Triple positive solutions for a class of two-point boundary-value problems. A fundamental approach - Palamides, P.K.; Palamides, A.P. (Нелінійні коливання, 2012)
In this paper, we prove the existence of three positive and concave solutions, by means of an elementary simple approach, to the 2th order two-point boundary-value problem x''(t) = α(t)f(t, x(t), x'(t)), 0 < t < 1, x(0) = x(1) = 0,. We rely on a combination of the analysis of the corresponding vector field on the phase-space along with Kneser’s type properties of the solutions funnel and the Schauder’s fixed point theorem. The obtained results justify the simplicity and efficiency (one could study the problem with more general boundary conditions) of our new approach compared to the commonly used ones, like the Leggett – Williams Fixed Point Theorem and its generalizations.