Problem for an Inhomogeneous Second-Order Evolutionary Equation with Homogeneous Integral Conditions

Abstract

We propose a method for the solution of the problem with homogeneous integral conditions for an inhomogeneous evolutionary equation with abstract operator in a Banach space H . The right-hand side of the evolutionary equation that belongs, for the fixed time variable, to a special subspace N ⊆ H can be represented as a Stieltjes integral with respect to a certain measure. The solution of this problem is also represented as a Stieltjes integral with respect to the same measure. We present examples of application of the method to the solution of the problem with integral conditions for the second-order partial differential equation in the time variable and, in general, an infinite-order partial differential equation in the space variable.