Multiple positive solutions for nth-order m-point boundary value problems with all derivatives

作者：Jiang Weihua^*

来源：Nonlinear Analysis-Theory Methods & Applications, 2008, 68(5): 1064-1072.

DOI：10.1016/j.na.2006.11.040

• Green's function
• the five-functional-fixed-point theorem
• m-point boundary value problem

We investigate the existence of positive solutions for the boundary value problem
u((n)) (t) + f (t, u (t), u' (t),..., u((n-2)) (t), u((n-1)) (t)) = 0, 0 < t < 1,
u(0) = u'(0) = ... = u((n-2)) (0) = 0, u((n-1)) (1) = Sigma (m-2)(i-1) k(i)u((n-1)) (xi(i)),
where f : [0, 1] x (R+)(n) -> R+ is continuous, k(i) > 0 (i = 1, 2,..., m - 2), 0 < Sigma(m-2)(i=1) k(i) < 1. We give at first the associated Green's function and some of its properties. Then, imposing growth conditions on f, we obtain the existence of at least three positive solutions for the boundary value problem by using the five-functional-fixed-point theorem. Finally, we give an example to demonstrate our result.

• 出版日期2008-3-1
• 单位河北科技大学; 河北师范大学