Assume that the right-hand side f of the differential equation (0.1) does depend on certain derivatives Du of the function u looked for, i. e., a partial differential equation of type (0.3) is to be investigated instead of an ordinary differential equation (0.1). Differential operators D entering into the right-hand side f of (0.3) are not bounded, in general. That is why in a fixed Banach space the initial value problem (0.3), (0.2) cannot be solved by using the method of successive approximations regarded in section 1.5.