# zbMATH — the first resource for mathematics

Positive solutions to singular boundary value problems with sign changing nonlinearities on the half-line via upper and lower solutions. (English) Zbl 1133.34017
The authors study the following boundary value problem on the half line
$x''-m^2x+\Phi(t)f(t,x)=0,\quad t\in(0,+\infty),$
$x(0)=0, x(\infty)=0,$ where $$f\in C(\mathbb{R}^+\times \mathbb{R}_0^+, \mathbb{R})$$ and $$m>0$$ is a constant, $$\Phi\in C(\mathbb{R}_0^+, \mathbb{R}^+)$$; here, $$\mathbb{R}^+=[0,\infty), \mathbb{R}_0^+=(0,\infty)$$. In this paper, $$f(t,x)$$ can change sign and may be singular at $$x=0$$. The lower and upper solution technique is presented for the above boundary value problem and some results on the existence of positive solutions are established.

##### MSC:
 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B16 Singular nonlinear boundary value problems for ordinary differential equations 34B40 Boundary value problems on infinite intervals for ordinary differential equations
Full Text:
##### References:
 [1] Agarwal, R. P., O’Regan, D.: Singular problems on the infinite interval modelling phenomena in draining flows. IMA Journal of Applied Mathematics, 66, 621–635 (2001) · Zbl 1073.34506 · doi:10.1093/imamat/66.6.621 [2] Agarwal, R. P., O’Regan, D., Lakshmikantham, V., Leela, S.: Existence of positive solutions for singular initial and boundary value problems via the classical upper and lower solution approach. Nonlinear Anal., 50, 215–222 (2002) · Zbl 1010.34011 · doi:10.1016/S0362-546X(01)00747-7 [3] Baxley, J. V.: Existence and uniqueness for nonlinear boundary value problems on in.nite intervals. J. Math. Anal. Appl., 147, 127–133 (1990) · Zbl 0719.34037 · doi:10.1016/0022-247X(90)90388-V [4] Bobisud, L. E.: Existence of solutions for nonlinear singular boundary value problems. Appl. Anal., 35, 43–57 (1990) · Zbl 0666.34017 · doi:10.1080/00036819008839903 [5] Bobisud, L. E.: Existence of positive solutions to some nonlinear singular boundary value problems on finite and infinite intervals. J. Math. Anal. Appl., 173, 69–83 (1993) · Zbl 0777.34017 · doi:10.1006/jmaa.1993.1053 [6] 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 · doi:10.1016/0362-546X(88)90070-3 [7] Chen, S. Z., Zhang, Y.: Singular boundary value problems on a half-line. J. Math. Anal. Appl., 195, 449–468 (1995) · Zbl 0852.34019 · doi:10.1006/jmaa.1995.1367 [8] Granas, A., Guenther, R. B., Lee, J. W., O’Regan, D.: Boundary value problems on in.nite intervals and semiconductor devices. J. Math. Anal. Appl., 116, 335–345 (1986) · Zbl 0594.34019 · doi:10.1016/S0022-247X(86)80002-6 [9] Guo, D. J., Lakshmikantham, V.: Nonlinear problems in abstract cones, Academic Press, Inc., New York, 1988 · Zbl 0661.47045 [10] O’Regan, D.: Theory of singular boundary value problems, World Scientific, Singapore, 1994 [11] Habets, P., Zanolin, F.: Upper and lower solutions for a generalized Emden-Fowler equation. J. Math. Anal. Appl., 181, 684–700 (1994) · Zbl 0801.34029 · doi:10.1006/jmaa.1994.1052 [12] Kelevedjiev, P.: Nonnegative solutions to some singular second-order boundary value problems. Nonlinear Anal., 36, 481–494 (1999) · Zbl 0929.34022 · doi:10.1016/S0362-546X(98)00025-X [13] 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 · doi:10.1016/S0362-546X(97)00719-0 [14] Okrasinski, W.: On a nonlinear ordinary differential equation. Ann. Polon. Math., 49, 237–245 (1989) [15] O’Regan, D.: Positive solutions for a class of boundary value problems on infinite intervals. Nonlinear Diff. Eqns. Appl., 1, 203–228 (1994) · Zbl 0823.34027 · doi:10.1007/BF01197747 [16] O’Regan, D.: Existence theory for ordinary differential equations, Kluwer, Dordrecht, 1997 [17] Zima, M.: On positive solutions of boundary value problems on the half-line. J. Math. Anal. Appl., 126, 127–136 (2001) · Zbl 1003.34024 · doi:10.1006/jmaa.2000.7399 [18] Zima, M.: On a certain boundary value problem. Comment. Math. Prace Mat., 29, 331–340 (1990) · Zbl 0724.34029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.