Cordial Volterra Integral Equations with Vanishing Delays*

Hongjiu Wang,Melusi Khumalo,Zhanwen Yang

doi:10.4236/jamp.2017.52026

Abstract

Cordial Volterra integral equations (CVIEs) from some applications models associated with a noncompact cordial Volterra integral operator are discussed in the recent years. A lot of real problems are effected by a delayed history information. In this paper we investigate some properties of cordial Volterra integral operators influenced by a vanishing delay. It is shown that to replicate all eigenfunctions , or , the vanishing delay must be a proportional delay. For such a linear delay, the spectrum, eigenvalues and eigenfunctions of the operators and the existence, uniqueness and solution spaces of solutions are presented. For a nonlinear vanishing delay, we show a necessary and sufficient condition such that the operator is compact, which also yields the existence and uniqueness of solutions to CVIEs with the vanishing delay.

Highlights

A kind of Volterra integral equations with weakly singular kernels arisen in 1975 [1] from some heat condition problems with mixed-type boundary conditions is transformed by Watson transforms [2] and the convolution theorem [3]
Cordial Volterra integral equations (CVIEs) from some applications models associated with a noncompact cordial Volterra integral operator are discussed in the recent years
In this paper we investigate some properties of cordial Volterra integral operators influenced by a vanishing delay

Summary

Introduction

A kind of Volterra integral equations with weakly singular kernels arisen in 1975 [1] from some heat condition problems with mixed-type boundary conditions is transformed by Watson transforms [2] and the convolution theorem [3]. Besides the existence and uniqueness of solutions to (2), it is more interesting how the eigenvalues and eigenfunctions of the operators are influenced by vanishing delays. For such a delay, we describe the spectrum, eigenvalues and eigenfunctions of the operator θ ,φ. Based on these discussions, we present the existence, uniqueness and the construction.

Propositional Delays

General Vanishing Delays

Concluding Remarks