On the theory of generalized ballistical problem for nonlinear abstract differential equations

Abstract

In the present work, we have obtained principally new results for solvability and non-solvability of one problem for second order abstract differential equations with the main linear part. The method, based on solving corresponding Dirichlet problem, lies on the basis of present investigations. In section 1, we present the formulation of the basic problem and reduce the short review of same investigations related to given problems. In section 2, we show the roll of the corresponding subsidiary boundary value problem for linear equation in the study of the basic problem. In section 3, we present the theorems of existence and nonexistence of the solution of the ballistical problem for nonlinear abstract, differential, equations with the main linear part, in the case when the initial and the latter points of the desired trajectory are not coinciding. Also we can prove the theorems of solvability and non-solvability of considering problem in the case, when the initial and the latter points of desired trajectory are coinciding.