Polyconvolutions and their applications to Fredholm and Barbashin-type integral equations

Abstract

This article introduces a new polyconvolution, which is studied here in the context of Lebesgue spaces with weights (under different possibilities and conditions). This means that we are looking at a convolution as an integral transform. Various properties of the new polyconvolution are deduced, and some of its relationships with the Fourier-sine and Fourier-cosine transforms become clear. In particular, we would like to emphasize the fact that we have obtained Young’s type theorem and Saitoh’s type inequality for this polyconvolution. Additionally, a Watson-type theorem is also deduced in the present context. Finally, using some of the previously deduced properties, the new polyconvolution is applied to obtain solvability results for classes of integral equations of Fredholm and Barbashin types. Several illustrative examples are provided to demonstrate the validity and applicability of the results.