×

Behavior of the solution of a Volterra equation as a parameter tends to infinity. (English) Zbl 0552.45010

Let \(u(t,\lambda)\) denote the solution of the scalar Volterra integrodifferential equation \[ u'(t)+\lambda \int^{t}_{0}a(t- s)u(s)ds=0\quad (0\leq t<\infty);\quad u(0)=1. \] The authors investigate the global behavior of \(u(t,\lambda)\) as a function of the parameter \(\lambda\), and prove the following theorem and corollary: Theorem: Let \(a\in C(0,\infty)\cap L^ 1(0,1)\) be nonnegative, non-increasing and convex on \((0,\infty)\) with \(0\leq a(\infty)<a(0+)\leq \infty\), and separate the Fourier transform A(\(\tau)\) of a(t) into real and imaginary parts as \(A(\tau)=\phi (\tau)-i\tau \Theta (\tau)\). (i) If \(\Theta(\tau)/\phi(\tau)\to 0\) as \(\tau\to 0\), then \(tu(t,\lambda)\to 0\) as \(\lambda\to \infty\), uniformly for \(t\in (0,\infty)\). (ii) If \(0< \limsup_{\tau \to \infty}\Theta (\tau)/\phi (\tau)<\infty\), then there exists a sequence \(\lambda_ n\to \infty\) such that the set \(\{t\in (0,\infty)| u(t,\lambda_ n)\nrightarrow 0\) as \(n\to \infty \}\) has positive Lebesgue measure. Corollary: If in addition \(-a'\) is convex on \((0,\infty)\), then (i) holds whenever \(a'(0+)=-\infty\) and (ii) holds whenever \(a'(0+)>-\infty\). An application is given in which the preceding result is used to classify a certain Volterra integrodifferential equation in Hilbert space as being of either parabolic or hyperbolic type.
Reviewer: O.Staffans

MSC:

45J05 Integro-ordinary differential equations
45M05 Asymptotics of solutions to integral equations
45N05 Abstract integral equations, integral equations in abstract spaces
PDFBibTeX XMLCite