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Two Positive Solutions for a Fourth-Order Three-Point BVP with Sign-Changing Green's Function

Year 2019, Volume: 2 Issue: 1, 60 - 68, 22.03.2019
https://doi.org/10.33434/cams.452839

Abstract

This paper concerns the fourth-order three-point boundary value problem (BVP) \[ u^{\left(4\right)}\left(t\right)=f\left(t,u\left(t\right)\right),\quad t\in\left[0,1\right], \] \[ u'\left(0\right)=u''\left(0\right)=u\left(1\right)=0,\;\alpha u''\left(1\right)-u'''\left(\eta\right)=0, \] where $f\in C\left(\left[0,1\right]\times\left[0,+\infty\right),\left[0,+\infty\right)\right)$, $\alpha\in\left[0,1\right)$ and $\eta\in\left[\frac{2\alpha+10}{15-2\alpha},1\right)$. Although the corresponding Green\textquoteright s function is sign-changing, we still obtain the existence of at least two positive and decreasing solutions under some suitable conditions on $f$ by applying the two-fixed-point theorem due to Avery and Henderson. An example is also given to illustrate the main results.

References

  • [1] A. Cabada, R. Enguica, L. Lopez-Somoza, Positive solutions for second-order boundary value problems with sign changing Green’s functions, Electron. J. Differential Equations 2017, No. 245, 1-17.
  • [2] A. R. Aftabizadeh, Existence and uniqueness theorems for fourth-order boundary value problems, J. Math. Anal. Appl. 116 (1986), 415-426.
  • [3] A. P. Palamides, A. Veloni, S. Alatsathianos, Positive solutions of a third-order BVP independent of the sign of the Green’s function, Differ. Equ. Dyn. Syst. 21(2013), No. 3, 261-279.
  • [4] A. P. Palamides, G. Smyrlis, Positive solutions to a singular third-order three-point boundary value problem with an indefinitely signed Green’s function, Nonlinear Anal. 68(2008), No. 7, 2104-2118.
  • [5] B.-W. Niu, J.-P. Sun, Q.-Y. Ren, Two positive solutions of third-order BVP with integral boundary condition and sign-changing Green’s function, J. Funct. Spaces 2015, Art. ID 491423, 8 pp.
  • [6] C. Gao, F. Zhang, R. Ma, Existence of positive solutions of second-order periodic boundary value problems with sign-changing Green’s function, Acta Math. Appl. Sin. Engl. Ser. 33(2017), No. 2, 263-268.
  • [7] D. Xie, H. Zhou, C. Bai, Y. Liu and X. Wu, Triple positive solutions for a third-order three-point boundary value problem with sign-changing Green’s function. Italian Journal of Pure and Applied Mathematics-N. 38(2017), 519-530.
  • [8] G. Infante, J. R. L.Webb, Three-point boundary value problems with solutions that change sign, J. Integral Equations Appl., 15 (2003), 37-57.
  • [9] J.-P. Sun, J. Zhao, Iterative technique for a third-order three-point BVP with sign-changing Green’s function, Electron. J. Differential Equations 2013, No. 215, 1-9.
  • [10] J.-P. Sun, J. Zhao, Positive solution for a third-order three-point boundary value problem with sign-changing Green’s function, Commun. Appl. Anal. 16 (2012), No. 2, 219-228.
  • [11] J. Wang, C. Gao, Positive solutions of discrete third-order boundary value problems with sign-changing Green’s function, Adv. Differ. Equ., 2015, 56 (2015).
  • [12] L.-J. Gao, J.-P. Sun, Positive solutions of a third-order three-point BVP with sign-changing Green’s function, Math. Probl. Eng. 2014, Art. ID 406815, 6 pp.
  • [13] N. Bouteraa, S. Benaicha, H. Djourdem and M-E. Benattia, Positive solutions of nonlinear fourth-order two-point boundary value problem with a parameter, Romanian journal of mathematics and computers science, 2018, volume 8, issue 1, p.17-30.
  • [14] N. Bouteraa, S. Benaicha, Triple positive solutions of higher-order nonlinear boundary value problems, Journal of Computer Science and Computational Mathematics, Volume 7, Issue 2, June 2017.
  • [15] R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, John Wiley and Sons, New York, NY, USA, 1976.
  • [16] R. I. Avery and J. Henderson, \Two positive fixed points of nonlinear operators on ordered Banach spaces," Communications on Applied Nonlinear Analysis, vol. 8, no. 1, pp. 27-36, 2001.
  • [17] R. Ma, Nonlinear periodic boundary value problems with sign-changing Green’s function, Nonlinear Anal. 74(2011), No. 5, 1714-1720.
  • [18] X. Li, J. Sun and F. Kong, Existence of positive solution for a third-order three-point BVP with sign-changing Green’s function, Electron. J. Qual. Theory Differ. Equ., 30 (2013), 1-11.
  • [19] Y. Zhang, J-P. Sun and J. Zhao, Positive solutions for a fourth-order three-point BVP with sign-changing Green’s function. Electron. J. Qual. Theory Differ. 2018, No. 5, 1-11.
  • [20] Y.H. Zhao, X.L. Li, Iteration for a third-order three-point BVP with sign-changing Green’s function, Journal of Applied Mathematics, vol. 2014, Article ID 541234, 6 pages, 2014.
  • [21] Z. Bai, The upper and lower solution method for some fourth-order boundary value problem, Nonlinear Anal. 67 (2007), 1704- 1709.
  • [22] Z. Bekri and S. Benaicha, Existence of positive of solution for a nonlinear three-point boundary value problem, Siberian Electronic Mathematical Reports. 14(2017).
Year 2019, Volume: 2 Issue: 1, 60 - 68, 22.03.2019
https://doi.org/10.33434/cams.452839

Abstract

References

  • [1] A. Cabada, R. Enguica, L. Lopez-Somoza, Positive solutions for second-order boundary value problems with sign changing Green’s functions, Electron. J. Differential Equations 2017, No. 245, 1-17.
  • [2] A. R. Aftabizadeh, Existence and uniqueness theorems for fourth-order boundary value problems, J. Math. Anal. Appl. 116 (1986), 415-426.
  • [3] A. P. Palamides, A. Veloni, S. Alatsathianos, Positive solutions of a third-order BVP independent of the sign of the Green’s function, Differ. Equ. Dyn. Syst. 21(2013), No. 3, 261-279.
  • [4] A. P. Palamides, G. Smyrlis, Positive solutions to a singular third-order three-point boundary value problem with an indefinitely signed Green’s function, Nonlinear Anal. 68(2008), No. 7, 2104-2118.
  • [5] B.-W. Niu, J.-P. Sun, Q.-Y. Ren, Two positive solutions of third-order BVP with integral boundary condition and sign-changing Green’s function, J. Funct. Spaces 2015, Art. ID 491423, 8 pp.
  • [6] C. Gao, F. Zhang, R. Ma, Existence of positive solutions of second-order periodic boundary value problems with sign-changing Green’s function, Acta Math. Appl. Sin. Engl. Ser. 33(2017), No. 2, 263-268.
  • [7] D. Xie, H. Zhou, C. Bai, Y. Liu and X. Wu, Triple positive solutions for a third-order three-point boundary value problem with sign-changing Green’s function. Italian Journal of Pure and Applied Mathematics-N. 38(2017), 519-530.
  • [8] G. Infante, J. R. L.Webb, Three-point boundary value problems with solutions that change sign, J. Integral Equations Appl., 15 (2003), 37-57.
  • [9] J.-P. Sun, J. Zhao, Iterative technique for a third-order three-point BVP with sign-changing Green’s function, Electron. J. Differential Equations 2013, No. 215, 1-9.
  • [10] J.-P. Sun, J. Zhao, Positive solution for a third-order three-point boundary value problem with sign-changing Green’s function, Commun. Appl. Anal. 16 (2012), No. 2, 219-228.
  • [11] J. Wang, C. Gao, Positive solutions of discrete third-order boundary value problems with sign-changing Green’s function, Adv. Differ. Equ., 2015, 56 (2015).
  • [12] L.-J. Gao, J.-P. Sun, Positive solutions of a third-order three-point BVP with sign-changing Green’s function, Math. Probl. Eng. 2014, Art. ID 406815, 6 pp.
  • [13] N. Bouteraa, S. Benaicha, H. Djourdem and M-E. Benattia, Positive solutions of nonlinear fourth-order two-point boundary value problem with a parameter, Romanian journal of mathematics and computers science, 2018, volume 8, issue 1, p.17-30.
  • [14] N. Bouteraa, S. Benaicha, Triple positive solutions of higher-order nonlinear boundary value problems, Journal of Computer Science and Computational Mathematics, Volume 7, Issue 2, June 2017.
  • [15] R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, John Wiley and Sons, New York, NY, USA, 1976.
  • [16] R. I. Avery and J. Henderson, \Two positive fixed points of nonlinear operators on ordered Banach spaces," Communications on Applied Nonlinear Analysis, vol. 8, no. 1, pp. 27-36, 2001.
  • [17] R. Ma, Nonlinear periodic boundary value problems with sign-changing Green’s function, Nonlinear Anal. 74(2011), No. 5, 1714-1720.
  • [18] X. Li, J. Sun and F. Kong, Existence of positive solution for a third-order three-point BVP with sign-changing Green’s function, Electron. J. Qual. Theory Differ. Equ., 30 (2013), 1-11.
  • [19] Y. Zhang, J-P. Sun and J. Zhao, Positive solutions for a fourth-order three-point BVP with sign-changing Green’s function. Electron. J. Qual. Theory Differ. 2018, No. 5, 1-11.
  • [20] Y.H. Zhao, X.L. Li, Iteration for a third-order three-point BVP with sign-changing Green’s function, Journal of Applied Mathematics, vol. 2014, Article ID 541234, 6 pages, 2014.
  • [21] Z. Bai, The upper and lower solution method for some fourth-order boundary value problem, Nonlinear Anal. 67 (2007), 1704- 1709.
  • [22] Z. Bekri and S. Benaicha, Existence of positive of solution for a nonlinear three-point boundary value problem, Siberian Electronic Mathematical Reports. 14(2017).
There are 22 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Habib Djourdem

Slimane Benaicha This is me

Noureddine Bouteraa

Publication Date March 22, 2019
Submission Date August 10, 2018
Acceptance Date January 21, 2019
Published in Issue Year 2019 Volume: 2 Issue: 1

Cite

APA Djourdem, H., Benaicha, S., & Bouteraa, N. (2019). Two Positive Solutions for a Fourth-Order Three-Point BVP with Sign-Changing Green’s Function. Communications in Advanced Mathematical Sciences, 2(1), 60-68. https://doi.org/10.33434/cams.452839
AMA Djourdem H, Benaicha S, Bouteraa N. Two Positive Solutions for a Fourth-Order Three-Point BVP with Sign-Changing Green’s Function. Communications in Advanced Mathematical Sciences. March 2019;2(1):60-68. doi:10.33434/cams.452839
Chicago Djourdem, Habib, Slimane Benaicha, and Noureddine Bouteraa. “Two Positive Solutions for a Fourth-Order Three-Point BVP With Sign-Changing Green’s Function”. Communications in Advanced Mathematical Sciences 2, no. 1 (March 2019): 60-68. https://doi.org/10.33434/cams.452839.
EndNote Djourdem H, Benaicha S, Bouteraa N (March 1, 2019) Two Positive Solutions for a Fourth-Order Three-Point BVP with Sign-Changing Green’s Function. Communications in Advanced Mathematical Sciences 2 1 60–68.
IEEE H. Djourdem, S. Benaicha, and N. Bouteraa, “Two Positive Solutions for a Fourth-Order Three-Point BVP with Sign-Changing Green’s Function”, Communications in Advanced Mathematical Sciences, vol. 2, no. 1, pp. 60–68, 2019, doi: 10.33434/cams.452839.
ISNAD Djourdem, Habib et al. “Two Positive Solutions for a Fourth-Order Three-Point BVP With Sign-Changing Green’s Function”. Communications in Advanced Mathematical Sciences 2/1 (March 2019), 60-68. https://doi.org/10.33434/cams.452839.
JAMA Djourdem H, Benaicha S, Bouteraa N. Two Positive Solutions for a Fourth-Order Three-Point BVP with Sign-Changing Green’s Function. Communications in Advanced Mathematical Sciences. 2019;2:60–68.
MLA Djourdem, Habib et al. “Two Positive Solutions for a Fourth-Order Three-Point BVP With Sign-Changing Green’s Function”. Communications in Advanced Mathematical Sciences, vol. 2, no. 1, 2019, pp. 60-68, doi:10.33434/cams.452839.
Vancouver Djourdem H, Benaicha S, Bouteraa N. Two Positive Solutions for a Fourth-Order Three-Point BVP with Sign-Changing Green’s Function. Communications in Advanced Mathematical Sciences. 2019;2(1):60-8.

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