Solutions of functional equations having bounded variation (Q5943221)

Let \(T\subset\mathbb{R}^s\), \(Z\subset\mathbb{R}^m\) be open sets, \(D\) an open subset of \(T\times T\), \(f:T\to Z\), \(g_i:D\to T\) analytic functions for \(1 \leq i\leq n\), \(h:D\times Z^n\to Z\) an analytic function and suppose that for each \(t\in T\) there exists a \(y\) for which \((t,y)\in D\) and \({\partial g_i \over \partial y}(t,y)\) has rank \(s\) for \(1\leq i\leq n\). The main result of the paper is the following. Suppose \(X_i\subset\mathbb{R}^{r_i}\), \(1\leq i\leq n\), \(T,Y\) and \(Z\) be open subsets of Euclidean spaces, \(X_i\) open in \(\mathbb{R}^{r_i}\), \(1\leq i\leq n\), and \(D\) be an open subset of \(T\times Y\). Consider the function \(f\) on \(T\), \(f_0\) continuous on \(Y\), \(f_i\) continuous and have locally bounded variation on \(X_i\) for \(1\leq i\leq n\), \(h\) continuously differentiable on \(Z\) and \(g_i:D\to X_i\) continuously differentiable on \(D\) for \(1\leq i\leq n\). If for each \(t\in T\) there exists an \(y\) such that \((t,y)\in D\) and the rank of \({ \partial g_i\over\partial y}(t,y)\) is \(r_i\) for \(1\leq i\leq n\) and for each \((t, y)\in D\) we have \((t,y,f_0(y), f_1(g_1(t,y)),\dots, f_n(g_n(t,y)))\in Z\) then the function \(f(t)=h(t,y, f_0(y), f_1 (g_1(t,y)),\dots, f_n(g_n(t,y)))\) is a locally Lipschitz function. As a corollary, if \(f\) has locally bounded variation then \(f\) is infinitely many times differentiable.