Nonlinear functional equations and their Baire category properties (Q1086439)

This paper is devoted to the study of the functional equation in a single variable (*) \(\phi (x)=h(x,\phi [f(x)])\), under specific assumptions on the function f. Denote by X a real interval and by \((Y,\|\|)\) a finite-dimensional Banach space. Let f be a function mapping X continuously into X and having at least one fixed point. For a positive integer p, put \(F_ p=\{x\in X\); \(f^ p(x)=x\}\) and note that, for every \(p\in N\), \(F_ p\neq \emptyset\) and \(f(F_ p)=F_ p\). For a fixed continuous function \(\phi_ 0:F_ p\to Y\), denote by \({\mathcal H}_ p(\phi_ 0)\) the set of all functions h mapping \(X\times Y\) continuously into Y in such manner that \(h(x,\phi_ 0[f(x)])=\phi_ 0(x)\), \(x\in F_ p\). The set \({\mathcal H}_ p(\phi_ 0)\), endowed with the metric of uniform convergence on all compact subsets of the space \(X\times Y\) is a complete metric space. Let \(R_ p(\phi_ 0)\) be the set of all elements \(h\in {\mathcal H}_ p(\phi_ 0)\) such that there exits an open set \(U\subset X\) and a continuous function \(\phi\) :U\(\mapsto Y\) satisfying: \(F_ p\subset U\), f(U)\(\subset U\), \(\phi /F_ p=\phi_ 0\), \(\phi (x)=h(x,\phi [f(x)])\), \(x\in U\). Such a function \(\phi\) is called a (local) continuous solution of (*). Assuming that the set \({\mathcal R}_ p(\phi_ 0)\) is of the second category in \({\mathcal H}_ p(\phi_ 0)\) and that \(\phi_ 0:F_ p\mapsto Y\) is a continuous map, the author studies the structure of the set \(F_ p\) and the behaviour of the function f in a neighbourhood of \(F_ p\). In this way, it follows that \(F_ p\) is an interval and, moreover, \(f^ p(x)>x\) for \(x<\inf F_ p\), \(x\in X\), and \(f^ p(x)<x\) for \(x>\sup F_ p\), \(x\in X\) (Theorem 1).