On the Weierstrass Theorem on extrema and its application for finding fixed points and zeros of mappings

In this paper, we examine the applicability of the Lagrange Multiplier Rule, specifically the Karush–Kuhn–Tucker Theorem, to investigate the existence of fixed points and zeros of certain potential mappings between finite and infinite spaces. Such zeros are investigated as constrained extrema, whose existence is established via the Weierstrass Theorem. By assuming the potentiality of the operator, we are able to relax the required invariance conditions. The paper is divided into two parts. In the first part, we provide simple proofs of the Lagrange Multiplier Rule and the Karush–Kuhn–Tucker Theorem in some special cases. In the second part, these results are used to investigate potential counterparts of several known theorems, such as the Brouwer Fixed Point Theorem and the Schauder Fixed Point Theorem.