An algebraic study of Volterra integral equations and their operator linearity

The algebraic study of special integral operators led to the notions Rota-Baxter and shuffle products which have found broad applications such as iterated integrals. This paper carries out an general equations, shows that there are rich structures underlying Volterra corresponding equations. First shown produce a matching twisted algebra satisfying integration-by-parts operator identities. In order provide universal space express free operated algebras then constructed in terms bracketed words rooted trees with decorations on vertices edges. Further explicit constructions objects category obtained by decorated generalization product, providing for separable As application these constructions, it is any equation kernels linear sense can be simplified combination integrals same kernels.