The Choice of the Regularization Parameter for Solving Linear Volterra-Stieltjes Integral Equations of the Third Kind

Abstract

For solving one class of Volterra linear integral equations of the third kind, the regularization parameter for the regularizing operator was chosen by Lavrentiev. In general case, the Volterra-Stieltjes integral equations do not always reduce to Volterra integral equations, since the Stieltjes integral equation does not always reduce to the Riemann integral or Lebesgue integral. Therefore, a Volterra-Stieltjes integral equation was of independent interest. The aim of the study was to construct a regularizing operator and prove the uniqueness theorem. In this study we used the notion of a derivative with respect to an increasing function, regularization method according to M.M. Lavrentiev, methods of functional analysis, methods for transforming equations, methods of integral and differential equations. Proposed methods can be used to study integral and integro-differential equations of the Volterra-Stieltjes type of high orders, as well as in qualitative study of some processes in fields of applied physics, ecology, medicine, and the theory of complex systems control.