Multiple positive solutions for the Dirichlet boundary value problems by phase plane analysis. (English) Zbl 1347.34042

Summary: We consider boundary value problems for the scalar differential equation \[ x''+ \lambda f(x)=0, \] subject to the boundary conditions \[ x(0)=0, x(1)=0, \] where \(f(x)\) is a seventh-degree polynomial and \(\lambda\) is 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.


34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
34B09 Boundary eigenvalue problems for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
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