On solvability of Barbashin's type equation and
initial value problem via the structure of
generalized convolutions

Abstract. The purpose of this article is to study the solvability of Barbashin-type integro-differential equation and initial value problem via the structure of well-known generalized convolutions. We give necessary and sufficient conditions for the unique explicit solution of this class of equations and estimate its boundedness in certain weighted \(L_1\) spaces. The illustrative examples of the obtained results are also considered.

Key words and phrases. Convolution, Hartley transform, Laplace transform, Fourier cosine transform, Barbashin integro-differential equations, initial value problem.