Existence of solutions for fourth order differential equation with four-point boundary conditions. (English) Zbl 1140.34308

The authors study the fourth order differential equation \[ u^{(4)}(t)=f(t,u(t)), \; t \in (0,1), \] subject to the boundary conditions
\[ \begin{gathered} u(0)=0,\quad au''(\xi_1)-bu'''(\xi_1)=0,\\ u(1)=0,\quad cu''(\xi_2)-du'''(\xi_2)=0, \end{gathered} \]
where \(0\leq \xi_1 \leq \xi_2\leq 1\). They prove that this boundary value problem (BVP) has a solution. The main ingredient of the proof is a nonlinear alternative theorem of Leray-Schauder-type. The authors also give some remarks on a paper by S. Chen, W. Ni and C. Wang [ Appl. Math. Lett. 19, 161-168 (2006; Zbl 1096.34009)], where this particular BVP has been studied previously.


34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations


Zbl 1096.34009
Full Text: DOI


[1] Anderson, D. R.; Davis, J. M., Multiple solutions and eigenvalues for a third-order right focal boundary value problem, J. Math. Anal. Appl., 267, 135-157 (2002) · Zbl 1003.34021
[2] Agarwal, R. P., Focal Boundary Value Problems for Differential and Difference Equations (1998), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0914.34001
[3] Agarwal, R. P.; Meehan, M.; O’Regan, D., Fixed Point Theory and Applications (2001), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0960.54027
[4] Bai, Z.; Wang, H., On positive solutions of some nonlinear fourth-order beam equations, J. Math. Anal. Appl., 270, 357-368 (2002) · Zbl 1006.34023
[5] Chen, S.; Ni, W.; Wang, C., Positive solution of fourth order ordinary differential equation with four-point boundary conditions, Appl. Math. Lett., 19, 161-168 (2006) · Zbl 1096.34009
[6] Graef, J. R.; Yang, B., On a nonlinear boundary value problem for fourth order equations, Appl. Anal., 72, 439-448 (1999) · Zbl 1031.34017
[7] Gupta, C. P.; Trofimchuk, S. I., Existence of a solution of a three-point boundary value problem and the spectral radius of a related linear operator, Nonlinear Anal., 34, 489-507 (1998) · Zbl 0944.34009
[8] Hao, Z.; Liu, L.; Debnath, L., A necessary and sufficient condition for the existence of positive solutions of fourth-order singular boundary value problems, Appl. Math. Lett., 16, 279-285 (2003) · Zbl 1055.34047
[9] Khan, A. R.; Webb, J. R.L., Existence of at least three solutions of a second-order three-point boundary value problem, Nonlinear Anal., 64, 1356-1366 (2006) · Zbl 1101.34005
[10] Liu, B., Positive solutions of second-order three-point boundary value problems with change of sign, Comput. Math. Appl., 47, 1351-1361 (2004) · Zbl 1060.34015
[11] Xu, X., Three solutions for three-point boundary value problems, Nonlinear Anal., 62, 1053-1066 (2005) · Zbl 1076.34011
[12] Zill, D. G.; Cullen, M. R., Differential Equations with Boundary-Value Problems (2001), Brooks/Cole
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.