Zhang, Xinguang Eigenvalue of higher-order semipositone multi-point boundary value problems with derivatives. (English) Zbl 1154.34016 Appl. Math. Comput. 201, No. 1-2, 361-370 (2008). The authors consider the higher order \(m\)-point boundary value problem \[ (x^{(n)})(t)+\lambda f(t, x(t), x'(t), \cdots, x^{(n-2)}(t))=0, \quad t\in (0,1), \]\[ x^{(i)}(0)=0, \quad 0\leq i\leq n-3,\quad x^{(n-1)}(0)=\sum^{m-2}_{i=1}a_i x^{(n-1)}(\xi_i), \]\(x^{(n-2)}(1)=\sum^{m-2}_{i=1}b_i x^{(n-2)}(\xi_i)\), where \(0<\xi_1<\xi_2<\cdots<\xi_{m-2}<1\) with \(0<\sum^{m-2}_{i=1}a_i<1\) and \(0<\sum^{m-2}_{i=1}b_i<1\), \(f\in C[(0,1)\times \mathbb{R}^{n-1}, \mathbb{R}]\) satisfies \[ f(t,u_1, \dots, u_{n-1})\geq p(t) \text{ and } p\in L^1[(0,1), (0,+\infty)], \] and \(f\) may be singular at \(t=0\) and/or \(t=1\). He proves the existence of positive solutions by applying a fixed point theorem in cones. For earlier results, see R. Ma and N. Castaneda [J. Math. Appl. Anal. 256, 556–567 (2001; Zbl 0988.34009)]. Reviewer: Ruyun Ma (Lanzhou) Cited in 6 Documents MSC: 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B16 Singular nonlinear boundary value problems for ordinary differential equations 47N20 Applications of operator theory to differential and integral equations 34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators Keywords:Multi-point boundary value problem; semipositone; positive solutions; cone cone Citations:Zbl 0988.34009 PDFBibTeX XMLCite \textit{X. Zhang}, Appl. Math. Comput. 201, No. 1--2, 361--370 (2008; Zbl 1154.34016) Full Text: DOI References: [1] Moshinsky, M., Sobre los problems a la frontiera en una dimension de caracteristicas discontinuas, Bol. Mat. Mexicana, 7, 1-25 (1950) [2] Timoshenko, S., Theory of Elastic Stability (1961), McGraw-Hill: McGraw-Hill New York [3] II’in, V. A.; Moiseev, E. I., Nonlocal boundary value problem for second kind for a Sturm-Liouville operator, Different. Equat., 23, 8, 979-987 (1987) · Zbl 0668.34024 [4] Gupta, C. P., A generalized multi-point boundary value problem for second order ordinary differential equations, Appl. Math. Comput., 89, 133-146 (1998) · Zbl 0910.34032 [5] Ma, R.; Castaneda, N., Existence of solutions of nonlinear \(m\)-point boundary value problems, J. Math. Anal. Appl., 256, 556-567 (2001) · Zbl 0988.34009 [6] Guo, Y.; Shan, W.; Ge, W., Positive solutions for second-order \(m\)-point boundary value problems, J. Comput. Appl. Math., 151, 415-424 (2003) · Zbl 1026.34016 [7] Chen, S.; Hu, J.; Li, L.; Wang, C., Existence result for \(n\)-ponit boundary value problem of second order ordinary differential equations, J. Comput. Appl. Math., 180, 425-432 (2005) · Zbl 1069.34011 [8] Ma, R., Multiple positive solutions for nonlinear \(m\)-point boundary value problem, Appl. Math. Comput., 148, 249-262 (2004) · Zbl 1046.34030 [9] Zhang, Z.; Wang, J., On existence and multiplicity of positive solutions to singular multi-point boundary value problems, J. Math. Anal. Appl., 295, 502-512 (2004) · Zbl 1056.34018 [10] Ma, R.; Thompson, B., Positive solutions for nonlinear \(m\)-point eigenvalue problems, J. Math. Anal. Appl., 297, 24-37 (2004) · Zbl 1057.34011 [11] Aris, R., Introduction to the Analysis of Chemical Reactors (1965), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ [12] Ma, R.; Ma, Q., Positive solutions for semipositone \(m\)-point boundary value problems, Acta Math. Sinica., 20, 2, 273-282 (2004) · Zbl 1063.34009 [13] Guo, D.; Lakshmikantham, V., Nonlinear Problems in Abstract Cone (1988), Academic Press Inc.: Academic Press Inc. New York · Zbl 0661.47045 [14] Castro, A.; Maya, C.; Shivaji, R., Nonlinear eigenvalue problems with semipositone, Electron. J. Diff. Equat. Conf., 05, 33-49 (2000) · Zbl 0959.35045 [15] Anuradha, V.; Hai, D. D.; Shivaji, R., Existence results for superlinear semipositone BVP’s, Proc. Am. Math. Soc., 124, 746-757 (1996) · Zbl 0857.34032 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.