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singular nonlinear boundary value problem; positive solution; Krasnosel'skii fixed point theorem; multi-point; half-line
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.
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