Three solutions for a discrete fourth-order boundary value problem with three parameters. (English) Zbl 07905295
Summary: This paper presents several sufficient conditions for the existence of at least three classical solutions of a boundary value problem for a fourth-order difference equation. Fourth-order boundary value problems act as models for the bending or deforming of elastic beams. In different fields of research, such as computer science, mechanical engineering, control systems, artificial or biological neural networks, economics and many others, the mathematical modelling of important questions leads naturally to the consideration of nonlinear difference equations. Our technical approach is based on variational methods. An example is included in the paper. Numerical computations of the example confirm our theoretical results.
MSC:
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
| 46E39 | Sobolev (and similar kinds of) spaces of functions of discrete variables |
| 39A70 | Difference operators |
Keywords:
discrete boundary value problem; fourth-order boundary value problem; three solutions; variational methods; critical point theoryReferences:
| [1] | Douglas R. Anderson and Feliz Minhós. A discrete fourth-order Lidstone problem with parameters. Appl. Math. Comput., 214(2):523-533, 2009. · Zbl 1172.39019 |
| [2] | Gabriele Bonanno and Pasquale Candito. Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities. J. Differential Equations, 244(12):3031-3059, 2008. · Zbl 1149.49007 |
| [3] | Gabriele Bonanno and Pasquale Candito. Infinitely many solutions for a class of discrete non-linear boundary value problems. Appl. Anal., 88(4):605-616, 2009. · Zbl 1176.39004 |
| [4] | Gabriele Bonanno and Pasquale Candito. Nonlinear difference equations investigated via critical point methods. Non-linear Anal., 70(9):3180-3186, 2009. · Zbl 1166.39006 |
| [5] | Gabriele Bonanno and Beatrice Di Bella. A boundary value problem for fourth-order elastic beam equations. J. Math. Anal. Appl., 343(2):1166-1176, 2008. · Zbl 1145.34005 |
| [6] | Haïm Brezis and Louis Nirenberg. Remarks on finding critical points. Comm. Pure Appl. Math., 44(8-9):939-963, 1991. · Zbl 0751.58006 |
| [7] | Alberto Cabada, Antonio Iannizzotto, and Stepan Tersian. Multiple solutions for discrete boundary value problems. J. Math. Anal. Appl., 356(2):418-428, 2009. · Zbl 1169.39008 |
| [8] | Silvia Frassu, Tongxing Li, and Giuseppe Viglialoro. Improvements and generalizations of results concerning attraction-repulsion chemotaxis models. Math. Methods Appl. Sci., 2022. Published online 1 June 2022. |
| [9] | Silvia Frassu and Giuseppe Viglialoro. Boundedness criteria for a class of indirect (and direct) chemotaxis-consumption models in high dimensions. Appl. Math. Lett., 132:108108, 2022. · Zbl 1491.35078 |
| [10] | Marek Galewski and Szymon G lab. On the discrete boundary value problem for anisotropic equation. J. Math. Anal. Appl., 386(2):956-965, 2012. · Zbl 1233.39004 |
| [11] | John R. Graef, Shapour Heidarkhani, Lingju Kong, and Min Wang. Existence of solutions to a discrete fourth order boundary value problem. J. Difference Equ. Appl., 24(6):849-858, 2018. · Zbl 1391.39006 |
| [12] | John R. Graef, Lingju Kong, and Qingkai Kong. On a generalized discrete beam equation via variational methods. Commun. Appl. Anal., 16(3):293-308, 2012. · Zbl 1278.39005 |
| [13] | John R. Graef, Lingju Kong, and Xueyan Liu. Existence of solutions to a discrete fourth order periodic boundary value problem. J. Difference Equ. Appl., 22(8):1167-1183, 2016. · Zbl 1466.39009 |
| [14] | John R. Graef, Lingju Kong, and Min Wang. Multiple solutions to a periodic boundary value problem for a nonlinear discrete fourth order equation. Adv. Dyn. Syst. Appl., 8(2):203-215, 2013. |
| [15] | Shapour Heidarkhani, Ghasem A. Afrouzi, Giuseppe Caristi, Johnny Henderson, and Shahin Moradi. A variational approach to difference equations. J. Difference Equ. Appl., 22(12):1761-1776, 2016. · Zbl 1375.39013 |
| [16] | Shapour Heidarkhani, Ghasem A. Afrouzi, Shahin Moradi, and Giuseppe Caristi. A variational approach for solving p(x)-biharmonic equations with Navier boundary conditions. Electron. J. Differential Equations, pages 1-15, Paper No. 25, 2017. · Zbl 1381.35036 |
| [17] | Shapour Heidarkhani, Ghasem A. Afrouzi, Amjad Salari, and Giuseppe Caristi. Discrete fourth-order boundary value problems with four parameters. Appl. Math. Comput., 346:167-182, 2019. · Zbl 1428.46027 |
| [18] | Shapour Heidarkhani, Anderson L. A. De Araujo, Ghasem A. Afrouzi, and Shahin Moradi. Multiple solutions for Kirchhoff-type problems with variable exponent and nonhomogeneous Neumann conditions. Math. Nachr., 291(2-3):326-342, 2018. · Zbl 1390.35083 |
| [19] | Jun Ji and Bo Yang. A boundary value problem for the discrete beam equation. In Dynamic systems and applications. Vol. 6, pages 191-196. Dynamic, Atlanta, GA, 2012. · Zbl 1332.39009 |
| [20] | Lingju Kong. Existence of solutions to boundary value problems arising from the fractional advection dispersion equation. Electron. J. Differential Equations, pages 1-15, Paper No. 106, 2013. · Zbl 1291.34016 |
| [21] | Lingju Kong. Solutions of a class of discrete fourth order boundary value problems. Minimax Theory Appl., 3(1):35-46, 2018. · Zbl 1384.39003 |
| [22] | Alexandru Kristály, Mihai Mihȃilescu, and Vicent ¸iu Rȃdulescu. Discrete boundary value problems involving oscillatory nonlinearities: small and large solutions. J. Difference Equ. Appl., 17(10):1431-1440, 2011. · Zbl 1236.39010 |
| [23] | Tongxing Li, Nicola Pintus, and Giuseppe Viglialoro. Properties of solutions to porous medium problems with different sources and boundary conditions. Z. Angew. Math. Phys., 70(3):1-18, Paper No. 86, 2019. · Zbl 1415.35156 |
| [24] | Tongxing Li and Giuseppe Viglialoro. Boundedness for a nonlocal reaction chemotaxis model even in the attraction-dominated regime. Differential Integral Equations, 34(5-6):315-336, 2021. · Zbl 1474.35004 |
| [25] | Haihua Liang and Peixuan Weng. Existence and multiple solutions for a second-order difference boundary value problem via critical point theory. J. Math. Anal. Appl., 326(1):511-520, 2007. · Zbl 1112.39008 |
| [26] | To Fu Ma. Positive solutions for a beam equation on a nonlinear elastic foundation. Math. Comput. Modelling, 39(11-12):1195-1201, 2004. · Zbl 1060.74035 |
| [27] | Mihai Mihȃilescu, Vicent ¸iu Rȃdulescu, and Stepan Tersian. Eigenvalue problems for anisotropic discrete boundary value problems. J. Difference Equ. Appl., 15(6):557-567, 2009. · Zbl 1181.47016 |
| [28] | Mohamed Ousbika and Zakaria El Allali. Existence of three solutions to the discrete fourth-order boundary value problem with four parameters. Bol. Soc. Parana. Mat. (3), 38(2):177-189, 2020. · Zbl 1431.35009 |
| [29] | Jingping Yang. Sign-changing solutions to discrete fourth-order Neumann boundary value problems. Adv. Difference Equ., pages 1-11, Paper No. 10, 2013. · Zbl 1368.39007 |
| [30] | Yue Yue, Yu Tian, Min Zhang, and Jianguo Liu. Existence of infinitely many solutions for fourth-order impulsive differential equations. Appl. Math. Lett., 81:72-78, 2018. · Zbl 1393.34039 |
| [31] | Binggen Zhang, Lingju Kong, Yijun Sun, and Xinghua Deng. Existence of positive solutions for BVPs of fourth-order difference equations. Appl. Math. Comput., 131(2-3):583-591, 2002. · Zbl 1025.39006 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
