ON VOLTERRA INTEGRAL OPERATORS WITH HIGHLY OSCILLATORY KERNELS

. We study the high-oscillation properties of solutions to integral equations associated with two classes of Volterra integral operators: compact operators with highly oscillatory kernels that are either smooth or weakly singular, and noncompact cordial Volterra integral operators with highly oscillatory kernels. In the latter case the focus is on the dependence of the (uncountable) spectrum on the oscillation parameter. It is shown that the results derived in this paper merely open a window to a general theory of solutions of highly oscillatory Volterra integral equations, and many questions remain to be an- swered.

1. Introduction. We consider linear Volterra integral equations (of the first and second kind) corresponding to the following two classes of Volterra integral operators on C[0, T ] with highly oscillatory kernels that are either smooth or weakly singular.
Second-kind Volterra integral equations with the kernel singularity were studied in [11] and [7] (see also for additional references).
The highly oscillatory version of the cordial Volterra integral operator V ϕ in (1.7) is where K ∈ C(D), with K(0, 0) = 0, the oscillator g is smooth and nonnegative on D, and K and g do not depend on ω. We will assume throughout this paper that the given functions K and g are real-valued. This paper is concerned with the following questions: • Let u ∈ C(I) be a solution of the second-kind VIE (1.9) or of the analogous first-kind VIE where the highly oscillatory kernel K ω,α (0 < α ≤ 1) is given by (1.3) or (1.4). Is u necessarily highly oscillatory?
• Does the spectrum of the cordial Volterra operator (1.8) with highly oscillatory kernel (1.3) always (i.e. for all oscillators g(t, s)) depend on ω ?
• If µ ∈ R is not in the spectrum of the cordial Volterra integral operator V ϕ defined in (1.7), does the highly oscillatory cordial VIE with K ω (t, s) as in (1.3), have a unique solution u ∈ C(I) for all ω > 0, and is it highly oscillatory? An analogous question can be asked about the solution of the cordial VIE of the first kind, The functions f in (1.11) and (1.12) are assumed to be independent of ω. The content of the paper is as follows. In Section 2 we shall look at the representation of solutions to highly oscillatory VIEs with smooth or weakly singular kernels. The results will yield some first insight into the answer of the first of the above questions; they will also reveal that the oscillatory behaviour of solutions to a first-kind VIE can be rather different from the one of the analogous second-kind VIE. Section 3 is concerned with the spectra of cordial VIEs with highly oscillatory kernels. It turns out that for certain oscillators the spectra do not depend on ω. Section 4 is devoted to a brief description of some open problems: it will be seen that a complete understanding of the oscillatory behaviour of solutions to VIEs corresponding to general oscillators is still lacking.

2.1.
VIEs of the second kind. In order to obtain some first insight into the nature of the solution of a second-kind VIE with highly oscillatory kernels (1.3) or (1.4) we consider the 'separable' oscillator with g 0 smooth, strictly increasing and g 0 (0) ≥ 0. The linear oscillator corresponds to g 0 (t) = t and represents the most obvious special case.
The following theorem reveals that the resolvent kernel R ω,α (t, s) inherits the highly oscillatory term of the kernel K ω,α (t, s) for all 0 < α ≤ 1) in (1.4). (ii) the oscillator g is separable, i.e. g(t, s) = g 0 (t) − g 0 (s), satisfies the conditions stated in (2.1) and does not depend on ω. Then the following statements are true: (a) For any α ∈ (0, 1], the resolvent kernel associated with the kernel K ω,α (t, s) in (1.4) is given by where the resolvent kernel R α (t, s) of the kernel (t − s) α−1 K(t, s) does not depend on ω.
This yields a second-kind VIE for u ω , namely It follows from the classical Volterra theory (see for example [3, Section 6.1.2]) that its unique solution u ω ∈ C(I) has the form where R α (t, s) denotes the resolvent kernel of the given kernel (t − s) α−1 K(t, s). If 0 < α < 1 the resolvent kernel corresponding to (t − s) α−1 K(t, s) inherits the weak singularity and is given by for some Q α ∈ C(D). The assertions (a) and (b) then follow immediately from the definitions (2.6).
We use two examples to illustrate the statements (a) and (b) of the above theorem for the case of the linear oscillator (2.2) (corresponding to g 0 (t) = t in (2.1)), and to obtain insight into the damping of highly oscillatory solutions.
. Thus, by (2.5) the solution of the corresponding highly oscillatory VIE (1.9) with α = 1 (continuous kernel) is readily seen to be It is highly oscillatory for large ω but tends to the value 1 as ω tends to ∞. (For ω = 0 we have of course u(t) = e λt .) We observe that the highly oscillatory term in the expression for u(t) contains the damping factor (λ + iω) −1 . This raises the question as to whether stronger damping of the highly oscillatory terms in the corresponding solutions of (1.9) is possible for certain classes of smooth (nonconstant) functions f . The following theorem provides the answer.
Proof. The resolvent kernel of the constant kernel K(t, s) = λ is R 1 (t, s) = λe λ(t−s) . Hence, according to Theorem 2.1 (α = 1) the solution of the VIE is given by The integral in the above expression is highly oscillatory. In order to derive an asymptotic expansion in terms of powers of (λ + iω) −1 we adapt the well-known technique used for highly oscillatory integrals of the form 1 0 e iωs f (s)ds (see for example [14,8]), with the difference that now the upper limit of integration is variable. Thus, integration by parts leads to the desired result.

2.2.
VIEs of the first kind. We shall see that the oscillation properties of solutions to highly oscillatory second-kind VIEs of the form (1.9) may be rather different from the ones of solutions to analogous first-kind VIEs (1.10). Before presenting some general results, we look at the solution of a simple singularly perturbed VIE (compare also the review paper [15]) linking the two types of VIEs.
We now show that for a certain class of first-kind VIEs with smooth highly oscillatory kernels, the solutions are non-oscillatory.
with highly oscillatory kernel with f ∈ C r (I) not depending on ω and satisfying f (j) (0) = 0 (j = 0, . . . , r − 1), is non-oscillatory: (2.14) Proof. The result is trivial when r = 0. If r ≥ 1, we differentiate the VIE (2.11) r + 1 times to obtain the desired result (2.13). (We note that the VIE (2.11) with kernel (2.12) does not have a unique continuous solution on the closed interval [0, T ] when f (j) (0) = 0 for j = 0, . . . , q (q < r − 1) but f (r−1) (0) = 0.) Remark 2.2. We observe that the solution (2.13) reduces to the well-known solution u(t) = f (r+1) (t) of (2.11) when ω = 0 in (2.12). When ω > 0 this result is modified by a non-oscillatory 'ω-perturbation' involving lower-order derivatives of f (t). This raises the obvious question as to whether this remains true (i) for first-kind VIEs with more general (continuous) kernels, and (ii) for first-kind VIEs with weakly singular kernels? As a first step towards answering this question, we consider a first-kind VIE with linear oscillator, where k(t − s) is a smooth, non-constant convolution kernel satisfying k(0) = 0 and f ∈ C 1 (I), with f (0) = 0. If we resort again to the substitutions (2.6) we see that the differentiated form of (2.15) may be written as (where we have assumed without loss of generality that k(0) = 1).
Theorem 2.4. Assume that the resolvent kernel associated with the kernel −k (t−s) in (2.16) is r 1 (t − s). Then the (unique) solution of the first-kind VIE (2.15) is given by Proof. The proof is straightforward, since by using again the notation of (2.6) and the fact that the solution of (2.16) is we obtain the desired result (2.17).
Hence the solution of the first-kind VIE (2.15) is highly oscillatory, in contrast to (2.13) and (2.14). Moreover, the highly oscillatory term contains the damping factor (1 − iω) −2 . This is similar to the damping observed in Example 2.1 for second-kind VIEs.
It turns out that stronger damping of the highly oscillatory component of the solution is possible for certain classes of smooth functions f in the first-kind VIE (2.15), in analogy to the result of Theorem 2.2. The following theorem makes this precise.
Theorem 2.5. Assume that f ∈ C q+1 (I) (q ≥ 1), with f (0) = 0. Then the unique solution u ∈ C(I) of the first-kind VIE (2.18) can be written in the form Proof. We proceed as in the proof of Theorem 2.2, using integration by parts for the two highly oscillatory integrals in the solution representation (2.17), where the resolvent kernel is r 1 (t − s) = −e −(t−s) . The asymptotic expansion of the second integral is the one we used in that proof; this then yields the expansion of the first integral, by replacing f (s) by f (s). We leave the details to the reader.
We conclude this section by stating the analogue of Theorem 2.3 for the weakly singular first-kind VIE t 0 (t − s) α−1 K(t, s)e iωg(t,s) u(s) ds = f (t), t ∈ I (0 < α < 1), (2.19) with linear oscillator g(t, s) = t − s. Here, K, g and f are assumed to be smooth and independent of ω, with K(t, t) = 0 for t ∈ I.

ON VOLTERRA INTEGRAL OPERATORS 923
For r = 0 the solution reduces to Proof. If we multiply both sides of (2.19) by e −iωt and resort once more to the substitutions (2.6), we obtain the first-kind VIE for u ω (t) = e −iωt u(t). It then follows from the classical inversion formula for Abel integral equations (cf. [3, Section 6.1.4]) and our assumptions on f (j) (0) (j = 0, . . . , r) that its solution is given by  is It is clearly not highly oscillatory.
Example 2.6. Consider the weakly singular first-kind VIE Using again Laplace transform techniques it is seen that, under the assumptions f ∈ C 2 (I) and f (0) = f (0) = 0, its unique solution in C(I), (see also [19, p.81]) is highly oscillatory. For ω = 0 it reduces to the well-known inversion formula for the non-oscillatory version of (2.23): we have for any f ∈ C 1 (I) with f (0) = f (0) = 0.

2.4.
Fredholm integral equations with highly oscillatory kernels. Ursell [21] studied the asymptotic behaviour (as ω → ∞) of the solution of the Fredholm integral equation K(x, y)e iωg(x,y) u(y) dy, (ω 1), (2.25) and oscillator g(x, y) = |x − y|. He showed that when ν −1 is not in the spectrum of F ω , and if f and K are continuous and independent of ω, the solution u = u(x; ω) of (2.24) behaves like Ursell's results for the above oscillator g(x, y) were refined in [4]. The papers [5] and [2] address the analogous problem for the highly oscillatory Fredholm integral operator (2.25) with the so-called 'Fox-Li' oscillator g(x, y) = (x − y) 2 which arises in laser and maser engineering ( [16], [20, Section 60], [2], and their references). Since these bounded linear Fredholm integral operators are compact, they have at most a countable number of (complex) eigenvalues that accumulate at the origin. However, the derivation of rigorous results on their existence and asymptotic behaviour is rather difficult since these Fredholm integral operators are complex-symmetric, but not Hermitian. As Landau says in [16], "this presents a major obstacle to a theoretical understanding of the equation [(2.24)] -indeed, even the existence of eigenvalues is difficult to prove".
We note in passing that, to the author's knowledge, the analysis of the asymptotic behaviour of the eigenvalues of (2.25) with weakly singular kernel |x − y| −α (0 < α < 1) and oscillators |x − y| or (x − y) 2 remains open.
As we shall show in Section 3.2, the analogous problem for highly oscillatory cordial (non-compact) Volterra integral operators (and corresponding second-kind VIEs) is much more simple: here, the eigenvalues, as well as their asymptotic properties as ω → ∞, are now completely known (see Theorem 3.2 and Corollary 3.3).
is called a cordial Volterra integral operator if its core ϕ is in L 1 (0, 1), and K ∈ C m (D) for some m ≥ 0, with K(0, 0) = 0. It will be seen in Lemma 3.1 that under these assumptions V ϕ is a non-compact operator. (We note that if K(0, 0) = 0 then V ϕ is compact.) We shall occasionally state results for the basic cordial Volterra integral operator It corresponds to the operator V ϕ in (3.1) with K(t, s) ≡ 1.
We first review some relevant basic properties of the cordial Volterra integral operators (3.1) and (3.2). The results are due to Vainikko ([22,23]). Lemma 3.1. Under the assumptions stated above, the following statements are true: (a) V ϕ is a bounded linear operator that maps C(I) into itself, and we have The spectrum of V ϕ is uncountable; it is related to the spectrum of the basic cordial Volterra operator V 0 ϕ by is an eigenvalue of V 0 ϕ .

3.2.
The spectra of highly oscillatory cordial Volterra operators. Suppose that the cordial Volterra integral operator V ϕ defined by (3.1) possesses the spectrum σ(V ϕ ) given by (3.3). How is the spectrum σ(V ω,ϕ ) of the corresponding highly oscillatory cordial Volterra operator with K ω (t, s) := K(t, s)e iωg(t,s) , related to σ(V ϕ ) ? Theorem 3.2. Assume that the oscillator g(t, s) defining the highly oscillatory kernel K ω (t, s) of the cordial Volterra operator V ω,ϕ in (3.9) is smooth, nonnegative, and independent of ω, and let K ∈ C(D) be independent of ω and satisfy K(0, 0) = 0. Then the spectrum V ω,ϕ depends on ω if, and only if, g(0, 0) = 0: we have σ(V ω,ϕ ) = e iωg(0,0) σ(V ϕ ). (3.10) The eigenvaluesφ ω (λ) of V ω,ϕ satisfy Proof. Since the kernel K ω (t, s) of V ω,ϕ satisfies with K(0, 0) = 0, Lemma 3.1(c) implies that the spectrum of V ω,ϕ is given by Hence the result of Theorem 3.2 follows. As we have indicated in Section 2.4, the analysis of the asymptotic behaviour of the spectra of the highly oscillatory Fredholm integral operators F ω in (2.25) with oscillators g(x, y) = (x − y) 2 or g(x, y) = xy is quite difficult (in fact, it appears that the analysis for the oscillator g(x, y) = xy has not yet been carried out). The situation is completely different for the highly oscillatory cordial Volterra integral operators (3.9) with the analogous oscillators, as the following corollary shows. Proof. If the oscillator g(t, s) has the property that g(0, 0) = 0, the statement (3.10) of Theorem 3.2 becomes our assertion.
The above analysis shows, as we have already mentioned in Section 2.4, that the derivation of the asymptotic behaviour of the spectra of highly oscillatory cordial Volterra integral operators is in sharp contrast to the one of the behaviour of the spectra of analogous highly oscillatory Fredholm integral operators with oscillators like g(x, y) = (x − y) 2 . (3.12) where the highly oscillatory cordial Volterra integral operator V ω,ϕ is defined in (3.9) and µ = 0. We assume as always that K(0, 0) = 0, and that f, K and the oscillator g do not depend on ω.

Cordial VIEs of the second kind. Consider the cordial VIE
Theorem 3.4. Assume that the given functions in the highly oscillatory cordial VIE (3.12) do not depend on ω and are subject to the conditions f ∈ C(I), K ∈ C(D), K(0, 0) = 0, with g smooth and nonnegative on D. If µ ∈ σ(V ϕ ) (that is, the non-oscillatory cordial VIE has a unique solution u ∈ C(I)), then the highly oscillatory cordial VIE (3.12) has a unique solution u ∈ C(I) for all ω > 0, regardless of whether g(0, 0) = 0 or g(0, 0) = 0.

3.4.
Cordial VIEs of the first kind. In [24] Vainikko extended his results [22] on the theory of second-kind cordial VIEs to analogous first-kind cordial VIEs of the form t 0 t −1 ϕ(s/t)K(t, s)u(s) ds = f (t), t ∈ I, (3.14) with K(t, s) ≡ 1; here, ϕ ∈ L 1 (0, 1) and f is at least in C 1 (I) (or in an analogous weighted space). Under certain boundedness conditions on the moments of ϕ and ϕ and the assumption that ϕ(1) = 0 ([24, Section 4]), (3.14) can be transformed into an equivalent cordial VIE of the second kind.
4. Future work. While the results of Sections 2 and 3 have yielded considerable insight into the oscillatory or non-oscillatory behaviour of solutions to highly oscillatory VIEs with smooth or weakly singular kernels whose oscillators g(t, s) are either linear (cf. (2.2)) or separable ((2.3)), much more work needs to be done before we have a complete understanding of the oscillation properties of solutions corresponding to general nonlinear oscillators. These observations raise also a challenging problem for the numerical analyst. Suppose we do not know a priori whether or not the solution of first-or second-kind VIE with a highly oscillatory kernel is highly oscillatory. Can we detect a highly oscillatory solution numerically, by means of some suitably designed computational scheme (e.g. collocation or discontinuous Galerkin) which employs highly accurate approximations of the underlying highly oscillatory moment integrals (cf. [12], [13], [14], and [8] and its references)?