关键词: 正解, 周期边值问题, 不动点定理

Abstract:

The fourth order periodic boundary value problem u⁽⁴⁾-m⁴u+F(t, u(τ(t)))=0, 0<t<2π, with u⁽ⁱ⁾(0)=u⁽ⁱ⁾(2π), i=0,1, 2, 3,  is studied by using the fixed point theorem in cones, where F:[0, 2π]×R⁺→ R⁺ and τ: [0, 2τi]→[0, 2π] are continuous and  0<m<1. Under suitable conditions on F, it is proved  that the problem  has at least two positive solutions if m ∈  (0, M), where M is the smallest positive root of the equation tan mπ=-tanh mπ, which takes the value 0.7528094 with an error of ±10^-7.

Key words: Positive solutions, Periodic boundary value problem, Fixed point theorem