## Monotone positive solutions for a fourth order equation with nonlinear boundary conditions.(English)Zbl 1177.34030

Consider the boundary value problems $u^{(4)}(t)=f(t,u,u'),\;0<t<1,\tag{1}$
$u(0)=u'(0)=0,\;u'''(1)=g(u(1)),\tag{2}$
$u''(1)=0, \tag{3}$
$u'(1)=0\tag{4}$ describing bending equilibra of elastic beams. Using the method of lower and upper solutions coupled with monotone iteration, the authors derive conditions on the functions $$f$$ and $$g$$ such that the boundary value problems $$(1),(2),(3)$$ and $$(1),(2),(4)$$ have monotone positive solutions. They also present numerical simulations of the problem $$(1),(2),(3)$$.

### MSC:

 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 34A45 Theoretical approximation of solutions to ordinary differential equations 74K10 Rods (beams, columns, shafts, arches, rings, etc.)
Full Text:

### References:

 [1] Amster, P.; Mariani, M.C., A fixed point operator for a nonlinear boundary value problem, Journal of mathematical analysis and applications, 266, 160-168, (2002) · Zbl 1007.34015 [2] Amster, P.; Cárdenas Alzate, P.P., A shooting method for a nonlinear beam equation, Nonlinear analysis, 68, 2072-2078, (2008) · Zbl 1146.34013 [3] Barari, A.; Omidvar, M.; Ganji, D.D.; Poor, A.T., An approximate solution for boundary value problems in structural engineering and fluid mechanics, Mathematical problems in engineering, 2008, 1-13, (2008) · Zbl 1151.74428 [4] Cabada, A.; Minhós, F.M., Fully nonlinear fourth-order equations with functional boundary conditions, Journal of mathematical analysis and applications, 340, 239-251, (2008) · Zbl 1138.34008 [5] Franco, D.; O’Regan, D.; Perán, J., Fourth-order problems with nonlinear boundary conditions, Journal of computational and applied mathematics, 174, 315-327, (2005) · Zbl 1068.34013 [6] Grossinho, M.R.; Ma, T.F., Symmetric equilibria for a beam with a nonlinear foundation, Portugaliae Mathematica, 51, 375-393, (1994) · Zbl 0815.34014 [7] Grossinho, M.R.; Tersian, St.A., The dual variational principle and equilibria for a beam resting on a discontinuous nonlinear elastic foundation, Nonlinear analysis, 41, 417-431, (2000) · Zbl 0960.34013 [8] Ma, T.F., Existence for a model of nonlinear beams on elastic bearings, Applied mathematics letters, 13, 11-15, (2000) · Zbl 0965.74030 [9] Ma, T.F., Existence results and numerical solutions for a beam equation with nonlinear boundary conditions, Applied numerical mathematics, 47, 189-196, (2003) · Zbl 1068.74038 [10] Ma, T.F., Positive solutions for a beam equation on a nonlinear elastic foundation, Mathematical and computer modelling, 39, 1195-1201, (2004) · Zbl 1060.74035 [11] Ma, T.F.; da Silva, J., Iterative solutions for a beam equation with nonlinear boundary conditions of third order, Applied mathematics and computation, 159, 11-18, (2004) · Zbl 1095.74018 [12] Xiaoping Zhang, Existence and iteration of monotone positive solutions for an elastic beam with a corner, Nonlinear Analysis: Real World Applications, in press (doi:10.1016/j.nonrwa.2008.03.017) [13] Yao, Qinliu, Monotonically iterative method of nonlinear cantilever beam equations, Applied mathematics and computation, 205, 432-437, (2008) · Zbl 1154.74021 [14] Agarwal, R.P.; O’Regan, D.; Wong, P.J.Y., Positive solutions of differential difference and integral equations, (1999), Kluwer Academic Publishers Dordrecht · Zbl 0923.39002 [15] Amann, H., Fixed point equations and nonlinear eigenvalue problem in ordered Banach spaces, SIAM review, 18, 620-709, (1976) · Zbl 0345.47044 [16] Zeidler, E., () [17] De Coster, C.; Habets, P., () [18] Agarwal, R.P.; Chow, Y.M., Iterative methods for a fourth order boundary value problem, Journal of mathematical analysis and applications, 10, 203-217, (1984) · Zbl 0541.65055 [19] Bai, Zhanbing; Huang, Bingjia; Ge, Weigao, The iterative solutions for some fourth-order $$p$$-Laplacian equation boundary value problem, Applied mathematics letters, 19, 8-14, (2006) · Zbl 1092.34510 [20] Cabada, A.; Ángel Cid, J.; Sanchez, L., Positivity and lower and upper solutions for fourth order boundary value problems, Nonlinear analysis, 67, 1599-1612, (2007) · Zbl 1125.34010 [21] Noor, M.A.; Mohyud-Din, S.T., An efficient method for fourth-order boundary value problems, Computers and mathematics with applications, 54, 1101-1111, (2007) · Zbl 1141.65375
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.