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Positive solutions to a semipositone superlinear elastic beam equation

  • Received: 11 October 2020 Accepted: 04 January 2021 Published: 07 February 2021
  • MSC : 34K10, 37C25

  • A semipositone fourth-order two-point boundary value problem is considered. In mechanics, the problem describes the deflection of an elastic beam rigidly fastened on the left and simply supported on the right. Under some conditions concerning the first eigenvalue corresponding to the relevant linear operator, the existence of nontrivial solutions and positive solutions to this boundary value problem is obtained. The main results are obtained by using the topological method and the fixed point theory of superlinear operators.

    Citation: Haixia Lu, Li Sun. Positive solutions to a semipositone superlinear elastic beam equation[J]. AIMS Mathematics, 2021, 6(5): 4227-4237. doi: 10.3934/math.2021250

    Related Papers:

  • A semipositone fourth-order two-point boundary value problem is considered. In mechanics, the problem describes the deflection of an elastic beam rigidly fastened on the left and simply supported on the right. Under some conditions concerning the first eigenvalue corresponding to the relevant linear operator, the existence of nontrivial solutions and positive solutions to this boundary value problem is obtained. The main results are obtained by using the topological method and the fixed point theory of superlinear operators.



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    [1] R. P. Agarwal, Y. M. Chow, Iterative method for fourth order boundary value problem, J. Comput. Appl. Math., 10 (1984), 203–217. doi: 10.1016/0377-0427(84)90058-X
    [2] Z. B. Bai, H. Y. Wang, On positive solutions of some nonlinear four-order beam equations, J. Math. Anal. Appl., 270 (2002), 357–368. doi: 10.1016/S0022-247X(02)00071-9
    [3] Z. B. Bai, The upper and lower solution method for some fourth-order boundary value problems, Nonlinear Anal.-Theor.,, 67 (2007), 1704–1709.
    [4] G. Bonanno, B. D. Bella, A boundary value problem for fourth-order elastic beam equations, J. Math. Anal. Appl., 343 (2008), 1166–1176. doi: 10.1016/j.jmaa.2008.01.049
    [5] G. Bonanno, B. D. 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
    [6] R. Graef, B. Yang, Positive solutions of a nonlinear fourth order boundary value problem, Communications on Applied Nonlinear Analysis, 14 (2007), 61–73.
    [7] C. P. Gupta, Existence and uniqueness results for the bending of an elastic beam equation, Appl. Anal., 26 (1988), 289–304. doi: 10.1080/00036818808839715
    [8] P. Korman, Uniqueness and exact multiplicity of solutions for a class of fourth-order semilinear problems, P. Roy. Soc. Edinb. A, 134 (2004), 179–190. doi: 10.1017/S0308210500003140
    [9] B. D. Lou, Positive solutions for nonlinear elastic beam models, International Journal of Mathematics and Mathematical Sciences, 27 (2001), 365–375. doi: 10.1155/S0161171201004203
    [10] R. Y. Ma, L. Xu, Existence of positive solutions of a nonlinear fourth-order boundary value problem, Appl. Math. Lett., 23 (2010), 537–543. doi: 10.1016/j.aml.2010.01.007
    [11] Q. L. Yao, Positive solutions for eigenvalue problems of four-order elastic beam equations, Appl. Math. Lett., 17 (2004), 237–243. doi: 10.1016/S0893-9659(04)90037-7
    [12] Q. L. Yao, Existence of $n$ solutions and/or positive solutions to a semipositone elastic beam equation, Nonlinear Anal.-Theor., 66 (2007), 138–150. doi: 10.1016/j.na.2005.11.016
    [13] Q. L. Yao, positive solutions of nonlinear elastic beam equation rigidly fastened on the left and simply supported on the right, Nonlinear Anal.-Theor., 69 (2008), 1570–1580. doi: 10.1016/j.na.2007.07.002
    [14] C. B. Zhai, R. P. Song, Q. Q. Han, The existence and the uniqueness of symmetric positive solutions for a fourth-order boundary value problem, Comput. Math. Appl., 62 (2011), 2639–2647. doi: 10.1016/j.camwa.2011.08.003
    [15] X. P. Zhang, Existence and iteration of monotone positive solutions for an elastic beam equation with a corner, Nonlinear Anal.-Real, 10 (2009), 2097–2103. doi: 10.1016/j.nonrwa.2008.03.017
    [16] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin-Heidelberg-Newyork, 1985.
    [17] D. J. Guo, V. Lakshmikanthan, Nonlinear Problems in Abstract Cones, Academic press, San Diego, 1988.
    [18] D. J. Guo, Nonlinear Functional Analysis, second edn., Shandong Science and Technology Press, Jinan, 2001 (in Chinese).
    [19] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620–709. doi: 10.1137/1018114
    [20] G. W. Zhang, J. X. Sun, Positive solutions of $m$-point boundary value problems, J. Math. Anal. Appl., 291 (2004), 406–418. doi: 10.1016/j.jmaa.2003.11.034
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