On differential-operator partial differential equations in locally convex spaces

We consider the differential operator equation of the first order in partial derivatives with respect to vectorvalued analytical vector function of two variables with values in the locally convex space. The relevance of the study is due to the complexity, and sometimes the impossibility of transferring existing methods for the study of differential-operator partial differential equations from normalized spaces to locally convex spaces. The theorem on the existence and uniqueness of the solution of the first-order differential operator equation in partial derivatives is formulated and proved. This statement essentially uses the concepts of order and type of operator proposed and studied by V. P. Gromov. On the basis of the obtained results we get solutions of two specific operator-differential equations.