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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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[1] Agarwal, R.P.; O’Regan, D., Infinite interval problems for differential, difference and integral equations, (2001), Kluwer Academic Publisher · Zbl 1003.39017
[2] Agarwal, R.P.; O’Regan, D., Nonlinear boundary value problems on the semi-infinite interval: an upper and lower solution approach, Mathematika, 49, 129-140, (2002) · Zbl 1049.34033
[3] Agarwal, R.P.; O’Regan, D., Infinite interval problems modeling phenomena which arise in the theory of plasma and elesrical potential theory, Stud. appl. math., 111, 339-358, (2003) · Zbl 1141.34313
[4] Chen, S.W.; Zhang, Y., Singular boundary value problems on a half-line, J. math. anal. appl., 195, 449-468, (1995) · Zbl 0852.34019
[5] Ehme, J.; Eloe, P.W.; Henderson, J., Upper and lower solution methods for fully nonlinear boundary value problems, J. differential equations, 180, 51-64, (2002) · Zbl 1019.34015
[6] Eloe, P.W.; Kaufmann, E.R.; Tisdell, C.C., Multiple solutions of a boundary value problem on an unbounded domain, Dynam. systems appl., 15, 1, 53-63, (2006) · Zbl 1108.34024
[7] Gomes, J.M.; Sanchez, J.M., A variational approach to some boundary value problems in the half-line, Z. angew. math. phys., 56, 192-209, (2005) · Zbl 1073.34026
[8] Guseinov, G.Sh.; Yaslan, I., Boundary value problems for second order nonlinear differential equations on infinite intervals, J. math. anal. appl., 290, 620-638, (2004) · Zbl 1054.34045
[9] Liu, Y.S., Boundary value problems for second order differential equations on infinite intervals, Appl. math. comput., 135, 211-216, (2002)
[10] Thomopson, H.B., Second order ordinary differential equations with fully nonlinear two point boundary conditions, Pacific J. math., 172, 255-276, (1996)
[11] Khan, R.A.; Webb, J.R.L., Existence of at least three solutions of a second-order three-point boundary value problem, Nonlinear anal., 64, 1356-1366, (2006) · Zbl 1101.34005
[12] Yan, B.Q.; O’Regan, D.; Agarwal, R.P., Unbounded solutions for singular boundary value problems on the semi-infinite interval: upper and lower solutions and multiplicity, J. comput. appl. math., 197, 365-386, (2006) · Zbl 1116.34016
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