×

Existence of positive solutions of a nonlinear fourth-order boundary value problem. (English) Zbl 1195.34037

Summary: We study the existence of positive solutions of fourth-order boundary value problem
\[ u^{(4)}(t) = f(t, u(t), u^{\prime \prime }(t)), t \in (0, 1). \]
\[ u(0) = u(1) = u^{\prime \prime }(0) = u^{\prime \prime }(1) = 0, \]
where \(f: [0, 1] \times [0, \infty ) \times (-\infty , 0] \to [0, \infty )\) is continuous. The proof of our main result is based upon the Krein-Rutman theorem and the global bifurcation techniques.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
47J15 Abstract bifurcation theory involving nonlinear operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Gupta, C. P., Existence and uniqueness theorems for a bending of an elastic beam equation, Appl. Anal., 26, 289-304 (1988) · Zbl 0611.34015
[2] Gupta, C. P., Existence and uniqueness results for the bending of an elastic beam equation at resonance, J. Math. Anal. Appl., 135, 208-225 (1988) · Zbl 0655.73001
[3] Agarwal, R. P., On fourth order boundary value problems arising in beam analysis, Differential Integral Equations, 2, 1, 91-110 (1989) · Zbl 0715.34032
[4] Del Pino, M. A.; Mansevich, R. F., Existence for a fourth-order boundary value problem under a two-parameter nonresonance condition, Proc. Amer. Math. Soc., 112, 1, 81-86 (1991) · Zbl 0725.34020
[5] Ma, Ruyun; Wang, Haiyan, On the existence of positive solutions of fourth-order ordinary differential equations, Appl. Anal., 59, 225-231 (1995) · Zbl 0841.34019
[6] Bai, Zhanbing; Wang, Haiyan, On positive solutions of some nonlinear fourth-order beam equations, J. Math. Anal. Appl., 270, 2, 357-368 (2002) · Zbl 1006.34023
[7] Li, Yongxiang, Positive solutions of fourth-order boundary value problems with two parameters, J. Math. Anal. Appl., 281, 477-484 (2003) · Zbl 1030.34016
[8] Chai, Guoqing, Existence of positive solutions for fourth-order boundary value problem with variable parameters, Nonlinear Anal., 26, 289-304 (2007) · Zbl 1113.34008
[9] Ma, Ruyun, Existence of positive solutions of a four-order boundary value problem, Appl. Math. Comput., 168, 1219-1231 (2005) · Zbl 1082.34023
[10] Ma, Ruyun, Nodal solutions of boundary value problems of fourth-order ordinary differential equations, J. Math. Anal. Appl., 319, 424-434 (2006) · Zbl 1098.34012
[11] Ma, Ruyun, Nodal solutions for a fourth-order two-point boundary value problem, J. Math. Anal. Appl., 314, 254-265 (2006) · Zbl 1085.34015
[12] Dancer, E. N., Global solution branches for positive mappings, Arch. Ration. Mech. Anal., 52, 181-192 (1973) · Zbl 0275.47043
[13] Zeidler, E., Nonlinear Functional Analysis and its Applications, I. Fixed-Point Theorems (1985), Springer-Verlag: Springer-Verlag New York
[14] Deimling, K., Nonlinear Functional Analysis (1985), Springer-Verlag: Springer-Verlag Berlin · Zbl 0559.47040
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.