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Multiple Nonnegative Solutions for a Class of Fourth-Order BVPs Via a New Topological Approach

Year 2022, Volume: 6 Issue: 3, 390 - 404, 30.09.2022
https://doi.org/10.31197/atnaa.977625

Abstract

In this paper, we study a class of fourth-order boundary value problems with integral boundary conditions.
The nonlinearity may have time-singularity and change sign. Moreover, it satisfies general polynomial growth conditions.
A new topological approach is applied to prove the existence of at least two nonnegative classical solutions.
An example of application illustrates the existence result.

References

  • [1] R.T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc. 149 (1970), 75-88.
  • [2] R. P. Agarwal, On fourth order boundary value problems arising in beam analysis, Differential and integral equations 2 (1989), 91-110.
  • [3] R. P. Agarwal, S. Kelevedjiev, On the solvability of fourth-order two-point boundary value problems, Mathematics 2020, 8, 603.
  • [4] K. Bachouche, A. Benmezai, S. Djebali, Positive solutions to semi-positone fourth-order ϕ-Laplacian BVPs, Positivity 21 (2017), 193-212.
  • [5] S. Benslimane, S. Djebali, K. Mebarki, On the ?xed point index for sums of operators, Fixed Point Theory, 23(2022), no. 1, 143-162.
  • [6] S. Djebali, T. Moussaoui, R. Precup, Fourth order p-laplacian nonlinear systems via the vector version of the Krasnosel'skii's ?xed point theorem, Mediterr. J. Math 6 (2009), no 4, 447-460.
  • [7] S. Djebali, K. Mebarki, Fixed point index theory for perturbation of expansive mappings by k-set contraction, Top. Meth. Nonli. Anal., 54 (2019), no 2A, 613-640.
  • [8] D. Guo, V. Lakshmikantham, Nonlinear problems in abstract cones, Academic Press, Boston, Mass, USA, vol. 5, (1988).
  • [9] C. Gupta, Existence and uniqueness results for the bending of an elastic beam equation at resonnance, Journal of Mathe- matical Analysis and Applications, 135(1988), 208-225.
  • [10] L. Lin, Y. Liu, D. Zhao, Multiple solutions for a class of nonliner fourth-order boundary value problems, Symmetry 2020, 12, 1989.
  • [11] B. Liu, Positive solutions of the fourth-order two point boundary value problems, Appl. Math. Comput, 148 (2004), no. 2, 407-420.
  • [12] Y. Liu, D. O'Regan, Multiplicity results for a class of fourth order semipositone m-point boundary value problems, Appl. Anal. 91(2012), 911-921.
  • [13] R. Ma, H. Wang, On the existence of positive solutions of fourth-order ordinary di?erential equation, Anal. Appl. 59(1- 4)(1995), 225-231.
  • [14] S. Reich, Fixed points of condensing functions, J. Math. Anal. Appl. 41 (1973) 460-467.
  • [15] Q. Wang, Y. Guo, Y. Ji, Positive solutions for fourth?order nonlinear differential equation with integral boundary condi- tions, Discrete Dynamics in Nature and Society, Vol. 2013, Article ID 684962, 10 pages.
  • [16] T. Xiang, R. Yuan, A class of expansive-type Krasnosel'skii fixed point theorems, Nonlinear Anal. 71 (2009), no. 7-8, 3229-3239.
  • [17] C. Zhai, C. Hiang, Existence of nontrivial solutions for a nonlinear fourth-order boundary value problem via iterative method, J. Nonlinear Sci. Appl. 9 (2016), 4295-4304.
  • [18] Y. Zhu, P. Weng, Multiple positive solutions for a fourth-order boundary value problem, Bol. Soc. Parana. Mat, 21(2003), 9-19.
Year 2022, Volume: 6 Issue: 3, 390 - 404, 30.09.2022
https://doi.org/10.31197/atnaa.977625

Abstract

References

  • [1] R.T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc. 149 (1970), 75-88.
  • [2] R. P. Agarwal, On fourth order boundary value problems arising in beam analysis, Differential and integral equations 2 (1989), 91-110.
  • [3] R. P. Agarwal, S. Kelevedjiev, On the solvability of fourth-order two-point boundary value problems, Mathematics 2020, 8, 603.
  • [4] K. Bachouche, A. Benmezai, S. Djebali, Positive solutions to semi-positone fourth-order ϕ-Laplacian BVPs, Positivity 21 (2017), 193-212.
  • [5] S. Benslimane, S. Djebali, K. Mebarki, On the ?xed point index for sums of operators, Fixed Point Theory, 23(2022), no. 1, 143-162.
  • [6] S. Djebali, T. Moussaoui, R. Precup, Fourth order p-laplacian nonlinear systems via the vector version of the Krasnosel'skii's ?xed point theorem, Mediterr. J. Math 6 (2009), no 4, 447-460.
  • [7] S. Djebali, K. Mebarki, Fixed point index theory for perturbation of expansive mappings by k-set contraction, Top. Meth. Nonli. Anal., 54 (2019), no 2A, 613-640.
  • [8] D. Guo, V. Lakshmikantham, Nonlinear problems in abstract cones, Academic Press, Boston, Mass, USA, vol. 5, (1988).
  • [9] C. Gupta, Existence and uniqueness results for the bending of an elastic beam equation at resonnance, Journal of Mathe- matical Analysis and Applications, 135(1988), 208-225.
  • [10] L. Lin, Y. Liu, D. Zhao, Multiple solutions for a class of nonliner fourth-order boundary value problems, Symmetry 2020, 12, 1989.
  • [11] B. Liu, Positive solutions of the fourth-order two point boundary value problems, Appl. Math. Comput, 148 (2004), no. 2, 407-420.
  • [12] Y. Liu, D. O'Regan, Multiplicity results for a class of fourth order semipositone m-point boundary value problems, Appl. Anal. 91(2012), 911-921.
  • [13] R. Ma, H. Wang, On the existence of positive solutions of fourth-order ordinary di?erential equation, Anal. Appl. 59(1- 4)(1995), 225-231.
  • [14] S. Reich, Fixed points of condensing functions, J. Math. Anal. Appl. 41 (1973) 460-467.
  • [15] Q. Wang, Y. Guo, Y. Ji, Positive solutions for fourth?order nonlinear differential equation with integral boundary condi- tions, Discrete Dynamics in Nature and Society, Vol. 2013, Article ID 684962, 10 pages.
  • [16] T. Xiang, R. Yuan, A class of expansive-type Krasnosel'skii fixed point theorems, Nonlinear Anal. 71 (2009), no. 7-8, 3229-3239.
  • [17] C. Zhai, C. Hiang, Existence of nontrivial solutions for a nonlinear fourth-order boundary value problem via iterative method, J. Nonlinear Sci. Appl. 9 (2016), 4295-4304.
  • [18] Y. Zhu, P. Weng, Multiple positive solutions for a fourth-order boundary value problem, Bol. Soc. Parana. Mat, 21(2003), 9-19.
There are 18 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Salim Benslimane This is me

Svetlin Georgiev 0000-0001-9320-8860

Karima Mebarki 0000-0002-6679-5059

Publication Date September 30, 2022
Published in Issue Year 2022 Volume: 6 Issue: 3

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