×

Solutions of 2\(n\)th Lidstone boundary value problems and dependence on higher order derivatives. (English) Zbl 1029.34019

The authors consider the Lidstone boundary value problem \[ \begin{gathered} (-1)^nu^{2n}(x)= f(x, u(x), u''(x),\dots, u^{2(n-1)}(x)),\quad 0< x< 1,\\ u^{(2i)}(0)= u^{(2i)}(1)= 0,\quad 0\leq i\leq n-1.\end{gathered} \] They develop a monotone iteration method in presence of upper and lower solutions to the above problem. For an earlier result, see R. Ma, J. Zhang and S. Fu [ibid. 215, No. 2, 415-422 (1997; Zbl 0892.34009)].
Reviewer: Ruyun Ma (Lanzhou)

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
47H10 Fixed-point theorems

Citations:

Zbl 0892.34009
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aftabizadeh, A. R., Existence and uniqueness theorems for fourth-order boundary value problems, J. Math. Anal. Appl., 116, 415-426 (1986) · Zbl 0634.34009
[2] Agarwal, R. P., On fourth-order boundary value problems arising in beam analysis, Differential Integral Equations, 2, 91-110 (1989) · Zbl 0715.34032
[3] Agarwal, R. P.; O’Regan, D.; Wong, P. J.Y., Positive Solutions of Differential (1999), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0923.39002
[4] Bai, Z. B., The method of lower and upper solutions for a bending of an elastic beam equation, J. Math. Anal. Appl., 248, 195-202 (2000) · Zbl 1016.34010
[5] Bai, Z. B.; Wang, H. Y., On positive solutions of some nonlinear fourth-order beam equations, J. Math. Anal. Appl., 270, 357-368 (2002) · Zbl 1006.34023
[6] Cabada, A., The method of lower and upper solutions for second, third, fourth and higher order boundary value problems, J. Math. Anal. Appl., 185, 302-320 (1994) · Zbl 0807.34023
[7] De Coster, C.; Sanchez, L., Upper and lower solutions, Ambrosetti-Prodi problem and positive solutions for fourth-order O.D.E., Riv. Mat. Pura Appl., 14, 1129-1138 (1994) · Zbl 0979.34015
[8] Davis, J.; Henderson, J., Uniqueness implies existence for fourth-order Lidstone boundary value problems, Panamer. Math. J., 8, 23-25 (1998) · Zbl 0960.34011
[9] Davis, J.; Henderson, J.; Wong, P. J.Y., General Lidstone problems: multiplicity and symmetry of solutions, J. Math. Anal. Appl., 251, 527-548 (2000) · Zbl 0966.34023
[10] Del Pino, M. A.; Manasevich, R. F., Existence for a fourth-order boundary value problem under a two parameter nonresonance condition, Proc. Amer. Math. Soc., 112, 81-86 (1991) · Zbl 0725.34020
[11] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1977), Springer-Verlag: Springer-Verlag New York · Zbl 0691.35001
[12] Ma, R. Y.; Zhang, J. H.; Fu, S. M., The method of lower and upper solutions for fourth-order two-point boundary value problems, J. Math. Anal. Appl., 215, 415-422 (1997) · Zbl 0892.34009
[13] Protter, M. H.; Weinberger, H. F., Maximum Principles in Differential Equations (1967), Prentice Hall: Prentice Hall Englewood Cliffs, NJ · Zbl 0153.13602
[14] Schroder, J., Fourth-order two-point boundary value problems, estimates by two side bounds, Nonlinear Anal., 8, 107-114 (1984) · Zbl 0533.34019
[15] Wong, P. J.Y.; Agarwal, R. P., Eigenvalues of Lidstone boundary value problems, Appl. Math. Comput., 104, 15-31 (1999) · Zbl 0933.65089
[16] Yao, Q. L.; Bai, Z. B., Existence of positive solutions of boundary value problems for \(u^{(4)}(t)\)−\( λh (t)f(u(t))=0\), Chinese Ann. Math. Ser. A, 20, 575-578 (1999), (in Chinese) · Zbl 0948.34502
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.