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An equivalent formulation of Sonine condition

Xiangcheng Zheng

Abstract

Sonine kernel is characterized by the Sonine condition (denoted by SC) and is an important class of kernels in nonlocal differential equations and integral equations. This work proposes a SC with a more general form (denoted by gSC), which is more convenient than SC to accommodate complex kernels and equations. A typical kernel is given, and the first-kind Volterra integral equation under gSC is accordingly transformed and then analyzed. Based on these results, it is finally proved that the gSC is indeed equivalent to the original SC, which indicates that the Sonine kernel may be essentially characterized by the behavior of its convolution with the associated kernel at the starting point.

An equivalent formulation of Sonine condition

Abstract

Sonine kernel is characterized by the Sonine condition (denoted by SC) and is an important class of kernels in nonlocal differential equations and integral equations. This work proposes a SC with a more general form (denoted by gSC), which is more convenient than SC to accommodate complex kernels and equations. A typical kernel is given, and the first-kind Volterra integral equation under gSC is accordingly transformed and then analyzed. Based on these results, it is finally proved that the gSC is indeed equivalent to the original SC, which indicates that the Sonine kernel may be essentially characterized by the behavior of its convolution with the associated kernel at the starting point.
Paper Structure (3 sections, 5 theorems, 28 equations)

This paper contains 3 sections, 5 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. A more general form
  3. Equivalence via integral equation

Key Result

Theorem 2.1

For $k(t)$ defined in (kt), there exists an associated kernel satisfying the gSC (mh2). Here $\Gamma(\cdot)$ denotes the Gamma function.

Theorems & Definitions (9)

  • Theorem 2.1
  • proof
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • Remark 3.4
  • Theorem 3.5