Stable critical points of even functionals on the Banach space and applications

Let f and g be two even functionals defined on the Banach space X. For any r>0, we establish the existence of a denumerable number of critical values of g on Mr(f):={x∈X:f(x)=r}. These critical values are stable–they do not disappear under small perturbation by functionals which may not be even. Our results extend the corresponding ones of Krasnoselskii's in the Hilbert space. Some applications of our abstract theorems are also presented.