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has been cited by the following article:

• TITLE: Cordial Volterra Integral Equations with Vanishing Delays*

  AUTHORS: Hongjiu Wang, Zhanwen Yang, Melusi Khumalo

  KEYWORDS: Cordial Volterra Integral Equations, Vanishing Delay, Propositional Delay, Compactness, Existence and Uniqueness

  JOURNAL NAME: Journal of Applied Mathematics and Physics, Vol.5 No.2, February 15, 2017

  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.