A note on improved contraction methods for discrete boundary value problems

This work investigates a two-point boundary value problem (BVP) involving a first-order difference equation, known as the ‘discrete’ BVP. Some sufficient conditions are formulated under which the discrete BVP will possess a unique solution. The innovation herein involves a strategic choice of metric and utilization of Hölder's inequality. This approach enables the associated mappings to be contractive, which were previously non-contractive in traditional settings. This consequently enables an improved application of the fixed-point theorem of Stefan Banach by addressing a wider range of problems than those covered by the current literature. A YouTube video presentation by the author designed to complement this work is available at http://www.youtube.com/watch?v=luLuQ1KyXy8.