Journal of Inequalities and Applications
Volume 6 (2000), Issue 3, Pages 261-285
doi:10.1155/S1025583401000169
On a minimax problem of ricceri
Dipartimento di Matematica, Università di Messina, Sant'Agata, Messina 98166, Italy
Received 18 August 1999; Revised 20 October 1999
Copyright © 2000 Giuseppe Cordaro. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Let E be a real separable and reflexive Banach space, X⊆E weakly closed and unbounded, Φ and Ψ two non-constant weakly sequentially lower sernicontinuous functionals defined on X, such that Φ+λΨ is coercive for each λ≥0. In this setting, if
supλ≥0infx∈X(Φ(x)+λ(Ψ(x)+ρ))=infx∈Xsupλ≥0(Φ(x)+λ(Ψ(x)+ρ))
for every ρ∈R, then, one has
supλ≥0infx∈X(Φ(x)+λΨ(x)+h(λ))=infx∈Xsupλ≥0(Φ(x)+λΨ(x)h(λ)),
for every concave function h:[0,+∞[→R.