Positive solutions for fourth-order differential equations with deviating arguments and integral boundary conditions. (English) Zbl 1200.34072
A certain non-local boundary value problem for a fourth-order nonlinear differential equation with a deviation is considered. The author investigates the problem both for advanced and delayed argument, and proves two multiplicity results by using the Avery-Peterson fixed-point theorem. An illustrative example is given as well.
Reviewer: Jiří Šremr (Brno)
MSC:
| 34K10 | Boundary value problems for functional-differential equations |
| 47N20 | Applications of operator theory to differential and integral equations |
Keywords:
fourth-order nonlinear differential equation; integral boundary condition; positive solution; multiplicity; advanced argument; delayReferences:
| [1] | Bai, D.; Xu, Y., Positive solutions of second-order two-delay differential systems with twin parameter, Nonlinear Anal., 63, 601-617 (2005) · Zbl 1091.34034 |
| [2] | Bai, D.; Xu, Y., Existence of positive solutions for boundary value problem of second-order delay differential equations, Appl. Math. Lett., 18, 621-630 (2005) · Zbl 1080.34048 |
| [3] | Boucherif, A., Second-order boundary value problems with integral boundary conditions, Nonlinear Anal., 70, 364-371 (2009) · Zbl 1169.34310 |
| [4] | Du, B.; Hu, X.; Ge, W., Positive solutions to a type of multi-point boundary value problem with delay and one-dimensional \(p\)-Laplacian, Appl. Math. Comput., 208, 501-510 (2009) · Zbl 1170.34046 |
| [5] | Feng, M.; Ji, D.; Ge, W., Positive solutions for a class of boundary-value problem with integral boundary conditions in Banach spaces, J. Comput. Appl. Math., 222, 351-363 (2008) · Zbl 1158.34336 |
| [6] | Jankowski, T., Positive solutions of three-point boundary value problems for second order impulsive differential equations with advanced arguments, Appl. Math. Comput., 197, 179-189 (2008) · Zbl 1145.34355 |
| [7] | Jankowski, T., Three positive solutions to second order three-point impulsive differential equations with deviating arguments, Int. J. Comput. Math., 87, 215-225 (2010) · Zbl 1198.34180 |
| [8] | Jiang, D., Multiple positive solutions for boundary value problems of second order delay differential equations, Appl. Math. Lett., 15, 575-583 (2002) · Zbl 1011.34055 |
| [9] | Wang, W.; Sheng, J., Positive solutions to a multi-point boundary value problem with delay, Appl. Math. Comput., 188, 96-102 (2007) · Zbl 1125.34333 |
| [10] | Wang, Y.; Zhao, W.; Ge, W., Multiple positive solutions for boundary value problems of second order delay differential equations with one-dimensional \(p\)-Laplacian, J. Math. Anal. Appl., 326, 641-654 (2007) · Zbl 1119.34050 |
| [11] | Yang, C.; Zhai, C.; Yan, J., Positive solutions of the three-point boundary value problem for second order differential equations with an advanced argument, Nonlinear Anal., 65, 2013-2023 (2006) · Zbl 1113.34048 |
| [12] | Bai, C., Triple positive solutions of three-point boundary value problems for fourth-order differential equations, Comput. Math. Appl., 56, 1364-1371 (2008) · Zbl 1155.34311 |
| [13] | 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 |
| [14] | 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 |
| [15] | Li, F.; Zhang, Q.; Liang, Z., Existence and multiplicity of solutions of a kind of fourth-order boundary value problem, Nonlinear Anal., 62, 803-816 (2005) · Zbl 1076.34015 |
| [16] | Liu, B., Positive solutions of fourth-order two point boundary value problems, Appl. Math. Comput., 148, 407-420 (2004) · Zbl 1039.34018 |
| [17] | Yang, D.; Zhu, H.; Bai, C., Positive solutions for semipositone fourth-order two-point boundary value problems, Electronic. J. Differential Equations, 2007, 16, 1-8 (2007) · Zbl 1118.34019 |
| [18] | Zhang, X.; Liu, L.; Zou, H., Positive solutions of fourth-order singular three point eigenvalue problems, Appl. Math. Comput., 189, 1359-1367 (2007) · Zbl 1126.34321 |
| [19] | Zhang, M.; Wei, Z., Existence of positive solutions for fourth-order m-point boundary value problem with variable parameters, Appl. Math. Comput., 190, 1417-1431 (2007) · Zbl 1141.34018 |
| [20] | Avery, R. I.; Peterson, A. C., Three positive fixed points of nonlinear operators on ordered Banach spaces, Comput. Math. Appl., 42, 313-322 (2001) · Zbl 1005.47051 |
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