# zbMATH — the first resource for mathematics

Positive solutions for nonlinear $$n$$th-order singular nonlocal boundary value problems. (English) Zbl 1148.34015
Summary: We study the existence and multiplicity of positive solutions for a class of $$n$$th-order singular nonlocal boundary value problems
$u^{(n)}(t)+a(t)f(t,u) = 0,\quad t\in (0,1),\quad u(0) = 0,\quad u'(0) = 0,\dots,u^{(n-2)}(0) =0,\quad \alpha u(\eta) =u(1),$
where $$0<\eta <1$$, $$0<\alpha\eta^{n-1}<1$$. The singularity may appear at $$t=0$$ and/or $$t = 1$$. The Krasnosel’skii-Guo theorem on cone expansion and compression is used in this study. The main results improve and generalize the existing results.

##### 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
Full Text:
##### References:
 [1] Gupta, CP, Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation, Journal of Mathematical Analysis and Applications, 168, 540-551, (1992) · Zbl 0763.34009 [2] Il’in, VA; Moiseev, EI, A nonlocal boundary value problem of the first kind for the Sturm-Liouville operator in differential and difference interpretations, Differential Equations, 23, 803-810, (1987) · Zbl 0668.34025 [3] Eloe, PW; Henderson, J, Positive solutions for higher order ordinary differential equations, Electronic Journal of Differential Equations, 1995, 8 pages, (1995) [4] Ma, RY, Positive solutions for a nonlinear three-point boundary-value problem, Electronic Journal of Differential Equations, 1999, 8 pages, (1999) [5] O’Regan, D, Singular second order boundary value problems, Nonlinear Analysis, 15, 1097-1109, (1990) · Zbl 0732.34021 [6] Hao, Z-C; Mao, AM, A necessary and sufficient condition for the existence of positive solutions to a class of singular second-order boundary value problems, Journal of Systems Science and Mathematical Sciences, 21, 93-100, (2001) · Zbl 0990.34027 [7] Agarwal, RP; O’Regan, D, Positive solutions to superlinear singular boundary value problems, Journal of Computational and Applied Mathematics, 88, 129-147, (1998) · Zbl 0902.34017 [8] Ma, RY, Positive solutions of singular second-order boundary value problems, Acta Mathematica Sinica, 41, 1225-1230, (1998) · Zbl 1027.34025 [9] Hao, Z-C; Liang, J; Xiao, T-J, Positive solutions of operator equations on half-line, Journal of Mathematical Analysis and Applications, 314, 423-435, (2006) · Zbl 1086.47035 [10] Kiguradze, IT; Lomtatidze, AG, On certain boundary value problems for second-order linear ordinary differential equations with singularities, Journal of Mathematical Analysis and Applications, 101, 325-347, (1984) · Zbl 0559.34012 [11] Lomtatidze, AG, A boundary value problem for second-order nonlinear ordinary differential equations with singularities, Differential Equations, 22, 416-426, (1986) [12] Lomtatidze, AG, Positive solutions of boundary value problems for second-order ordinary differential equations with singularities, Differential Equations, 23, 1146-1152, (1987) · Zbl 0714.34029 [13] Eloe, PW; Ahmad, B, Positive solutions of a nonlinear[inlineequation not available: see fulltext.]th order boundary value problem with nonlocal conditions, Applied Mathematics Letters, 18, 521-527, (2005) · Zbl 1074.34022 [14] Guo DJ, Lakshmikantham V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering. Volume 5. Academic Press, Boston, Mass, USA; 1988. · Zbl 0661.47045
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.