A variational approach to a class of nonlinear eigenvalue problems.
Peter Heß · Proceedings of the American Mathematical Society · 1971
Let f be a real-valued differentiable function defined on the real reflexive Banach space X . The problem of minimizing f over a subset of X is investigated under the following mild monotonicity assumption on the derivative f ′ f’ of f : if { u n } \{ {u_n}\} is a sequence in X converging weakly to some u ∈ X u \in X , then lim sup ( f ′ u n , u n − u ) ≧ 0 \lim \sup (f’{u_n},{u_n} - u) \geqq 0 . The eigenvalue problem f ′ u = λ g ′ u f’u = \lambda g’u for some λ ∈ R 1 \lambda \in {R^1} , with g ′ g’ being the derivative of a further function g , is then reduced to that first question.