beta >0\) and\N\[\N\begin{array}{cc}\N\beta <\frac{1}{4}\alpha ^{2}K^{2} & \text{if }\alpha <2(1-\frac{1}{K}), \\\N\beta <(K-1)(\alpha K-K+1) & \text{if }\alpha \geq 2(1-\frac{1}{K})\N\end{array}.\N\]\NThen Equation (1) has a continuously differentiable solution on \(\mathbb{R}\) with bounded derived function.