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Existence results and numerical solutions for a beam equation with nonlinear boundary conditions. (English) Zbl 1068.74038
Summary: We study the existence of solutions of the nonlinear fourth-order equation of Kirchhoff type \[ u^{(iv)}- m\Biggl(\int^L_0|u'(s)|^2 ds\Biggr) u''= f(x, u), \] under nonlinear boundary conditions which models the deformations of beams on elastic supports.

74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74G20 Local existence of solutions (near a given solution) for equilibrium problems in solid mechanics (MSC2010)
Full Text: DOI
[1] Arosio, A., A geometrical nonlinear correction to the Timoshenko beam equation, Nonlinear anal., 47, 729-740, (2001) · Zbl 1042.74531
[2] Ball, J., Initial boundary value problem for an extensible beam, J. math. anal. appl., 42, 61-90, (1973) · Zbl 0254.73042
[3] Choo, S.M.; Chung, S.K., Finite difference approximate solutions for the strongly damped extensible beam equations, Appl. math. comput., 112, 11-32, (2000) · Zbl 1026.74079
[4] Dickey, R.W., Free vibrations and dynamic buckling of the extensible beam, J. math. anal. appl., 29, 443-454, (1970) · Zbl 0187.04803
[5] Ekeland, I., On the variational principle, J. math. anal. appl., 47, 324-353, (1974) · Zbl 0286.49015
[6] Fečkan, M., Free vibrations of beams on bearings with nonlinear elastic responses, J. differential equations, 154, 55-72, (1999) · Zbl 0927.35071
[7] Feireisl, E., Non-zero time periodic solutions to an equation of petrovsky type with nonlinear boundary conditions: slow oscillations of beams on elastic bearings, Ann. scuola norm. super. Pisa, 20, 133-146, (1993) · Zbl 0794.73029
[8] Grossinho, M.R.; Ma, T.F., Symmetric equilibria for a beam with a nonlinear elastic foundation, Portugal. math., 51, 375-393, (1994) · Zbl 0815.34014
[9] Grossinho, M.R.; Tersian, S., The dual variational principle and equilibria for a beam resting on a discontinuous nonlinear elastic foundation, Nonlinear anal., 41, 417-431, (2000) · Zbl 0960.34013
[10] Grossinho, M.R.; Tersian, S., An introduction to minimax theorems and their applications to differential equations, (2001), Kluwer Dordrecht · Zbl 0997.34034
[11] Kirchhoff, G., Vorlesungen über mathematiche physik: mechanik, (1876), Teubner Leipzig · JFM 08.0542.01
[12] Ma, T.F., Existence results for a model of nonlinear beam on elastic bearings, Appl. math. lett., 13, 11-15, (2000) · Zbl 0965.74030
[13] Ma, T.F., Boundary stabilization for a non-linear beam on elastic bearings, Math. methods appl. sci., 24, 583-594, (2001) · Zbl 0981.35034
[14] Perla Menzala, G.; Zuazua, E., The beam equation as a limit of a 1-d nonlinear von Kármán model, Appl. math. lett., 12, 47-52, (1999) · Zbl 0946.74035
[15] Woinowsky-Krieger, S., The effect of axial force on the vibration of hinged bars, J. appl. mech., 17, 35-36, (1950) · Zbl 0036.13302
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