A singular third-order 3-point boundary-value problem with nonpositive Green's function

Abstract

We find a Green's function for the singular third-order three-point BVP $$ u'''(t)=-a(t)f(t,u(t)),quad u(0)=u'(1)= u''(eta )=0 $$ where $0leq eta <1/2$. Then we apply the classical Krasnosel'skii's fixed point theorem for finding solutions in a cone. Although this problem Green's function is not positive, the obtained solution is still positive and increasing. Our techniques rely on a combination of a fixed point theorem and the properties of the corresponding vector field.

Keywords

• Three-point singular boundary-value problem
• fixed point in cones
• third-order differential equation
• positive solution
• Green's function
• vector field