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Unbounded solutions of the singular boundary value problems for second order differential equations on the half-line. (English) Zbl 1045.34009
The authors consider the following singular boundary value problem for a second-order differential equation on the half-line \[ {1\over p(t)} (p(t)x'(t))'+ f(t, x(t))= 0,\quad t\in (0,+\infty), \] \(x(0)= 0\), \(\lim_{t\to+\infty} p(t) x'(t)= b> 0\), where \(p\in C([0,+\infty), \mathbb R)\cap C'(0,\infty)\), \(p(t)> 0\) for \(t\in (0,\infty)\), and \(f\in C(0,\infty)\times (0,\infty))\). They show that under certain additional conditions imposed on the functions \(p\) and \(f\), the above problem has at least two nonnegative unbounded solutions.

MSC:
34B40 Boundary value problems on infinite intervals for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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[1] Chen, S.; Zhang, Y., Singular boundary value problems on a half-line, J. math. anal. appl., 195, 449-468, (1995) · Zbl 0852.34019
[2] Kawano, N.; Yanagida, E.; Yotsutani, S., Structure theorems for positive radial solutions to δu+K(|x|)up=0 in rn, Funkcialaj ekvaciaj, 36, 557-579, (1993) · Zbl 0793.34024
[3] Junyu, T., The radial solutions of 2-order semilinear elliptic equations, Acta math. appl. sinica, 19, 1, 57-64, (1996) · Zbl 0864.35040
[4] O’Regan, D., Theory of singular boundary value problems, (1994), World Scientific Singapore · Zbl 0808.34022
[5] Baxley, J.V., Existence and uniqueness for nonlinear boundary value problems on infinite intervals, J. math. anal. appl., 147, 127-133, (1990) · Zbl 0719.34037
[6] Guo, D.; Liu, X.Z., Impulsive integra-differential equations on unbounded domain in a Banach space, Nonlinear studies, 3, 1, 49-57, (1996) · Zbl 0864.45009
[7] Guo, D., Boundary value problems for impulsive integro-differential equation on unbounded domains in a Banach space, Appl. math. comput., 99, 1-15, (1999) · Zbl 0929.34058
[8] Liu, X., Some existence and nonexistence principles for a class of singular boundary value problems, Nonlinear anal., 27, 10, 1147-1164, (1996) · Zbl 0860.34010
[9] Dajun, G.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Acdemic Press New York · Zbl 0661.47045
[10] Corduneanu, C., Integral equations and stability of feedback systems, (1973), Academic Press New York · Zbl 0268.34070
[11] Meehan, M.; O’Regan, D., Existence theory for nonlinear Fredholm and Volterra integral equations on half-open intervals, Nonlinear anal., 35, 355-387, (1999) · Zbl 0920.45006
[12] Agarwal, R.P.; O’Regan, D., Twin solutions to singular Dirichlet problems, J. math. anal. appl., 240, 433-445, (1999) · Zbl 0946.34022
[13] Kelevedjiev, P., Nonnegative solutions to some singular second-order boundary value problems, Nonlinear anal., 36, 481-494, (1999) · Zbl 0929.34022
[14] Bobisud, L.E., Existence of solutions for nonlinear singular boundary value problems, Appl. anal., 35, 43-57, (1990) · Zbl 0666.34017
[15] Bosbisud, L.E.; O’Regan, D.; Royalty, W.D., Solvability of some nonlinear boundary value problems, Nonlinear anal., 12, 855-869, (1988) · Zbl 0653.34015
[16] O’Regan, D., Existence theory for ordinary differential equations, (1997), Kluwer Dordrecht · Zbl 1077.34505
[17] Agarwal, R.P.; O’Regan, D., Singular boundary value problems for suplinear second order ordinary and delay differential equations, J. diff. eqs., 130, 333-355, (1996) · Zbl 0863.34022
[18] Erbe, L.H.; Hu, S.C.; Wang, H.Y., Multiple positive solutions of some boundary value problems, J. math. anal. appl, 184, 640-648, (1994) · Zbl 0805.34021
[19] Guo, Z.M., On the number of positive solutions for quasilinear elliptic eigenvalue problems, Nonlinear anal., 27, 229-247, (1996) · Zbl 0860.35090
[20] Kisik, H.; Lee, Y.H., Existence of multiple positive solutions of singular boundary value problems, Nonlinear anal., 28, 1429-1438, (1997) · Zbl 0874.34016
[21] Liu, Z.L.; Li, F.Y., Multiple positive solutions of nonlinear two-boundary value problem, J. math. anal. appl., 203, 610-625, (1996) · Zbl 0878.34016
[22] Ubilla, R., Multiplicity results for the 1-dimensional generalized p-Laplacian, J. math. anal. appl., 190, 611-623, (1995) · Zbl 0831.34032
[23] Wang, J.Y.; Gao, W.J., A singular boundary value problem for the one-dimensional p-Laplacian, J. math. anal. appl., 201, 851-866, (1996) · Zbl 0860.34011
[24] Gao, D., Fixed point of mixed monotone operators and applications, Appl. anal., 31, 215-224, (1988)
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