Geodesic
The method of upper and lower solutions for a Lidstone boundary value problem
Czechoslovak Mathematical Journal, Tome 55 (2005) no. 3, pp. 639-652
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In this paper we develop the monotone method in the presence of upper and lower solutions for the $2$nd order Lidstone boundary value problem \[ u^{(2n)}(t)=f(t,u(t),u^{\prime \prime }(t),\dots ,u^{(2(n-1))}(t)),\quad 0
In this paper we develop the monotone method in the presence of upper and lower solutions for the $2$nd order Lidstone boundary value problem \[ u^{(2n)}(t)=f(t,u(t),u^{\prime \prime }(t),\dots ,u^{(2(n-1))}(t)),\quad 01, u^{(2i)}(0)=u^{(2i)}(1)=0,\quad 0\le i\le n-1, \] where $f\:[0,1]\times \mathbb{R}^{n}\rightarrow \mathbb{R}$ is continuous. We obtain sufficient conditions on $f$ to guarantee the existence of solutions between a lower solution and an upper solution for the higher order boundary value problem.
Classification : 34B15
Keywords: $n$-parameter eigenvalue problem; Lidstone boundary value problem; lower solution; upper solution 
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Guo, Yanping; Gao, Ying. The method of upper and lower solutions for a Lidstone boundary value problem. Czechoslovak Mathematical Journal, Tome 55 (2005) no. 3, pp. 639-652. http://geodesic.mathdoc.fr/item/CMJ_2005_55_3_a5/

[1] A.  R.  Aftabizadeh: Existence and uniqueness theorems for fourth-order boundary value problems. J.  Math. Anal. Appl. 116 (1986), 415–426. | DOI | MR | Zbl

[2] C. De Coster, C.  Fabry and F.  Munyamarere: Nonresonance conditions for fourth-order nonlinear boundary problems. Internat. J.  Math. Sci. 17 (1994), 725–740. | DOI | MR

[3] M. A. Del Pino and R. F.  Manasevich: Existence for a fourth-order boundary value problem under a two parameter nonresonance condition. Proc. Amer. Math. Soc. 112 (1991), 81–86. | DOI | MR

[4] C. P.  Gupta: Existence and uniqueness theorem for a bending of an elastic beam equation. Appl. Anal. 26 (1988), 289–304. | DOI | MR

[5] R. A.  Usmani: A uniqueness theorem for a boundary value problem. Proc. Amer. Math. Soc. 77 (1979), 327–335. | DOI | MR | Zbl

[6] R.  Agarwal: On fourth-order boundary value problems arising in beam analysis. Differential Integral Equations 2 (1989), 91–110. | MR | Zbl

[7] A.  Cabada: The method of lower and upper solutions for second, third, fourth and higher order boundary value problems. J.  Math. Anal. Appl. 185 (1994), 302–320. | DOI | MR | Zbl

[8] C.  De Coster and L.  Sanchez: Upper and lower solutions, Ambrosetti-Prodi problem and positive solutions for fourth-order O. D. E. Riv. Mat. Pura. Appl. 14 (1994), 1129–1138. | MR

[9] P.  Korman: A maximum principle for fourth-order ordinary differential equations. Appl. Anal. 33 (1989), 267–273. | DOI | MR | Zbl

[10] J.  Schröder: Fourth-order two-point boundary value problems. Nonlinear Anal. 8 (1984), 107–114. | DOI

[11] Zhanbing Bai: The method of lower and upper solutions for a bending of an elastic beam equation. J.  Math. Anal. Appl. 248 (2000), 195–402. | DOI | MR

[12] R. Y.  Ma, J. H. Zhang and S. M.  Fu: The method of lower and upper solutions for fourth-order two-point boundary value problems. J.  Math. Anal. Appl. 215 (1997), 415–422. | DOI | MR

[13] J. M.  Davis and J.  Henderson: Triple positive symmetric solutions for a Lidstone boundary value problem. Differential Equations Dynam. Systems 7 (1999), 321–330. | MR

[14] J. M.  Davis, P. W.  Eloe and J.  Henderson: Triple positive solutions and dependence on higher order derivatives. J.  Math. Anal. Appl. 237 (1999), 710–720. | DOI | MR

[15] J. M.  Davis, J.  Henderson and P. J. Y.  Wong: General Lidstone problems: multiplicity and symmetry of solutions. J. Math. Anal. Appl. 251 (2000), 527–548. | DOI | MR

[16] D. Gilbarg and N. S.  Trudinger: Elliptic Partial Differential Equations of Second Order. Springer-Verlag, New York, 1977. | MR