Positive Solutions for Semipositone ( p , n − p ) Right Focal Boundary Value Problems

We consider the ( p , n m p ) right focal boundary value problem: $${\matrix{{(- 1)^{n - p} u^{(n)} \! = \lambda \;f(t, u), } \hfill & \ {{\rm for }\ 0 \lt t \lt 1, } \hfill \cr \quad \quad \,{u^{(i)} (0) = 0, } \hfill & {0 \le i \le p - 1, } \hfill \cr \quad \quad \,{u^{(i)} (1) = 0, } \hfill & {p \le i \le n - 1, } \hfill \cr}} $$ where 1 h p h n m 1 is fixed and u > 0. Using a fixed point theorem for operators on a cone, we develop criteria for the existence of positive solutions of the boundary value problem for u on a suitable interval.

Keywords:

• Right Focal Boundary Value Problem
• Positive Solution
• Krasnosel'skii Fixed Point Theorem