On a minimax problem of ricceri

Abstract

Let Ebe a real separable and reflexive Banach space, Xc_ E weakly closed and unbounded, ,I and two non-constant weakly sequentially lower semicontinuous functionals defined on X, such that + A is coercive for each A > 0. In this setting, if inf sup((I)(x) + A((x) + p)) sup inf((x) + A((x) + p)) xX >_o A>O xX for every p E R, then, one has inf sup((x)+ A(x) + h(A)), sup, x_>o xexinf((x) + A(x) + h(A)) xex ,>_o for every concave function h'[O, +[ R.

Further results related to a minimax problem of Ricceri

“…The proof is similar to that of [7,Lemma 3]. So, here we omit some passages that can be find in the cited article.…”

“…The proof is divided into four steps. We prove only the first step and refer to [7] for the others. By Corollary 2.3, we have…”

“…We want to stress out that the proof of Theorem 1.1 is not a straightforward consequence of the ideas developed in [7]. In fact, the proof of [7,Lemma 3] does not work when the interval I is upper-bounded.…”

“…In fact, the proof of [7,Lemma 3] does not work when the interval I is upper-bounded. For this reason, different arguments are needed in order to prove Lemma 2.4 which is a key technical preliminary result.…”

Existence of three solutions for a quasilinear two point boundary value problem

“…For a thorough account on the subject, we refer to [1], [3], [4], [7], [8], [9], [11], [12]. For basic notations and definitions we refer to [13].…”