A deformation lemma and its applications
The author proves a new deformation lemma and uses it to improve several well-known critical point theorems. Let X be a real Banach space. For Palais-Smale condition [\textit{R. S. Palais} and \textit{E. Smale}, Bull. Am. Math. Soc. 70, 165-172 (1964; Zbl 0119.092)] satisfying differentiable functional f and real number c let \(f^ c=\{x\in X:\) f(x)\(\geq c\}\), \(f_ c=\{x\in X:\) f(x)\(\leq c\}\) and \(K_ c=\{x\in X:\) \(f'(x)=0\), \(f(x)=c\}\). The lemma, which is the main result of this work, asserts that if there exist two closed sets \(A\subset f^ c\) and \(B\subset f_ c\) satisfying \(A\cap B=\emptyset\) and \(A\cap K_ c=\emptyset\) then there exists an \(\epsilon >0\) and a homeomorphism \(\eta\) of X onto X such that (i) f(\(\eta\) x)\(\geq f(x)\forall x\in X\), (ii) \(f(\eta z)\geq c+\epsilon\forall x\in A\) and (iii) \(\eta x=x\forall x\in B\).
scientific article; zbMATH DE number 4214976