Multiple Positive Solutions for the Dirichlet Boundary Value Problems by Phase Plane Analysis

Abstract

We consider boundary value problems for scalar differential equationx′′+λfx=0,x(0)=0,x(1)=0, wheref(x)is a seventh-degree polynomial andλis a parameter. We use the phase plane method combined with evaluations of time-map functions and make conclusions on the number of positive solutions. Bifurcation diagrams are constructed and examples are considered illustrating the bifurcation processes.