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Unbounded upper and lower solutions method for Sturm-Liouville boundary value problem on infinite intervals. (English) Zbl 1167.34320

The authors consider the Sturm-Liouville boundary value problem of second order differential equations on the half line

u '' (t)+φ(t)f(t,u(t))=0,t(0,)u(0)-au ' (0)=B,u ' (+)=C,

where φ:(0,)(0,), f:[0,)× 3 are continuous, a>0,B,C· The general unbounded upper and lower solution theory is established. By using the upper and lower solution method, the Schauder’s fixed point theorem and Nagumo’s conditions sufficient conditions are given for the existence of solutions as well as for the existence of unbounded positive solutions. An example illustrating the main results is also presented.

MSC:
34B40Boundary value problems for ODE on infinite intervals
34B15Nonlinear boundary value problems for ODE
34B24Sturm-Liouville theory
47N20Applications of operator theory to differential and integral equations
34B18Positive solutions of nonlinear boundary value problems for ODE