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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
\begin{aligned} &u''(t)+\phi(t)f(t,u(t))=0, \quad t\in (0,\infty)\\ &u(0)-au'(0)=B,\quad u'(+\infty)=C, \end{aligned} where $$\phi: (0,\infty)\to (0,\infty),$$ $$f: [0,\infty)\times {\mathbb R}^3\to {\mathbb R}$$ are continuous, $$a>0, B, C\in {\mathbb R}.$$ 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:
 34B40 Boundary value problems on infinite intervals for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 34B24 Sturm-Liouville theory 47N20 Applications of operator theory to differential and integral equations 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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