Cordial Volterra Integral Equations with Vanishing Delays *

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 t , 0 λ = or 0 λ R > , 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.


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].In [4], the author generalizes such kind of equations into cordial Volterra integral equations (CVIEs) with the form  ), and so on.
It is shown that the cordial Volterra integral operator ϕ  in the Banach space ( ) C I is noncompact and its spectrum is a non-countable set, i.e., In [5], the author describes the eigenvalues and eigenfucntions of the operator ϕ  on the space ( ) C I when (0,1) with some 1 p > : 1) the point spectrum of ϕ  is exactly the set ( ) { ( ) : 0} 2) the dimension of the null space ( )    is the sum of the multiplici- ties of the roots of ( ): ( ) 0 3) the linearly independent eigenfunctions are given by ( ) , , ln ,
The pure Volterra integral equations with vanishing delay (VIEwND) are initially studied in [6] and a special form of VIEwND, proportional delay differential equations, is widely used in practical applications, for example, electrodynamics [7] [8], nonlinear dynamical systems [9] [10], and also the survey papers [11] [12].In this paper, we consider the CVIEs with a vanishing delay, where ( ) t θ is a continuous delay function such that (0) 0 θ = and ( ) θ < for all 0 t I < ∈ and the operator with delay is similarly defined by 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.In Section 2, we show that the proportional delay ( ) t qt θ = , 0 1 q < < , is the only one that replicates all eigenfunctions t λ , 0 λ = or 0 λ ℜ > .For such a delay, we describe the spectrum, eigenvalues and eigenfunc- tions of the operator

Propositional Delays
For a vanishing delay ( ) The cordial Volterra integral operator with a vanishing delay (3) is also written as a cordial Volterra integral operator with a variable kernel, i.e., where the discontinuous kernel ( , ) a t s is defined by The properties of the operator ,a ϕ  with continuous kernels are investigated in [13] such as it is compact if and only if (0, 0) 0 a = .From the above defini- tion, the discontinuous function a always satisfies (0, 0) 0 a = , but the com- pactness of the operator Theorem 2.1.Assume that the function 2) If all power-functions , 0 , the integration is defined by since u is uniformly continuous on the closed interval.The uniform continuity of ξ implies that there exists a We, without loss of generality, assume that 2) Without loss of generality, suppose that . Then similarly to the approach in [4], there exists a polynomial 0 ( ) This contradiction implies the proof is complete. Remark 2.2.In [4], the author shows that an operator  mapping ( ) C I to ( ) C I has the two properties: 1)  is a bounded operator; 2) all power-functions t λ , 0 λ = or 0 λ ℜ > , are eigenfunctions of  ; if and only if ϕ =   is a cordial Volterra integral operator.While including vanishing delays, the two properties only hold for a proportional delay ( ) , we define an integration function of the core by 0 ( ) ( ) d .
CVIEs naturally reduce to a proportional delay form where the corresponding operator has the form q ϕ ϕ =   , some more detailed properties on cordial Volterra integral operators with a proportional delay are presented in the following theorem.
Both the existence and uniqueness of solutions to (5) are valid when the parameter µ does not lie in the spectrum of the corresponding operators.On the other hand, for µ lying in the spectrum, by the same technique in [5], we are also able to construct solutions to (5).For convenience, we review some notations in [5]: , there exists at most one solution to (5), and there exists exactly one solution to (5) when where u ⊥ is linearly combined by such functions 1 (if

General Vanishing Delays
For a more general vanishing delay, the compactness of the cordial Volterra integral operators is influenced by the value of (0) θ ′ . with such a vanishing delay term is shown in Lemma 3.7.The proof will be completed, when the non-compactness of the operator is proved in Lemma 3.8. The simplest compact condition according to Theorem 3.1 is (0) 1 and that the delay function ( ) t θ satisfies the assumptions (D1), (D2), (D3).Then the operator , Remark 3.3.Consider the constant core ( ) 1 x ϕ ≡ .Then 1) ,1 q  , 0 1 q < < , are non-compact in ( ) C I .
2) For ( ) sin   has a bounded inverse in ( ) C I (see in [14]).Hence the proof is complete. Example 3.5.Consider the following CVIEs with a vanishing delay 2) (the linear form of Lighthill's equations) and ( ) sin Then the corresponding operators are compact and there exists a unique solu- the condition in this lemma yields that for all ( ) In the following, we let

Concluding Remarks
In this paper, we consider CVIEs with a vanishing delay: 1) a proportional delay, 2) a nonlinear vanishing delay ( ) t θ .
The first case reduces to a classical CVIE with a core limited to a subinterval.


CVIEs appear in a lot of application models, such as Diogo core


. In Section 3, we present a necessary and sufficient condition for the compactness of the operator , θ ϕ  with a vanishing delay.Based on these discussions, we present the existence, uniqueness and the construction of solutions to (2).


is influenced not only by the core but also by the value of (0) θ ′ (see in Corollary 2.3 and Theorem 3.1).

5 )
The range of the operator , θ ϕ µ −   is the whole space ( ) C I if and only if

∈
 is a root of , ( ) 0 there exists at most one solution * delay function ( ) t θ satisfies the assumptions (D1), (D2), (D3).Then the operator , the definition of the function ξ , it is known that (0Ascoli-Arzela theorem, the compactness of the cordial Volterra integral operator , θ ϕ The existence and uniqueness of solutions to (2) is similar to the classical second kind of VIEs when the corresponding operator is compact.Theorem 3.4.Assume that 1 (0,1) L ϕ ∈ and that the delay function ( ) t θ satisfies the assumptions (D1), (D2), (D3) and that supp [0, (0)] ϕ θ′ ⊆ .Then for all 0 µ ≠ and all ( ) f C I ∈, there exists a unique solution to (2).Proof.In Lemma 3.9, it is shown that the null space of the operator , C I is {0} , which together with the compactness of ,

=
and the proof is complete.