Perturbations of vectorial coverings and systems of equations in metric spaces

E. R. Avakov, A. V. Arutyunov, S. E. Zhukovskiĭ, and E. S. Zhukovskiĭ studied the problem of Lipschitz perturbations of conditional coverings of metric spaces. Here we propose some extension of the concept of conditional covering to vector-valued mappings; i.e., the mappings acting in products of metric spaces. The idea is that, to describe a mapping, we replace the covering constant by the matrix of covering coefficients of the components of the vector-valued mapping with respect to the corresponding arguments. We obtain a statement on the preservation of the property of conditional and unconditional vectorial coverings under Lipschitz perturbations; the main assumption is that the spectral radius of the product of the covering matrix and the Lipschitz matrix is less than one. In the scalar case this assumption is equivalent to the traditional requirement that the covering constant be greater than the Lipschitz constant. The statement can be used to study various simultaneous equations. As applications we consider: some statements on the solvability of simultaneous operator equations of a particular form arising in the problems on $n$-fold coincidence points and $n$-fold fixed points; as well as some conditions for the existence of periodic solutions to a concrete implicit difference equation.