Existence and uniqueness of solutions for a fourth-order boundary value problem. (English) Zbl 1169.34308

Summary: We study the fourth-order two-point boundary value problem
\[ \begin{aligned} & x''''(t)-f(t,x(t),x'(t),x'',x'''(t))=0,\quad t\in(0,1),\\ & x(0)=x'(1)=0,\quad ax''(0)-bx'''(0)=0,\quad cx''(1)+dx'''(1)=0.\end{aligned} \]
By means of lower and upper solution method, growth conditions on the nonlinear term \(f\) which guarantee the existence of solutions for the above boundary value problem are given. In particular, we obtain the uniqueness of the solution by imposing a monotonicity condition of the term \(f\).


34B15 Nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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[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., Focal Boundary Value Problems for Differential and Difference Equations (1998), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0914.34001
[3] Agarwal, R. P.; Regan, D. O.; Wong, P. J.Y., Positive Solutions of Differential, Difference, and Integral Equations (1999), Kluwer Academic: Kluwer Academic Dordrecht
[4] Anderson, D. R.; Davis, J. M., Multiple solutions and eigenvalues for third-order right focal boundary value problem, J. Math. Anal. Appl., 267, 135-157 (2002) · Zbl 1003.34021
[5] Bai, Z., The upper and lower solution method for some fourth-order boundary value problems, Nonlinear Anal., 67, 1704-1709 (2007) · Zbl 1122.34010
[6] Baxley, J.; Haywood, L. J., Nonlinear boundary value problems with multiple solutions, Nonlinear Anal., 47, 1187-1198 (2001) · Zbl 1042.34517
[7] De Coster, C.; Sanchez, L., Upper and lower solutions, Ambrosetti-Prodi problem and positive solutions for fourth-order O.D.E, Riv. Math. Pura. Appl., 14, 1129-1138 (1994) · Zbl 0979.34015
[8] 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
[9] Dulcska, E., Soil settlement effects on buildings, (Developments in Geotechnical Engineering, vol. 69 (1992), Elsevier: Elsevier Amsterdam)
[10] Ehme, J.; Eloe, P. W.; Henderson, J., Upper and lower solution methods for fully nonlinear boundary value problems, J. Differential Equations, 180, 51-64 (2001) · Zbl 1019.34015
[11] Gupta, C. P., Existence and uniqueness theorem for a bending of an elastic beam equation, Appl. Anal., 26, 289-304 (1988) · Zbl 0611.34015
[12] Hao, Z.; Liu, L., A necessary and sufficiently condition for the existence of positive solution of fourth-order singular boundary value problems, Appl. Math. Lett., 16, 279-285 (2003) · Zbl 1055.34047
[13] Palamides, P. K., Uniqueness of monotone positive solutions for singular boundary value problems, Comm. Appl. Nonlinear Anal., 9, 79-89 (2002) · Zbl 1023.34019
[14] Palamides, P. K., Positive solutions for higher-order Sturm-Liouville problems. A new approach via vector field, Differential Equations Dynam. Systems, 9, 83-103 (2002) · Zbl 1231.34026
[15] Soedel, W., Vibrations of Shells and Plates (1993), Dekker: Dekker New York · Zbl 0865.73002
[16] Timoshenko, S. P., Theory of Elastic Stability (1961), McGraw-Hill: McGraw-Hill New York
[17] Usmani, R. A., A uniqueness theorem for a boundary value problem, Proc. Amer. Math. Soc., 77, 327-335 (1979) · Zbl 0424.34019
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