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Exact solutions of a class of second-order nonlocal boundary value problems and applications. (English) Zbl 1189.34005

The author considers the second-order linear differential equation

$-{u}^{\text{'}\text{'}}+{k}^{2}u=f\left(t\right),\phantom{\rule{1.em}{0ex}}t\in \left[a,b\right]$

subject to different multipoint boundary conditions. He presents the exact solution by means of Green’s function. As applications he studies uniqueness and iteration of the positive solutions for a nonlinear singular second-order $m$-point boundary value problem.

##### MSC:
 34A05 Methods of solution of ODE 34B05 Linear boundary value problems for ODE 34B10 Nonlocal and multipoint boundary value problems for ODE 34B27 Green functions 34A45 Theoretical approximation of solutions of ODE 34B18 Positive solutions of nonlinear boundary value problems for ODE
##### References:
 [1] Pokornyi, Yu V.; Borovskikh, A. V.: The connection of the Green’s function and the influence function for nonclassical problems, Journal of mathematical sciences 119, No. 6, 739-768 (2004) · Zbl 1079.34016 · doi:10.1023/B:JOTH.0000012754.03115.4f [2] Zhao, Zengqin: Solutions and greens functions for some linear second-order three-point boundary value problems, Computers and mathematics with applications 56, 104-113 (2008) · Zbl 1145.34324 · doi:10.1016/j.camwa.2007.11.037 [3] Bai, Chuanzhi; Xie, Dapeng; Liu, Yang; Wang, Chunli: Positive solutions for second-order four-point boundary value problems with alternating coefficient, Nonlinear analysis 70, 2014-2023 (2009) · Zbl 1173.34012 · doi:10.1016/j.na.2008.02.099 [4] Chen, Haibo; Li, Peiluan: Three-point boundary value problems for second-order ordinary differential equations in Banach spaces, Computers and mathematics with applications 56, 1852-1860 (2008) · Zbl 1152.34362 · doi:10.1016/j.camwa.2008.04.024 [5] Han, Xiaoling: Positive solutions for a three-point boundary value problem at resonance, Journal of mathematical analysis and applications 336, 556-568 (2007) · Zbl 1125.34014 · doi:10.1016/j.jmaa.2007.02.069 [6] Liu, Xiping; Jia, Mei: Existence of monotone positive solutions to a type of three-point boundary value problem, Acta mathematicae applicatae sinica 30, No. 1, 111-119 (2007) · Zbl 1125.34008 [7] Pang, Huihui; Feng, Meiqiang; Ge, Weigao: Existence and monotone iteration of positive solutions for a three-point boundary value problem, Applied mathematics letters 21, 656-661 (2008) · Zbl 1152.34313 · doi:10.1016/j.aml.2007.07.019 [8] Sun, Jingxian; Xu, Xian; Donal, O’regan: Nodal solutions for m-point boundary value problems using bifurcation methods, Nonlinear analysis 68, 3034-3046 (2008) · Zbl 1141.34009 · doi:10.1016/j.na.2007.02.043 [9] Zhang, Kemei; Xie, Xue-Jun: Existence of sign-changing solutions for some asymptotically linear three-point boundary value problems, Nonlinear analysis 70, 2796-2805 (2009) · Zbl 1165.34005 · doi:10.1016/j.na.2008.04.004 [10] Feng, Hanying; Ge, Weigao; Jiang, Ming: Multiple positive solutions for m-point boundary-value problems with a one-dimensional p-Laplacian, Nonlinear analysis 68, 2269-2279 (2008) · Zbl 1138.34005 · doi:10.1016/j.na.2007.01.052 [11] Feng, Meiqiang; Zhang, Xuemei; Ge, Weigao: Exact number of solutions for a class of two-point boundary value problems with one-dimensional p-Laplacian, Journal of mathematical analysis and applications 338, 784-792 (2008) · Zbl 1147.34008 · doi:10.1016/j.jmaa.2007.04.058 [12] Guo, Yanping; Ji, Yude; Liu, Xiujun: Multiple positive solutions for some multi-point boundary value problems with p-Laplacian, Journal of computational and applied mathematics 216, 144-156 (2008) · Zbl 1141.34017 · doi:10.1016/j.cam.2007.04.023 [13] Ji, Dehong; Ge, Weigao: Existence of multiple positive solutions for Sturm – Liouville-like four-point boundary value problem with p-Laplacian, Nonlinear analysis 68, 2638-2646 (2008) · Zbl 1145.34309 · doi:10.1016/j.na.2007.02.010 [14] Lian, Hairong; Ge, Weigao: Positive solutions for a four-point boundary value problem with the p-Laplacian, Nonlinear analysis 68, 3493-3503 (2008) · Zbl 1151.34019 · doi:10.1016/j.na.2007.03.042 [15] Xu, Fuyi: Multiple positive solutions for nonlinear singular m-point boundary value problem, Applied mathematics and computation 204, 450-460 (2008) · Zbl 1159.34018 · doi:10.1016/j.amc.2008.05.139 [16] Zhu, Yanling; Wang, Kai: On the existence of solutions of p-Laplacian m-point boundary value problem at resonance, Nonlinear analysis 70, 1557-1564 (2009) · Zbl 1165.34314 · doi:10.1016/j.na.2008.02.035