Positive solutions of fourth-order two point boundary value problems. (English) Zbl 1039.34018

Liu, B.

Appl. Math. Comput. 148, No. 2, 407-420 (2004).

Summary: By using the Krasnoselskii fixed-point theorem, we study the existence of one or multiple positive solution of the fourth-order two-point boundary value problem \(y^{(4)}(t)=f(t,y(t),y''(t))\), \(y(0)=y(1)=y''(0)=y''(1)=0\). We also give some examples to illustrate our results.

MSC:

34B18

Positive solutions to nonlinear boundary value problems for ordinary differential equations

Keywords:

Positive solution; Boundary value problems; Cone

References:

[1]	Aftabizadeh, A. R., Existence and uniqueness theorems for fourth-order boundary problems, J. Math. Anal. Appl., 116, 415-426 (1986) · Zbl 0634.34009
[2]	Yang, Yisong, Fourth-order two-point boundary value problem, Proc. Am. Math. Soc., 104, 175-180 (1988) · Zbl 0671.34016
[3]	Del Pino, M. A.; Manasevich, R. F., Existence for fourth-order boundary value problem under a two-parameter nonresonance condition, Proc. Am. Math. Soc., 112, 81-86 (1991) · Zbl 0725.34020
[4]	Gupta, C. P., Existence and uniqueness theorem for a bending of an elastic beam equation, Appl. Anal., 26, 289-304 (1988) · Zbl 0611.34015
[5]	Gupta, C. P., Existence and uniqueness results for some fourth order fully quasilinear boundary value problem, Appl. Anal., 36, 169-175 (1990)
[6]	Ma, Ruyun; Wang, Haiyai, On the existence of positive solutions of fourth order ordinary differential equations, Appl. Anal., 59, 225-231 (1995) · Zbl 0841.34019
[7]	Ma, Ruyan, Positive solutions of fourth-order two point boundary value problems, Ann. Differen. Equations, 15, 305-313 (1999) · Zbl 0964.34021
[8]	Deimling, K., Nonlinear Functional Analysis (1985), Springer: Springer New York · Zbl 0559.47040
[9]	Krasnoselskii, M. A., Postive solution of Operator Equations (1964), Noordhoff: Noordhoff Gronignen