Coincidence preservation for a one-parameter family of pairs of Zamfirescu-type multi-valued mappings

Recently, a concept of a pair of multi-valued Zamfirescu type mappings between metric spaces was introduced by the authors. As well, a coincidence existence theorem was proved for such pairs of mappings. It was shown that this theorem is a generalization of the fixed point theorem for a multi-valued Zamfirescu mapping by Kritsana Neammanee and Annop Kaevkhao (2010). In this paper, the main result is the theorem on the preservation of coincidence point existence in some open set, for a parametrized family of pairs of multi-valued Zamfirescu type mappings. It is shown that this result follows from the theorem on the preservation of zero existence, for a family of $(\alpha,\beta)$-search functionals introduced earlier by T. N. Fomenko. In addition the connection of this result with the theorem by Granas and Frigon (1994) on the preservation of fixed point existence, for a contracting family of multi-valued mappings, is considered.