Cyclic classes and an ergodic theorem in dynamic fuzzy systems (Q1292080)

This study is concerned with a dynamic fuzzy system governed by the equation \[ S_{n+1}(y)=\sup_{x\in E}\biggl[ \min\bigl( S_n(x), Q(x,y) \bigr)\biggr],\;y\in E \] where a state space \(E\) is a complete metric space and \(Q\) is a normal semi-continuous fuzzy relation defined in \(E\times E\). \(\{S_n\}\) is a sequence of fuzzy states. The paper provides a detailed discussion of the limiting behavior of fuzzy states of the system and analyzes the cyclic behavior of the system. Finally, an ergodic theorem is formulated. A brief illustrative example is provided.