Stability and differentiability with respect to small parameter of mixed value problem for a nonlinear partial differential equation of eighth order

Paper deals with continuous dependence and differentiability with respect to small parameter of generalized solution of mixed value problem for nonlinear partial differential equation of the eighth order, left-hand side of which is superposition of two operators of fourth order. By the aid of Fourier method the mixed problem is reduced to the study of countable system of nonlinear Volterra integral equations of the second kind with small parameter. We proved the continuous dependence of generalized solution of considered mixed value problem with respect to small positive parameter. Also we proved the differentiability of the solution with respect to small positive parameter. While proofing the existence of derivative of countable system of nonlinear Volterra integral equations of the second kind the method of successive approximations is used. The results obtained in this paper play important role in construction of the asymptotic expansions with respect to small parameter of solution of mixed value problem for considered nonlinear partial differential equation of the eighth order.