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The measure of non-compactness and approximation numbers of certain Volterra integral operators. (English) Zbl 0788.45013

This paper studies Volterra integral operators \(K:L^ p(\mathbb{R}^ +) \to L^ q (\mathbb{R}^ +)\) of the form \[ Kf(x)=v(x) \int^ x_ 0k(x,y) u(y)f(y) dy, \] where \(1<p \leq q<\infty\) and \(k\), \(u\) and \(v\) are prescribed functions. Under appropriate conditions we give sharp upper and lower estimates of the distance of \(K\) from the compact operators; and when \(K\) is compact, we provide upper and lower estimates for its approximation numbers when \(p=q =2\) and \(k(x,y)=p_ n(x-y)\), where \(p_ n\) is a polynomial of degree \(n\).
As an example we show that when \(u(x)=e^{Dx}\) and \(v(x)=e^{-Bx}\), with \(0<D<B\), and \(k(x,y)=x-y\), then the \(m\)th-approximation number of \(K:L^ 2(\mathbb{R}^ +) \to L^ 2(\mathbb{R}^ +)\) is bounded above and below by multiples of \(m^{-2}\).

MSC:

45P05 Integral operators
45G10 Other nonlinear integral equations
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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