On the solvability in the class of distributions of differential equations with derivatives of functionals in Banach spaces

The paper considers the initial value problem for a differential equation with the derivatives of the functionals in Banach spaces. The operator of the elder derivative has the structure of projector, i.e. its core is infinite-dimensional. The solution is constructed in the space of generalized functions with the support bounded on the left in the form of convolution of the fundamental solution of the differential operator with the right-hand side of the equation, which includes a free function and some initial conditions of the initial problem. The process of construction of the fundamental solution is realized with the aid of a fundamental operator function for a specially constructed matrix differential operator with an irreversible (generally speaking) matrix in the derivative, i.e. with the operator of finite index. Analysis of the generalized solution constructed by this technique allows one to obtain the sufficient conditions of solvability for our initial-value problem in the classes of finite smoothness functions, and also propose constructive formulas needed to restore such a solution. An illustrative example is given.