A fundamentally simple approach to a singular boundary value problem

✍ Scribed by Panos K. Palamides; Alex P. Palamides

• Publisher: Elsevier Science
• Year: 2008
• Tongue: English
• Weight: 385 KB
• Volume: 68
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2007.03.039

✦ Synopsis

Existence results, by means of a simple approach, for the 3rd order differential equation

satisfying the three-point boundary value conditions

are given, where 0 < η < 1 is a fixed point in contrast to the usual case 1/2 < η < 1, f (t, x, y, z) ≥ 0 for any t ∈ (0, 1), x ≥ 0, y, z ∈ R, and the map α(t) is positive on (0, 1) and suitable singular at the end points. Furthermore the obtained solution x(t) is positive and concave on (0, 1).

In addition, without any monotonicity assumption on the nonlinearity, we prove the existence of a sequence of such solutions with lim n→∞ x n = 0.

Our principal tools are very simple applications on the plane of the well-known from combinatorial topology Sperner's Lemma, in comparison with other classical approaches, like fixed point or degree theorems.