A third-order 3-point BVP. Applying Krasnosel'skii's theorem on the plane without a Green's function

作者：Palamides Panos K^*; Palamides Alex P

来源：Electronic Journal of Qualitative Theory of Differential Equations, 2008, (14): 1-15.

• three point boundary value problem
• third order differential equation
• positive solution
• vector field
• fixed point in cones

Consider the three-point boundary value problem for the 3(rd) order differential equation:
{x''' (t) = alpha (t) f (t, x (t), x' (t), x '' (t)), 0 < t < 1, x (0) = x' (eta) = x '' (1) = 0,
under positivity of the nonlinearity. Existence results for a positive and concave solution x (t), 0 <= t <= 1 are given, for any 1/ 2 < eta < 1. In addition, without any monotonicity assumption on the nonlinearity, we prove the existence of a sequence of such solutions with
lim(n) (-->) (infinity) parallel to x(n)parallel to = 0.
Our principal tool is a very simple applications on a new cone of the plane of the well-known Krasnosel' skii's fixed point theorem. The main feature of this aproach is that, we do not use at all the associated Green's function, the necessary positivity of which yields the restrictio eta is an element of 2 (1/2, 1). Our method still guarantees that the solution we obtain is positive.

• 出版日期2008
• 单位中国人民解放军海军大连舰艇学院