Local existence and uniqueness of increasing positive solutions for non-singular and singular beam equation with a parameter. (English) Zbl 1495.34047
Summary: This paper is concerned with a class of beam equations with a parameter. By using the fixed point theorems of mixed monotone operator and the properties of cone, we study the non-singular and singular case, respectively, and obtain the sufficient conditions which guarantee the local existence and uniqueness of increasing positive solutions. Also, we present an iterative algorithm that converges to the solution. Moreover, we get some pleasant properties of the solutions for the boundary value problem dependent parameter. At last, two examples are given to illustrate the main results.
MSC:
| 34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |
| 34A45 | Theoretical approximation of solutions to ordinary differential equations |
| 47N20 | Applications of operator theory to differential and integral equations |
| 74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |
| 34B16 | Singular nonlinear boundary value problems for ordinary differential equations |
Keywords:
existence and uniqueness; positive solution; non-singular and singular beam equation; fixed point theorem of mixed monotone operatorReferences:
| [1] | Amaster, P.; Mariani, M. C., A fixed point operator for a nonlinear boundary value problem, J. Math. Anal. Appl., 266, 160-168 (2002) · Zbl 1007.34015 · doi:10.1006/jmaa.2001.7722 |
| [2] | Ma, T. F., Existence results and numerical solutions for a beam equation with nonlinear boundary conditions, Appl. Numer. Math., 47, 189-196 (2003) · Zbl 1068.74038 · doi:10.1016/S0168-9274(03)00065-5 |
| [3] | Grossinho, M. R.; Tersian, S. A., 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 · doi:10.1016/S0362-546X(98)00285-5 |
| [4] | Barari, A.; Omidvar, M.; Ganji, D. D.; Poor, A. T., An approximate solution for boundary value problems in structural engineering and fluid mechanics, Math. Probl. Eng., 2008 (2008) · Zbl 1151.74428 · doi:10.1155/2008/394103 |
| [5] | Franco, D.; O’Regan, D.; Perán, J., Fourth-order problems with nonlinear boundary conditions, J. Comput. Appl. Math., 174, 315-327 (2004) · Zbl 1068.34013 · doi:10.1016/j.cam.2004.04.013 |
| [6] | Cabada, A.; Minhós, F. M., Fully nonlinear fourth-order equations with functional boundary conditions, J. Math. Anal. Appl., 340, 239-251 (2008) · Zbl 1138.34008 · doi:10.1016/j.jmaa.2007.08.026 |
| [7] | Ma, T. F., Positive solutions for a beam equation on a nonlinear elastic foundation, Math. Comput. Model., 39, 1195-1201 (2004) · Zbl 1112.78021 · doi:10.1016/S0895-7177(04)90526-2 |
| [8] | Amster, P.; Cárdenas Alzate, P. P., A shooting method for a nonlinear beam equation, Nonlinear Anal., 68, 2072-2078 (2008) · Zbl 1146.34013 · doi:10.1016/j.na.2007.01.032 |
| [9] | Bai, Z., The upper and lower solution method for some fourth-order boundary value problems, Nonlinear Anal., 67, 1704-1709 (2007) · Zbl 1122.34010 · doi:10.1016/j.na.2006.08.009 |
| [10] | Caballero, J.; Harjani, J.; Sadarangani, K., Uniqueness of positive solutions for a class of fourth-order boundary value problems, Abstr. Appl. Anal., 2011 (2011) · Zbl 1222.34027 |
| [11] | Graef, J. R.; Kong, L. J.; Kong, Q. K.; Yang, B., Positive solutions to a fourth order boundary value problem, Results Math., 59, 141-155 (2011) · Zbl 1218.34024 · doi:10.1007/s00025-010-0068-7 |
| [12] | Li, S. Y.; Zhang, Q. X., Existence and uniqueness of monotone positive solutions for an elastic beam equation with nonlinear boundary conditions, Comput. Math. Appl., 63, 1355-1360 (2012) · Zbl 1247.74035 · doi:10.1016/j.camwa.2011.12.065 |
| [13] | Pei, M. H.; Chang, S. K., Monotone iterative technique and symmetric positive solutions for a fourth-order boundary value problem, Math. Comput. Model., 51, 1260-1267 (2010) · Zbl 1206.65188 · doi:10.1016/j.mcm.2010.01.009 |
| [14] | Li, S. Y.; Zhai, C. B., New existence and uniqueness results for an elastic beam equation with nonlinear boundary conditions, Bound. Value Probl., 2015 (2015) · Zbl 1341.34031 · doi:10.1186/s13661-015-0365-x |
| [15] | Cabada, A.; Tersian, S., Multiplicity of solutions of a two point boundary value problem for a fourth-order equation, Appl. Math. Comput., 219, 5261-5267 (2013) · Zbl 1294.34016 |
| [16] | Alves, E.; Ma, T. F.; Pelicer, M. L., Monotone positive solutions for a fourth order equation with nonlinear boundary conditions, Nonlinear Anal., 71, 3834-3841 (2009) · Zbl 1177.34030 · doi:10.1016/j.na.2009.02.051 |
| [17] | Yao, Q. L., Positive solutions for eigenvalue problems of fourth-order elastic beam equations, Appl. Math. Lett., 17, 237-243 (2004) · Zbl 1072.34022 · doi:10.1016/S0893-9659(04)90037-7 |
| [18] | Wang, W. X.; Zheng, Y. P.; Yang, H.; Wang, J. X., Positive solutions for elastic beam equations with nonlinear boundary conditions and a parameter, Bound. Value Probl., 2014 (2014) · Zbl 1307.34054 · doi:10.1186/1687-2770-2014-80 |
| [19] | Zhai, C. B.; Jiang, C. R., Existence and uniqueness of convex monotone positive solutions for boundary value problems of an elastic beam equation with a parameter, Electron. J. Qual. Theory Differ. Equ., 2015 (2015) · Zbl 1324.92048 · doi:10.1186/s13662-015-0426-6 |
| [20] | Yuan, C. J.; Jiang, D. Q.; O’Regan, D., Existence and uniqueness of positive solutions for fourth-order nonlinear singular continuous and discrete boundary value problems, Appl. Math. Comput., 203, 194-201 (2008) · Zbl 1169.34315 |
| [21] | He, Y., Existence theory for single positive solution to fourth-order boundary value problems, Appl. Math. Comput., 4, 480-486 (2014) |
| [22] | Zhai, C. B.; Jiang, C. R.; Li, S. Y., Approximating monotone positive solutions of a nonlinear fourth-order boundary value problem via sum operator method, Mediterr. J. Math., 14, 1-12 (2017) · Zbl 1369.34042 · doi:10.1007/s00009-017-0844-7 |
| [23] | Cabrera, I. J.; Lo’Pez, B.; Sadarangani, K., Existence of positive solutions for the nonlinear elastic beam equation via a mixed monotone operator, J. Comput. Appl. Math., 327, 306-313 (2018) · Zbl 1485.34090 · doi:10.1016/j.cam.2017.04.031 |
| [24] | Guo, D.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), New York: Academic Press, New York · Zbl 0661.47045 |
| [25] | Zhai, C. B.; Zhang, L. L., New fixed point theorems for mixed monotone operators and local existence uniqueness of positive solutions for nonlinear boundary value problems, J. Math. Anal. Appl., 382, 594-614 (2011) · Zbl 1225.47074 · doi:10.1016/j.jmaa.2011.04.066 |
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