Research article Special Issues

Solvability for some fourth order two-point boundary value problems

  • Received: 22 April 2020 Accepted: 04 June 2020 Published: 09 June 2020
  • MSC : 34A08, 34B15, 35J05

  • Some fourth-order two-point boundary value problems are considered in this paper. Firstly, the Green's function is obtained by the use of the Laplace transform. Secondly, the first eigenvalue is given by using Ritz method. Then, by the use of the properties of self-adjoint operators and the fixed point index theory, the existence of positive solutions is obtained. Finally, an example is given to illustrate the main results.

    Citation: Zhanbing Bai, Wen Lian, Yongfang Wei, Sujing Sun. Solvability for some fourth order two-point boundary value problems[J]. AIMS Mathematics, 2020, 5(5): 4983-4994. doi: 10.3934/math.2020319

    Related Papers:

  • Some fourth-order two-point boundary value problems are considered in this paper. Firstly, the Green's function is obtained by the use of the Laplace transform. Secondly, the first eigenvalue is given by using Ritz method. Then, by the use of the properties of self-adjoint operators and the fixed point index theory, the existence of positive solutions is obtained. Finally, an example is given to illustrate the main results.


    加载中


    [1] Z. Bai, The method of lower and upper solutions for a bending of an elastic beam equation, J. Math. Anal. Appl., 248 (2000), 195-202. doi: 10.1006/jmaa.2000.6887
    [2] G. Bonanno, B. Bella, D. O'Regan, Non-trivial solutions for nonlinear fourth-order elastic beam equations, Comput. Math. Appl., 62 (2011), 1862-1869. doi: 10.1016/j.camwa.2011.06.029
    [3] Y. Li, Positive solutions of fourth-order boundary value problems with two parameters, J. Math. Anal. Appl., 281 (2003), 477-484. doi: 10.1016/S0022-247X(03)00131-8
    [4] R. Ma, J. Zhang, S. Fu, The method of lower and upper solutions for fourth-order two-point boundary value problems, J. Math. Anal. Appl., 215 (1997), 415-422. doi: 10.1006/jmaa.1997.5639
    [5] Y. Wei, Q. Song, Z. Bai, Existence and iterative method for some fourth order nonlinear boundary value problems, Appl. Math. Lett., 87 (2019), 101-107. doi: 10.1016/j.aml.2018.07.032
    [6] Q. Yao, The positive solution of singular beam equation with simple support at both ends, Adv. Math. China, 5 (2009), 590-598.
    [7] Q. Yao, Existence and multiplicity of positive solutions to a class of elastic beam equations, J. Shandong Univ., 5 (2004), 64-67. (in Chinese)
    [8] Q. Yao, Y. Li, Existence theorem for a class of nonlinear elastic beam equations, J. South China Univ. Tech., 37 (2006), 124-127. (in Chinese)
    [9] D. Zhao, H. Wang, J. Wang, Existence of three positive solutions for a class of singular beam equations with corner angles and bending moments, Acta. Math. Appl. Sinica, 34 (2011), 813-821. (in Chinese)
    [10] Z. Bai, Z. Du, S. Zhang, Iterative method for a class of fourth-order p-Laplacian beam equation, J. Appl. Anal. Comput., 9 (2019), 1443-1453.
    [11] F. Zhu, L. Liu, Y. Wu, Positive solutions for systems of a nonlinear fourth-order singular semipositone boundary value problems, Comput. Math. Appl., 15 (2010), 448-457.
    [12] R. Agarwal, Y. Chow, Iterative methods for a fourth order boundary value problem, J. Comput. Appl. Math., 10 (1984), 203-217. doi: 10.1016/0377-0427(84)90058-X
    [13] R. Ma, X. Wu, Existence of multiple positive solutions for a class of fourth-order two-point boundary value problems, Acta Math. Sci., 22A (2002), 244-249. (in Chinese)
    [14] X. Wu, R. Ma, Existence of multiple positive solutions for a class of fourth-order two-point boundary value problems, Acta Anal. Funct. Appl., 2 (2000), 342-348. (in Chinese)
    [15] J. Caballero, J. Harjani, K. Sadarangani, Uniqueness of positive solutions for a class of fourth-order boundary value problems, Abstr. Appl. Anal., 2011 (2011), 1-13.
    [16] D. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, New York, Academic Press, 1988.
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3035) PDF downloads(357) Cited by(2)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog