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

Yıl 2019, Cilt: 2 Sayı: 1, 60 - 68, 22.03.2019

Habib Djourdem Slimane Benaicha Noureddine Bouteraa

https://doi.org/10.33434/cams.452839

Cited By: 5

Öz

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.

Anahtar Kelimeler

two positive solutions, Completely continuous, fourth-order boundary value problem, Green\textquoteright s function, two positive solutions

Kaynakça

• [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).