Ma, Ruyun; Zhang, Jihui; Fu, Shengmao The method of lower and upper solutions for fourth-order two-point boundary value problems. (English) Zbl 0892.34009 J. Math. Anal. Appl. 215, No. 2, 415-422 (1997). The method of upper and lower solutions coupled with the monotone iterative technique is used to guarantee the existence of a couple of monotone sequences converging to the extremal solutions in a sector of the following fourth-order boundary value problem: \[ u^{(IV)}(x)=f(x,u(x),u''(x)), \qquad u(0)=u(1)=u''(0)=u''(1)=0, \] where \(f:[0,1]\times \mathbb{R}^{2}\rightarrow \mathbb{R}\) is continuous. In order to demonstrate the main result, the existence of a lower solution \(\beta\) and an upper solution \(\alpha\) with \(\beta\leq\alpha\) and \(\beta''\geq\alpha''\) on \([0,1]\) is assumed. As a previous result, the authors also prove a maximum principle. Reviewer: Eduardo Liz (Vigo) Cited in 2 ReviewsCited in 123 Documents MSC: 34B15 Nonlinear boundary value problems for ordinary differential equations 34B27 Green’s functions for ordinary differential equations Keywords:upper and lower solutions; monotone method; maximum principle; Green functions PDFBibTeX XMLCite \textit{R. Ma} et al., J. Math. Anal. Appl. 215, No. 2, 415--422 (1997; Zbl 0892.34009) Full Text: DOI Link 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., On fourth-order boundary value problems arising in beam analysis, Differential Integral Equations, 2, 91-110 (1989) · Zbl 0715.34032 [3] 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 [4] De Coster, C.; Fabry, C.; Munyamarere, F., Nonresonance conditions for fourth-order nonlinear boundary value problems, Internat. J. Math. Math. Sci., 17, 725-740 (1994) · Zbl 0810.34017 [5] 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 [6] 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 [7] Dunninger, D., Existence of positive solutions for fourth-order nonlinear problems, Boll. Un. Mat. Ital., 7, 1129-1138 (1987) · Zbl 0643.34020 [8] Gupta, C. P., Existence and uniqueness results for some fourth order fully quasilinear boundary value problem, Appl. Anal., 36, 169-175 (1990) [9] Gupta, C. P., Existence and uniqueness results for the bending of an elastic beam equation at resonance, J. Math. Appl. Appl., 135, 208-225 (1988) · Zbl 0655.73001 [10] Gupta, C. P., Existence and uniqueness theorem for a bending of an elastic beam equation, Appl. Anal., 26, 289-304 (1988) · Zbl 0611.34015 [11] Korman, P., A maximum principle for fourth-order ordinary differential equations, Appl. Anal., 33, 267-273 (1989) · Zbl 0681.34016 [12] Ma, R. Y., Some multiplicity results for an elastic beam equation at resonance, Appl. Math. Mech., 14, 193-200 (1993) · Zbl 0776.73037 [13] Ma, R. Y.; Wang, H. Y., On the existence of positive solutions of fourth-order ordinary differential equations, Appl. Anal., 59, 225-231 (1995) · Zbl 0841.34019 [14] Njoku, F. I., Positive solutions for two-point BVP’s: Existence and multiplicity results, Nonlinear Anal., 13, 1329-1338 (1989) · Zbl 0704.34020 [15] Sadyrbaev, F., Two-point boundary value problem for fourth-order, Acta Univ. Latviensis, 553, 84-91 (1990) [16] Schroder, J., Fourth-order two-point boundary value problems; estimates by two side bounds, Nonlinear Anal., 8, 107-114 (1984) · Zbl 0533.34019 [17] Usmani, R. A., A uniqueness theorem for a boundary value problem, Proc. Amer. Math. Soc., 77, 327-335 (1979) · Zbl 0424.34019 [18] Yang, Y., Fourth-order two-point boundary value problem, Proc. Amer. Math. Soc., 104, 175-180 (1988) · Zbl 0671.34016 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.