Study of Simplicial Methods with Stability

This paper begins with the introduction of the concepts of standard triangulation, essential set, approximate essential set etc. This study presents some new findings about simplicial algorithms that take into account the continuities of approximate fixed point sets. The subsistence of finite essential connected components in approximate fixed point sets by vector-valued labels is obtained by proving the upper semi-continuity of a set-valued mapping of approximate fixed points with the help of simplicial methods that are vector-valued; examples are provided to demonstrate how this differs significantly from the property for integer-valued labeling simplicial methods. It is also demonstrated that essential sets exist by concentrating on both domain and function perturbations. Finally, the paper is concluded in section 5.