Existence and location results for hinged beam equations with unbounded nonlinearities

Abstract

a b s t r a c tThis work presents some existence, non-existence and location results for the problem composed by the fourth-order fully nonlinear equation+ are continuous functions and s is a real parameter, with the Lidstone boundary conditionsThis problem models several phenomena, such as, the bending of an elastic beam simply supported at the endpoints.The arguments used apply a lower and upper solutions technique, a priori estimations and topological degree theory. In this paper we replace the usual bilateral Nagumo condition by some one-sided conditions, which enables us to consider unbounded nonlinearities.

Solvability of higher-order BVPs in the half-line with unbounded nonlinearities

“…However the same estimations hold with a similar one-sided assumption, allowing that the boundary value problems can include unbounded nonlinearities. In this way it generalizes the two-sided condition, as it is proved in [7,10].…”

“…If condition (4) holds, then by (7) there are t * , t + ∈ [0, +∞) such that t * < t + , u (t * ) = r and u (t) > r, ∀t ∈ (t * , t + ]. Therefore…”

“…Now consider that f verifies (5). By (7), consider that there are t − , t * ∈ [0, +∞) such that t − < t * and u (t * ) = r, u (t) > r, ∀t ∈ [t − , t * ). Therefore, following similar steps as before…”

On higher order fully periodic boundary value problems

“…In fact, condition (10) in our main result (see Theorem 4) refers an, eventually, opposite monotony to (A 2 ) and improves the existent results in the literature for periodic higher order boundary value problems. In short, our technique is based on lower and upper solutions not necessarily ordered, in the topological degree theory, like it was suggested, for example, in [23,24], and has the following key points:…”