The authors introduce the families of generalized quasipseudodistances in quasigauge spaces and define three kinds of dissipative set-valued dynamic systems with these families of generalized quasi-pseudodistances and with some families of not necessarily lower semicontinuous entropies. Assuming that quasigauge spaces are left

$K$ sequentially complete (but not necessarily Hausdorff), they prove that for each starting point each dynamic process or generalized sequence of iterations of these dissipative set-valued dynamic systems left converges. They also show that if an iterate of these dissipative set-valued dynamic systems is left quasiclosed, then these limit points are periodic points. Examples illustrating ideas, methods, definitions, and results are constructed.