Constructive solutions for systems of iterative functional equations
This comprehensive paper studies the systems of iterative functional equations. Generally, they are of the following form. Let \(X,Y\) be nonempty sets and \(p\geq2\) an integer. For \(X_i\subset X\), \(\bigcup_{i=0}^{p-1}X_i=X\), \(W_i\subset X\) and given functions \(f_i: X_i\to W_i\) and \(F_i: X_i\times Y\to Y\) (\(i=0,1,\dots,p-1\)) we consider the system of functional equations \[ \varphi(f_i(x))=F_i(x,\varphi(x)),\quad x\in X_i, \,i=0,1,\dots,p-1,\tag{\(*\)} \] with an unknown function \(\varphi: X\to Y\). The above system is considered with some ``compatibility conditions''. The first part of the paper is devoted to providing some necessary and sufficient conditions for the existence and uniqueness of the solutions of \((*)\), in the general case, as well as for the the existence of bounded and continuous solutions (under suitable additional assumptions upon \(X\) and \(Y\)). Later, the authors consider some particular cases, with respect to the type of the functions \(f_i\) and \(F_i\). Finally, some applications are given including some earlier results covered by the general results presented in the considered paper.
scientific article; zbMATH DE number 6710483
scientific article