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Positive solutions of a nonlinear $n$th order boundary value problem with nonlocal conditions. (English) Zbl 1074.34022
Summary: We discuss the existence of positive solutions of a nonlinear $n$th-order boundary value problem $$u^{(n)}+ a(t) f(u)= 0,\quad t\in (0,1),$$ $$u(0)= 0,\quad u'(0)= 0,\ u'(0)= 0,\dots, u^{(n-2)}(0)= 0,\quad\alpha u(\eta)= u(1),$$ with $0< \eta< 1$, $0< \alpha\eta^{n-1}< 1$. In particular, we establish the existence of at least one positive solution if $f$ is either superlinear or sublinear by applying the fixed-point theorem in cones due to Krasnoselskij and Guo.

##### MSC:
 34B18 Positive solutions of nonlinear boundary value problems for ODE 34B15 Nonlinear boundary value problems for ODE
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##### References:
 [1] Il’in, V. A.; Moiseev, E. I.: Nonlocal boundary value problem of the first kind for a Sturm Liouville operator in its differential and finite difference aspects. Differ. equ. 23, No. 7, 803-810 (1987) [2] Il’in, V. A.; Moiseev, E. I.: Nonlocal boundary value problem of the second kind for a Sturm Liouville operator. Differ. equ. 23, No. 8, 979-987 (1987) · Zbl 0668.34024 [3] Gupta, G. P.: Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation. J. math. Anal. appl. 168, 540-551 (1992) · Zbl 0763.34009 [4] Eloe, P. W.; Henderson, J.: Positive solutions for higher order ordinary differential equations. Electron. J. Differential equations 03, 1-8 (1995) · Zbl 0814.34017 [5] Eloe, P. W.; Henderson, J.; Wong, P. J. Y.: Positive solutions for two point boundary value problems. Dynam. systems appl. 2, 135-144 (1996) · Zbl 0876.34016 [6] Lian, W. C.; Wong, F. H.; Yeh, C. C.: On existence of positive solutions of nonlinear second order differential equations. Proc. amer. Math. soc. 124, 1111-1126 (1996) · Zbl 0857.34036 [7] Ma, R.: Positive solutions for a nonlinear three-point boundary value problem. Electron. J. Differential equations 34, 1-8 (1998) [8] Ma, R.: Positive solutions of a nonlinear m-point boundary value problem. Comput. math. Appl. 42, 755-765 (2001) · Zbl 0987.34018 [9] Coppel, W.: Disconjugacy. Lecture notes in mathematics 220 (1971) · Zbl 0224.34003 [10] Guo, D.; Lakshmikantham, V.: Nonlinear problems in abstract cones. (1988) · Zbl 0661.47045 [11] Erbe, L. H.; Wang, H.: On the existence of positive solutions of ordinary differential equations. Proc. amer. Math. soc. 120, 743-748 (1994) · Zbl 0802.34018 [12] Bohner, M.; Peterson, A.:. (2003) [13] P.W. Eloe, Positive operators and maximum principles, Cubo (in press) [14] Eloe, P. W.; Henderson, J.: Inequalities based on a generalization of concavity. Proc. amer. Math. soc. 125, 2103-2108 (1997) · Zbl 0868.34008 [15] Eloe, P. W.: Maximum principles for a family of nonlocal boundary value problems. Adv. difference equations 3, 201-210 (2004) · Zbl 1083.39018