Problem with integral condition for evolution equation

We propose a method of solving the problem with nonhomogeneous integral condition for homogeneous evolution equation with abstract operator in a linear space $H$. For right-hand side of the integral condition which belongs to the special subspace $Lsubseteq H$, in which the vectors are represented using Stieltjes integrals over a certain measure, the solution of the problem is represented in the form of Stieltjes integral over the same measure.

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