# Article

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Keywords:
boundary-value problems; positive solutions; fixed-point theorem; cone
Summary:
The paper deals with the existence of multiple positive solutions for the boundary value problem $$\begin{cases} (\varphi (p(t)u^{(n-1)})(t))' + a(t)f(t, u(t), u'(t), \ldots , u^{(n-2)}(t)) = 0, \quad \ 0 < t < 1, \\ u^{(i)}(0) = 0, \quad i = 0, 1, \ldots , n - 3,\\ u^{(n-2)}(0) = \sum _{i=1}^{m-2}\alpha _iu^{(n-2)}(\xi _i),\quad u^{(n-1)}(1) = 0, \end{cases}$$ where $\varphi \colon \Bbb R \rightarrow \Bbb R$ is an increasing homeomorphism and a positive homomorphism with $\varphi (0) = 0$. Using a fixed-point theorem for operators on a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem.
References:
[1] Agarwal, R. P., O'Regan, D.: Multiplicity results for singular conjuate, focal, and $(N, P)$ problems. J. Differ. Equations 170 142-156 (2001). DOI 10.1006/jdeq.2000.3808 | MR 1813103
[2] Bai, C., Fang, J.: Existence of multiple positive solution for nonlinear $m$-point boundary value problems. Appl. Math. Comput. 140 (2003), 297-305. DOI 10.1016/S0096-3003(02)00227-8 | MR 1953901
[3] Baxley, J. V.: Existence and uniqueness of nonlinear boundary value problems on infinite intervals. J. Math. Anal. Appl. 147 (1990), 122-133. DOI 10.1016/0022-247X(90)90388-V | MR 1044690
[4] Deimling, K.: Nonlinear Functional Analysis. Springer, New York (1985). MR 0787404 | Zbl 0559.47040
[5] Feng, W., Webb, J. R. L.: Solvability of an $m$-point boundary value problems with nonlinear growth. J. Math. Anal. Appl. 212 (1997), 467-480. DOI 10.1006/jmaa.1997.5520 | MR 1464891
[6] Il'in, V. A., Moiseev, E. I.: Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator. Differ. Equations 23 (1987), 979-987.
[7] al, P. Kelevedjiev et: Another understanding of fourth-order four-point boundary-value problems. Electron. J. Differ. Equ., Paper No. 47 (2008), 1-15. MR 2392951
[8] Kong, L., Kong, Q.: Second-order boundary value problems with nonhomogeneous boundary conditions (II). J. Math. Anal. Appl. 330 (2007), 1393-1411. DOI 10.1016/j.jmaa.2006.08.064 | MR 2308449 | Zbl 1119.34009
[9] Lan, K. Q.: Multiple positive solutions of semilinear differential equations with singularities. J. London Math. Soc. 63 (2001), 690-704. DOI 10.1112/S002461070100206X | MR 1825983 | Zbl 1032.34019
[10] Liang, S. H., Zhang, J. H.: The existence of countably many positive solutions for nonlinear singular $m$-point boundary value problems. J. Comput. Appl. Math. 214 (2008), 78-89. DOI 10.1016/j.cam.2007.02.020 | MR 2391674 | Zbl 1183.34031
[11] Liu, B. F., Zhang, J. H.: The existence of positive solutions for some nonlinear boundary value problems with linear mixed boundary conditions. J. Math. Anal. Appl. 309 (2005), 505-516. DOI 10.1016/j.jmaa.2004.09.036 | MR 2154132 | Zbl 1086.34022
[12] Liu, Yuji: Non-homogeneous boundary-value problems of higher order differential equations with $p$-Laplacian. Electron J. Differ. Equ., Paper No. 20 (2008), 1-43. MR 2383384 | Zbl 1139.34012
[13] Wang, J. Y.: The existence of positive solutions for the one-dimensional $p$-Laplacian. Proc. Amer. Math. Soc. 125 (1997), 2275-2283. DOI 10.1090/S0002-9939-97-04148-8 | MR 1423340 | Zbl 0884.34032
[14] Wang, Y., Ge, W.: Existence of multiple positive solutions for multi-point boundary value problems with a one-dimensional $p$-Laplacian. Nonlinear Anal., Theory Methods Appl. 67 (2007), 476-485. MR 2317182
[15] Wang, Y., Hou, C.: Existence of multiple positive solutions for one-dimensional $p$-Laplacian. J. Math. Anal. Appl. 315 (2006), 144-153. DOI 10.1016/j.jmaa.2005.09.085 | MR 2196536 | Zbl 1098.34017
[16] Webb, J. R. L.: Positive solutions of some three point boundary value problems via fixed point index theory. Nonlinear Anal. 47 (2001), 4319-4332. DOI 10.1016/S0362-546X(01)00547-8 | MR 1975828 | Zbl 1042.34527
[17] Zhou, Y. M., Su, H.: Positive solutions of four-point boundary value problems for higher-order with $p$-Laplacian operator. Electron. J. Differ. Equ., Paper No. 05 1-14 (2007). MR 2278419 | Zbl 1118.34021

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