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Zbl 1167.34320
Lian, Hairong; Wang, Peiguang; Ge, Weigao
Unbounded upper and lower solutions method for Sturm-Liouville boundary value problem on infinite intervals. (English)
[J] Nonlinear Anal., Theory Methods Appl. 70, No. 7, A, 2627-2633 (2009). ISSN 0362-546X

The authors consider the Sturm-Liouville boundary value problem of second order differential equations on the half line $$\aligned &u''(t)+\phi(t)f(t,u(t))=0, \quad t\in (0,\infty)\\ &u(0)-au'(0)=B,\quad u'(+\infty)=C, \endaligned $$ 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.
[Sotiris K. Ntouyas (Ioannina)]

MSC 2000:: *34B40 Boundary value problems on infinite intervals
34B15 Nonlinear boundary value problems of ODE
34B24 Sturm-Liouville theory
47N20 Appl. of operator theory to differential and integral equations
34B18 Positive solutions of nonlinear boundary value problems

Keywords: boundary value problem; infinite intervals; upper and lower solutions; Nagumo-condition; fixed point theory

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