This article presents a list of sufficient conditions for the existence of fixed points of semi-closed 1-set-contractive operators in real Banach spaces. All results presented here are evident corollaries of the following analogue for semi-closed 1-set-contractive operators of the Leray-Schauder fixed point principle: If is a semi-closed 1-set-contractive operator such that
then has at least one fixed point in .
In turn, this result is an evident corollary of the statement that provided that is semi-closed 1-set-contractive operator satisfying the condition for some and for all , . The main part of the article is devoted to inequalities that imply (*); among others, the inequality
(, ) is treated.