An eigenvalue problem for the Stieltjes mean sigma-integral related to Gronwall’s inequality

Abstract:

This paper is concerned with the solution of the eigenvalue problem $\lambda f(t) = \smallint _0^t {f(s)\;dg(s),0 \leqq t \leqq T}$, where g is nondecreasing and right continuous on [0, T] and the integral is the Stieltjes mean sigma-integral. In addition to the solution of the general problem, the solution of the problem with positive eigenfunctions is given. The relationship of the eigenvalue problem to the failure of the Gronwall inequality for g is also established. Furthermore, the techniques used to solve this eigenvalue problem are readily adapted to solve the extended problem in which the function g is merely of bounded variation and right continuous.