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On the positive solutions of Lidstone boundary value problems. (English) Zbl 1093.34515

Summary: Existence, nonexistence and multiplicity of positive solutions are considered for the Lidstone boundary value problem \[ (-1)^n\omega^{(2n)}= \lambda f\bigl( t,\omega)\bigr),\;0\leq t\leq 1,\quad \omega^{(2i)}(0)=\omega^{(2i)} (1)=0, \;0\leq i \leq n-1, \] with \(\lambda>0\). By using Krasnoselski’s fixed-point theorem, some new results are obtained. Particularly, it is proved that the Lidstone boundary value problem has \(N\) positive solutions under suitable conditions, where \(N\) is an arbitrary natural number.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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[1] Guo, D.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press: Academic Press Boston · Zbl 0661.47045
[2] Erbe, L. H.; Hu, S.; Wang, H., Multiple positive solutions of some boundary value problems, J. Math. Anal. Appl., 184, 640 (1994) · Zbl 0805.34021
[3] Yao, Q.; Bai, Z., Existence of positive solutions of BVP for \(u^{(4)}(t)\)−\( λh (t)f(u(t))=0\), Chinese Ann. Math., 20(A), 575 (1999), in Chinese · Zbl 0948.34502
[4] Q. Yao, Positive radial solutions of semilinear elliptic equation Δ \(ufru\); Q. Yao, Positive radial solutions of semilinear elliptic equation Δ \(ufru\) · Zbl 1108.35066
[5] Agarwal, R. P., Boundary Value Problems for Higher Order Differential Equations (1986), World Scientific: World Scientific Singapore · Zbl 0598.65062
[6] Wong, P. J.Y.; Agarwal, R. P., Eigenvalues of Lidstone boundary value problems, Appl. Math. Comput., 104, 15 (1999) · Zbl 0933.65089
[7] Henderson, J.; Thompson, H. B., Multiple symmetric positive solutions for a second order boundary value problem, Proc. Amer. Math. Soc., 128, 2373-2379 (2000) · Zbl 0949.34016
[8] Yao, Q., Monotone iterative technique and positive solutions of Lidstone boundary value problems, Appl. Math. Comput., 131, 477-485 (2002) · Zbl 1024.34019
[9] Krasnosel’skii, M. A., Positive Solutions of Operator Equations (1964), Noordhoff Groningen: Noordhoff Groningen Netherland · Zbl 0121.10604
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