Uniform l1 Behavior of a Time Discretization Method for a Volterra Integrodifferential Equation with Convex Kernel; Duality of the Weak Parallelogram Laws on Banach Spaces

The first chapter of this thesis concerns the stability and convergence of a numerical method in which the backward Euler method is combined with order one convolution quadrature for approximating the integral term of the linear Volterra integrodifferential equation u'(t) + ∫t0 β (t--s) Au(s) ds = 0, t ≥ 0, u( 0) = u0, which arises in the theory of linear viscoelasticity. Here A is a positive self-adjoint densely defined linear operator in a real Hilbert space and β(t) is locally integrable, nonnegative, nonincreasing, convex, with -- β'(t) and β''(t) convex. We establish stability and convergence of the method under these hypotheses on β(t). Thus, the method is stable and convergent for a wider class of kernel functions β(t), than was previously known. We also extend the class of operators A for which the method is stable and convergent. The second chapter of this thesis explores a family of weak parallelogram laws for Banach spaces. Some basic properties of such spaces are obtained. The main result is that a Banach Space satisfies a lower weak parallelogram law if and only if its dual satisfies an associated upper weak parallelogram law, and vice-versa. Connections are established between the weak parallelogram laws and the following: subspaces, quotient spaces, Cartesian products, and the Rademacher type and co-type properties.