Existence and Iteration of Monotone Positive Solution for a Fourth-Order Nonlinear Boundary Value Problem

Abstract

This paper is concerned with the following fourth-order three-point boundary value problem BVP \[ u^{\left(4\right)}\left(t\right)=f\left(t,u\left(t\right)\right),\quad t\in\left[0,1\right], \] \[ u'\left(0\right)=u''\left(0\right)=u\left(1\right)=0,\;u'''\left(\eta\right)+\alpha u\left(0\right)=0, \] where $f\in C\left(\left[0,1\right]\times\left[0,+\infty\right),\left[0,+\infty\right)\right)$ , $\alpha\in\left[0,6\right)$ and $\eta\in\left[\frac{2}{3},1\right)$. Although corresponding Green\textquoteright s function is sign-changing, we still obtain the existence of monotone positive solution under some suitable conditions on $f$ by applying iterative method. An example is also given to illustrate the main results.

Keywords

• Boundary value problem,Green’s function,Positive solution,Iterative method,Sign-changing

References

1. [1] A. R. Aftabizadeh, Existence and uniqueness theorems for fourth-order boundary value problems, J. Math. Anal. Appl., 116 (1986), 415-426.
2. [2] A. Cabada, S. Tersian, Existence and multiplicity of solutions to boundary value problems for fourth-order impulsive differential equations, Bound. Value Probl., 105 (2014).
3. [3] D. R. Anderson, R. I. Avery, A fourth-order four-point right focal boundary value problem, Rocky Mountain J. Math., 36 (2006), 367-380.
4. [4] E. Alves, T. F. Ma, M. L. Pelicer, Monotone positive solutions for a fourth order equation with nonlinear boundary conditions, Nonlinear Anal., 71 (2009), 3834-3841.
5. [5] J. R. Graef, B. Yang, Positive solutions for fourth-order focal boundary value problem, Rocky mountain journal of mathematics, 44(3) (2014), 937-951.
6. [6] N. Bouteraa, S. Benaicha, H. Djourdem, M. E. Benattia, Positive solutions of nonlinear fourth-order two-point boundary value problem with a parameter, Romanian J. Math. Comput. Sci., 8(1) (2018), 17-30.
7. [7] N. Bouteraa, S. Benaicha, Triple positive solutions of higher-order nonlinear boundary value problems, J. Comput. Sci. Comput. Math., 7(2) (2017).
8. [8] R. P. Agarwal, On fourth-order boundary value problems arising in beam analysis, Differ. Integral Equ., 2(1) (1989), 91–110.

9. [9] S. Lia, X. Zhanga, Existence and uniqueness of monotone positive solutions for an elastic beam equation with nonlinear boundary conditions, Comput. Math. Appl., 63 (2012), 1355–1360.
10. [10] W. Wang, Y. Zheng, H. Yang, J. Wang, Positive solutions for elastic beam equations with nonlinear boundary conditions and a parameter, Bound. Value Probl., 80 (2014), 1-17.
11. [11] Y. Li, Positive solutions of fourth-order boundary value problems with two parameters, Journal of Mathematical Analysis and Applications, 281(2) (2003), 477–484.
12. [12] Z. Bai, The upper and lower solution method for some fourth-order boundary value problem, Nonlinear Anal., 67 (2007), 1704- 1709.
13. [13] Z. Bekri, S. Benaicha, Existence of positive of solution for a nonlinear three-point boundary value problem, Sib. ‘Elektron. Mat. Izv., 14 (2017), 1120–1134.
14. [14] A. P. Palamides, G. Smyrlis, Positive solutions to a singular third-order three-point BVP with an indefinitely signed Green’s function, Nonlinear Anal., 68 (2008), 2104-2118.
15. [15] D. J. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, 5, Academic Press, New York, NY, USA, 1988.
16. [16] M. A. Krasnoselskii, Positive Solutions of Opearator Equations, Noordhoff, Groningen, The Netherlands, 1964.
17. [17] Y. Zhang, J. P. Sun, J. Zhao, Positive solutions for a fourth-order three-point BVP with sign-changing Green’s function, Electron. J. Qual. Theory Differ. Equ., 5 (2018), 1-11.
18. [18] 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, 245 (2017), 1–17.
19. [19] G. Infante, J. R. L. Webb, Three-point boundary value problems with solutions that change sign, J. Integral Equations Appl., 15(1) (2003), 37–57.
20. [20] J. P. Sun, X. Q.Wang, Existence and iteration of monotone positive solution of BVP for an elastic beam equation, Mathematical Problems in Engineering, 2011, Article ID 705740, 10 pages.
21. [21] 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(215) (2013), 1-9.
22. [22] Y. H. Zhao, X. L. Li, Iteration for a third-order three-point BVP with sign-changing green’s function, J. Appl. Math., (2014), Article ID 541234, 6 pages.