On the solution of an integral-functional equation with a parameter
This paper studies the Fredholm integral equation \[ \phi(t) = (L\phi)(t) = b \int_{at - a + 1}^{at} \phi(\tau) d\tau, \tag{\(*\)} \] where \(a > 1\) and \(b = a/(a-1)\) (for \(a \leq 1\) no nonzero solutions of this equation exist). This equation can also be written in differential form \( \phi'(t) = ab[\phi(at) - \phi(at - a + 1)] \). It is shown that this equation has a solution supported on \([0,1]\), which is unique under the normalization \(\int_0^1 \phi(\tau) d\tau = 1\). This solution is \(C^\infty\) smooth, symmetric with respect to the point \(t = \frac 12\), nonnegative, nondecreasing in \([0,\frac 12]\), nonincreasing in \([\frac 12,1]\), and constant in \([1/a,1-1/a]\) if \(a > 2\). Several other properties of this solution are also established about its Laplace transform, convergence of successive approximations, and its Fourier expansion. Furthermore, a complete description is given of the spectrum and eigenfunctions of the compact operator \(L\) in \((*)\).
scientific article; zbMATH DE number 1129195
scientific article