# Article

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Keywords:
third order; half-line; $\phi$-Laplacian; singular problem; positive solution; derivative dependance; upper and lower solution
Summary:
This work is devoted to the existence of solutions for a class of singular third-order boundary value problem associated with a $\phi$-Laplacian operator and posed on the positive half-line; the nonlinearity also depends on the first derivative. The upper and lower solution method combined with the fixed point theory guarantee the existence of positive solutions when the nonlinearity is monotonic with respect to its arguments and may have a space singularity; however no Nagumo type condition is assumed. An example of application illustrates the applicability of the existence result.
References:
[1] Agarwal, R.P., Meehan, M., O’Regan, D.: Fixed Point Theory and Applications. Cambridge Tracts in Mathematics, vol. 141, Cambridge University Press, 2001. DOI 10.1017/CBO9780511543005.008 | MR 1825411 | Zbl 0960.54027
[2] Agarwal, R.P., O’Regan, D.: Infinite Interval Problems for Differential, Difference, and Integral Equations. Kluwer Academic Publishers, Dordrecht, 2001. MR 1845855 | Zbl 0988.34002
[3] Corduneanu, C.: Integral Equations and Stability of Feedback Systems. vol. 104, Academic Press, New York, 1973. 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 (112) (2008), 2940–2952. DOI 10.1016/j.camwa.2007.11.023 | MR 2401442 | Zbl 1142.34316
[5] Djebali, S., Mebarki, K.: On the singular generalized Fisher-like equation with derivative depending nonlinearity. Appl. Math. Comput. 205 (1) (2008), 336–351. DOI 10.1016/j.amc.2008.08.009 | MR 2466638 | Zbl 1183.34039
[6] Djebali, S., Mebarki, K.: Multiple unbounded positive solutions for three-point bvps with sign-changing nonlinearities on the positive half-line. Acta Appl. Math. 109 (2) (2010), 361–388. DOI 10.1007/s10440-008-9322-3 | MR 2585794 | Zbl 1195.34042
[7] Djebali, S., Saifi, O.: Positive solutions for singular $\phi$-Laplacian BVPs on the positive half-line. EJQTDE (56) (2009), 24pp. MR 2546349 | Zbl 1201.34040
[8] Djebali, S., Saifi, O.: Positive solutions for singular BVPs on the positive half-line with sign changing and derivative depending nonlinearity. Acta Appl. Math. 110 (2) (2010), 639–665. DOI 10.1007/s10440-009-9466-9 | MR 2610584
[9] Djebali, S., Saifi, O.: Upper and lower solution method for singular $\phi -$Laplacian BVPs with derivative depending nonlinearity on $[0,+\infty )$. Commun. Appl. Anal. 14 (4) (2010), 463–480. MR 2757411
[10] Djebali, S., Saifi, O.: Third order BVPs with $\phi$-Laplacian operators on $[0,+\infty )$. Afr. Diaspora J. Math. 16 (1) (2013), 1–17. MR 3091711 | Zbl 1283.34019
[11] Djebali, S., Saifi, O.: Upper and lower solutions for $\phi$-Laplacian third-order BVPs on the half-line. Cubo 16 (1) (2014), 105–116. DOI 10.4067/S0719-06462014000100010 | MR 3185792 | Zbl 1319.34038
[12] Guo, Y., Yu, C., Wang, J.: Existence of three positive solutions for $m$-point boundary value problems on infinite intervals. Nonlinear Anal. 71 (3–4) (2009), 717–722. DOI 10.1016/j.na.2008.10.126 | MR 2527493 | Zbl 1172.34310
[13] Han, G., Li, F.: Multiple solutions of some fourth-order boundary value problems. Nonlinear Anal. 66 (11) (2007), 2591–2603. DOI 10.1016/j.na.2006.03.042 | MR 2312608 | Zbl 1126.34013
[14] 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 (1) (2010), 27–43. DOI 10.1007/s10440-009-9528-z | MR 2653048 | Zbl 1203.34038
[15] Tian, Y., Ge, W., Shan, W.: Positive solutions for three-point boundary value problem on the half-line. Comput. Math. Appl. 53 (7) (2007), 1029–1039. DOI 10.1016/j.camwa.2006.08.035 | MR 2331357 | Zbl 1131.34019
[16] Yan, B., Liu, Y.: Unbounded solutions of the singular boundary value problems for second order differential equations on the half-line. Appl. Math. Comput. 147 (3) (2004), 629–644. DOI 10.1016/S0096-3003(02)00801-9 | MR 2011077 | Zbl 1045.34009
[17] Yan, B., O’Regan, D., Agarwal, R.P.: Positive solutions for second order singular boundary value problems with derivative dependence on infinite intervals. Acta Appl. Math. 103 (1) (2008), 19–57. MR 2415171 | Zbl 1158.34011
[18] Yang, Y., Zhang, J.: Existence of solutions for some fourth-order boundary value problems with parameters. Nonlinear Anal. 69 (4) (2008), 1364–1375. DOI 10.1016/j.na.2007.06.035 | MR 2426697 | Zbl 1166.34012
[19] Yang, Y., Zhang, J.: Nontrivial solutions for some fourth order boundary value problems with parameters. Nonlinear Anal. 70 (11) (2009), 3966–3977. DOI 10.1016/j.na.2008.08.005 | MR 2515313 | Zbl 1171.34006

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