Parameterization Method for a Fredholm Integro-Differential Equation in Locally Convex Lattice Algebra

Аннотация

We study a two-point boundary value problem for a Fredholm integro-differential
equation with a supremum term in a complete Hausdorff unital locally convex lattice algebra.
The topology of the underlying algebra is generated by a directed family of continuous monotone
seminorms, which makes it possible to replace norm estimates by seminormwise estimates while
retaining the order structure required by the supremum operator. After rewriting the differential
equation in variation-of-constants form, we establish a uniform contraction estimate in the locally convex function space BD([0, T], A n). Under a contraction bound that is uniform over the
defining seminorms, the associated functional-integral equation has a unique solution. To treat
the boundary condition constructively, the interval is partitioned and the values of the solution
at the left endpoints are introduced as additional parameters. This reduces the original problem to special Cauchy problems on the subintervals together with a block operator equation for
the parameters. Topological invertibility of the parameter operator and a uniform contraction
condition for the coupled parameter-solution iteration yield unique solvability of the boundary
value problem and convergence of the proposed successive approximation scheme in every defining seminorm. The result extends the classical Banach-space parameterization framework to a
locally multiplicatively convex lattice-algebra setting; the Banach lattice algebra case is recovered
when the defining family consists of a single norm.