This is to recollect the equivalent formulations of the Brouwer xed point theorem. We collect a large number of recently known sources of such equivalents. More recently, Jinlu Li obtained two xed point theorems on newly dened quasi-point-separable topological vector spaces. His theorems extend the Tychono xed point theorem on locally convex t.v.s. However, we note that his new theorems are logically equivalent to the Brouwer xed point theorem. Consequently, we add up our large list of such equivalents.

Certain maximum principles can be reformulated to various types of xed point theorems and conversely, based on Metatheorem due to ourselves. Such principles are Zorn's lemma, Banach contraction principle, Nadler's xed point theorem, Brézis-Browder principle, Caristi's xed point theorem, Ekeland's variational principle, Takahashi's nonconvex minimization theorem, some others and their variants, generalizations or equivalent formulations. Consequently, we have many new theorems equivalent to known results on xed point, common xed point, stationary point, common stationary point, and others. We show that such points are all maximal elements of certain ordered sets. Further we introduce our earlier related works as a history of our Metatheorem.

For a quite long period, the so-called L-structure or L-spaces have been studied by some authors. They have several trivial misconceptions such as their L-spaces extend the well-known generalized convex (G-convex) spaces. In order to clarify this matter and others, we show that our KKM theory on abstract convex spaces improves typical results in L-spaces. Main topics in this paper are related to extensions of the Himmelberg fixed point theorem. Since such studies are beyond of L-spaces, we cordially claim that now is the proper time to give up the useless study on L-spaces and their variants FC-spaces.

Recently, we improved our long-standing Metatheorem in Fixed Point Theory. In this paper, as its applications, some well-known order theoretic fixed point theo- rems are equivalently formulated to existence theorems on maximal elements, com- mon fixed points, common stationary points, and others. Such theorems are the ones due to Banach, Nadler, Browder-Göhde-Kirk, Caristi-Kirk, Caristi, Brøndsted, and possibly many others.