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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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References:
 [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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