Equations Describing Multidimensional Causal Systems

This paper contains an abstract treatment of functional equations which involve non-linear causal operators. We consider equations on linear spaces that are certain extensions of Banach or Hilbert spaces. The causality is defined in the traditional way, but by a “past” we mean a set from a given family of subsets of a generalized “time domain.”An operator A is said to have a resolvent operator Q, if for each u from a certain space, the equation $x = A(x,u)$ possesses a unique solution x ; then $x = Qu$. We prove a theorem giving conditions for the existence of Q, and for continuity and boundedness of Q in a certain sense.Further results are obtained for the case $A(x,u) = Bx + u$, i.e., for an equation of Hammerstein’s type. In particular, it is shown that some monotonicity properties of operators involved guarantee the existence and continuity of the resolvent Q. The theory is illustrated by several concrete examples.