On the measure of noncompactness and approximation numbers of a class of Volterra integral operators. (English. Russian original) Zbl 0815.45006

Stepanov, V. D.; Edmunds, D. E.

Russ. Acad. Sci., Dokl., Math. 47, No. 3, 618-623 (1993); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 330, No. 6, 700-703 (1993).

This paper gives sharp estimates on the measure of noncompactness of integral operators of the form \(Kf(x) = v(x) \int_ 0^ x k(x,y)f(y)u(y)dy\), acting from the Lebesgue space \(L^ p (\mathbb{R}^ +)\) to the Lebesgue space \(L^ q (\mathbb{R}^ +)\), with \(1 < p \leq q < \infty\). In addition, the behavior of the approximation numbers \(\alpha_ m (K)\) is investigated in the case where \(p = q = 2\) and \(K(x,y) = P_ n(x - y)\) with \(P_ n\) a polynomial of degree \(n\).
For \(n = 1\) sharp results are obtained, and an example is given where \(v\) and \(u\) are exponentials. The kernel \(k(x,y)\) is supposed to be nondecreasing with respect to \(x\) and nonincreasing with respect to \(y\), and to satisfy certain global bounds.

Reviewer: O.Staffans (Espoo)

MSC:

45P05	Integral operators
47G10	Integral operators
47B07	Linear operators defined by compactness properties
45D05	Volterra integral equations

Keywords:

Volterra integral operators; estimates; measure of noncompactness; Lebesgue space; approximation numbers