# Article

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Keywords:
singular nonlinear boundary value problem; positive solution; Krasnosel'skii fixed point theorem; multi-point; half-line
Summary:
We establish not only sufficient but also necessary conditions for existence of solutions to a singular multi-point third-order boundary value problem posed on the half-line. Our existence results are based on the Krasnosel'skii fixed point theorem on cone compression and expansion. Nonexistence results are proved under suitable a priori estimates. The nonlinearity $f=f(t,x,y)$ which satisfies upper and lower-homogeneity conditions in the space variables $x, y$ may be also singular at time $t=0$. Two examples of applications are included to illustrate the existence theorems.
References:
[1] Agarwal, R. P., O'Regan, D.: Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer Academic Publishers, Dordrecht (2001). DOI 10.1007/978-94-010-0718-4 | MR 1845855 | Zbl 0988.34002
[2] Chen, S., Zhang, Y.: Singular boundary value problems on a half-line. J. Math. Anal. Appl. 195 (1995), 449-468. DOI 10.1006/jmaa.1995.1367 | MR 1354555 | Zbl 0852.34019
[3] Corduneanu, C.: Integral Equations and Stability of Feedback Systems. Mathematics in Science and Engineering 104. Academic Press, New York (1973). DOI 10.1016/s0076-5392(08)x6099-0 | MR 0358245 | Zbl 0273.45001
[4] Djebali, S., Mebarki, K.: Multiple positive solutions for singular BVPs on the positive half-line. Comput. Math. Appl. 55 (2008), 2940-2952. DOI 10.1016/j.camwa.2007.11.023 | MR 2401442 | Zbl 1142.34316
[5] Djebali, S., Saifi, O.: Third order BVPs with $\phi$-Laplacian operators on $[0,+\infty)$. Afr. Diaspora J. Math. 16 (2013), 1-17. MR 3091711 | Zbl 1283.34019
[6] Guo, D. J., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Notes and Reports in Mathematics in Science and Engineering 5. Academic Press, Boston (1988). DOI 10.1016/c2013-0-10750-7 | MR 0959889 | Zbl 0661.47045
[7] Liang, S., Zhang, J.: Positive solutions for singular third-order boundary value problem with dependence on the first order derivative on the half-line. Acta Appl. Math. 111 (2010), 27-43. DOI 10.1007/s10440-009-9528-z | MR 2653048 | Zbl 1203.34038
[8] Liu, Y.: Existence and unboundedness of positive solutions for singular boundary value problems on half-line. Appl. Math. Comput. 144 (2003), 543-556. DOI 10.1016/S0096-3003(02)00431-9 | MR 1994092 | Zbl 1036.34027
[9] Wei, Z.: A necessary and sufficient condition for the existence of positive solutions of singular super-linear $m$-point boundary value problems. Appl. Math. Comput. 179 (2006), 67-78. DOI 10.1016/j.amc.2005.11.077 | MR 2260858 | Zbl 1166.34305
[10] Wei, Z.: Some necessary and sufficient conditions for existence of positive solutions for third order singular super-linear multi-point boundary value problems. J. Appl. Math. Comput. 46 (2014), 407-422. DOI 10.1007/s12190-014-0756-7 | MR 3252121 | Zbl 1311.34052
[11] Yan, B., O'Regan, D., Agarwal, R. P.: Unbounded positive solutions for second order singular boundary value problems with derivative dependence on infinite intervals. Funkc. Ekvacioj, Ser. Int. 51 (2008), 81-106. DOI 10.1619/fesi.51.81 | MR 2427544 | Zbl 1158.34011

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