Statkevich V. Essentially infinite-dimensional equations

We investigate differential equations, systems of differential equations and boundary-value problems with essentially infinite-dimensional operators (of the Laplace-Levy type). Abstract Picard theorems are proved thus the parallel between the theory of nonlinear differential equations (and systems) with nonregular and infinite-dimensional elliptic operators and the theory of common nonlinear differential equations is obtained. For linear differential equations (and systems) theorems on the existence and uniqueness of solutions are proved, explicit formulas are given. We propose a linear system of differential equations with operator coefficients (nonregular elliptic operator polynom) and obtain the Cramer's rule analog. For nonlinear boundary-value problems the existence and uniqueness of solutions is proved, for linear boundary-value problems explicit formulas are given.