Pseudo-Brayton cycle is defined as an ideal Brayton cycle admitting the difference between heat capacities of working fluid during heating and cooling processes. The endo-pseudo-Brayton cycle which is a pseudo-Brayton cycle with heat transfer processes is analyzed with the consideration of maximum power conditions and the results were compared with those of the endo-Carnot cycle and endo-Brayton cycle. As results, the maximum power is an extremum with respect to the cycle temperature and the flow heat capacities of heating and cooling processes. At the maximum power condition, the heat capacity of the cold side is smaller than that of heat sink flow. And the heat capacity of endo-Brayton cycle is always between those of heat source and sink flows and those of the working fluids of pseudo-Brayton cycle. There is another optimization problem to decide the distribution of heat transfer capacity to the hot and cold side heat exchangers. The ratios of the capacies of the endo-Brayton and the endo-pseudo-Braton cycles at the maximum power condition are just unity. With the same heat source and sink flows and with the same total heat transfer caqpacities, the maximum power output of the Carnot cycle is the least as expected, but the differences among them were small if the heat transfer capacity is not so large. The thermal efficiencies of the endo-Brayton and endo-Carnot cycle were proved to be 1-.root.(T$_{7}$/T$_{1}$) but it is not applicable to the pseudo-Brayton case, instead it depends on comparative sizes of heat capacities of the heat source and sink flow.w.