Positive Solutions for Systems of Fourth Order Two-Point Boundary Value Problems with Parameter

Year 2019, Volume: 2 Issue: 1, 30 - 38, 20.04.2019

Noureddine Bouteraa , Slimane Benaicha Habib Djourdem

https://doi.org/10.33187/jmsm.432678

Cited By: 11

Abstract

This paper deals with the existence of positive solutions for a system of nonlinear singular fourth-order differential equations with a parameter $\lambda$ subject two-point boundary conditions. Our analysis relies on the Krasnoselskii fixed point theorem and under suitable conditions, we derive explicit eigenvalue intervals of $\lambda$ for the existence of at least one positive solution for the system.

Keywords

Green's function , Positive solutions , System of differential equations , Two-point boundary value problem

References

• [1] R. B. Agarwal, D. O’Regan, P. J. Wang, Positive solutions of differential, Difference, and integral equations, Kluwer Academic, Boston, Ma, 1999.
• [2] D. R. Anderson, R. I. Avery, A fourth-order four-point right focal boundary value problem, Rocky Mountain J. Math., 36 (2006), 367-380.
• [3] N. Bouteraa, S. Benaicha, Triple positive solutions of higher-order nonlinear boundary value problems, J. Comput. Sci. Comput. Math., 7 (2017), 25–31.
• [4] N. Bouteraa, S. Benaicha, Nonlinear boundary value problems for higher-order ordinary differential equation at resonance, Romanian J. Math. Comput. Sci., 2 (2018), 83–91.
• [5] N. Bouteraa, S. Benaicha, Existence of solutions for third-order three-point boundary value problem, Mathematica, 60(83) (2018), 12–22.
• [6] A. Cabada, S. Heikkila, Uniqueness, comparison and existence results for third-order functional initial-boundary value problems, Comput. Math. Appl., 41 (2001), 607–618.
• [7] A. Cabada, S. Tersian, Existence and multiplicity of solutions to boundary value problems for fourth-order impulsive differential equations, Bound. Value Probl., 2014, 2014:105.
• [8] J. Davis, J. Henderson, Uniqueness implies existence for fourth-order Lidstone boundary value problems, Panamer. Math. J., 8 (1998), 23–35.
• [9] H. Djourdem, S. Benaicha, Existence of positive solutions for a nonlinear three-point boundary value problem with integral boundary conditions, Acta Math. Univ. Comenianae, 87 (2018), 167–177.
• [10] L. H. Erbe, H. Wang, On -the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc. 120 (1994) 743–748.
• [11] C. P. Gupta, Existence and uniqueness theorem for the bending of an elastics beam equation, Appl. Anal., 26 (1988), 289–304.
• [12] G. Infante, P. Pientramala, A cantilever equation witht nonlinear boundary conditions, Electron. J. Qual. Theory Differe. Equ., Spec. Ed. I., 15 (2009), 1–14.
• [13] R. Ma, L. Xu, Existence of positive solutions for a nonlinear fourth-order boundary value problem, Appl. Math. Lett. 23 (2010), 537–543.
• [14] Y. Sun, C. Zhu, Existence of positive solutions for singular fourth-order three-point boundary value problems, Adv. Differ. Equ., 2013, 2013:51. doi:10.1186-1847-2013-51.
• [15] A. R. Aftabizadeh, Existence and uniqueness theorems for fourth-order boundary value problems, J. Math. Anal. Appl. 116 (1986), 415–426.
• [16] E. Alves, T.F.Ma., M.L. Policer, Monotone positive solutions for a fourth order equation with nonlinear boundary conditions value, Nonlinear Anal. 71 (2009), 3834–3841.
• [17] C. P. Gupta, Existence and uniqueness theorem for the bending of an elastics beam equation at resonance, J. Math. Anal Appl. 135 (1988), 208–225.
• [18] R. Ma, B. Thompson, Nodal solutions for a nonlinear fourth-order eigenvalue problem, Acta Math. Sin. Engl. Ser. 24 (2008), 27–34.
• [19] Q. Yao, Local existence of multiple positive solutions to a singular cantiliver beam equation, J. Math. Anal. Appl. 363 (2010), 138–154.
• [20] N. Bouteraa, S. Benaicha, Positive periodic solutions for a class of fourth-order nonlinear differential equations, Numerical Analysis and Applications., 22(1) (2019), 1–14.
• [21] N. Bouteraa, S. Benaicha, H. Djourdem, M.E. Benattia, Positive solutions for fourth-order two-point boundary value problem with a parameter, Romanian J. Math. Comput. Sci., 8 (2018), 17–30.
• [22] H. Djourdem, S. Benaicha, N. Bouteraa, Existence and iteration of monotone positive solution for a fourth-order nonlinear boundary value problem, Fundam. J. Math. Appl., 1 (2018), 205–211.
• [23] L. Hu, L. L. Wang, Multiple positive solutions of boundary value problems for systems of nonlinear second order differential equation, J. Math. Anal. Appl., 355 (2007), 1052–1060.
• [24] B. Liu, L. Liu, Y. Wu, Positive solutions for singular systems of three-point boundary value problems, Computers Math. Appl., 53 (2007), 1429–1438.
• [25] Y. Zhou, Y. Xu, Positive solutions of three boundary value problems for systems of nonlinear second order ordinary differential equation, J. Math. Anal. Appl., 320 (2006), 578–590.
• [26] M. A. Krasnosel’skii, Positive solutions of operator equations, Noordhoff, Groningen, 1964.