09-09E

We consider nonlinear-mediating-field generalizations of the Wick-Cutkosky model. Using an expansion in the nonlinearity parameter and eliminating the mediating field by means of the covariant Green function we derive a Lagrangian with many-point time-nonlocal interaction terms. In low-order approximations of $\varphi ^3$ theory we obtain the usual two-current interaction and a three-current confining one. The same result is obtained exactly for a version of the dipole model. The Hamiltonization and canonical quantization is performed with time non-locality taken into account approximately. Relativistic two- and three-particle wave equations are derived variationally by using many-particle Fock space trial states. The non-relativistic limits of these equations are obtained and their properties are analyzed and discussed briefly.