On a singular multi-point third-order boundary value problem on the half-line

Abstract

We establish not only sufficient but also necessary conditions for existence of solutions to a singular multi-point third-order boundary value problem posed on the half-line. Our existence results are based on the Krasnosel'skii fixed point theorem on cone compression and expansion. Nonexistence results are proved under suitable a priori estimates. The nonlinearity $f=f(t,x,y)$ which satisfies upper and lower-homogeneity conditions in the space variables $x, y$ may be also singular at time $t=0$. Two examples of applications are included to illustrate the existence theorems.

Keywords

• singular nonlinear boundary value problem
• positive solution
• krasnosel'skii fixed point theorem
• multi-point
• half-line