A Generalised SOR Framework for Fredholm Integral Equations on Large Intervals in Banach Spaces

Classical numerical approaches to Fredholm integral equations often rely on an implicit orthodoxy: discretise first, iterate afterwards. While effective in numerous settings, this paradigm becomes less transparent when the equation is defined on large intervals, where the infinite-dimensional structure of the problem quickly perturbs finite-dimensional approximations and obscures much of the anticipated benefit of classical schemes.

In this work, we recalibrate the compass by exploring a reversed paradigm. Rather than allowing discretisation to dominate the analysis from the outset, we begin by reformulating the equation as a system of coupled operator equations acting on a product Banach space, induced by a subdivision of the underlying domain. This naturally leads to an operator matrix whose entries are bounded linear operators, allowing notions familiar from linear algebra to be revisited within an infinite-dimensional framework.

Within this setting, we introduce a generalised successive over-relaxation (GSOR) scheme tailored to such operator matrices. A central theoretical result establishes that, under suitable conditions, the operator matrix is strictly diagonally dominant with respect to the operator norm. This structural property governs the convergence analysis of the method: it ensures the invertibility of the operator system and yields convergence of the GSOR iteration in the Banach space setting.

Rather than being merely a numerical device, our contribution clarifies how classical relaxation ideas can migrate, with minimal distortion, from finite-dimensional linear systems to systems of bounded linear operators. This perspective highlights a genuine theoretical bridge between linear algebra and functional analysis and suggests that iterating at the operator level offers a structurally sound and conceptually natural way to approach Fredholm integral equations on large intervals.