Weighted Sonine conditions and application

TL;DR

This paper extends classical Sonine conditions to include weights, enabling better modeling of inhomogeneous systems in integral and differential equations, and demonstrates their application in proving well-posedness.

Contribution

It introduces weighted Sonine conditions, characterizes their relation to classical conditions, and applies them to reformulate and analyze integral and nonlocal differential equations.

Findings

Weighted Sonine conditions generalize classical kernels.

Weighted conditions do not significantly alter the set of Sonine kernels.

Application to integral and differential equations proves their well-posedness.

Abstract

The Sonine kernel described by the classical Sonine condition of convolution form is an important class of kernels used in integral equations and nonlocal differential equations. This work extends this idea to introduce weighted Sonine conditions where the non-convolutional weight functions accommodate the inhomogeneity in practical applications. We characterize tight relations between classical Sonine condition and its weighted versions, which indicates that the non-degenerate weight functions may not introduce significant changes on the set of Sonine kernels. To demonstrate the application of weighted Sonine conditions, we employ them to derive equivalent but more feasible formulations of weighted integral equations and nonlocal differential equations to prove their well-posedness, and discuss possible application to corresponding partial differential equation models.