Abstract: In this paper, some results of pansymmetry and fixed pansy stems theorems are extended to the case of a class of binary relations under the framework of pansystems methodology. It is concretely discussed when a class of binary relations has a common fixedsubset on a finite universal set. Several simple but comprehensive decisive theorems are obtained which can be determined if a class of binary relations has a common fixed subset. As a result. the mum decisive theorem-Markov-Kakutani Theorem-is extended, which is in traditionally fixed point theory and can be determined if a class of mappings has a common fixed point.

Key words: thermopiezoelectric elastodynamics, variational principle, region-wise variational principle