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Existence of positive solutions for second-order semipositone differential equations on the half-line. (English) Zbl 1117.34033

The authors are concerned with the existence of positive solutions of a semi-positone Sturm-Liouville boundary value problem on the half-line of the form
\[ (p(t)x'(t))'+f(t,x)+q(t)=0, \]
\[ \alpha_1 x(0)-\beta_1\lim_{t\rightarrow 0^+}p(t)x'(t)=0, \]
\[ \alpha_2 \lim_{t\rightarrow +\infty}x(t)-\beta_2\lim_{t\rightarrow +\infty}p(t)x'(t)=0, \]
where \(\alpha_i\geq 0,\; \beta_i>0\; (i=1,2)\) with \[ \alpha_2\beta_1+\alpha_1\beta_2 +\alpha_1\alpha_2\int^{+\infty}_0 \frac{ds}{p(s)}>0;\quad q:(0,\infty)\to R \]
is Lebesgue integrable. Their main results generalize those obtained in Yansheng Liu [Appl. Math. Comput. 144, 543–556 (2003; Zbl 1036.34027)].
Reviewer: Ruyun Ma (Lanzhou)

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations

Citations:

Zbl 1036.34027
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References:

[1] 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
[2] Liu, Y.S., Existence and unboundedness of positive solutions for singular boundary value problems on half-line, Appl. math. comput., 144, 543-556, (2003) · Zbl 1036.34027
[3] Chen, S.Z.; Zhang, Y., Singular boundary value problems on a half-line, J. math. anal. appl., 195, 449-468, (1995) · Zbl 0852.34019
[4] Liu, X.Y., Solutions of impulsive boundary value problems on the half-line, J. math. anal. appl., 222, 411-430, (1998) · Zbl 0912.34021
[5] Yan, B.Q., Boundary value problems on the half-line with impulses and infinite delay, J. math. anal. appl., 259, 94-114, (2001) · Zbl 1009.34059
[6] Aris, R., Introduction to the analysis of chemical reactors, (1965), Prentice-Hall Englewood Cliffs, NJ
[7] Zhang, X.G.; Liu, L.S., Positive solutions of superlinear semipositone singular Dirichlet boundary value problems, J. math. anal. appl., 316, 525-537, (2006) · Zbl 1097.34019
[8] Xu, X.A., Positive solutions for singular semi-positone boundary value problems, J. math. anal. appl., 273, 480-491, (2002) · Zbl 1028.34020
[9] X.A. Xu, Existence and multiplicity of positive solutions for multi-parameter three point differential equations system, J. Math. Anal. Appl., in press. · Zbl 1113.34017
[10] X.A. Xu, Positive solutions for singular semi-positone three point systems, Nonlinear Anal., in press. · Zbl 1114.34024
[11] Lian, H.R.; Ge, W.G., Existence of positive solutions for Sturm-Liouville boundary value problems on the half-line, J. math. anal. appl., 321, 781-792, (2006) · Zbl 1104.34020
[12] Guo, D.J.; Lakshmikantham, V., Nonlinear problems in abstract cone, (1988), Academic Press Inc. New York
[13] Agarwal, R.P.; O’Regan, D., Infinite interval problems for differential, difference and integral equations, (2001), Kluwer Academic · Zbl 1003.39017
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